Giovanni Gallavotti ( E d.)
Statistical Mechanics Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 21-27, 1976
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11107-5 e-ISBN: 978-3-642-11108-2 DOI:10.1007/978-3-642-11108-2 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Liguori, Napoli 1976 With kind permission of C.I.M.E.
Printed on acid-free paper
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C m T R O INTERNAZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
I Ciclo
- Bressanone d a l
21 giugno a1 2 4 giugno
1976
STATISTICAL MECHANCIS
Coordinatore: Prof. Giovanni Gallavotti
P. Cartier:
Theorie de la mesure. Introduction B la mecanique statistique classique (Testo non pervenuto)
C. Cercignani:
A sketch of the theory of the Boltzmann equation.-
O.E. Lanford:
Qualitative and statistical theory of dissipative systems.-
E.H. Lieb:
many particle Coulomb systems.-
B. Tirozzi:
Report on renormalizationgroup.-
A. Wehrl:
Basic properties of entropy in quantum mechanics.
2 ENTRO INTERN AZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
A SKETCH OF THE THEORY OF THE BOLTZMANN EQUATION
C . CERCIGNANI
Istituto d i M a t e m a t i c a ,
Politecnico d i Milano
C o r s o tenuto a B r e s s a n o n e d a l 21 giugno a1 24 giugno 1976
A Sketch of t h e Theory of t h e
Bolt zmann equation C a r l o Cercignani I s t i t u t o d i Matemeticz P o l i t e c n i c o d i Tfiilano Milano, I t a l y
I n t h i s seminar, I s h a l l b r i e f l y review t h e t h e o p of t h e Boltzmann equation. How t h e l a t t e r a r i s e s from t h e 1 , i o u v i l l e equation has been d i s c u s s e d i n 0. Lanf ord 's l e c t u r e s . We s h a l l w r i t e t h e Boltzmann e q u z t i o n i n t h i n form
where t , 2 ,
5
r i a b l e s , while
4
denote t h e time, space and v e l o c j t y vni s t h e d i s t r i b u t i o n f u n c t i o n , normalized
i n such a way t h a t
where
M
i s t h e mass contained i n t h e r e ~ i o nover which t h e
i n t e a r a t i o n with r e s p e c t t o
Q({,{) i s t h e
2 extends.
so c a l l e d c o l l i s i o n term, e x p l i c i t l y obtai-
n a b l e from t h e f o l l o w i n g d e f i n i t i o n
where
$
i s an a u s i l i a r y v e l o c i t y v e c t o r , V is t h e re-
l a t i v e speed, i.e..
t h e mamitude of t h e v e c t o r
#'=Q&~),$=
-f
to
and
etc.,
where
'
and
-5;
y = f -I*, are releted
f , through t h e r e l a t i o n s e x p r e s s i n g conservation
of momentum and energy i n a c o l l i s i o n
where
2 in
i s a u n i t v e c t o r , whose p o l a r a n g l e s a r e
2 R
p o l a r c o o r d i n a t e system with
Y as p o l a r a x i s .
T n t ~ r r ~ t i oextends n t o ~ l vla l u e s of and
rr/2 with r e r o e c t t o 6 , from
t o 6 . Finally
B ( ~ v )i s
4 ma
and between
6
to
0
21 w i t h respect
related t o the d i f f e r e n t i a l cross
section
q(qvby
and
i s t h e mass of a g a s molecule. For f u r t h e r d e t a i l s
m
the relation
one should c o n s u l t one of my books [I ,2]. Eq. (1 ) i s v a l i d f o r monatomic molecules
and i s more Ke-
n e r n l t h a n t h e Boltzrnann e q u a t i o n considered by Lanford i n
-
h i s l e c t u r e s , because i t i s n o t but a l l o w s molecules
r e s t r i c t e d t o r i g i d spheres, with any d i f f e r e n t i a l c r o s s s e c t i o n .
The c a s e of r i g i d spheres i s obtained by s p e c i a l i z i n g as f o l l o w s
B(~v)
where
d
i s t h e sphere diameter. Another importa?t cF.se i s
o f f e r e d by t h e so c a l l e d 1~;axwellmolecules. The l a t t e r a r e c l z s s i c a l point masses i n t e r a c t i n r w i t h s c e n t r a l f o r c e i n v e r s e l y p r o p o r t i o n a l t o t h e f i f t h Dower of t h e i r mutual d i s t e n c e ; a s a consequence, it t u r n s out t h a t
B(6,V)
i s independent of
V.
It i s c l e a r t h a t i n i t i a l and boundary c o n d i t i o n s a r e required i n o r d e r t o s o l v e t h e Boltzmann e q u a t i o n , s i n c e t h e l a t t r r contain3 t h e time and space d e r i v a t i v e s of
f.
The bound:-
4 C O~? I -
d i t i o n s a r e p a r t i c u l a r l y important s i n c e t h e y d e s c r i b e t h e int e r a c t i o n of t h e g a s molecules with s o l i d w a l l s , but p a r t ) cul a r d i f f i c u l t t o e s t a b l i s h ; t h e d i f f i c u l t i e s a r e due, mainly, t o o u r l a c k of b o w l e d g e of t h e s t r u c t u r e of t h e s u r f a c e l c y ~ r - . of s o l i d bodies and hence of t h e i n t e r a c t i o n p o t e n t i a l of tlir g a s molecules with molecules of t h e s o l i d . \'!hen a molecule i::p i n c e s upon s s u r f a c e , it i s adsorbed and may form chemi cn.1 bonds, d i s s o c i a t e , become i o n i z e d o r d i s p l a c e s u r f a c e atoms. The simplest p o s s i b l e model of t h e pas-surface
interaction
i s t o assume t h a t t h e molecules a r e s p e c u l a r l y r e f l e c t e d a t
t h e s o l i d boundary. T h i s assumption i s extremely u n r e n l i s t l c i n e e n e r a l and can be used only i n p a r t i c u l a r c a w s . Tn peneral, a molecule s t r i k i n g a s u r f a c e a t a v e l o c i t y from it a t a v e l o c i t y
-
9 /
reflects
which i s s t r i c t l y determined only
i f t h e path of t h e molecule within a w b l l can be computed exac-
t l y . T h i s computation i s impossible because i t depends upon a g r e a t number of d e t a i l s , such as t h e l o c a t i o n s and v e l o c i t i e s of a l l t h e molecules of t h e wall. Hence vze m&y only hope t o compute t h e p r o b a b i l i t y d e n s i t y
R
&' - 4 5 )
thrt
2
no1 cc:r? e
e' f. +df .
s t r i k i n g t h e surface with v e l o c i t y between emerces with v e l o c i t y between
-t
and
and If
-fi+dY -
re-
R i s hown,
@
i t i s easy t o w r i t e t h e boundary condition f o r
where g i s t h e u n i t v e c t o r nonnal t o t h e wall and we assumed must be replaced t h e wall t o be at r e s t (otherwise
2,
by
f-%,tL% , denot i n s t h e w a l l 's v e l o c i t y . ) I n ~ e n e r a l ,R w i l l be d i f f e r e n t a t d i f f e r e n t p o i n t s o f t h e
w ? 1 1 and d i f f e r e n t times; t h e dependence on 5 and t i s not shown
e x n l i c i t l y t o make t h e equations shorter. If t h e wall r e s t i t u t e s all the e a s molecules ( i . e .
i t i s non-
porous m d nonadsorbing 1, t h e t o t a 1 p r o b a b i l i t y f o r an impinginp aolecule t o be re-emitted,
with no matter what v e l o c i t y
-I
is
mity:
A n obvious property of t h e k e r n e l
Kt!?!) i s
t h a t i t cannot
assume negative values
Another b a s i c property of t h e kernel
R
,
which can be cal-
l e d t h e " r e c i p r o c i t y laww o r t h e " d e t a i l e d balance", a s follows
where
$&)
[I,
21
is
written
:
is pmportlond t o
u*p[-.!!y(2KQJ,where
To i s
{(S)
t h e temperature of t h e w e l l ( i n o t h e r wonls,
j5 E ;:F.Y-~?-
l i a n d i s t r i b u t i o n f o r a pas a t r e s t a t t h e t e n n e r a t u r e of t h o w a l l 1. We n o t e a simple consequence of r e c i p r o c i t y ; i f t l - e
4
d i s t r i b u t i o n i s t h e w a l l ilaxwellian at t h e w a l l according t o E q . ( l l ) ,
md aess
IS
Y
--rln u j
- .-
ror.t.rne6
t h e n t h e d i s t r i b u t i o n fimctior,
of t h e emerging molecules i s a g a i n
4
or, i n o t h e r words, t h e
w a l l >Iaxwellian s a t i s f i e s t h e boundary c o n d i t i o n s . I n f a c t ,
we i n t e p r a t e Eq. ( 1 3 ) with r e s p e c t t o obtain
-8'
1f
and u s e Eq. (1 1 ) we
and t h i s equation proves o u r statement, according t o Eq. (10). It i s t o be remarked t h a t Eq. (14 1, although a consenuence o f
Eq. ( 1 3 ) (when Eq. ( 1 1 ) h o l d s ) i s l e s s r e s t r i c t i v e t h m Xq.
(1')
and could be s a t i s f i e d even i f Eq. ( 1 3 ) f a i l e d . A s a consequence of t h e above p r o p e r t i e s , one can p m v e [2]
t h e f o l l o w i n g remarkable theorem: Let
C ( I ) be
arpment
2.
a s t r i c t l y convex continuous f u n c t i o n of i t s
Then f o r any s c a t t e r i n p k e r n e l
R(k'd3)
sati-
s f y i n e Eqs. (11 ), (121, ( 1 4 ) , t h e follow in^ i n e q u a l i t y h o l d s
where
g
i s t h e w a l l Idaxwellian,
3 = $/fo
and i n t e ~ r e t i o n
extends t o t h e f u l l ranges of v a l u e s of t h e components of' t h e v a l u e s of
e=
4
through Eq. (1.6).
for
Zq9Jbeing
r e l a t e d t o those f o r ) . 9 C ~
E q u a l i t y i n Eq. ( 1 5 ) h o l d s i f and only if
almost everywhere, u n l e s s
R (EL5 )
i s proportionzl
t o a d e l t a function. d s a corollary, t h e following inequality holds
- h e r e [q*%Jd
denotes t h e normal h e a t f l u x fed.
:
[2)
i n t o the g a s
by t h e s o l i d c o n s t i t u t i n g t h e w a l l and R i s t h e g a s c o n s t m t . We want t o g e n e r a l i z e t h e H-theorem,
considered
in
0. L a -
f o r d ' s l e c t u r e s , t o t h e c a s e of a g a s bounded by s o l i d w a l l s which may o r may n o t be a t r e s t . To t h i s end we d e f i n e
and observe t h a t
{A
RUZ
w i t h r e s p e c t t o i from 1 t o 3 i s understood).
Mow, t h e f o l l o w i n g i d e n t i t y h o l d s f o r any t h e i n t e g r a l s make sense:
~ , fa,
provided
This i d e n t i t y f o l l o w by s t r a i g h t f o r w a r d manipulations; f o r details, see
rl
,q
.
Applying Eq. ( 2 0 ) t o t h e c a s e obtain
P = (D8f ,$I{
(430)) we
where t h e i n e q u a l i t y f o l l o w s from t h e f a c t t h a t
(1-A) &8 ;\
is
always n e g a t i v e , except f o r )( = 1 , where i t i s zero. Hence eq u a l i t y i n Eq. ( 2 1 ) i s v a l i d i f and only i f
or letting
denote
(P+E =P*?'!* I
1
Q = 1 and, a s R y = r ; ( i = 1,2,2) and
T h i s equation i s s a t i s f i e d t r i v i a l l y by consequence of Eqs. ( 4 ) and (5),by
p ;it can be shown
Qr
[2
]
t h e r e a no o t h e r li-
that
n e a r l y independent c o l l i s i o n i n v a r i a n t s (such i s t h e neme f o r the solutions
of Eq. (23)). A s a consequence, t h e most qene-
r a l d i s t r i b u t i o n f u n c t i o n s a t i s f y i n g Eq. ( 2 2 ) i s given by
where a,
2 ,c
a r e constant. Eq. (24)
b e rewritten i n the
f o l l o w i n g form
where
9 , y,T
- -
a r e new c o n s t a n t s r e l a t e d t o t h e previous
onesand have t h e meaning of d e n s i t y , mass v e l o c i t y a d temper a t u r e a s s o c i a t e d with t h e d i s t r i b u t i o n f u n c t i o n t o well-known formulas
El ,2]
distribution. Eqs. (19) and ( 2 1 ) imply t h a t
.
Eq.
f
according
(25 ) gives a I~axwellian
where t h e e q u a l i t y s i g n a p p l i e s i f ar.d only i f l i e n , i.e.
f
i s Xaxwel
-
i s given by Eq. (25 ).
I f we i n t e g r a t e , both s i d e s of Ea.
( 2 6 ) with resnect t o 2
nver a region R b o u d e d by s o l i d walls, we have, i f t h e boun3 9 of R moves with v e l o c i t y .k,:
dary
(27)
where
dS
i s a surface element of t h e boundary
and
9R
;
t h e inward. normal. The second term i n t h e i n t e g r a l comes from the f a c t t h a t , i f t h e boundary i s moving, when forming t h e time d e r i v a t i v e of H we have t o t a k e i n t o account t h a t t h e region of in.tegration changes with time. I f we u s e Eq. (16),
when we replaced
Eq. (27) becomes:
f
by J-%. i n Eq. ( 1 6 ) as required. Eq. ( 2 6 ) ~ e n e r a l i z e st h e H-theorem, showing t h a t H decreases with
time if t h e r e i s no heat exchange walls.
~ l s o ,e q u a l i t y i n Eq. ( 2 8 ) a p p l i e s i f and only i f
lr-ellian. where
between t h e gas and t h e
1
4
is
-
Eq. ( 2 8 ) suggests t h a t
H,. be i n t e r p r e t e d a s ?/? i s t h e entropy of t h e gas, s i n c e it s a t i s f i e s t h e
i n e q u a l i t y (Clausius-Duhem inequality). This i d e n t i f i c a t i o n i s v a l i d a t e d by evaluatinq H a t equilibrium, when h ~ v et h e form indicated i n Eq. ( 2 5 ) ; i n such a case
turns out t o have t h e same dependence on t ropy i n ordinary thermodynamics.
P
and
f m~st
)LC- RH as the en-
Let u s now b r i e f l y examine t h e problem of solving t h e Boltzrnann equation; because of t h e nonlinear n a t u r e of t h e c o l l i s i o n
term
,
t h i s i s a d i f f i c u l t problem. A very p a r t i c u l a r
c l a s s of s o l u t i o n s i s o f f e r e d by ? f a x w e l l i e ~dist ri.bt:tj.ons, So.
(25 1, which d e s c r i b e s t a t e s c h a r a c t e r i z e d by t h e f a c t that r . ~ i t b e r hea.t f l u x n o r s t r e s s e s o t h e r tha? i s o t r o p i c pressurn are n r e s e n t . i f we want t o d e a c r i b e more r e a l i s t i c nonsquili'oriur:! s i t u a t i o n s , vie have t o rely upon approxi.cata methods, t y n i c e l 1 . v -perturbation t e c h n i q u e s , The sj.molest approach i s
where
4
which
mpy
ti0
virj.te
i s a Maxwellian end
g i s a "small pnraneter", o r nay n o t appear i n t h e Eoltzmzrm equ?ti.on. I n t h c w i l l appear i n t h e i n i t i a l and hour,dnxy concli
second case,
t i o n s and t h e e q u a t i o n f o r
kL
-
w i l l be
where
i s c a l l e d t h e l i n e a r i z e d Boltzmann operator. 3q. ( 3 0 ) , j.n t u r n ,
i s c a l l e d l i n e a r i z e d Boltzmann equation.
H i l b e r t space
then
L
'p
where t h e s c a l a r product i s given bv
i s a symmetric o p e r a t o r io
I n addition,
L
If one i n t r o d u c e s a
i s non-nepative
2:
pad t h e equzllt,!r
in.vsrimt. : n
i.e.
s i g h o l d s '1
4.n
a collisio~
zu?h a c s s e
t h e c o l l l s r 3 n l n v a r i m t s are eigenfunctlons assoclafed
with t h e f i v e f o l d tor
m d onl;.. i f h
t .,411 t h e s e
degenerate e i r e n v a l u e
= 0 of t h e open=-
propertjeJfollow inmediately f mrn Eq. ( ? ?
(20), i f t h e circumstance t h a t
fe
)
and
s a t i s f i e s Eq. ( 2 2 ) i s pro-
n e r l y taken i n t o account. 39
(35) su(~(re8-t. i n v e s t i g a t i n g t h e spectrum of
p r ~ b l e ma r i s e s when we look f o r t h e s o l u t i o n of Eq. t h e space homogeneo~sc a s e
(aR/aa = 0) .
Eq.
L ;t h i s (30) i n
(34 ) shows t h a t
t h e speotrun i s contained i n t h e n e r a t i v e r e a l s e n i a x i s of t h e
x-?lane;
it t u r n s out t h a t t h e spectrum i s extremely d e v n d e n t
u m n t h e form o f t h e choice of t h e f u n c t i o n
5 (0,V)
appea-
r i n ~i n Eq. ( 3 ) . It i s completely d i s c r e t e f o r t h e c a s e of Hax::~11 molecules, while i t i s p a r t l y d i s c r e t e an3 p a r t l y continuous i:~ t h e c a s e of r i g i d spheres. F o r f u r t h e r d e t a i l s , one should
[7,23
c o n s u l t Ref s.
.
An i n t e r e s t i n g ~ r o h l e nc r i s e s when one i n v e s t i q a t e e t h e so-
l u t i o n s which do n o t d e ~ e n don t i n e t and two space c o o r d i n e t s s , ; i n this c z s e one has t o s o l v e t h e equation ~ ? yx2 and x,
i n t h e unknown
k = &. (x,, b,
between t h i s e ~ a t i o nand Eq.
,k,,h)= (,(x~, - ). The similzr!ty (30) with
t h a t we look f o r s o l u t i o n s of t h e form
a & , / a= ~0
sup~ests
{*ere
satisfies
g
which i s t h e a n a l o g j e of
U=;\k.The f i r s t
question i s i-.rtho-
t h e sol-utions of Eq. (381 a r e s u f f i c i e n t t o c o n s t r u c t t h e p n e r z l s o l u t i o n of Eq.
( 3 6 ) by euperposition. Next c o m e s a study of
t h e s e t of vfilues of (different fron
h
f o r ~vhick!Eq.
( 3 8 ) has a s o l u t i o n
9 = 0).
The problem h e r e i s more d i f f i c u l t beczuse t h e r e i s w play between
L
and t h e m u l t i p l i c a t i v e o n e r a t o r
4 . .1
-
:~.-.r-r.
1-5-
t i o n t h e e x i s t e n c e of t h e c o l l i s i o n i n v a r i m t s s ~ t i s f y i n y :T r .
( 3 5 ) prevents
L
from beinp a s t r i c t l y n e ~ a t i v eoperator. In
s p i t e of t h i s , it i s p o s s i b l e t o show [2]
t h ~ tth e p e n e r a l
s o l u t i o n of Eq. (36) can be w r i t t e n a s follows:
a=O are the five a r e t h e e i g e n s o l u t i o n s of Eq. 8 c o l l i s i o n i n v a r i a n t s yo=f q
0
0 i s e x a c t l y what i s required t o describe
and
a decay e i t h e r f o r
x > y,
or
X,
4 yl
, where
i s the
C
x,
location of a boundary, The general s o l u t i o n given by Eq. (39) then shows t h a t , i f the region where t h e gas i s contafned ( e i t h e r a h a l f space o r a s l a b of thickness
d , because of t h e assumption t h a t
h is
independent of two space coordinates) i s much t h i c k e r than t h e mean f r e e path
l , then %
w i l l be n e g l i g i b l e except i n boun-
dary l a y e r s a few mean f r e e paths thick. These l a y e r s receive the name of "Knudsen l a y e r s " o r "Kinetic boundary layers". Outs i d e them t h e s o l u t i o n i s accurately described by t h e asymptot i c p a r t hA
, defined
by Eq. ( 4 0 ) ; i t can be shown [2]
that i f
we conpute t h e s t r e s s t e n s o r and heat f l u x v e c t o r s a r i s i n g from hA
, they
t u r n out t o be r e l a t e d t o t h e v e l o c i t y and temperature
r r a d i e n t s by t h e NavierStokes-Fourier
r e l a t i o n s ,with t h e f ol-
lowing expressions f o r t h e v i s c o s i t y c o e f f i c i e n t heat conduction coefficient
k
:
and t h e
These r e s u l t s can be extended t o more ~ e n e r ? lr r o t l e - .-
r.2
Very i n t e r e s t i n q problems z r l s e when he j n ~ o : > r l , ' ; - ~r:>>< 1s riot s a t i s f i e d , i . e .
the s l a b thiclmess
t h e mean f r e e math 1 s c o r m - r - P I P rlth
n, 2 1
; t h e i r t r e e t n e n t I.:;, l-onevrr,
1.3-
yond t h e l i m i t s of t h e present eemina,r.
REFERENCES 1
- C.
Cercignani
- **Blathematical! k t h o d s Plenum P r e s s , N.Y.
2
- C.
Cercignani
in K i n e t i c Theor;",
(1969)
- "Theory and Applicztion
of t h e fioltzmsml
Equation1*, S c o t t i s h Academic Press, E d i n burgh (1975).
CENTRO INTERNAZIONALE MATWATICO ESTIVO (c.I.M.E.)
QUALITATIVE AND S T A T I S T I C A L THEORY OF D I S S I P A T I V E SYSTEMS
O s c a r E.
LANFORD I11
D e p a r t m e n t of M a t h e m a t i c s , U n i v e r s i t y of C a l i f o r n i a B e r k e l e y , C a l i f o r n i a 94720
C o r s o tenuto a B r e s s a n o n e d a l 21 a1 2 4 g i u g n o 1 9 7 6
Qualitative
Statistical Theory
of Dissi~ativeSystems -
Oscar E. Lanford I11 Department of Mathematics University of California Berkeley, California 94720
Preparation of these notes was supported in part by NSF Grant MCS 75-05576. A01.
26 Chapter I.
Elementary Qualitative
Theory of Mfferential Eauations.
This series of l e d w e s w i l l be concerned with t h e s t a t i s t i c a l theory of dissipative system and, a t l e a s t metaphorically, with i t s applications t o hydrodynamics.
The priacipal objective will be t o t r y t o
clariFy the question of how t o construct the appropriate ensemble f o r the s t a t i s t i c a l theory of turbulence. t h i s point f o r some time.
We w i l l not, however, come t o
It should be noted a t the outset t h a t the
relevance of our discussion t o t h e theory of turbulence i s dependent on the guess t h a t , despite the fact t h a t f l u i d flow problems have i n f i n i t e dimensional s t a t e spaces, t h e important phenomena are essentially f i n i t e dimensional.*
This point of view i s ncrt universally accepted [ h ] . On
the other hand, the theory i s not restricted t o fluid flow problems; it also applies t o a large number of model systems arising, f o r example, i n mathematical biology 171. The methods we w i l l discuss a r e limited i n that they appear not t o have anything t o say about such traditionally central issues as the characteristic s p a t i a l properties of turbulent flow, t h e d y n d c s of vorticity, etc.
Instead, they attempt t o clarify t h e apparently stoch-
a s t i c character o f t h e f l o w and i t s peculiar dependence-independence on i n i t i a l conditions.
To explain what t h i s means, l e t us look b r i e f l y a t
two important but not completely precise distinctions
- between consenra-
t i v e and dissipative systems end between stable and unstable ones. Intuitively, when we say t h a t a system i s conservative, we mean t h a t , once it has been started i n motion, it w i l l keep going forever without
*It
may be t h a t t h i s ceases t o be t r u e for "fully developed turbulence"
wid that what we say here applies t o turbulence a t r e l a t i v e l y low Rey-
nolds numbers and not a t high Reynolds numbers.
f u r t h e r external driving.
Mathematically t h i s is usually reflected i n
t h e f a c t t h a t t h e equations of motion may b e written i n Hamiltonian f o m , with t h e consequent conservation of energy and phase space volume.
Amon@:
numerous examples, l e t u s note a.
t h e Newtonian two-body problem
b.
t h e motion of a f i n i t e number of f r i c t i o n l e s s and p e r f e c t l y e l a s t i c b i l l i a r d b a l l s on a rectangular table.
These examples i l l u s t r a t e t h e d i s t i n c t i o n between s t a b l e and unstable systems.
The N&onian
two-body system i s s t a b l e i n t h e sense t h a t t h e
e f f e c t s of small perturbations of t h e i n i t i a l conditions grow slowly i f a t a l l and hence t h a t long-term predictions about t h e s t a t e of t h e system are possible on t h e b a s i s of approximate information about t h e i n i t i a l state.
In t h e b i l l i a r d system, on t h e other hand, even very small changes
i n t h e initial s t a t e a r e soon amplified so t h a t they have l a r g e e f f e c t s . I f t h e system i s s t a r t e d out repeatedly, in almost but not exactly t h e same way, t h e long-term h i s t o r i e s w i l l almost c e r t a i n l y be t o t a l l y different.
I n t h i s sense, although t h e motion i s s t r i c t l y speaking
deterministic, it i s from a p r a c t i c a l point of view effectively random; t h e coarse features of t h e s t a t e of t h e system a t large times depend on unobservably f i n e d e t a i l s of t h e s t a t e a t time zero. Consider next d i s s i p a t i v e systems.
I n t u i t i v e l y , these have some
s o r t of f r i c t i o n a l mechanism which tends t o damp out motion and must therefore be driven by external forces i f they a r e not simply t o stop. A mathematical t r a n s c r i p t i o n of t h i s notion which i s as general as t h e
corresgondence "conservative
r ~amiltonian" does not seem t o e x i s t , but
it is ,-ruerally not d i f f i c u l t t o agree on whether a given dynamical system i s dissipative o r not.
We w i l l consider systems driven
~ time-
independent forces, such as a viscous f l u i d flowing through a pipe or e l e c t r i c c i r c u i t s driven by batteries.
I n many cases these systems
display behavior which i s simpler than that of conservative systems
--
they may tend, independent of how they are started out, t o a aynamical equilibrium i n which driving forces a r e exactly balanced by dissipation. A system which tends t o t h e same equilibrium, no matter where i n its
s t a t e space it starts, appears t o forget its i n i t i a l conditions and hence t o be "even more stable" than t h e conservative Newtonian two-body system considered above.
Long-term predictions can be made which don't depend
on the i n i t i a l s t a t e but only on t h e parameters appearing i n t h e equations of motion. The next simplest possible behavior i s t h e existence of a globally
attracting periodic solution or l i m i t cycle.
In t h i s case the equations
) x0(0) of motion admit a s o l ~ i o n x o ( t ) with x o ( ~ =
f o r some r > 0,
and every solution of t h e equations of motion converges t o the s e t
{xo(t): 0
< t < .r)
as t
+ m.
What usually happens i n t h i s situation
is in fact something more special: tl(xl)
nth
0
tl < T
For each i n i t i a l
xl there exists
such t h a t
Although t h e long-time behavior is no longer completely independent of the initial point, the r o l e of t h e i n i t i a l point i s simply t o determine the phase t
1'
Again t h e motion s a t i s f i e s our i n t u i t i v e criterion f o r
s t a b i l i t y ; t h e long-term effect of a s d l change in t h e i n i t i a l point is simply a small change i n the phase.
It is natural t o ask what comes next a f t e r periodic o r b i t s in t h e
hierarchy of complexity f o r dissipative systems.
One plausible guess,
advocated by Landau among others, i s t h a t instead of having a single period, t h e system may have two o r more independent periods
-- i .e .,
t h a t the s t a t e space may contain a t o r u s of dimension two o r greater which i s invariant under t h e solution flow, which a t t r a c t s a t l e a s t nearby solution curves, and on which t h e solution flow reduces i n approp r i a t e co-ordinates t o uniform velocity flow.
Although t h i s c e r t a i n l y
can happen, it i s not l i k e l y t o be common since it i s destroyed by most small perturbations when it does occur.
What t u r n s out t o be much more
l i k e l y i s t h e presence of what have come t o be called "strange a t t r a c t o r s " s e t s invariant under t h e solution flow and a t t r a c t i n g nearby o r b i t s but which, instead of being smooth manifolds l i k e t o r i , have a complicated Cantor-set -like s t r u c t u r e
.
We w i l l present shortly a simple example of
a system with such a strange a t t r a c t o r , but before doing so we need t o introduce some notions from t h e q u a l i t a t i v e theory of d i f f e r e n t i a l equations. The s t a t e s of t h e physical system we a r e considering w i l l be assumed t o form a menifold M which we w i l l t a k e t o b e finite-dimensional (although much of t h e formal theory extends e a s i l y t o infinite-dimensional manifolds).
The equations of motion w i l l be taken t o be first-order
ordinary d i f f e r e n t i a l equations on M which we w i l l write i n t h e c l a s s i -
c a l co-ordinate form
where n
is t h e dimension of
M.
To avoid uninteresting complications
we w i l l assume t h a t the right-hand side is an i n f i n i t e l y differentiable f b c t i o n of
xl,.
..
,X
t
n*
We w i l l denote t h e solution mappings by Tt ,
so T x i s the solution curve passing through x
will assume t h a t , for any x, {T~x:t > 0 )
t
T x
a t time zero.
We
exists for all t > 0 and There a r e many interesting cases in
is relatively campact.
which t h i s condition i s s a t i s f i e d but in which solution curves do not in general exist f o r a l l t < 0; i f t h e s t a t e space M
t h e condition i s automatically satisfied
i s compact.
The mathematical transcription of t h e existence of a dynamic equilibrium t o which the system tends no matter how it i s s t a r t e d out i s as follows:
There e x i s t s a stationary solution xo of t h e equations of
m t i o n such t h a t
t l i m T x = xo
tfor all x E M. tionary solution.
Such an xo is said t o be a Gobally attractinn staMore generally, a stationary solution xo i s locally
attracting i f
for a l l x
in some neighborhood of
x,,.
While it i s generally d i f f i c u l t
t o determine whether a stationary solution is globally attracting, there
i s a simple sufficient condition for a stationary solution t o be a local attractor:
It suffices t h a t the matrix of p a r t i a l derivatives
giving t h e linearized equation of motion st xo have ell i t s eigenvalues in t h e open l e f t half -plane. '
We have already defined what we mean by saying t h a t a periodic solution t o t h e equations of motion is globally attracting; we w i l l similarly say t h a t a periodic solution locally a t t r a c t i n g i f f o r all x set
{z(t): 0
< t < T)
(g(t)
with period
T
is
in some neighborhood of t h e points
we have
To give a linear c r i t e r i o n f o r periodic solution t o be .locally a t t r a c t ing
which i s analogous t o t h e one given above for stationary solutions,
we introduce t h e notion of t h e Poincard map associated with a periodic solution.
Take a
gmall
piece
t o t h e periodic solution.
E of n-1 dimensional surface transverse
For each y
t h e periodic solution define @(y) solution curve
on
C and sufficiently near t o
t o be t h e f i r s t point on t h e forward
{TtY: t > 0) which i s again in
point where t h e periodic solution crosses
E,
t. I f
then
yo
denotes t h e
@(yo)= yo.
In
order t h a t the periodic solution be l o c a l l y attracting it is sufficient t h a t the derivative matrix
have all i t s eigenvalues i n t h e open unit disk. some s e t
yl,.
..,ynml
ere, we have
of l o c a l co-ordinates f o r C
chosen
and expressed
@
i n terms of these co-ordinates.) The two simple situations described above solutions and attracting periodic solutions related v i a t h e
biiurcation.
- attracting stationary
- t u r n out t o be closely
suppose t h a t our d i f f e r e n t i d equation
r which may indicate, f o r example, how hard
depends on a parameter
t h e system i s being driven.
Suppose also t h a t f o r some value
a stationary solution x = xr
rc of
r
changes from stable t o unstable by having
a ccanplex conjugate pair of eigenvalues for t h e linearization
at
xr
cross from t h e l e f t t o t h e right half-plane at non-zero speed.
It turns out t h a t , under these circumstances, i f a certain complicated
combination of the f i r s t , second, end t h i r d p a r t i a l derivatives of with respect t o x
at x = xr,
s l i g h t l y larger than
F
r = rc, is positive, then for r
rc there is an attracting periodic solution which
can be regarded as making a small c i r c l e around t h e now-unstable station-
ary solution xr.
As
r decreases t o rc t h e periodic orbit shrinks
down t o t h e single point
xr
solution undergoes a normal
.
In t h i s case we say t h a t t h e stationary bifurcation t o a periodic solution.
is also possible f o r t h e above-mentioned complicated combination of
p a r t i a l derivatives of
F t o be negative.
In t h i s case no attracting
It
periodic solution i s formed.
Instead, f o r r s l i g h t l y smaller
r
as
increases t o
rc,
xr which shrinks d m t o
t h e r e e x i s t s an unstable periodic solution near x r
than
rc. In t h i s case we s a y t h a t t h e stationary
C
solution undergoes an inverted bifurcation.
(Other, more complicated,
things can happen i f t h e combination of partial derivatives i s zero, o r i f nore than two eigenvalues cross t h e imaginary axis simultaneously. ) For a detailed discussion of t h e Hopf b i f u r c a t i o n end r e l a t e d phenomena, see [ 6 1. We next need some more general notions which apply even i n t h e absence of stationary and periodic solutions. s t a t e space we define t h e
tn
+
-.
implies t h a t
s e t of
The assumption t h a t w(x)
i s not empty;
t {T x: t w(x)
t invariant under t h e s o l u t i o n flow T verges as
t
x,
denoted by
x
dx),
in the t o be
t ~ ( x ) i s t h e s e t of a l l c l u s t e r points of sequences T "x
Alternatively, with
*limit
For any point
+
-
.
> O}
i s r e l a t i v e l y compact
i s a l s o evidently closed and I f t h e solution curve
t o a s t a t i o n a r y solution
xo
conversely, if ~ ( x )contains only one point
then xo
t Tx
con-
w(x) = {x0} ;
then
xo
is a s t a t i o n -
ary solution and
l i m T'X = xo. Similar statements hold f o r solution tcurves converging t o a periodic solution. Let
t Then every forward solution c w e T x
converges as t
-+
-
to
5,
and
5
i s t h e smallest closed s e t with t h i s ~ropert'ty. I n order t o under-
stand t h e behavior of solution curves Tor large positive times, it is enouqh t o study t h e solution flow on and near
6,
i-e.,
is t h e
essential part of t h e s t a t e space from t h e long-tern point of view.
One
important difference between dissipative and non-dissipative systems i s "
that
fl tends t o be sma3.1 f o r dissipative systems and t o be t h e whole To see the l a t t e r fact we f i r s t
s t a t e space f o r conservative systems. note:
Proposition. T~
Any finite measure
ha8 support in
Proof.
)1
invariant under the s o t u t i a flow
6.
We want t o show t h a t
ydp = 0 for any continuous f b e t i o n
whose support i s compact and d i s j o i n t from of
p
under
fi.
By t h e assumed invariance
t T ,
Now, since t h e support of
(p
is disjoint from
lim ~ ( T ~ =x o)
t*
for
6, x
so by t h e dominated convergence theorem
I f , f o r example, we consider a Hroniltonian system with t h e property
that
{x: ~ ( x ) 9 E)
i s compact f o r each
E, then Licuville's Theorem
implies t h a t every point of t h e s t a t e space i s i n t h e support of some invariant measure and hence, by t h e proposition, t h a t
-
fi i s t h e whole
s t a t e space. We w i l l say t h a t a point for
T~
x
of t h e s t a t e space i s a wanderinq point
i f t h e r e i s a neighborhood
Tt U n U =
The non-wandering
U
of
x
such t h a t
f o r a l l suffliciently large t
R is t h e s e t of
.
all points which a r e not wander-
It follows a t once from t h e definition t h a t t h e s e t of wandering
ing.
points i s open and hence t h a t
Q i s closed; it i s also easy t o see t h a t
f o r all x.
.
Hence,
5
6 $ Q,
equality does hold i n most i n t e r e s t i n g cases (including flows
CQ
While it i s not d i f f i c u l t t o construct flows f o r which
which s a t i s f y h a l e ' s Axiom A, t o be described below).
If t h e solution
flaw has a globally a t t r a c t i n g stationary solution xo, then
a similar statement holds when t h e r e i s a globally a t t r a c t i n g periodic solution. The simplest asymptotic behavior a d i f f e r e n t i a l equation can have
i s f o r all solution curves t o converge t o t h e same stationary or periodic solution.
One trivial way in which t h e s i t u a t i o n can become more compli-
cated i s to have several locally attracting stationary and periodic solutions.
If
xo
i s a locally attracting stationary solution define
the basin of attraction
of
t o be
t B ( I X ~ I=') {x: lim T x = x0} t-
.
One can define in a similar way t h e basin of attraction of a periodic solution.
A basin of attraction i s open and invariant under t h e solution
('The fact that it i s open may not be quite obvious.
flow.
case of a l o c a l l y attracting stationary solution xo the basin of attraction of
xo.
Consider t h e
and l e t x be i n
Because x 0 is l o c a l l y attracting there
i s a neighborhood V of
xo such t h a t every solution curve beginning t in V converges t o %, and because T x converges t o xo there t exists t such that T x E V. Then (T '1-4 i s an open neighborhood 0 of
x
contained i n the basin of attraction of
x,,. ) Note t h a t , since
basins of attraction are open and disjoint, the s t a t e space M,
if
connected, cannot be written as t h e union of two o r more basins.
Hence,
i f there a r e a t l e a s t two locally attracting stationary o r periodic solution, there must be some solution curves (lying on the boundaries of t h e basins of attraction) which do not converge t o any of them. We are going t o investigate a t t r a c t o r s which are more complicated than single points and periodic solutions, and we should therefore define precisely and generally what we mean by an a t t r a c t o r .
Unfortunate-
l y , no such definition seems t o be agreed upon, so we w i l l improvise by
l i s t i n g a number of properties which an a t t r a c t o r ought t o have, being careful t h a t t h e conditions are indeed s a t i s f i e d in t h e special case of Axiom A a t t r a c t o r s , where t h e r e does exist an accepted definition ( t o be
discussed below).
To begin with, an a t t r a c t o r should be a closed
(compact?) subset
of t h e s t a t e space, invariant under t h e solution
X
f l a w , which a t t r a c t s nearby o r b i t s i n t h e sense t h a t there exists an
open s e t U
containing X
such t h a t , for any x
in U
t l i m d ( x~, ~ )= 0 tta,
(or equivalently,
.
w(x) C X )
then
t T x remains near X
B(x)
of
X
Second, we require t h a t i f
for all t > 0.
x i s near X
The basin of attraction
i s now defined t o be
The argument given above f o r attracting stationary solutions i s e a s i l y adapted t o show t h a t
B(X) i s open.
We also want t o put i n t o the defini-
tion some condition which prevents an a t t r a d o r from being decomposable i n t o a f i n i t e number of other a t t r a c t o r s ; a good way t o do t h i s i s t o require t h a t some solution curve contained i n t h e attractor i s dense i n t h e a t t r a c t o r , i.e.,
that t h e solution flox T~
t o p o l o ~ i c a l l yt r a n s i t i v e .
restricted t o
X
is
Nothing i n t h e above l i s t of conditions pre-
vents t h e whole s t a t e space from being an a t t r a c t o r ; it w i l l be one i f t h e r e i s a single solution curve dense in t h e whole s t a t e space, and t h i s frequently happens f o r conservative systems i s taken t o be a single energy surface).
( i f t h e s t a t e space
For dissipative systems, on the
other hand, a t t r a c t o r s w i l l generally be s m a l l a t l e a s t in t h e sense of having empty interiors. The preceding discussion has considered only continuous flows
T~.
For many purposes it is useful t o have a p a r a l l e l s e t of definitions f o r the corresponding discrete situation, the s e t of powers single smooth transformation
T of a manifold.
{ T ~ )of a
The task of adapting our
discussion t o t h i s s l i g h t l y different context i s straightforward; we leave it t o t h e reader. Up t o t h i s point, we have been dealing with elementarg general considerations.
Although necessary i n order t o get s t a r t e d , t h e ideas
developed so f a r do not seem t o be sufficiently specific t o lead t o any very interesting analysis.
In order t o go further we must impose addi-
t i o n a l r e s t r i c t i o n s on the systems we study.
In recent years, it has
turned out t o be particularly f r u i t f u l t o impose some s o r t of hyperboli-
city condition.
The fundamental reference i n this area is [ 1d; we w i l l
sketch here a few of t h e basic ideas. W e expleined above t h a t a stationary solution xo
i s a t least a
local a t t r a c t o r i f all eigenvalues of t h e matrix
are i n t h e open l e f t half-plane.
More generally, we say that
hyperbolic stationary solution i f no eigenvalues f o r DF on the imaginary axis.
xo i s a
are precisely
In this case. Efn s p l i t s into two complementary
subspaces
ES and EU, each invariant under DF,
values of
have s t r i c t l y negative real p&s while t h e eigenvalues ES have s t r i c t l y positive r e a l parts ; and E~ are called
of
such that the eigen-
DF~
DF~ En respectively t h e stable and unstable eigenspaces f o r DF. ?enote the linearization of
F a t xo, i .e.
I f we l e t
,. F
then a solution x"(t)
as t
converges t o xo to
xo
at
t
+
of t h e linearized equation
-
-
+
-
i f and only i f
i f and only i f
i ( 0 ) € xo + E'
and converges
;(o) € xO + E ~ . Going back t o t h e
f u l l (non-linear) equation we define t h e s t a b l e and unstable manifolds a t xO ( f ' ( x o )
wU(x0) respectively)
by
t wU(rO) = {x EM: T x
-+
xo
as t
+
- -1
From t h e definition it i s not apparent t h a t these s e t s are submanifolds, o r even t h a t they contain any points other than
xo,
Stable Manifold Theorem f o r Hyperbolic Fixed Points.
submmifolde* of
M,
but we have:
f' (xO) wd
w-ith dimensions equal ~espectiveZyt o
wU(x0) are
dim ~'(x,,)
and dim EU(rO). These ~ u b n m i f o Z Lcontain xo and are tangent a t xo t o E ' ( X ~ ) and
respectiveZy.
*
There i s a technical d i s t i n c t i o n which needs t o be noted here. The s t a b l e and unstable manifolds are immersed but not i n general imbedded submanifolds of M. This means t h a t , although made up of countably many smooth pieces, t h e y can fold back a r b i t r a r i l y near themselves. A simple example of an immersed one-dimensional submanifold of ~3 i s t h e "Lissajous figure"
whce
u1,u2,
end ul/u2
a r e all i r r a t i o n a l .
With these elementary examples for motivation, we will now give a general definition of hyperbolic set.
There are i n f a c t two definitions,
t
one for transformations 4 and one f o r flows T the definition f o r transformations. and invertible mapping and l e t by A.
Thus, l e t
.
We w i l l give only
9 be a differentiable
A be a compact s e t mapped onto i t s e l f
For t h e sake of concreteness we w i l l assume t h a t a single s e t of
co-ordinates can be chosen f o r an open s e t containing A ( i .e., we w i l l a d as i f
A
is contained in Eln); there i s , however, no d i f f i c u l t y in
eliminating t h i s assumption by giving a coordinate f r e e version of t h e definition.
We define t h e derivative of
4
at x
t o be t h e n
x
n
matrix
end similarly define
D O ~ ( X )f o r any integer m
( ~ o s i t i v eor negative).
EJy t h e chain r u l e
We are going t o define A t o be a hyperbolic set f o r
cP
i f , roughly
speaking, any infinitesimal displacement from a point
x
belonging t o A
can be decomposed as t h e sum of two infinitesimal displacements, o n e p f
which contracts exponentially under positive powers of
O
of which contracts exponentially under negative powers of
2recisely: x E h
h
i s a hyperbolic
set
for O
and the other O.
More
i f t h e r e e x i s t s f o r each
a s p l i t t i n g of B~ i n t o a direct sum of complementary subspaces
z 3 ( x ) , E~(X) such that:
Fur some c > 0, 1 < 1, which do not depend on
x
or
m,
I n addition t h e s p l i t t i n g i a required t o be invariant under 8 :
and t o vary continuously with x. ing:
For every xo € A,
such t h a t
ns = aim ~ ' ( x ) and n
,. ..,&(XI
ESb)
5
t h e r e e x i s t s an open neighborhood U
such t h a t t h e r e e x i s t El(x)
This l a s t condition means t h e follow-
n
of
U
xo
= dim EU(x) a r e constant on U
and
cont inuoua IRn-valued functions
defined on
is spanned b y c1(x)
u nA
such t h a t , f o r each x E A
,...,cn,(x)
n U,
and EU(x) is spanned by
b),...,$,(XI. Alternatively, a s i t t m r r o u t , i t i s e n o u g h t o
ns+l require t h a t
i ( x , ~ ) :E E E'(x))
a r e both closed subsets of
A x R";
and
'
{ ( x , ~ ) :5 E ~ ~ ( x ) )
continuity as formulated above then
follows automatically. It i s p a r t i c u l a r l y i n t e r e s t i n g t o apply t h i s definition with
mn-wandering s e t of
8.
We say t h a t
8 s a t i s f i e s Axiom A
if
h
the
1.
2.
i s a hyperbolic s e t
The periodic points f o r 8
are dense in
n.
This condition has proved very f r u i t f u l for mathematical analysis.
It i s ,
on the other hand, hard t o verify in practical applications and non-trivi a l examples axe r e l a t i v e l y scarce.
It i s a t t h i s time s t i l l too early
t o decide whether Axiom A as it stands is too r e s t r i c t i v e t o apply t o cases of i n t e r e s t , but e i t h e r it o r some weakened version of it seems l i k e l y t o p l w en important r o l e i n future developments. It may be helpful t o note here one difference between hyperbolic
fixed points and more general hyperbolic sets. x
cannot be an a t t r a c t o r unless
A hyperbolic fixed point
E ~ ( S )is t r i v i a l .
r e s t r i c t i o n f o r general hyperbolic s e t s .
There i s no such
It frequently happens t h a t a
hyperbolic a t t r a d o r is made up l o c a l l y of i n f i n i t e l y many smooth "leaves"
- lower-dimensional
surfaces which are everywhere tangent t o
~ ~ ( x )Two . nearby points on t h e same leaf mve apart under t h e action of the transformation, but t h e whole assembly of leaves is attracting.
Chapter 11. The Lorenz System
We t u r n now from generalities t o a discussion o f a p a r t i c u l a r system of equations.
This system could hardly be simpler
-- t h e
state
space i s three-dimensional and t h e equations a r e
with b ,o,
r
positive constants
- but
it displays a bewildering
assortment of non-trivial mathematical phenomena.
SO far as I know, t h i s
system of equations was f i r s t seriously investigated by E. N. Lorenz [ 5
1
some f i f t e e e n years ago; i n recent times it has been studied intensively by Yorke , Guckenheimer [ 3 ] , Martin and McLaughlin McCracken [ 6
1, and
Williams [ 123, emong crthers
[ 81, Marsden and
.
One of t h e appealing aspects of t h e Lorenz system is t h e fact t h a t
it was not constructed f o r t h e purpose of proving t h e p o s s i b i l i t y of complicated behavior; r a t h e r , it turned up i n t h e course of a p r a c t i c a l investigation.
In h i s o r i g i n a l paper, Lorenz was l e d t o t h i s system
by t h e following considerations:
Consider t h e equations of motion f o r two-
dimensional convection i n a container of height
H
and length
L.
These
equations can be viewed, h e u r i s t i c a l l y a t l e a s t , as a f i r s t order differe n t i a l equation on an i n f i n i t e dimensional s t a t e space; t h e points of t h e
s t a t e space are p a i r s consisting of a stream fbnction temperature f i e l d T (x,z)
+(x,z)
and a
subject t o appropriate boundary conditions.
Look f o r solutions of t h e form
where T-, T+
denote t h e temperatures at t h e bottom and t h e t o p of t h e
container respectively.
(Such solutions correspond t o free o r no-stress
boundary conditions on the velocity f i e l d and t o t h e absence of heat flow through t h e ends of t h e container.)
Express t h e equations of
motion d i r e c t l y i n terms of t h e Fourier coefficients
€lm,n(t), I J J ~ , ~ ( ~ )
and d r a s t i c a l l y truncate t h e resulting i n f i n i t e s e t of coupled differe n t i a l equations by putting a l l except
IJJm,n,8m,n identically equal t o zero
1111,l'81,lS 8 Now put 0,2-
and choose
cl, c2, c3 so as t o simplify the d i f f e r e n t i a l equation; t h e
result i s t h e Lorenz system. With t h i s derivation canes a physical interpretation f o r t h e parameters
b,U,r.
Specificslly,
b
is a simple geometric constant
(4/(1+ ( H / L ) ~ ) ) ,U is t h e Prandtl number (i.e., t h e r a t i o of viscosity t o thermal conductivity) and r
i s a numerical constant times t h e Rayleigh
number, i.e., difference
i s a dimensionless rider proportional t o t h e tenperature (T-
- T+).
and 10 respectively.
The values of
b
and o will be fixed a t 8 / 3
We w i l l f i r s t discuss schematically how t h e
behavior of t y p i c a l solutions changes with what happens f o r a p a r t i c u l a r value of
r ; then describe i n d e t a i l
r.
We begin with a number of elementary observations about t h e Lorenz system: i)
he
The equations are invariant under t h e transformation
physical origin of t h i s symmetry i s invariance of t h e equations of
m t i o n unde'r r e f l e c t i o n through a v e r t i c a l l i n e a t t h e center of t h e
.
container ) ii)
The solution flow
T~
volumes i n t h e s t a t e space lR3
generated by t h e Lorenz system shrinks a t a uniform r a t e .
This follous from
t h e equation
This r a t e i s in f a d quite l a r g e ; a s e t of s t a t e s occuping unit volume a t time zero occupies only t h e v o l ~e iii) values of
A l l solutions are bomded f o r
X,Y,Z
= t > 0,
a r e damped by t h e motion.
at time one. and very l a r g e i n i t i d
To show t h i s we introduce
An elementary computation gives
with constants on
r ,U,b.
cl, c2 which do not depend on
X,Y ,Z,
but may depend
he essential point is t h a t , despite t h e quadratic terms
i n t h e equations of motion, there are no cubic terms i n
du at .)
It
follows e a s i l y t h a t every solution curve eventually gets and stays i n t h e i n t e r i o r of t h e b a l l
B where u ( 2 c2/c1. This b a l l i s mapped t into i t s e l f by T , and by ii) t h e volume of its image under T~
goes t o zero as t
goes t o i n f i n i t y .
TbX converges t o large t
Hence, every solution curve
t o the set
which i s closed and has Lebesgue measure zero. f o r T~
The non-wandering s e t
is contained in t h i s intersection and therefore also has
measure zero. We now describe what happens as r is varied s t a r t i n g from zero. Recall t h a t , in t h e derivation f%m t h e convection equations,
r was
yroportional t o t h e imposed v e r t i c a l temperature difference and is therefore a measure of how hard t h e system i s being driven.
For
r
between zero and one, inclusive, it i s not hard t o show t h a t
2
is
a globally attracting stationary solution.
As
r
is increased past
one, t h i s solution becomes unstable and bifurcates i n t o a pair of locally a t t r a c t i n g stationary solutions
C=
(m)
, r-1) ,
.
$2
-C '
= (-~bm -A=) ), ,r-1).
These are easily checkecl t o be t h e
only stationary solutions aside from
2
; they remain present, but
not necessarily stable, f o r all r > 1. Physically, they reuresent steady convectim.
Also for a l l r > 1 t h e stationary solution
is
hyperbolic, with a two-dimensional stable manifold and a one-dimensional unstable manifold.
For
r s l i g h t l y greater than one, nearly a l l solution
curves converge e i t h e r t o C or t o
C) f o r large time; t h e only exceptions
are those on t h e two-dimensional stable manifold of The two steady convection solutions
r = 470/19
3
before t h m .
A t a special value of
f o r s l i g h t l y l a r g e r values of orbits. to
< remain stable u n t i l
but various interesting things happen
24.74 ( f o r u = 101,
sional unstable manifold for
and
E.
r
around r = 13.9 the one dimen-
returns t o
2
(homoclinic o r b i t ) , end
r there are two unstable hyperbolic periodic
It is not known whether t h e non-wandering s e t
{c,c',~a l l the way up t o t h i s value
of
r o r whether periodic
solutions or other kinds of recurrence appear e a r l i e r . t h e appearances are t h a t , u n t i l
R remains equal
r i s nearly equal t o
Nevertheless, 470/19, a l l
solution curves except f o r a s e t of measure zero converge t o one of This does not remain true a l l t h e way up t o
or
what happens f o r
r's
C
r = 470/19, but
s l i g h t l y below t h a t c r i t i c a l value i s most easily
understood i n terms of what happens above it. As
r passes t h e c r i t i c a l value of
470/19, both
C and
become unstable through having a complex-conjugate p a i r of non-real eigenvalues cross i n t o t h e right half plane.
This does n o t lead, v i a
a normal Hopi bifurcation, t o stable periodic orbits near
C
and
5'
for
r
s l i g h t l y above
470/19.
Instead (see [ 8
1, [ 6 1),
what
happens i s an inverted Hopf bifurcation i n which an unstable periodic solution contracts t o each of to
470/19.
c , ~ 'and
disappears as
The behavior above t h e c r i t i c a l v a u e of
r
increases
r seems not
t o be accessible t o analysis by "infinitesimal" b i f u r c a t i o n theory but requires a global investigation of t h e behavior of solutions which has so f a r been possible t o carry out only by following solutions nlnnerically on a computer.
The next stage of our discussion w i l l be
a description of t h e r e s u l t s of such a numerical investigation, carried
r = 28.
out f o r t h e a r b i t r a r i l y chosen value
In t h e v i c i n i t y of any one of t h e t h r e e s t a t i o n a r y solution
2,C,Cf
,
t h e m t i o n i s similar t o t h a t given by t h e l i n e a r i z a t i o n a t t h e s t a t i o n a r y solution.
For each of t h e
of complex eigenvalues
2
.094
+
and
C' ,
10.2 i
t h e l i n e a r i z a t i o n hes a p a i r
and a negative eigenvalue -13.85.
Hence, i n t h e l i n e a r i z e d motion, t h r e e things a r e going on a t quite d i f f e r e n t speeds: i ) t h e compnent in t h e negative eigendirection damps out
rapidly ii )
t h e component i n t h e two-dimensional r e a l eigenspace associated
with t h e complex eigenvalue p a i r r o t a t e s a t a moderate speed and d s o iii)
expands slowly.
More s p e c i f i c a l l y t h e "rotation period'' i s component i s multiplied by about 2 x
lo4
2nh0.2 = .62; t h e ccmtracting f o r each r o t a t i o n ; and the'
r o t a t i n g component expands by about 6% with each r o t ation.
The same
q u a l i t a t i v e picture holds f o r t h e correct (not linearized) motion near
-C
and
2'
.
Passing through each of these points i s a two dimensional
surface, i t s unstable manifold, which strongly a t t r a c t s nearby solution curves and along which solutions s p i r a l slowly outward.
The approximate
appearance of a t y p i c a l solution curve i s shown in figure 1.
Figure 1
The normal t o t h i s surface a t
C
has polar angle 70' with respect t o t h e
Z axis and azimuthal angle 153O with respect t o t h e X
axis.
A s it turns out, t h e r e i s quite a l a r g e domain around each of
where t h i s picture is q u a l i t a t i v e l y correct.
2, C'
This does not yet t e l l us
much about t h e asymptotic behavior o f t y p i c a l solutions since t h e steady growth of t h e r o t a t i n g component eventually drives t h e solution curve
out of t h i s domain, and we have t o look a t where it goes.
The key t o
understanding t h e recurrent behavior of t h e Lorenz system i s t h e fact t h a t it usually goes i n t o t h e corresponding d d n around t h e other steady convection solution, where it i s again a t t r a c t e d t o t h e unstable manifold, eventually pushed out again, returns t o t h e o r i g i n a l domain,
and proceeds t o repeat t h e whole process.
The r e p e t i t i o n i s , however,
typically only approximate and may d i f f e r quite a l o t from t h e f i r s t cycle i n d e t a i l .
Although most o r b i t s continue forever s h u t t l i n g back
and forth between
and
C' ,
they are only exceptionally periodic
o r even asymptotic t o periodic solutions. To form a more precise picture of t h i s process we take a section with t h e horizontal plane
Z = 27 containing
C
and
c'.
Solution curves
then become simply discrete s e t s of points, and we w i l l i n fact keep track only of these crossing points where We thus define a "~oincar; map"
4
dz
< 0 (downcrossings).
of the plane t o i t s e l f which takes
each point t o t h e next downcrossing on i t s solution curve. see,
4
i s not defined everywhere, but it does t u r n out t o be defined
almost every-where.) when
8
(AS we s h d
Figure 2 shows what happens t o a domain around
C
i s applied t o it a few times; t h e transverse scale is grossly
exaggerated and t h e s t r i p s are r e a l l y much thinner than indicgted:
Figure 2
The figure r a i s e s t h e question of what i r o n t h e domain around
2
i s attached t o
c',
of
C
t o t h e domain around
2'.
Since one end
and t h e other i s a t t r a c t e d t o t h e unstable manifold
continuity considerations would seem t o suggest t h a t t h e s t r i p
w i l l get pulled diagonally across from
not t h e case: map
O does t o t h e s t r i p running
to
Such i s , however,
2'.
Although t h e solution flow i s continuous, t h e Poincar6
@ need not be, and it i n f a d undergoes a ,jump discontinuity
between
2
and
2'
.
The source of t h i s discontinuity i s t h e t h i r d stationary salution at
2.
Recall t h a t
2
is a hyperbolic stationary solution with a two-
dimensional stable manifold. motion t h a t t h e
It i s easy t o see from t h e equations of
Z axis i s contained i n t h e s t a b l e manifold of
we should expect t h e s t a b l e manifold t o i n t e r s e c t t h e plane a curve passing through
X = Y
3
w i l l c a l l t h e curve i n question
so
Z = 27 i n
0.
This t u r n s out t o be correct; we
Z;
it is shown i n Figure 5
running from upper l e f t t o luwer right.
.
2,
as
( ~ c t u a l l y t, h i s is only one
of i n f i n i t e l y many pieces of t h e intersection of t h e s t a b l e manifold of
-0
with t h e plane
Z = 27.)
Solution curves s t a r t i n g on C
proceed monutonically t o
2
i s not defined along Z.
Let us investigate what
approaches
Z
and never return t o t h e plane
from t h e upper r i g h t by t r a c i n g t h e orbit
very s l i a t l y above C.
For a long time
2
Z = 27; as
@
z
Tt& f o r
&
t T & tracks i t s neighbors on
t h e s t a b l e manifold and hence gets very close t o to
@(x)does
simply
2
.
While it i s close
i t s motion is well approximated by t h e l i n e a r i z a t i o n of t h e
equations of m t i o n a t zero.
The l i n e a r i z a t i o n has two negative eigen-
values and one positive one; t h e eigenvector corresponding t o t h e posit i v e eigenvalue i s horizontal ( i .e. has
Z-component zero).
In the
linearized motion, t h e components in t h e negative eigendirections decay steadily t o zero while t h e component in the positive eigendirection grows.
I n i t i d l y t h e negative eigencomponents s r e much l a r g e r than
t h e positive one (since t h e t r a j e c t o r y comes in near t h e s t a b l e manifold), but, uuless t h e positive eigencomponent i s exactly zero, it w i l l eventually dominate t h e others and t h e t r a j e c t o r y w i l l move awq from zero along t h e positive eigendirectiorl.
The modifications introduced i n t o t h i s picture
by t h e non-linear terms i n t h e interaction are simple:
A trajectory
entering t h e v i c i n i t y of zero near t h e stable manifold leaves along t h e unstable manifold; t h e closer it is t o t h e stable manifold i n i t i a l l y , t h e closer it will get t o zero and t h e closer it w i l l be t o t h e unstable manifold when it leaves.
The unstable manifold consists of two solution
curves, growing out of t h e origin i n opposite directions; which of these branches w i l l be i d l o v e d is determined by which side of t h e stable manifold t h e t r a j e c t o r y l i e s on.
See figure 3.
Figure 3.
Since t h e solution we are following s t a r t s s l i g h t l y above
C in
figure 5, it w i l l eventually be picked up by t h e branch of t h e unstable manifold of
2 along which
X
and Y
i n i t i a l l y increase.
,
of t h e unstable manifold makes a large loop around
This piece
as shown
schematically i n figure 4, end makes i t s f i r s t downcrossing of t h e plane
A,
2 = 27 a t t h e paint
with co-ordinates
comparison, t h e co-ordinates of
2'
Figure
x i s any point @(x)i s near
Thus, if above
z,
below
2
,
A
@(x) i s near
.
A' ,
-C -A
X
+
-X, Y
+
-Y,
Z
+
o or
-8.3).
(-8.5, -9.5))
4
on t h e plane
2 = 27
lying near but s l i g h t l y
On t h e other hand,
if
is slightly
t h e f i r s t downcrossing of t h e other
branch of t h e unstable manifold of under
are
(-5.2,
2
(i.e., t h e symmetric image of A_
z).
The picture, as developed so f a r , i s shown in figure 5.
The arc
i s part of the intersection of t h e unstable manifold of
2 with
t h e plane object for
Z = 27;
C'
.
i t s synnnetric image
A'
i s t h e corresponding
We have indicated by arrows t h e images of a few
important points under
a,
and we have
put
B)
= @(A),
Br
@(A').
This f i g u r e , unlike t h e others i n t h i s c h a ~ t e r ,is drawn c a r e f u l l y t o scale.
Note t h a t , although
B'
appears t o l i e on
f a c t be s l i g h t l y t o t h e right of it.
Figure 5
c'A',
it must in
We are now able t o form a f a i r l y comprehensive image of t h e behavior of a t y p i c a l solution which s t a r t s , say, near
2.
quickly a t t r a c t e d t o t h e unstable manifold of
C; then proceeds t o s p i r a l
C exactly, it will eventually
out along it. Unless it happens t o h i t land somewhere on t h e part of ed as s p i r a l l i n g around
2'
w i l l be somewhere very near
neighborhood of
2,
It should then be regard-
and i t s next downcrossing
Since points near
B' ,
C go t o t h e
t h e general point can land
and A_', and what happens next depends on whether
B'
it lands above o r below
u n t i l it is above C lands above C
r a t h e r than
CIA_'.
.
below C
while A_ goes t o
A_'
anywhere between
C A_
Such a solution is
C.
I f it lands below
C,
it s p i r a l s around
B A_ ; i f it t o B A_ . The
and then makes a t r a n s i t i o n back t o
it immediately makes t h e t r a n s i t i o n back
motion continues i n t h i s way forever, a l t e r n a t e l y moving around
-
C';
t h e only way it can stop is f o r t h e solution curve t o h i t
and
Z
exactly, and t h i s is extremely =likely. A few other features of t h e motion should be noted here.
ell, although t h e motion may s t a r t a r b i t r a r i l y near t o
has gotten away it can never return t o
c5
(or t o
C9 once it
B'.)
thus gaps of non-zero s i z e between t h e stationary solutions
and t h e region where t h e o r b i t i s recurrent. t h e stationary solution a t very close t o
2
The a r c irom A_ t o i t s preimage under
e, a t y p i c a l o r b i t A
to
Then are
C
and
This i s not t h e case with
(but only very infrequently).
s t r e t c h e s out t h e a r c from
F i r s t of
can be expected t o approach Second, r e c a l l t h a t
1 along t h e full length of
g' A'
C i s t h e image under @ of t h e arc from C t o 8,
i .e
., of a s i n g l e cycle under
8 of
C' A
.
8
.
Tracing back through a few applications of
4
from
to
C
is t h e image mder
of a rather smell piece of t h e arc
x
with
just s l i g h t l y outside
know t h a t t h e distance from than the distance from
= to
t h e position of a pint along
@ we find t h a t the arc
8 (n 'about 24, C 4 running t r o m
as it happens)
(O(5) ,
to
g. From t h e l i n e a r approximation we
@(x) t o
i s only &out 6%greater
2, so an C near
uncertainty of about 6% i n leads t o complete uncertainty
about where i t s orbit will land on t h e arc from complete uncertainty about where on
B' A'
A
to
C and hence t o
it w i l l go next.
Thus,
although t h e moticm is completely deterministic, it is unstable in t h e sense t h a t s m a l l changes in t h e i n i t i a l position are amplified rapidly. This means t h a t t h e behavior is effectively random;
t o determine where
an orbit w i l l be a f t e r making a number of transitions from
-B'
A'
BA
to
and back requires unreasonably precise knowledge about i t s i n i t i a l
position.
F i n a l l y , we need t o c l a r i f y an apparently contradictory aspect
of the above description.
Points on
C are on the s t a b l e manifold of O_
and t h e corresponding solution curves approach zero f a i r l y directly. i s natural t o visualize t h e s t a b l e manifold of
as remaining more or
C
leas f l a t all t h e way t o i n f i n i t y and hence a s separating Since a r d u t i o n curve cannot croes t h e stable manifold of
C'
from
2
2'.
.
it would
seem t o be impossible for solutione t o cycle back and forth between and
It
C
The fallacy in t h i s argument l i e s in an incorrect guess about
the global structure of the stable manifold of passing through t h e upper left-hand part of s p i r a l arotlnd
C(
2.
Solution curves
C, when followed backwards,
and those n e i r t h e intersection of
A'
with
C
s p i r a l around it a r b i t r a r i l y often. manifold of
2
Thus, one p a r t of t h e s t a b l e
wraps i n f i n i t e l y often around
i n f i n i t e l y often around
c'.
Another p a r t *aw
C , but i n t h e opposite direction.
The global
i s quite complicated, and it
s t r u c t u r e of t h e s t a b l e manifold of
manages t o stay out of t h e way of t h e m t i o n as described
above.
Raving seen what a t y p i c a l solution c w e looks l i k e , we w i l l next t r y t o construct a more comprehensive view of beginning in a neighborhood of as sketched i n Figure
A'
A
to
O(B),
T: i s mapped t o t h e narrow sh&d
but s l i g h t l y below
two similar s t r i p s .
Consider a neighborhood S
and 6':
cD t h e part of S lying above C Is mapped
t o t h e narrow shaded s t r i p f r o m
to
l. ((B' JA').
6, consisting of two enlongated ovals
Under one application of
lying below
(BA)
& solution curves
C' A'.
while t h e part of s t r i p m i n g from
The image of
S'
S
B'
i s t h e union of
We can a t t h i s point simplify t h e picture substanti-
a l l y by exploiting t h e symmetry of t h e Lorenz system.
We will i d e n t i f y
points i n pairs
(-X, -Y,
27)
and
( ~ , ~ , 2 7 and ) represent each such
pair by i t s member i n the half-plane'
X
>Y
(with some appropriate
convention on t h e l i n e X = Y ) ; correspondingly, we replace CI
the quotient mapping
@
it l i e s in t h e half-plane in figure 6:
sh-
obtained by reflecting t h e image of Y > X.
8
,
to
and
A
maps
A
o r near
A' ,
a l l points near
S. Points near
t o a continuous mapping of a l l of
C
S t o the point
f~
A..
S
The extended
t o A_' i s
- it permits
I: are mapped by
cb
A)
so
but t h e reflection sends
C t o t h e v i c i n i t y of
B'
S into itself.
is a n o t h v advantage t o making the identification be defined on all of
cb whenever
This leaves only half of t h e picture
The shaded reglm running f m m
reflected t o run froln
cb by
.
to
A,
There
6
to
either near
i
sends
may therefore be extended
into i t s e l f , sending the arc may be visualized as obtained
by t h e following three steps: 1. Stretch S
out t o roughly twice i t s original length, while
shrinking it l a t e r a l l y . 2.
Pinch the resulting set along the image of
3.
Fold back i n t o S , w i t h the pinch going t o
Figure 7
C.
A.
Straightening out and broadening t h e picture a b i t , we obtain t h e shaded region below
Here,
,.
@
C goes t o
8s
A
t h e image 'of
S under
with pinching and
&
goes smoothly t o B_. Applying
again gives a s e t consisting of four long t h i n pieces, two inside t h e
upper shaded region of Figure 8 and two inside t h e lower. s t r i p s are pinched together at pinched together at
B
A
All four
and t h e two upper ones are a l s o
Similmly
A,
6 3 ( ~ ) consists o f e i g h t s t r i p s , all pinched together at
four pinched a t
B,
transverse l i n e
P
and two pinched a t
If we i n t e r s e c t with a
we f i n d successively
Figure 10
A
Continuing t h e process,
=
"
P(s)
consists of uncountably n=l many longitudinal arcs end i n t e r s e c t s any transverse a r c l i k e 3 in
Q = n
P
a Cantor s e t .
fi
The longitudinal arcs making up
are Joined together
in t h e following complicated ww:
Each arc i s pinched together a t each
end with uncountably many others.
h s t (but not quite a l l ) arcs pinch
together at
^3
$'(A) ,@ ( A )
A_
,
with t h e other pinches occurring a t
B = $(A),
,... . Note t h a t t h e pinching points a r e exactly t h e successive
downcrossings of t h e unstable manifold of
g.
There w i l l normally be
i n f i n i t e l y many of these pinching points scattered densely through
$,
although i s it also possible f o r t h e s e t of pinching points t o be f i n i t e . This l a t t e r s i t u a t i o n happens i f and only i f
6
A_ i s a periodic point
o r equivalently i f t h e unstable manifold f o r
2
i s contained i n t h e
s t a b l e menifold; we should expect t h i s t o be t h e case f o r a countable dense s e t of values of
for
r in any small neighborhood of 28.
Whether t h e s e t of pinching points i s f i n i t e o r not, it i s easy t o see t h a t
some orbit
$ has
which i s dense i n
i s contained in t h e non-wandering set f o r A
Q
a t t r a c t s all o r b i t s s t a r t i n g in. S,
A
8.
6
and hence t h a t OD
Since
6=
A in(s), n=l so it meets a l l the requirements
of our e a r l i e r provisional definition of a t t r a c t o r , (and it certainly deserves t h e epithet "strange.") solution flow T~
The corresponding a t t r a c t o r for t h e
i s now not hard t o visualize.
I n t h e vicinity of
t h e plane Z = 27 it consists l o c a l l y of stacks of uncountably many two dimensional pieces which intersect transverse arcs i n Cantor sets. Globally these two-dimensional leaves all pinch together along t h e unstable manifold of
g
consisting of two solution curves which we
should expect t o be dense in the attractor.
Although t h e question has
not been carefiilly investigated, it appears t h a t t h e basin of attraction
g, g'
f i l l s all of I R ~ except f o r
and t h e i r respective one-dinensionel
s t able manifolds. We now return, b r i e f l y , t o t h e behavior of t h e Lorenz system f o r
r
s l i g h t l y l e s s than t h e c r i t i c a l value of 470/19.
Because of t h e f i n i t e
gap between t h e stationary solutions
and t h e a t t r a c t o r , it
i s not r e a l l y necessary f o r
2
and
C and
2'
t o be unstable in order f o r t h e
a t t r a c t o r t o e x i s t ; all that i s necessary is t h a t the unstable manifold of
2
which forms t h e outside edge of t h e a t t r a c t o r not f a l l i n t o t h e
C
basin of attraction of periodic orbits near
-C
and
C'
C
or
and
2'. The existence of small unstable g' shows t h a t t h e basins of attraction
of
are not very large for r s l i g h t l y below 470/19, and it
turns out i n fact t h a t t h e unstable m i f o l d of
2
i s not attracted t o
-C
C' unless r i s l e s s than about 24.1.
and
Thus, f o r
24.1 < r < 24.74,
the system has ( a t l e a a t ) three d i s t i n c t a t t r a c t o r s , t h e point attractors and a strange a t t r a c t o r between them.
C, C'
Which a t t r a c t o r traps a
given orbit depends on where t h e o r b i t s t a r t s , but o r b i t s s t a r t i n g near
-0
go t o t h e strange attractor.
F'hysicaJly, t h e system d i s p l w s
hysteresis; it has several possible behaviors depending on i t s past I f we imagine increasing t h e temperature gradient slowly from
history.
zero t h e solution w i l l simply track one of the two stationary solutions
r = 470/19.
up t o
r
I f , on the other hand, a temperature gradient making
s l i g h t l y l e s s than 470/19 is turned on suddenly with t h e system i n i t i a l l y
at r e s t
,a
s t a t e of permanent chaotic mot ion results.
In t h e above discussion, nothing has been s a i d about t h e behavior of t h e Lorenz system f o r
r larger than 28.
Preliminary numerical experi-
ments indicate t h a t several further changes occur i n t h e qualitative behavior of t y p i c a l o r b i t s , but, t o
knowledge, a detailed analysis has
not yet been made. It may be interesting t o note t h a t the general structure of t h e Lorenz a t t r a c t o r
- t h e fact that
it is made frum two-dimensional unstable
manifolds of a p a i r of stationary solutions folded back on themselves i n f i n i t e l y often space.
A
-
does not depend on t h e dimensionality of t h e s t a t e
similar a t t r a c t o r can e a s i l y be constructed in a space of
an a r b i t r a r y number of dimensions, and s t i l l consists l o c a l l y of an uncountable family of two-dimensional sheets, stacked up i n a Cantor-setl i k e way.
It is thue a t l e a s t possible t h a t analogues of t h e Lorenz
a t t r a c t o r e x i s t f o r r e a l i s t i c approximations t o the equations of hydroQnamics (or even f o r these equations themselves).
Chapter 111. Ergodic Theory of Dissipative Systems
Let us now t r y t o see what physical conclusions could be drawn i f we knew t h a t t h e f u l l convection equations
- o r some f i n i t e
dimensional approximation t o them which i s s u f f i c i e n t l y detailed t o give an accurate description of t h e physical phenomena behavior similar t o t h a t of t h e Lorenz system.
-- had
Thus, consider a
system of equations with an a t t r a c t o r on which t h e motion depends i n a s e n s i t i v e way on i n i t i a l conditions and whose basin of a t t r a c t ion contains some physically relevant i n i t i a l s t a t e s .
If t h e system
i s s t a r t e d out i n t h e basin of a t t r a c t i o n , its s t a t e a t l a r g e positive times i s not arbitrary:
one can a t l e a s t predict with confidence
t h a t it w i l l be very near t o t h e a t t r a c t o r , which w i l l normally occupy a small f r a c t i o n of t h e whole basin of a t t r a c t i o n .
On t h e
other hand, because of t h e i n s t a b i l i t y of t h e motion on t h e a t t r a c t o r i t s e l f , we cannot reasonably hope t o be able t o make accurate pred i c t i o n s about where near t h e a t t r a c t o r t h e system w i l l be found. In other words, t h e s t a t e a t l a r g e positive times i s somewhat r e s t r i c t e d a s it must be near t h e a t t r a c t o r but otherwise appears t o be "random," i. e.
, not
t o depend i n a predictable way on t h e i n i t i a l s t a t e .
A s a p r a c t i c a l matter, t h e main objective of t h e theory of
convection i s t h e computation of such quantities a s t h e thermal conductivity of t h e convective layer, and these quantities are supposed t o depend on t h e physical parameters of t h e system (viscos-
i t y , temperature gradient, etc. ) but not on the i n i t i a l state.
At
f i r s t glance it appears t h a t these computations are impossible in principle i f t h e asyntptotic behavior i s determined by something l i k e t h e Lorenz attractor;
The instantaneoue r a t e of heat t r a n s f e r
can be expected t o depend both on the time and on t h e initial s t a t e and i s not l i k e l y t o approach a limiting value a t t
goes t o
On closer examination, however, t h e s i t u a t i o n i s not as
inflnity.
bad aa it seems.
What i s usually required f o r applications i s not,
f o r example, t h e instantaneous r a t e of heat t r a n s f e r , but rather t h e average of t h i s quantity over a long period of time, and it is only t h e limiting value of t h i s time average which needs t o be independent of initial conditions.
This suggests t h a t it would be
useful t o have some s o r t of ergodic theorem f o r dissipative systems. We w i l l now o u t l i n e one possible version of such a theorem, motivated on t h e one hand by its intended applications and on t h e other hand by what has been proved in special cases. Let
be a flow, A
Tt
a t t r a c t ion
B.
an a t t r a c t o r f o r T~ with basin of
By an ergodic theorem f o r
(Tt ,A)
we mean a theorem
asserting t h e e d s t e n c e of t h e following objects:
a) A probability measure U1\ on A, t solution flow T b)
A subset
X
of
B,
invariant under t h e
of Lebesgue measure zero such t h a t :
For any continuous Anction
f
on B
and any x i n B but not in
This formulation has a number of related aspects; we o f f e r t h e following remarks t o c l a r i e what it i s intended t o mean.
The main
thing being asserted i s t h a t forward time averages of "general" functions on t h e basin of attraction exist and are independent of the i n i t i a l state.
Independence of i n i t i a l conditions cannot be
expected t o be t m e e n t i r e l y without qualification.
For example,
most non-trivial a t t r a c t o r s contain i n f i n i t e l y many unstable periodic o r b i t s ; the time average s t a r t i n g at a point exactly on one of these orbits w i l l simply be t h e average over t h e o r b i t , which w i l l not be a t all l i k e t h e time average f o r a typical i n i t i a l point.
We must
therefore be prepared t o throw out an exceptional s e t of i n i t i a l conditions
- i n our formulation. the s e t - which ought t o be X
negligible from t h e physical point of view.
We have taken as our
c r i t e r i o n of physical n e g l i g i b i l i t y t h a t t h e s e t of exceptianal points have Lebesgue measure zero.
Note t h a t Lebesgue measure i t s e l f t has r e l a t i v e l y l i t t l e connection w i t h t h e flow T , and i n p a r t i c u l a r
is
supposed t o be invariant under Tt ; it has rather been pulled
in a r t i f i c i a l l y t o provide an elementary way of s t a t i n g t h a t a certain
set is negligible.
This criterion f o r n e g l i g i b i l i t y has a number of
drawbacks
-- notably,
it applies only t o flows on finite-dimensional
manifolds cmd not t o t h e convection equations themselves
- and
t h e r e are indications t h a t it could be replaced by a sharper condition f o m d a t e d i n terms of Hausdorff dimension. Next:
We are considering time averages only f o r continuous
functions and not, say, f o r general bomded Borel fur~ctions. Some such r e s t r i c t i o n is necessary t o avoid t r i v i a l counterexamples arising from t h e fact t h a t t h e flow is non-recurrent on B \ A. if
A
For example,
consists simply of an attracting stationary solution, it is
easy t o construct a bounded Borel function f such t h a t
does not e x i s t f o r any x
i n t h e basin of a t t r a c t i o n other than t h e
stationary solution i t s e l f . Third:
In our formulation, the exceptional s e t i s taken t o be
independent of t h e flmction able t o a l l o w it t o vary with
whereas it might seem more reason-
f,
f.
It turns out, however, t o be no
more r e s t r i c t i v e t o require t h e existence of a single exceptional set.
To see t h i s , assume t h a t time averages exist and are indepen-
dent of i n i t i a l condition f o r each continuous m c t i o n , but allowthe exceptional s e t t o depend on the function.
Choose a countable
set of continuous functions whose restrictions t o A t h e space of a l l continuous functions on A; provided t h a t
i s compact.
Let
are dense in
t h i s w i l l be possible,
X be t h e union of the exceptional
s e t s f o r these couutably many functions;
X w i l l again be a set
of Lebesgue measure zero and it is easy t o see t h a t time averages exist and are independent of initial condition i n B \ X for all continuous functions Finally:
f.
Our formulation of a general ergodic theorem requires
t h a t time averages be obtained as mean values with respect t o a probability measure yA on A.
If
A i s compact, t h i s is automatic
once time averages are known t o exiat and t o be essentially indepen-
1c
dent of i n i t i a l condition. value of
lim T-
;
To see t h i s , l e t
dt ~ ( T ~ xfor ) almost
is defined f o r a l l functions
f
? denote the common x.
continuous on B,
he quantity
?
but i s i s easily
seen t h a t two functions which are equal on A have t h e same average, so f
I+
? can be regarded as a f b c t i o n a l defined on the space of
continuolzs functions on A.
This functional is lineas, positive,
and takes t h e constant function 1t o 1, and hence, by the Riesz Representation Theorem, has t h e form
for a uniquely determined probability measure yA on A.
I n s p i t e of
t h e fact that i t s existence is automatic, t h e measure y,,
i s interest-
ing and important since it ought t o be possible t o describe it i n t r i n s i c a l l y and hence t o give a procedure f o r computing time averages other than by applying t h e definition.
We have here a close analogy
t o t h e usual view of t h e r o l e of t h e microcanonical ensemble i n c l a s s i c a l s t a t i s t i c a l mechanics, and t h e measure
uA
may t h e r e
fore be viewed as an equilibrium ensemble f o r t h e dissipative system.
One important practical difference from c l a s s i c a l s t a t i s t i -
c a l mechanics should be noted:
The microcanonical ensemble for a
Hamiltonisn system can be written down directly i n terms of t h e Hamiltonian.
To construct pA, on t h e other hand, it i s necessary
f i r s t t o locate t h e a t t r a c t o r
A
and then t o analyze exhaustively
t h e behavior of t h e solution f l o w on and n e w
A.
So f a r , t h i s
process appears t o require detailed informat ion about t h e solutions t o the equations of motion, as opposed t o simply knowing the different ial equations themselves. To get a complete picture of the behavior of t y p i c a l solutions of a s e t of d i f f e r e n t i a l equations, we would want t o do something l i k e t h e following: a.
show t h a t , except f o r a s e t of Lebesgue measure zero, t h e
s t a t e space s p l i t s into t h e basins of a f i n i t e number of attractors.
b.
prove an ergodic theorem f o r each of these attractors.
The asymptotic properties of a solution curve will then depend on which basin of a t t r a c t i o n it l i e s in, but essentially all solution curves i n a given basin w i l l have the same s t a t i s t i c a l properties over long periods of time.
This program has been completely carried
out by Ruelle and Bowen [ l o ] , [l]f o r flows on compact manifolds which
setis* Smale's Axiom A.
Rather than describe t h e proof of t h e Ruelle-Bowen theorem, we will t r y t o i l l u s t r a t e t h e idea of t h e proof by showing how
it miefit be adapted t o prove an ergodic theorem for the Lorenz system.
This procedure has the advantage of concreteness and
r e l a t i v e simplicity; it has t h e disadvantage t h a t it i s not r e a l l y a proof of anything as : a.
t h e argument s t a r t s from some qualitative features of
t h e Lorenz a t t r a c t o r which are strongly suggested by numerical experiments but which are certainly not proved b.
even assming these qualitative properties t o hold, t h e
proof of an ergodic theorem for t h e Lorenz system involves some algebraic and analytic complexities not present i n t h e Axiom A case snd not yet completely overcome. What we w i l l therefore actually do is t o reduce t h e proof of t h e
ergodic theorem of t h e Lorenz a t t r a c t o r t o a question about a onedimensional transformation and then suggest how t o t r e a t the onedimensional problem by considering a model problem with a number of technical simplif icatf OM. The f i r s t s t e p i n our proposed proof of an ergodic theorem
t f o r t h e Lorenz a t t r a c t o r is a reduction from t h e solution f l o w T t o t h e Poincard map
4 discussed in the preceding chapter,
is, we assme we have an ergodic theorem f o r
t get one f o r T
.
For
x
That
4 and show how t o
in t h e plane Z = 27, l e t Z(E) denote
t h e time required for t h e solution curve through 1~ t o return t o
i t s f i r s t damcrossing of t h e plane; i f t h e solution curve never
,qyturns, we put
For any
~ ( 5= )w.
_q
in the basin of attraction
whose solution curve eventually makes a damcrossing at a point time averages s t a r t i n g a t
c,
have t h e same l i m i t as time averages
s t a r t i n g a t & s o we may a s well consider only time averages s t a r t i n g at points 5 = ( x , Y , ~ T ) Mere X*Y < 72.
of
9
< 0. i . e . , where
We w i l l exclude immediately 5's
on t h e stable manifold
; as the stable manifold is a s e t of measure zero, t h i s w i l l
not affect t h e proof of an ergodic theorem. function
Then, f o r any continuous
fl
lim T-
f
[
dt fl(T
t
lim
M-l
1 fCBd
,=o 3 = wm. iTN-1 lim
**
1 rCPx)
n=o
providhd bath t h e l i m i t in t h e numerator and l i m i t in t h e denominator exist.
If
f and T
were continuous, an ergodic theorem for
ip
would say that both numerrrtor and denominator exist f o r almost a l l 5 and a r e essentially independent of f o r t h e l i m i t on t h e l e f t . it approaches i n f i n i t y as
x;
t h e same would then follow
Unfortunately, r
x
approaches
(x) is not
continuous;
Z. To complete t h e reduction
properly therefore requires an approximation argument using some
s p e c i d properties of t h e equilibrium ensemble f o r 8.
This argu-
ment i s i n e s s e n t i a l t o t h e main o u t l i n e of t h e proof; we w i l l not
give it. Next, and purely t o simplify t h e exposition, we w i l l r e s t r i c t cons ideration t o those continuous functions t h e symmetry (X,Y,z)
+
(-x,-Y ,z)
.
8
.
invariant under
This permits us t o consider only
one of t h e two p a r t s of t h e a t t r a c t o r f o r 4, A
f
end t o replace
8 by
General continuous functions can be handled by a straightforward
extension of t h e argument.
For t h e remainder of t h i s chapter we w i l l
A
always consider
4 r a t h e r than
8,
and we w i l l drop t h e
We must next examine in d e t a i l t h e action of
A
.
8 on and near t h e
a t t r a c t o r Q. The picture we want t o develop i s t h a t some neighborhood of
Q aecomposes i n t o a one-parameter family of non-intersecting
a r c s running transverse t o t h e a t t r a c t o r .
The arcs a r e characterized
by t h e property t h a t each of them contracts t o a point under repeated
application of distance from +m
.
8, i.e.,
en%+
to
if
n 4+ x
+, +x
are i n t h e same a r c then t h e
goes t o zero rapidly as n
goes t o
Accordingly, we w i l l r e f e r t o them as contracting arcs. The existence of contracting arcs i s suggested by t h e fact t h a t
8 compresses strongly i n a direction transverse t o t h e a t t r a c t o r . To see i n more d e t a i l what is happening, l e t us look a t a point 3~ on o r near t h e a t t r a c t o r , and a l i n e segment
passing e i t h e r
above o r below 3f and rouefily p a r a l l e l t o t h e a t t r a c t o r :
Figure 1 Applying
4 mves both & and
and a l s o s l i d e s attractor.
much closer t o t h e a t t r a c t o r
a and 8 s l i g h t l y
Since a and
away from
& along t h e
move away from 2 in opposite
d i r e c t i o n s , t h e r e must be points l i k e y on separation between the attractor. of
@,
4& and
such that t h e
remains a t a s u b s t a n t i a l angle t o
cby
Because of t h e strength of t h e transverse compression
t h i s condition locates
.
f a i r l y precisely along
y
apply
4 again and require t h a t t h e separation between
@2y
remain transverse t o t h e a t t r a c t o r ; t h i s w i l l locate
m r e precisely.
Now
425 and y
even
Continuing in t h i s w a y we construct a sequence of
successive approximations which ultimately y i e l d s a single point on
any
5
anx
with t h e property t h a t t h e separation between
i s transverse t o t h e a t t r a c t o r f o r all positive
of t h i s transversal separation end t h e fact t h a t t h e transverse direction, t h e distance from f o r each n
any
to
o t h e r than
anz y,
compresses in
@
5 to
is
For a fixed &,
any ,
decreases exponentially with
n.
on t h e other hand, w i l l eventually
be drawn away from 3~ by t h e stretching action of
attrador.
and
Because
a small f r a c t i o n of t h e distance from @nz t o
so t h e distance from Any point of
n.
y
t h e point8
y
4
along t h e
on t h e various possible
nearby
longitudinal segments
a6 s t r i n g together t o form a one-
dimensiand s e t which, by construction, is contracted under t h e action of
O.
Thus: Each point
t o l i e on a contracting arc.
=
s u f f i c i e n t l y near t o Si ought
Contracting arcs are uniquely deter-
mined l o c a l l y , and two contracting arcs which i n t e r s e c t must be continuations of each other.
There i s no apparent reason why cm-
t r a d i n g arcs must be unreasonably short; it ought t o be possible t o continue each of them a t l e a s t across t h e AiLl thickness of t h e attractor.
W e thus a r r i v e at a picture l i k e t h e following, where
t h e predominantly v e r t i c a l segments represent contracting arcs.
Figure 2 This figure i l l u s t r a t e s an important feature of t h e decomposition i n t o contracting arcs.
If
W
t h e ends as i l l u s t r a t e d , then another contracting arc.
i s a contracting arc, cut o f f a t @W w i l l be part but not rill of
Frequently, t h e r e w i l l be a second contract-
ing a r c W'
,
that
shares a contracting a r c with
@W'
running acrms t h e opposite end of t h e a t t r a c t o r , such @W.
I n t h i s case, i f
&E W
and &@E W'
then t h e distance from
exponentially as n
x,
x1
anz
and
anz'
goes t o zero
goes t o i n f l n i t y in s p i t e of t h e fact t h a t
are not in t h e same contracting arc.
For
t o be in t h e
same contracting a r c a s 5 it i s necessary but
sufficient t o
haw
an& and
anz'
approach each other as n
goes t o i n f i n i t y .
x
It i s i n f a c t t o be expected t h a t , f o r a t y p i c a l point
a t t r a c t o r , t h e s e t of points
&'
near t h e
near t h e a t t r a c t o r such t h a t
will consist of an i n f i n i t e (but countable) union o f contracting
arcs end w i l l b e dense i n a neighborhood of t h e a t t r a c t o r . We can now formulate a precise s e t of assumptions about t h e existence and properties of contracting arcs :
Asstmrptim:
Existence and Absolute Continuity of the Contmcting
Foliation.
There i s a continuous decomposition o f a neighborhood
of the a t t m c t o r R into a me-parameter family o f smooth arm ( a m t ~ ~ ~ u tarcs) i n g with the foZZ&g
a.
pmperties:
( @ n t r a c t i v i t y ) There e x i s t constants C,
0 < A < 1
, such that i f 5 , x2
A
,
with
are i n the same contracting arc
It would nar be natural t o construct a co-ordinate system f o r
a neighborhood of
such thet the contracting arcs are l i n e s
Cl
where one of t h e co-oranates i r constant.
It turns out ultimately
t o be more convenient t o do only part of the reparametrization, i .e. t o construct only the co-ordinate which i s constant on contracting arcs or, equivalently, t o parametrize t h e s e t of contracting arcs. To do t h i s we draw in some convenient way a smooth arc Y
running
t h e f u l l length of the a t t r a c t o r (but not necessarily in t h e attractor) which crosses each contracting arc exactly once and a t a non-zero angle. We l e t onto Y
(we w i l l refer t o such an arc Y
a s a lonaitudinal
arc).
denote t h e projection of a neighborhood of the a t t r a c t o r along contracting arcs, i .e
t h e unique point of
y
., f o r each
3 n
on t h e same contracting arc as
Figure 3
(x) denotes r.
We add t o our l i s t of assumptions a regularity property for r : c.
(Absolute continuity) If yl
arc, then n mrrtricted t o yl f2.om
yl
i s amj other longitudinal
is a differentiabZe mapping
t o y wiUl a Eb'lder continuous d e r i m t i v e .
This does nut complete t h e statement of t h e assumption, but a t t h i s point we digress t o r e l a t e these conditions t o l m o ~ f a d s about Axis A attractors.
( I t perhaps needs t o be mentioned t h a t
@
does
not s a t i s f y Adom A; it i s neither one-one nor differentiable on
2.
Its apparently mild f a i l u r e t o f u l f i l l the conditions turns out t o
have far-reaching consequences; t h e Lorenz a t t r a c t o r has a much w r e i n t r i c a t e and delicate structure than i s possible for an Axiom A attractor. )
For a general Axiom A a t t r a c t o r , each point of t h e
a t t r a c t o r has a neighborhood ( i n t h e manifold of s t a t e s ) which s p l i t s s continwusly i n t o an (n-n )-parameter family of smooth submaDifolds of dimension ns,
called contracting leaves and analogous t o the
contracting arcs of t h e above discussicn.
Each contracting leaf
shrinks exponentially under repeated applications of t h e transformat i o n ; moreover, wherever it passes through t h e a t t r a c t o r it is appearing in t h e s are two n-n dimensional sur-
tangent t o t h e infinitesimal stable space 'E statement of Axiom A.
If
Y and yl
faces each running transverse t o the contracting leaves, then proj e c t i m d o n g contracting leaves defines a continuous mapping from y1
t o y.
One of t h e unpleasant technical features of t h i s subject
is t h e fact t h a t , even i f t h e transformation i t s e l f is i n f i n i t e l y
differentiable, t h i s projection does not need t o b e continuously differentiable.
It i s , however, s u f f i c i e n t l y well behaved t o
send (n-ns)-dimensional Lebesgue measure on yl
t o t h e product of
Lebesgue measure on y with a ~ b ' l d e rcontinuous density.
This
property i s called absolute c o n t i n u i t ~of t h e contractina f o l i a t i o n ; in t h e special case where n s = n-1
( s o y , yl
are one-dimensional)
absolute continuity implies continuous d i f f e r e n t i a b i l i t y . We return now t o t h e problem of proving en ergodic theorem f o r @
.
The next s t e p in t h e argument i s t o show t h a t it suffices t o
prove t h e existence end e s s e n t i a l independence of i n i t i a l point f o r f'unct ions
f
which are continuous g&
const ant
contracting arcs.
To see t h i s , we assume t h e ergodic theorem f o r such functions and prove it for general continuous functions. continuous on a neighborhood
V
Thus, l e t
Then choose m
be uniformly
of t h e a t t r a c t o r which s p l i t s , as
i n t h e above assumption, i n t o contracting axcs. find 6 > 0 suchthat
f
d(x1,x2) 0,
and
implies If(xl) - f ( x 2 ) 1 a
and
\
k The prescription i s ambiguous i f Q (x) = a.
= 1 i f qk(x)
< a.
It would be easy t o
lift t h i s ambiguity by making one of t h e inequalities s t r i c t , but
f o r our purposes it i s b e t t e r t o allow such x ' s t o have more than one coding.
The inverse correspondence
- from sequences t o
x's
-
w i l l turn out t o be unambiguously defined and almost, but not
quite, one-one.
so
We s e t
,...) i s a c o m g of
(iO
x
it qk(x) E ~ ( i ~ f o r) k = 0,1,2,.
It follows readily from t h e assumed properties of
only element
.. .
0 i s the 3 x of A(1) with ~ ( x E) A(1) and, since Q (0) = 1,
t h a t there is no x with
x E ~ ( 1 ) .Q (x) E
CP
that
~ ( 1 1 q, ~ * ( x ) €
~(1).
Thus, no coding sequence has three successive ones and only x's such t h a t cpk ( x ) = 0 ones.
f o r some k
admit codings with two successive
I n t h e l a t t e r case, replacing t h e block ( 1 , ~ )wherever it
appears i n t h e coding sequence of t h e block (1,0) gives another coding of t h e same point, so:
Every point of
[O,I] &its
a coding without successive ones.
We w i l l say t h a t a f i n i t e or i n f i n i t e sequence of 0's and 1 ' s is admissible i f it contains no p a i r of successive l l s , and we w i l l let
denote t h e s e t of all admissible sequences.
In what foZla06
we c m i d e r only codings into acbnissible sequences. The next s t e p i n t h e analysis i s t o show t h a t every admissible sequence i s a coding of exactly one x. any f i n i t e admissible sequence (iO,il,.
To do t h i s , we define, f o r
..,ik),
.
A
(i.e.,
A(iO,il,..
begin with
i 1= : k
.,ih)
1
A
is t h e s e t of
for
x's
j = O,~,.+.,JS~
admitting codings which
( iO , . . . , ik).)
..,\I,
w m i t i a . For any finite acbnissibte sequence
...,ik)
A ( iO,
thun ak
Proof. for
is a non-enrpty closed intervat of length no larger
(where a-l
is the expansion o m t a n t of conhition dl
We argue by induction on
k = 0.
( t o ,il,.
k.
The statement i s c l e a r l y t r u e
For k > 0 we h a w by definition:
From t h e assumed properties of rp it follows t h a t continuously d i f f e r e n t i a b l e , snd expanding on cp(A(1)) = A(0); cp(A(0)) = [0,11.
...,
Thrs, since
q(Ai ) 3 A ( i l ) 3 A ( i l , i k ) s o q maps 0 A(il,. ,ik). Bp the induction hypothesis
..
A(iO), and t h a t (io,il)
A(iO,
...,\)
A(il,.
..,\)
non-empty closed i n t e r v a l , s o t h e same i s t r u e of
since
q ' > a
on A(io),
cp i s one-one,
# (l,l),
A(iO,.
is a
..,ik 1. .
where
X denotes length of an interval (or Lebesgue measure of
a more general s e t ) .
Again using t h e inductim hypothesis
as desired.
..
I f , now,
= ( i0,il,i2,. )
i s an admissible sequence,
is a decreasing sequence of closed intervals with length going
t o zero and hence t h e r e i s exactly one point x i n every A ( i O ,il,.
admits
..,ik).
i
We w i l l note t h i s
by n ( i ) ; it evidently
as a coding and is the only point which does.
evem admissible sequence point
x
Thus,
& is a coding of a uniquely defined
The mapping n is continuolrs from
n(&) i n [0,1].
A,
equigped with t h e to polo^ it i n h e r i t s as a subset of t h e compact product space
(0,lF
,
t o [0,1].
Although r i s not one-one, it
i s easy t o see t h a t it is a t most two-to-one and t h a t there a r e only
88 countably many x ' s with more than one admissible coding. a
denote t h e s h i f t mapping on
A,
is an admissible coding of
then i f
We l e t
02
x,
is an d i s s i b l e
coding of cp(x), i . e .
If a were exactly one-one
, it
between cp and t h e s h i f t mapping case,
a
0.
would s e t up an isomorphism Although t h i s i s not t h e
i s close enough t o a t r u e isomorphism f o r many purposes,
and t h i s i s t r u e i n p a r t i c u l a r f o r t h e analysis o f continuous* measures on [0,1]. Any probability measure under
rr t o a probability measure
The inverse operation
-
Tp
-- l i f t i n g measures
; on d
projects
on [0,1] defined by
from 10 ,1] t o
A-- i s
not q u i t e s o simple, but t o any continuous probability measure
u
on [0,1] t h e r e corresponds a unique continuous probability measurq Y
A measure i s continuous i f it assigns measure zero t o any s e t consisting of a s i n g l e point.
; on A
with
construct
TO
G = p .
;,
.. ,..., ik
we l e t
b(i0
denoke the ey&.der s e t
c
we define
'
on cylinder s e t s by
--
u(A(io,.
..,ikN
.
= u(A(io,*. , i k h
;t o
and we use standard measure theoly t o extend probability measure on
he
A,.,
a Bore1
construction does not work f o r s
completely general measure p because
i; as defined above will
,...,$)
not be f i n i t e l y additive i f t h e boundaries of t h e A(iO have non-zero
p-measure.
For continuous measures there i s no
problem since t h e boundary of points.)
Projection by
T,
A(iO,.
..,\)
contains only two
then, s e t s up an isomorphism between
t h e s e t s of continuous probability measures on
and on [0,1].
This isomorphism i s e a s i l y seen t o preserve most i n t e r e s t i n g r e l a t i o n s , e.g. and
1G2
only i f
ni;
c1
axe, and
and
"
u2
a r e equivalent i f and only i f
i s invariant (ergodic) under
61
a i f and
i s invariant (ergodic) under (9.
We can therefore adopt the following strategy f o r constructing a 9-invariant meesure equivalent t o Lebesgue meesure
A:
i ) L i f t Lebesgue measure t o a measure ii) Construct a s h i f t invariant measure
ergodic and equivalent t o iii)
Project on
under
-X
on j6
-. p on
.A
which i s
n t o obtain t h e desired measure p
[O,lI.
This strategy involves a trade-off.
It replaces t h e possibly compli-
cated mapping cp by t h e simple and standard s h i f t mapping a ,
but
it also replaces the simple and standard Lebesgue measure by t h e l e s s
.
simple measure 1 on
A.
The u t i l i t y of the trade-off depends on
whether o r not we can find methods t o control the behavior of key t o obtaining such control is t h e observation t h a t
x.
The
-,
X i s the
Gibbs s t a t e f o r a one-dimensional semi-infinite classical l a t t i c e system with a rapidly-decreasing s h i f t invariant many-body potential. b y semi-infinite we mean t h a t t h e l a t t i c e s i t e s m e labelled by t h e
non-negative integers rather than by all t h e integers.) configuration space i s
A
with no two consecutive 1's
- t h e s e t of sequences of 0's
- the l a t t i c e system
Since t h e and 1 ' s
w i l l have a nearest
n e i a b o r exclusion, but we w i l l argue shortly t h a t t h e potential i s othervise f i n i t e
.
Before Justiiying t h e claim t h a t f i r s t describe why t h i s fact i s useful.
is a Gibbs s t a t e , l e t us
Standard theorems about
one-dimensi onal classical st a t i s t i c a l mechanics can be applied t o show that t h e same interaction f o r t h e two-sided i n f i n i t e l a t t i c e
'
system has a unique Gibbs s t a t e and t h a t t h i s Gibbs s t a t e is trans-
lation (i.e.,
s h i f t ) invariant with very good ergodic properties.
(See Ruelle [9] for uniqueness, Gallavotti [ 2 ] f o r ergodic pro-
; be the measure on
perties. ) Let
d obtained from the invariant
Gibbs s t a t e by ignoring the part of t h e l a t t i c e system t o t h e l e f t of the origin (i.e., by projection).
Using the fact that the inter-
action between t h e part of the l a t t i c e system t o the l e f t of the origin and the part t o t h e right is bounded except for the e f f e c t s of the nearest neighbor exclusion, it i s easy t o show that absolutely continuous with respect t o
-1
$ is
with a Radon-Nikodp
derivative which i s both bomded above and bounded away from zero.
-1
Thus, once
has been identified as a Gibbs s t a t e , t h e standard
theory of Gibbs s t a t e s yields almost inmediately the existence and ergodicity of a shift-invariant measure
equivalent t o
and
hence of a cp-invariant measure equivalent t o Lebesgue measure.
X i s a Gibbs s t a t e , let us
To see why
..
sequence il ,i2,.
.
respect t o
A
.. .
il,i2,.
fix
an admissible
and compute t h e conditional probabilities with
of the two possible values
- 0 and 1 - of
i0 given
(TO complete t h e identification, we vill need t o
compute, more generally, conditional probabilities of the various values of
is trivial;
..
iO,. ,ik i0
given
has t o be
otherwise not be allwed.
%+l,.. . .)
If
i1 = 1 the computation
..
0 since t h e sequence iO,il,. would
We assume therefore t h a t
t h i s case the conditional probability is equal t o
il = 0.
In
ore
precisely: General theory assures us t h a t t h i s l i m i t e x i s t s
f o r almost all
(il,i2,.
..)
and i s equal almost everywhere t o the
desired conditional probability.
l i m i t e x i s t s f o r all
am
.
.
(ii,i2,..)
end give a formula for i t . )
i s obtained by transporting Lebesgue measure,
Because
Recall t h a t
We w i l l i n fact shm that t h e
A(iO,.
..,im )
is sn interval of length no greater than
Moreover, for each of t h e two possible values of
,...,im )
A(iO
write
xi
onto A(il
f o r n ( i O,il..
0 on all of
,...,i m ) .
h(iO,il,.
..,in)
..) then
If
m
9'
maps
i s large, and i f we
i s nearly equal t o
cf' (xi )
0 and we have
The approximation becomes exact as m +
Thus i f we define
iO,cp
w
so
we find t h a t t h e conditional probability of
i0 given
,...
il ,i2
is equal t o
Entirely similar arguments shcw t h a t t h e conditicaal probability of
..,\ given
iO,.
..
ik+l..
is equal t o
We can now construct t h e interaction f o r which state.
The i n t u i t i v e iaea is t h a t
h ( i O,il,.
-
X
i s a Gibbs
..) should represent
the contribution of t h e l a t t i c e s i t e zero t o t h e total energy.
This
i s not a well-defined concept, however, s o t h e r e will be some choices
t o be made in t h e construction of t h e i n t e r a c t i m .
We w i l l think of
our l a t t i c e system as a spin system r a t h e r than a l a t t i c e gas.
The
interaction i s then specified by giving, f o r each f i n i t e subset
X
of
Z,
a function
ax
defined on
{0,1)
X
,
forward translation-invarience requirements;
sub3ect t o a straight-
$
i s interpreted a s
t h e p o t e n t i a l energy due t o direct interaction among a l l t h e l a t t i c e s i t e s in X.
The t o t a l energy f o r a configuratian
I
defined on a f i n i t e s u b l a t t i c e A i s
We construct an interaction by defining: unless
X i s an interval (i.e., a s e t of t h e form
.
(il 7il+17. ,il+k)
90,1,, ,,7k)(io7"'7ik)
= $(io3. *.7ik)
- 5-l(i0,"'7ik-l)
where ~
(o,...,\ i
inf
f
7
t
\+29."
h(iO,-.. ,ik7$+17$+2s*..),k
= 0,1,.-.
=-1 = O
The function
ex
f o r an i n t e r v a l of length
zero is determined by t r a n s l a t i o n invariance:
k+l not s t a r t i n g a t
With these definitions it i s easy t o see t h a t
Taking the l i m i t of t h e second of these equations as m
+
with
k
held fixed gives
Comparing t h i s equation with the previously obtained formula f o r conditional probabilities relative t o
1 shows that
i s indeed
a Gibbs s t a t e f o r a semi-infinite l a t t i c e system with the interaction we have ccastructed. To apply standard results from s t a t i s t i c a l mechanics t o show t h a t t h i s interaction has a unique Gibbs s t a t e , we need t o know t h a t t h e interaction drops off rapidly at large separatians. that
It turns out
goes t o zero exponentially a s
k goes t o i n f i n i t y .
To see t h i s ,
observe:
I
.
..,\I1
, ,k) ( iO,.
minimum of loglq'(x)l
i s no l a r g e r than t h e maximum minus
..,\I A(iO,. ..,%)
on &(i0,.
The length of
k i s no l a r g e r t h a n a
is Hslder continuous on A(0) and on
C?'
A(1).
To conclude, l e t us s w e y b r i e f l y how t h e above development uould have t o be modified t o apply t o t h e Lorenz system.
The Lorenz
then decreasing from a
to
Q
which comes from t h e
cp i s again increasing t o some point
a;
1, and we have
Formally, we can approach t h i s mapping i n t h e same way as our model q : We code each point
x
of
[ O,1]
i n t o a sequence of ones and zeros
determined by whether t h e successive
ri&t of
a,
9k (XI'S are
t o the l e f t or the
and thus t r a n s l a t e t h e problen i n t o a s t a t i s t i c a l -
mechanical me.
The t e c h n i c a l complications are two-fold.
F i r s t of
all, t h e image of t h e coding is no longer as simple as before.
It
i s possible t o have more than two successive ones, but because
cp
moves points of
[O ,a] non-trivially t o t h e ri&t it is not possible
(The maximnu number is actually 25 for k A straightforward analysis shms t h a t , unless" q, ( 0 ) = a
t o have a r b i t r a r i l y many.
r = 28).
f o r some k ,
t h e image of t h e coding cannot be described by specify-
ing a f i n i t e number of excluded f i n i t e sequences.
In s t a t i s t i c a l
mechanical terms, t h i s means t h a t the corresponding classical l a t t i c e system has i n f i n i t e l y many "exclusions," of a r b i t r a r i l y long range, generalizing the nearest-nei&bor exclusion of t h e model
Q.
A
second difficulty is caused by the i n f i n i t e tangent t o the graph of 'p a t
a which means t h a t t h e contribution h
of a single l a t t i c e
s i t e t o the t o t a l energy i s not bounded above.
These features make
t h e s t a t i s t i c a l mechanical problem considerably mare d i f f i c u l t then t h e me we have considered.
References
[l] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Inventiones Math. 181-202 0 9 7 5 1.
a
[2] G. Gallavotti, Ising model and Bernoulli schemes in one dimension, Commun. Math. Phys. 2 (19731, 183-190.
[3]
J. Guckenheimer, A strange strange attractor, in [6], pp. 368-381.
[4] J. Leray, Sur le mowement dlun liquide visqueax emplissant llespace, Acta Math. 63 (1934)~193-248. [5] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. (19631, 130-141. [6] J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applied Mathematical Sciences 19, Springer-
[TI R. M.
May, Simple mathematical models with very complicated dynamics, Nature 261 (19761, 459-467.
[8] J. B. McLaughlin and P. C. Martin, Transition to turbulence of a statically stressed fluid system, Phys. Rev. A 12 (19751, 186-203. [9] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. ~ath.Phys. 2 (1968), 267-278. [lo] D. Ruelle, A measure associated with Axiom A attractors, Amer. Sour. Math., to appear. [ll] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc 12 (19671, 747-817.
.
[12] R. F. Williams, The structure of Lorenz attractors, Preprint, Northwestern University (1976).
CEN TRO INTEF@lAZIONALE MATPlATICO ESTIVO (c. I.M.E.)
MANY PARTICLE COULOMB SYSTEMS
E l l i o t t H. LIEB
D e p a r t m e n t s of M a t h e m a t i c s and P h y s i c s Princeton University
- Princeton,
N.J.
08540
C o r s o tenuto a B r e s s a n o n e dal 21 giugno a1 24 giu@o 1976
MANY PARTICLE COULOMB SYSTEMS
E l l i o t t H. Lieb
*
Departments of Mathematics and Physics Princeton University Princeton, N .J. 08540
Lectures presented a t the 1976 session on s t a t i s t i c a l mechanics of the International Mathematical Summer Center (C.I.M.E.)
Bressanone, Italy,
June 21-27.
* Work
partially supported by U.S. National Science Foundation grant
MCS 75-21684.
With t h e introduction of t h e Schroedinger equation i n 1926 i t became possible t o resolve one of t h e fundamental paradoxes of t h e atomic theory of matter (which i t s e l f had only become universally accepted a few decades earlier):
Why do the electrons not f a l l i n t o the nucleus?(Jeans, 1915).
Following t h i s success, more complicated questions posed themselves.
Why
is the lowest energy of bulk matter extensive, i.e. why i s i t proportional
t o N, the number of p a r t i c l e s , instead of t o some higher power of N? Next, why do the ordinary laws of thermodynamics hold?
Why, i n s p i t e of
the long range Coulomb force, can a block of matter be broken i n t o two pieces which, a f t e r a microscopic separation, a r e independent of each other? The aim of these l e c t u r e s is t o answer the above questions i n a simple and coherent way.
It is a summary of research I have been engaged
i n f o r the p a s t few years, and i t has been my good fortune t o have had the b e n e f i t of collaboration with J.L. Thirring.
Lebowitz, B. Simon and W.E.
Without t h e i r i n s i g h t s and stimulation probably none of t h i s
could have been carried t o f ~ i t i o n . The accompanying flow c h a r t might be helpful.
I n s e c t i o n I atoms
a r e shown t o be s t a b l e because of the Sobolev inequality, not the Heisenberg uncertainty principle.
A new inequality r e l a t e d t o Sobolev's
2 3 N f o r functions i n the antisymmetric tensor product L (1 ) i s presented i n s e c t i o n 11. Thomas-Fed theory (which was introduced i n 1927 j u s t a f t e r the Schroedinger equation) i s analyzed i n s e c t i o n 111. This subject i s i n t e r e s t i n g f o r three reasons:
( i ) As an application of
nonlinear functional analysis; ( i i ) It turns out t h a t i t agrees asymptotically with the Schroedinger equation i n a l i m i t i n which the
nuclear charges go t o i n f i n i t y ; ( i i i ) The no-binding theorem of ThomasF e d theory, when combined with t h e i n e q u a l i t y of s e c t i o n 11, y i e l d s a simple proof of the s t a b i l i t y of matter. IV.
The l a t t e r i s given i n s e c t i o n
The f i r s t proof of s t a b i l i t y is due t o Dyson and Lenard i n 1967, but
the proof i n s e c t i o n I V i s much simpler. dynamic problem.
Section V deals with t h e thermo-
The d i f f i c u l t y here i s not the one of collapse, which
was s e t t l e d i n s e c t i o n I V , b u t t h e p o s s i b i l i t y of explosion caused by the long range p a r t of the Coulomb p o t e n t i a l .
Newton's theorem t h a t a charged
sphere behaves from the outside as though a l l i t s charge were concentrated a t the center, together with some geometric f a c t s about the packing of b a l l s , i s used t o tame the l / r p o t e n t i a l .
Section VI on Hartree-Fock
theory i s r e a l l y outside t h e c e n t r a l theme, b u t i t has been added a s a f u r t h e r exercise i n functional analysis and because i t is, a f t e r a l l , the most common approximation scheme t o solve t h e Schroedinger equation. Chapters I1 and I V come from (Lieb-Thirring, 19751, Chapter I11 from (Lieb-Simon, 1976), Chapter V from (Lieb-Lebatitz,
1972) and Chapter V I
from (Lieb-Simon, 1973). A n attempt w a s made t o present the main ideas i n as simple and
readable a form as possible, and therefore t o amit many technical d e t a i l s . There were two reasons f o r t h i s .
The f i r s t was t o t r y t o make t h e
l e c t u r e s accessible t o p h y s i c i s t s as w e l l as t o mathematicians.
This a l s o
c r e a t e s n o t a t i o n a l and semantic problems which, i t is hoped, have been a t l e a s t p a r t i a l l y resolved.
With t h i s aim i n mind, I hope t h e i n c l u s i o n of
such things as an explanation of young's i n e q u a l i t y w i l l be excused.
The
second reason stems from the b e l i e f t h a t i f enough h i n t s of a proof a r e given then a competent analyst would as soon supply the d e t a i l s f o r
himself as read about them. The bibliography i s not scholarly, but I believe no theorem has been quoted without proper credit.
I am most grateful t o S.B.
Treiman who generously devoted much time
to reading the manuscript and who made many valuable suggestions t o improve i t s c l a r i t y .
. V.
Newton's Theorem, Screening, and the Geometry of Ball Packing
I
d
1 I. Atomic Stability and the Uncertainty
' 11.
.--.-
Principle
Uncertainty Principle f o r
IV. H-Stability
+'
of Matter
V.
4.
N Fermions *
Limit and Stability
L
C
Fl VI.
The numbers indicate sections.
Thermodynamic
I I
Hartree-Pock
I.
The S t a b i l i t y of Atoms
By t h e phrase " s t a b i l i t y of an atom" is meant t h a t t h e ground s t a t e energy of an atom is f i n i t e .
This i s a wealrer notion than the concept of
8 - a t a b i l i t y of matter, t o be discussed i n Section I V , which means t h a t the ground state energy of a many-body system i s not merely bounded below but is a l s o bounded by a constant times t h e number of p a r t i c l e s .
This,
i n turn, is d i f f e r e n t from thermodynamic s t a b i l i t y discussed i n Section
v. Coneider t h e Hamiltonian f o r t h e hydrogenic atom:
(using u n i t s i n which
62 12
2 3 = 1, m = 1 and [ e l = 1) E a c t s on L (1 ), the
square i n t e g r a b l e functions on 3-space.
Why i s t h e ground s t a t e energy
f i n i t e , i.e. why is
f o r some E
0
>
-m?
The obvious elementary quantum mechanics textbook
answer is t h e Heisenberg uncertainty p r i n c i p l e (Heisenberg, 1927) : I f the k i n e t i c energy is defined by
T,,, 5 $ l ~ $ ( x ) l ~ d x and i f
:'-en when
T <x2> > 914
4
4-
.
(4)
The i n t u i t i a n behind applying t h e Heisenberg uncertainty p r i n c i p l e
(4) t o t h e ground s t a t e problem (2) is t h a t i f the e l e c t r o n t r i e s t o g e t
within a distance R of the nucleus, the k i n e t i c energy TJ,i s a t l e a s t a s large as R-~.
Consequently 2 R - 2 - ~ / ~ , and t h i s has a minimum
-zL/4 f o r R = 212. The above argument is false!
The Heisenberg uncertainty p r i n c i p l e
says no such thing, despite the endless invocation of the argument. Consider a J, consisting of two p a r t s , J, =
J,
1
is a narrow wave
packet of radius R centered a t the o r i g i n with ~ 1 1 ~ 1 ~ - 1 / 2J,2 . is spherically symmetric and has support i n a narrow s h e l l of mean radius L
11$2 12-1/2. I f L i s 11x1-' lJ,(x)1 2dx - 1/2R.
and
l a r g e then, roughly, j x
IJ,(x) I 2dx
Thus, from ( 4 ) we can conclude
2 2 T$ > 9 / 2 ~ and hence t h a t <J,,HJ,>'9/2L -Z/2R. and using
2
- L212 whereas
only t h a t
With t h i s wave function,
only the Heisenberg uncertainty p r i n c i p l e , we can make Eo
a r b i t r a r i l y negative by l e t t i n g R
+
0.
A more c o l o r f u l way t o put t h e s i t u a t i o n i s t h i s :
an e l e c t r o n
cannot have both a sharply defined p o s i t i o n and momentum.
I f one is
w i l l i n g t o place the e l e c t r o n i n two widely separated packets, however, say here and on the moon, then the Heisenberg uncertainty p r i n c i p l e alone does not preclude each packet -
from having a sharp p o s i t i o n and
momentum. Thus, while (4) is correct i t is a p a l e r e f l e c t i o n of t h e power of the operator -A t o prevent collapse. (i.e.
A b e t t e r uncertainty p r i n c i p l e
a lower bound f o r t h e k i n e t i c energy i n terms of some i n t e g r a l of
$ which does not involve derivatives) is needed, one which r e f l e c t s more
accurately the f a c t t h a t i f one t r i e s t o compress a wave function qwhere then the k i n e t i c energy w i l l increase. -
This p r i n c i p l e was provided
by Sobolev (Sobolev, 1938) and f o r some unknown reason h i s inequality, which is simple and goes d i r e c t l y t o the h e a r t of t h e matter, has not
made its way i n t o the quantum mechanics textbooks where i t belongs. Sobolev's inequality i n three dimensions (unlike (4) i t s form is dimension dependent) i s TS
a
-~ ~ 1 ~ 1 ,
j ~ v S ( d l ~ d2xK ~ ~ $ P ( X ) ~ & I " ~
where
i a the density and
i s known t o be the best possible constant.
( 5 ) is non-linear i n p , but
t h a t i s unimportant. A rigorous derivation of (5) would take too long t o present but i t
can be made plausible as follows
(Rosen, 1971) : Ks i s the minimum of
Let us accept t h a t a minimizing JI e x i s t s ( t h i s is t h e hard p a r t ) and t h a t i t s a t i s f i e s t h e obvious v a r i a t i o n a l equation
with a > 0 .
Assrnning a l s o t h a t t h e r e is a minimizing $ which i s now
negative and spherically symnetric ( t h i s can be proved by a rearrangement inequality), one finds by inspection t h a t
-
2 -1/2 ~(x) (~/~I~/.~(I+I~I When t h i s is inserted i n t o the expression f o r K' K
-
t h e r e s u l t is
( 1 2 ) ~ ~Th . e minimizing $ is not square integrable, bur t h a t is-
of no concern.
Naw let us make a simple calculation t o show how good (5) r e a l l y is. For anp
J,
and hence ~ ( x LO, ) Ip
,minih(p):
.
1)
The l a t t e r calculation is t r i v i a l (for any potential) since gradients a r e One finds t h a t the solution t o the v a r i a t i o n a l equation is
not involved.
~(x= ) a[1x1'1-~-1]112 R - K s n -413~-1.
f o r 1x1 (R and p(x) = 0 f o r 1x1
hen
-
h ( ~ ) ~ ~ ( n / 2 =) - ~ ( 4 1~3 )~Z bq (Recall t h a t one Rydberg
Ry = 114 i n these units.)
2
R, with
. Thus, (5) leads
e a s i l y t o the conclusion
and t h i s is an excellent lower bound t o the correct especially since no d i f f e r e n t i a l equation had t o be solved. I n anticipation of l a t e r developments a weaker, but a l s o useful, By ~ & l d e r ' sinequality*
form of (5) can be derived.
and, since w e always take
1 T,
*
-
2 Ks
1, fp (x)S13dx
.
Hiilder 's inequality s t a t e s t h a t
when p-l+q-l q
2
-
312.
= 1 and p
2 1. To obtain (10) take f
p, g
-
p2I3,
p = 3,
Note t h a t there is now an exponent 1 outside the i n t e g r a l .
Although Ks
is the best constant i n (5) i t is not the b e s t constant i n (11).
the l a t t e r Kl. /p(x)dx
Call
K1 is the minimum of / ~ v ) ( x )12dx/1p(3S13dx subject t o
= 1. This leads t o a non-linear Schroedinger equation whose
numerical solution y i e l d s (J. Barnes, private communication) K1
=
9.578
.
In any. event and hence
KC is much bigger than Ks;
i t is t h e c l a s s i c a l value, and w i l l be
encountered again i n section I1 and i n section 111 where its significance w i l l be c l a r i f i e d .
We can repeat the minimization calculation analogous t o (8) using the bound (12) and the functional hC(p) = xC/p ( x ~ ~ / ~ d 1x1-' x -P ~ / (xldx (We could, of course, use the b e t t e r constant K1.)
for
I xl
5 R.
.
This time
R is determined by /p = 1 and one f i n d s t h a t
R = (Kc/Z) (4/n2) 'I3 and
3 l I 3 i s only 8.2% greater than 413. The Sobolev inequality (5) o r its v a r i a n t (12) is, f o r our purposes, a much b e t t e r uncertainty p r i n c i p l e than Heisenberg's a l s o f a i r l y accurate.
- indeed i t is
W= nov want t o extend (12) t o the N-particle case
i n order to establish the s t a b i l i t y of bulk matter.
The important new
fact that w i l l be invoked is that the N particles are fermions; that is to say the N-particle wave function is an antisymmetric function of the N-space, spin variables.
11.
Extension of t h e Uncertainty Principle t o Many Fermions A w e l l known elementary calculation is t h a t of t h e lowest k i n e t i c
v , of
energy, T
N fermions i n a cubic box of volume V. For l a r g e N one
finds that
where p
-
N/V and q is t h e number of s p i n s t a t e s a v a i l a b l e t o each
p a r t i c l e (q-2 f o r electrons).
(15) is obtained by merely adding up the
N/q lowest eigenvaluea of -A with Dirichlet ($60) boundary conditions on the walls of the box. proportional t o N~~~ were not fermions.
The important feature of (15) is t h a t i t is instead of N, as would be the case i f the p a r t i c l e s
The e x t r a f a c t o r N~~~ is e s s e n t i a l f o r the s t a b i l i t y
of matter; i f electrons were bosons matter would not be s t a b l e . -213 , ought t o (15) suggests t h a t (12), with a f a c t o r q extend t o t h e N-particle case i f p(x) is interpreted properly.
The idea
i s old, going back t o Lenz (Lenz, 1932) who got i t from Thomas-Fermi theory.
The proof t h a t something l i k e (12) is not only an appro-tion
but i s a l s o a lower bound is new. To say t h a t the N p a r t i c l e s a r e fermione with q s p i n s t a t e s means t h a t the N-particle wave function JI (xl, xi€ 13 and oiE { 1,2,.
...,x , ;
U1,
...,uN) defined f o r
..,¶I is antisymmetric i n the p a i r s (xi,ui).
norm is given by
Define
t o be the usual k i n e t i c energy of $ and define
The
t o be the s i n g l e p a r t i c l e density, i . e . t h e probability of finding a p a r t i c l e a t x. Theorem 1.
The analogue of (12) is the following.
If kq,
f o r ; i f t h e r e a r e k such l e v e l s and i f N < kq then Eo > q Ee
1'
the surplus p a r t i c l e s can be placed i n wave packets f a r away from the o r i g i n with a r b i t r a r i l y small k i n e t i c energy.) Eo
2
<J,,H#
On the other hand,
= T - a j ~(xl5I3dx by the v a r i a t i o n a l principle.
'4
J,
I f these
two i n e q u a l i t i e s a r e combined together with (20), which says t h a t
I e. 1
*
2 - ( 4 / 1 5 ~ ) a ~l/p~$ ( ~ ) ~ / ~ dthen x , (18) i s t h e r e s u l t .
I
Young's inequality s t a t e s t h a t
when p-l+q-l+r'l
= 2 and p,q,r
2 1. For (22) take p = ~ and 2 q=l.
It might not be too much o u t of place t o explain a t t h i s point why K'
is c a l l e d the c l a s s i c a l constant.
The name does not stem from its
a n t i q u i t y , a s i n the i d e a l gas k i n e t i c energy ( U ) , but r a t h e r from c l a s s i c a l mechanics
-- more p r e c i s e l y
quantum mechanics.
This i n t u i t i v e idea is valuable.
the semiclassical approximation t o
A s the proof of Theorem 1 shows, the constant i n (18) f o r T
J,
is
simply r e l a t e d t o t h e constant i n (20) f o r the sum of t h e eigenvalues. The point is t h a t the semiclassical approximation t o this sum i s
and t h i s , i n turn, would y i e l d (18) without t h e
factor.
The
semiclassical approximation i s obtained by saying t h a t a region of volume (2n13 i n the 6-dimensional phase space (p,x) can accommodate one eigenstate.
Hence, i n t e g r a t i n g over the s e t 0 (H)
, in
which ~ ( ,x) p = P~+v(x)
i s negative,
I f a coupling constant g i s introduced, and i f V is replaced by gV, then i t is a theorem t h a t the semiclassical approximation i s asymptotic a l l y exact as g
+
OD
f o r any
v
in L
~ (R~). / ~
Theorem 1 gives a lower bound t o t h e - k i n e t i c energy of fermions which i s c r u c i a l f o r the H-stability of matter a s developed i n Section
IV
.
To appreciate the significance of Theorem 1 it should be
compared with t h e one-particle Sobolev bound (12). outside some fixed domain, Q, of volume V.
Suppose t h a t p(x)
Then since
by Halder's inequality, one sees t h a t T grows a t l e a s t a s f a s t a s N513
'4
Using (12) alone,one would only be able t o conclude t h a t T
J,
This d i s t i n c t i o n stems from the Pauli principle, i.e. nature of the N-particle wave function.
0
.
grows as N.
the antisymmetric
As we s h a l l see, t h i s N~~~
growth
i s e s s e n t i a l f o r the s t a b i l i t y of matter because without i t the ground s t a t e energy of N p a r t i c l e s with Coulomb forces would grow a t l e a s t as f a s t a s -N7l5
instead of -N.
The Fermi pressure is needed t o prevent a collapse, but t o l e a r n how t o e x p l o i t i t we must f i r s t turn t o another chapter i n the theory of Coulomb systems, namely Thomas-Fed theory.
111.
Thomas-Fenni Theory The s t a t i s t i c a l theory of atoms and molecules was invented independ-
e n t l y by Thomas and F e d (Thomas, 1927, Fermi,1927).
For many years t h e
TF theory was regarded a s an uncertain approximation t o t h e N-particle Schroedinger equation and much e f f o r t was devoted t o t r y i n g t o determine i t s v a l i d i t y (e.g.
~ o m b i s , 1949)
.
It was eventually noticed numerically
(Sheldon, 1955) t h a t molecules d i d not appear t o bind i n t h i s theory, and then T e l l e r ( T e l l e r , 1962) proved t h i s t o be a g e n e r a l theorem. I t is now understood t h a t TF theory i s r e a l l y a l a r g e Z theory
(Lieb-Simon,
1976); t o b e p r e c i s e i t is exact i n the limit Z
+ m.
For
f i n i t e 2, TF theory i s q u a l i t a t i v e l y c o r r e c t i n t h a t i t adequately describes t h e bulk of an atom o r molecule. give binding.
It is n o t p r e c i s e enough t o
Indeed, i t should n o t do s o because binding i n TF theory
would imply t h a t the cores of atoms bind, and t h i s does not happen. Atomic binding is a f i n e quantum e f f e c t .
Nevertheless, TF theory
deserves t o be w e l l understood because i t i s e x a c t i n a l i m i t ; the TF theory i s t o the many e l e c t r o n system a s t h e hydrogen atom i s t o the few e l e c t r o n system.
For t h i s reason the main f e a t u r e s of t h e theory a r e
presented h e r e , mostly without proof. A second reason f o r our i n t e r e s t i n TF theory i s t h i s :
i n the
next s e c t i o n t h e problem of t h e H-stability of matter w i l l b e reduced t o a TF problem.
The knowledge t h a t TF theory is H-stable
(this is a
c o r o l l a r y of t h e no binding theorem) w i l l enable us t o conclude t h a t t h e t r u e quantum system is H-stable. The Hamiltonian f o r N e l e c t r o n s with k s t a t i c n u c l e i of charges zi > 0 and l o c a t i o n s Ri i s
where
and
The nuclear-nuclear repulsion U i s , of course, a e o n s f e ~ tterm i n
$ but
it is included f o r two reasons: (i)
W e wish t o consider the dependence on the Ri of EQ ({z ,R }C1) :the ground s t a t e energy of I$ N j j j
.
( i i ) Without U the energy w i l l not be bounded by N. The nuclear k i n e t i c energy i s not included i n
%.
For t h e H-
Q s t a b i l i t y problem we a r e only i n t e r e s t e d i n finding a lower bound t o EN, and t h e nuclear k i n e t i c energy adds a p o s i t i v e term. inf
C R 1~
Q z j ,Rj %({
I n other words,
i s smaller than the ground s t a t e energy of t h e t r u e
Hamiltonian (defined i n (58)) i n which t h e nuclear k i n e t i c energy is included.
Later on when we do the proper thermodynamics of t h e whole
system we s h a l l have t o include the nuclear k i n e t i c energy. The problem of estimating
%Q is
a s old a s t h e Schroedinger equation.
The TF theory, a s i n t e r p r e t e d by Lenz
(Lenz, 1932), reads a s follows:
For fermions having q s p i n s t a t e s (q= 2 f o r elecfrons) define t h e e n e r a functional:
i s t h e TF energy f o r A e l e c t r o n s (1 need not be an i n t e g e r , of course). When A = N the minimizing
p
is supposed t o approximate the p
JI
given by
is
(17), wherein 9 i s the t r u e ground s t a t e wave function, and supposed t o approximate
Q %.
The second and fourth tenne on t h e r i g h t s i d e of (26) a r e exact but the f i r s t and t h i r d a r e not.
The f i r s t is t o some e x t e n t j u s t i f i e d by
t h e k i n e t i c energy i n e q u a l i t y , Theorem 1; the t h i r d term w i l l be discussed later.
I n any event, (26) and (27) define TF theory.
It would b e too much t o t r y t o reproduce h e r e the d e t a i l s of our analysis of TF theory.
A s h o r t summary of some of t h e main theorems w i l l
have t o s u f f i c e . The f i r s t question is whether o r not
E~P
(which, by simple estimates
using Young's and Hglder's i n e q u a l i t i e s can b e shown t o b e f i n i t e f o r a l l A ) is a minimum as d i s t i n c t from merely an infimum.
The d i s t i n c t i o n is
c r u c i a l because the TF equatcon (the Euler-Lagrange equation f o r (26) and
with
has a s o l u t i o n with
i f and only i f there is a minimizing
Ip-x
The b a s i c theorem is Theorem 3. (i)
k A
IZ I
I zj
j-1
than
(0) haa a minimum on the s e t I P ( X ) ~ I=CA .
YI
TF
f o r EA
.
(ii) and ( 2 9 ) .
TF
This minimizing P ( c a l l i t p X ) i s unique and s a t i s f i e s (281 p is non-negative,
a n d - u i s t h e chemical p o t e n t i a l , i.e.
-u (iii)
-
aEy/ax
.
There i s no other s o l u t i o n t o (28) and (29) (for any
IP=X
TF other than pX (iv)
u
) with
.
X = 2, p = 0.
Otherwise Y > 0,
&E? is s t r i c t l y
decreasing i n A. (v)
&X
v a r i e s from 0
to 2,
p v a r i e s continuously from
+-
0.
(vi) p i s a convex, decreasing function of A. TF
(di) $A (x) > 0 f o r a l l x (5/3)ICC q-2'3 TF
I f X > Z then E ( I ) is not with Jp = X. Negative ions -
EP
exists and Ef
=
E?
X.
Hence when X = Z
P ? ( x ) ~ / ~=
(F(3 .
a minimum and (28) and (29) have no s o l u t i o n do not e x i s t i n TF theory.
Nevertheless,
@ X 2 2.
The proof of Theorem 3 is an exercise i n functional analysis. Basically, one f i r s t shows t h a t &(P) i s bounded below s o t h a t Ef
exists.
The Banach-Alaoglu theorem is used t o find an L " ~ weakly convergent TF sequence of 0's such t h a t E ( P ) converges t o EX
.
Then one notes t h a t
&(PI is weakly lower semicontinuous s o t h a t a minimizing p e x i s t s . uniqueness comes from an important property of $(p), convex. -
The
namely t h a t i t is
This a l s o implies t h a t the minimizing p s a t i s f i e s /p
-
A.
A
major point t o notice is t h a t a s o l u t i o n of t h e TF equation is obtained as a byproduct of minimizing. g ( p ) ; a d i r e c t proof t h a t the TF equation has a solution would be very complicated.
Apart from t h e d e t a i l s presented i n Theorem 3, the main point i s t h a t TF theory i s w e l l defined.
I n p a r t i c u l a r the density p i s unique
-
a s t a t e of a f f a i r s i n marked contrast t o t h a t of Hartree-Fock theory. The TF density p p has t h e following properties:
Lf
Theorem 4. (i)
12Z
then
-
(5/3)KC q-213 ~ 7 ( x ) ~Z ~ ~I X~-
near each Ri. (ii)
I n the n e u t r a l caae, 1 = Z =
R ~ I - ~
k
1
j-1
as
1x1
+
(iii)
OD,
i r r e s ~ e c t i v eof t h e d i s t r i b u t i o n of t h e nuclei.
TF
TF
p X (x) a r e r e a l a n a l y t i c i n x away from a l l the Ri,
(x)
on a l l of 3-space i n t h e n e u t r a l case and on ix:
TF (x) > 11) i n the
p o s i t i v e i o n i c case. (32) is especially remarkable:
a t l a r g e distances one loses a l l
knowledge of the nuclear charges and configuration.
Property (i)r e c a l l s
the s i n g u l a r i t y found i n the minimization of hC(p) (see (13)). (31) can be seen from (28) and (29) by inspection.
(32) is more
s u b t l e b u t i t is consistent with t h e observation t h a t (28) and (29) can be rewritten (when u=O) a s c 312 -(4n)-'~ @fP(x) = -i(3/5)q213 ( y ( x ) h C 1 away from the R
i*
I f i t is assumed t h a t 4
of 1x1 then (32) follows. (Sommerfeld, 1932).
TF
z (x) goes t o zero as a p w e r
This observation was f i r s t made by Sonnnerfeld
The proof t h a t a paver law f a l l o f f actually occurs
is somewhat s u b t l e and involves p o t e n t i a l t h e o r e t i c ideas such a6 t h a t used i n the proof of Leauna 8.
As pointed o u t e a r l i e r , t h e connection between TF theory and t h e Schroedinger equation i s b e s t seen i n t h e l i m i t Z +
k, of n u c l e i be h e l d f i x e d , b u t l e t N
+ m
and zi +
-.
-
Let t h e n u d e r ,
i n such a way t h a t
k
1
t h e degree of i o n i z a t i o n N/Z i s constant,where Z = we make the following d e f i n i t i o n :
To this end
j=l zj'
Fix {z R lk and X. I t is n o t j' j j = l For each N = 1,2, define aN by
...
necessary t o assume t h a t X ( 2 .
his means (23) replace z j by zj% and R by R -'I3. j j a ~ t h a t t h e nuclei come together a s N + I f they s t a y a t f i x e d p o s i t i o n s
1%
In
= N.
l$
-.
then t h a t is equivalent, i n t h e limit, t o i s o l a t e d atoms, i . e .
i t is
equivalent t o s t a r t i n g with a l l t h e n u c l e i i n f i n i t e l y f a r from.each other. F i n a l l y f o r the nuclear configuration {%z j, ground s t a t e wave function,
41 be
the
Q t h e ground s t a t e energy, and pN(x) be the
s i n g l e p a r t i c l e density a s defined by (17). J,
sN-113~jl:=l l e t
[Note:
If
Q
i s degenerate,
can be any ground s t a t e wave function a s f a r a s Theorem 5 is concerned.
If
Q
i s not an eigenvalue, but merely i n £ spec
l$,
then i t i s p o s s i b l e
Q s t i l l given by (17), i n t o d e f i n e an approximating sequence qN, with pN such a way t h a t Theorem 5 holds.
We omit t h e d e t a i l s of t h i s construc-
t i o n here.] It i s important t o note t h a t t h e r e i s a simple and obvious s c a l i n g
r e l a t i o n f o r TF theory, namely
and
f o r any a
2 0.
Hence, f o r t h e above sequence of systems parametrized by
f o r a l l N. I f , on the other hand, the n u c l e i a r e held f i x e d then one can prove that
where E?(Z) i s t h e energy of an i s o l a t e d atom of nuclear charge z. The j k 1 a r e determined by t h e condition t h a t X = X i f X ( Z (otherwise 1 j=1j
1
Xj = Z) and t h a t the chemical p o t e n t i a l s of the
1=1
lc at-
same.
Another way t o say t h i s i s t h a t t h e X
(37).
With t h e nuclei fixed, the analogue of (36) i s
l i m 0 , a s
w i l l be necessary i n Section I V , then (39) is c o r r e c t i f KC i s replaced
Theore.5.
maNEN/P&{z
,%'1/3~ 1)
0 l e t
..,%;Rdl, ...,%) , where the three E~
systems (i.e. p=O f o r a l l a ) .
a r e defined f o r n e u t r a l
The goal is t o show t h a t f ( 1 )
f (0) = 0, i t is enough t o show t h a t df(a)/da
2 0.
jsl
2
0.
Since
From (26) and (27) i t
is t r u e , and almost obvious, t h a t
This is the TF version of the Feynman-Hellman theorem; notice how the nuclear-nuclear repulsion comes i n here.
where na(x)
-
TP
TF (x) and
Thus,
(y is the p o t e n t i a l f o r
.., a ~ ~ , z , ~ ,...,%; Rl,. ..,\I and )? is t h e p o t e n t i a l f o r TF TF {az, ,...,a zm ; R 1,...,Rm 1. (x) ) +2 (x) f o r a l l x by Lenrma 8, and
carl,.
'
ria (x)
hence
2
0.
Theorem 6 has a n a t u r a l application t o the s t a b i l i t y of matter problem.
A s w i l l be shown i n the next section, the TF energy (27) is,
with s u i t a b l y modified constants, a lower bound t o the true quantum energy for Theorem 9.
2.
By Theorem 3 ( i v ) and Theorem 6 we have t h a t
Q { z
R
k
Ik
and let Z =
j' j j=l-
1 zj.
Jal
Then f o r ~ 1 1 1 1 ~ 0
The l a t t e r constant, 2.21, i s obtained by numerically solving t h e TF equation f o r a s i n g l e , n e u t r a l atom (J. Barnes, p r i v a t e cormnunication). By scaling, (43) holds f o r any choice of K'
i n the d e f i n i t i o n (26) of
$5(PI. Theorem 9 is what w i l l be needed f o r the H-stability because i t says t h a t the TF system is H-stable,
i.e.
of matter,
the energy is
bounded below by a constant times the nuclear p a r t i c l e number (assuming t h a t the z
1
a r e bounded, of course).
Another application of Theorem 6 t h a t w i l l be needed i s the following strange inversion of the r o l e of electrons and n u c l e i i n TF theory.
It
w i l l enable us t o give a l a s e r bound t o the t r u e quantum-mechanical electron-electron repulsion.
This theorem has nothing t o do with quantum
mechanics per se; i t i s r e a l l y a theorem purely about e l e c t r o s t a t i c s even though i t i s derivqd from the TF no binding theorem.
Theorem 10. Suppose t h a t xl,
...,%
and define
a r e any N d i s t i n c t points i n 3-space
-1 N
V,(y)
lY-xj 1-l j =l
Let y -
> 0 and l e t p(x) be any nownegative function such t h a t Ip(x)dx
0
Notice how the f i r s t and second terms on the r i g h t s i d e of (45) combine t o give
+ 1/2
since
To control the k i n e t i c energy i n (23) Theorem 1 is used; the t o t a l r e s u l t is then
Q 2 a ~ ~ , ( x ) ~ / ~ d x - ~ ~ ( x )dx s , (+x$ )
E,
1 (x) ~ ~
111Y -0 '
J, (yldxdy
with
R e s t r i c t y, which w a s a r b i t r a r y , s o t h a t a > 0. Then, a p a r t from the constant term -(2.21)~y-l, (49) i s j u s t g a ( p ), the Thomas-Fenni J,
energy functional ga(p,)
-> E"a,N
6 applied
to p
$'
but with q'2'3~C
replaced by a.
Since
:inf { F a ( p ) : I p = ~ }(by d e f i n i t i o n ) , and since the n e u t r a l
case always has the lowest TF energy, a s shown i n Theorem 9, we have t h a t
Thus we have proved the following: Theorem 11. f
+
i s a normalized, antisymmetric function of space and
spin of N variables, and i f there a r e q s p i n s t a t e s associated with each
p a r t i c l e then, f o r any y > 0 such t h a t a defined by (50) is p o s i t i v e ,
The optimm choice f o r y
i n which case
# , -(2.21)
k
1/2 2
This is the desired r e s u l t , but some a d d i t i o n a l remarks a r e i n order. (1)
-
Since [l+a 'l2l2 < 2+2a,
a r e bounded above by some f i x e d z, Thus, provided t h e nuclear charges z j is indeed bounded below by a constant tfoes t h e t o t a l p a r t i c l e number
#
N+k
. (2)
Theorem 11 does not presuppose n e u t r a l i t y .
(3)
For electrons, q=2 and t h e prefactor i n (53) is -(2.08)N.
As
remarked a f t e r Theorem 1, t h e unwanted constant ( 4 ~ ) ~ has ' ~been improved t o [4n/ (1.83)
1'I3.
Using t h i s , the prefactor becomes
- (1.39)N.
If
zj = 1 (hydrogen atoms) and N = k ( n e u t r a l i t y ) then
# 2 -(5.56)N (4)
= -(22.24)N
By
.
The power law z7I3 cannot be improved upon f o r l a r g e z because
Theorem 5 a s s e r t s t h a t the energy of an atom is indeed proportional to z7I3 f o r l a r g e z.
(5)
It is a l s o possible t o show t h a t matter i s indeed bulky.
This
w i l l be proved f o r any J, and any nuclear configuration (not j u s t the minimum energy configuration) f o r which E~ < 0.
d
-
The minimizing nuclear
configuration is, of course, included i n t h i s hypothesis.
where
N . 8;1 i s (23) but with a f a c t o r 112 multiplying 1-1 I Ai.
,jy> 2 2%, where EN is
Then
By Theorem (11).
the ri&t s i d e of (53) (replace K'
by KC/2
Therefore, the f i r s t important f a c t i s t h a t
there).
and t h i s is bounded above by the t o t a l p a r t i c l e number.
2 0,
i t is easy t o check t h a t there is a C > 0 such P t h a t f o r any nonnegative p(x),
Next, f o r any p
{ ~ (x)5/3dx~p12 p j
1 XI
P (x) dx
2
cp{Jp (x)dx) 1+5p/6
It is easy t o f i n d a minimizing p f o r t h i s , and t o calculate C : p(xl2I3 = 1-1x1'
P
Since T
4'
-
f o r 1x1 < 1; ~ ( x )= 0, otherwise.
s a t i s f i e s (18) we have t h a t
-
with C' C (KC/4)*I2 (4ns)-~/3. P P I f it is assumed t h a t ~ Z ! / ~ / N is bounded, and hence t h a t J ( N ~ ' ~ / )'I2 > d l 3 f o r some A, we reach the conclusion t h a t the radius of the system i s a t l e a s t of t h e order N " ~ ,
a s i t should be.
The above analysis did not use any s p e c i f i c property of t h e Coulomb p o t e n t i a l , such as the v i r i a l theorem. general Hamiltontan H i n (58). n,k
It i s a l s o applicable t o t h e more
(6)
The q dependence was purposely retained i n (53) i n order t o say
something about bosom.
I f q=N, then i t is easy t o s e e t h a t the' require-
ment of antisymmetry i n $ is no r e s t r i c t i o n a t a l l . one has simply
2 3N over a l l of L (1 )
I n t h i s case then,
# = inf spec %
.
Therefore
It was shown by Dyson and Lenard (Dyson-Lenard, 1967) t h a t EQ (bosons) N
2 -(constant)N 5/3 ,
and by Dyson (Dyson, 1967) t h a t
Q %(bosone) Proving (57) was not easy.
5 -(constant) N 715
.
Dyson had t o construct a r a t h e r complicated
v a r i a t i o n a l function r e l a t e d t o the type used i n the BCS theory of superconductivity.
Therefore bosom a r e not s t a b l e under the a c t i o n of
Coulomb forces, but the exact power law is not yet known.
Dyson has
conjectured t h a t i t is 715. I n any event, the e s s e n t i a l point has been made t h a t F e d s t a t i s t i c s is
e s s e n t i a l f o r the s t a b i l i t y of matter.
The uncertainty principle
f o r one p a r t i c l e , even i n the strong form (5), together with i n t u i t i v e notions t h a t the e l e c t r o s t a t i c energy ought not t o b e very great, a r e insufficient for stability.
The additional physical f a c t t h a t is needed
is t h a t the k i n e t i c energy increases a s the 513 power of the fermion density.
-
V.
The Thermodynamic Limit
Q i s bounded below by t h e t o t a l p a r t i c l e Having e s t a b l i s h e d t h a t EN number, the next question t o consider i s whether, under appropriate conditions, $/N
has a l i m i t a s N +
-, a s
expected.
More generally, the
same question can be asked about the f r e e energy per p a r t i c l e when t h e t e m p e r a t u r e is not zero and t h e p a r t i c l e s a r e confined t o a box. It should be appreciated t h a t the d i f f i c u l t y i n obtaining t h e lower
bound t o
4
came almost e n t i r e l y from t h e r
t h e Coulomb p o t e n t i a l .
-1
s h o r t range s i n g u l a r i t y of
Other p o t e n t i a l s , such a s t h e Yukawa p o t e n t i a l ,
with t h e same s i n g u l a r i t y would present t h e same d i f f i c u l t y which would be resolved i n the same way.
The s i n g u l a r i t y was tamed by the
p
5/3
behavior of t h e fermion k i n e t i c energy. The d i f f i c u l t y f o r t h e thermodynamic l i m i t i s d i f f e r e n t .
-1 caused by t h e long range r behavior of t h e Coulomb p o t e n t i a l .
It i s
I n other
words, we a r e faced with t h e problem of explosion r a t h e r than implosion. Normally, a p o t e n t i a l t h a t f a l l s off with d i s t a n c e more slowly t h a n r'3'E f o r some
E
> 0 does = h a v e
a thermodynamic l i m i t .
Because the charges
have d i f f e r e n t signs, however, t h e r e i s hope t h a t a c a n c e l l a t i o n a t l a r g e d i s t a n c e s may occur. An a d d i t i o n a l physical hypothesis w i l l be needed, namely n e u t r a l i t y . To appreciate t h e importance of n e u t r a l i t y consider t h e case t h a t the e l e c t r o n s have p o s i t i v e , instead of negative charge. every term i n (23) would be p o s i t i v e .
Q > 0 because Then EN
While the H-stability question i s
t r i v i a l i n t h i s case, the thermodynamic l i m i t is not.
I f the p a r t i c l e s
a r e constrained t o be i n a domain R whose volume IRl is proportional t o N , the p a r t i c l e s w i l l r e p e l each other so s t r o n g l y t h a t they w i l l a l l go
t o t h e boundary of R i n order t o minimize t h e e l e c t r o s t a t i c energy.
The
minimum e l e c t r o s t a t i c energy w i l l be of the order +N
2
-1/3
_
,$i/3
Hence no thermodynamic l i m i t w i l l e x i s t . When the system is n e u t r a l , however, the energy can be expected t o be extensive, i.e. O(N).
For t h i s t o be so, d i f f e r e n t p a r t s of the
system f a r from each other must be approximately independent, despite the long range nature of the Coulomb force.
The fundamental physical, o r
r a t h e r e l e c t r o s t a t i c , f a c t t h a t underlies t h i s is screening;
the distri-
bution of the p a r t i c l e s must be s u f f i c i e n t l y n e u t r a l and i s o t r o p i c l o c a l l y s o t h a t according t o Newton's theorem (13 below) the e l e c t r i c p o t e n t i a l f a r away w i l l be zero.
The problem i s t o express t h i s idea i n precise
mathematical form. We begin by defining the Hamiltonian f o r the e n t i r e system consisting 2 of k nuclei, each of charge z and mass M, and n electrons (6 /2 = 1, m = 1,
The f i r s t and second terms i n (58) a r e , respectively, the k i n e t i c energies of the e l e c t r o n s and the nuclei.
The l a s t three terms are, respectively,
the electron-nuclear, electron-electron and nuclear-nuclear Coulomb interactions. a r e yi.
The electron coordinates a r e xi and the nuclear coordinates
The electrons a r e fermions with s p i n 1/2; the nuclei may be
e i t h e r bosons o r fermians. The b a s i c n e u t r a l i t y hypotheses is t h a t n and k a r e r e l a t e d by n-kz. It is assumed t h a t z i s r a t i o n a l .
The thermodynamic 1Mt t o be discussed here can be proved under more general assumptions, i.e. we can have s e v e r a l kinds of negative p a r t i c l e s (but they must a l l b e fermions i n order t h a t the b a s i c s t a b i l i t y estimate of Section I V holds) and s e v e r a l kinds of n u c l e i with Neutrality must always hold,
d i f f e r e n t s t a t i s t i c s , charges and masses. however.
Short range forces and hard cores, i n addition t o t h e Coulomb
forces, can a l s o be included with a considerable s a c r i f i c e i n s i m p l i c i t y of the proof.
H
n,k a s well).
a c t s on square integrable functions of n+k v a r i a b l e s (and s p i n
To complete t h e d e f i n i t i o n of H we must specify boundary n,k conditions: choose a domain Q (an open s e t , which need not be connected) and require t h a t JI = 0 i f xi o r yi a r e on t h e boundary of Q. For each non-negative integer j , choose an n
1
k
j
determined by ( 5 9 ) , and choose a domain Q
j'
and a corresponding
The symbol N
j
will
henceforth stand f o r the pair (n k ) and j' j
We require t h a t the d e n s i t i e s
be such t h a t
.
l i m pj = p jp
is then the density i n the thermodynamic l i m i t .
the Q
j
t o be a sequence of b a l l s of r a d i i R
It can be shown t h a t the
j
and s h a l l denote them by B
same thermodynamic
f r e e energy holds f o r any sequence N
1'
Q
j
Here we s h a l l choose j'
l i m i t f o r t h e energy and
and depends
9 on
the limiting
p and 6, and not on the "shape" of the
Q
j
, provided
the Sl
go t o i n f i n i t y
J
i n some reasonable way. The b a s i c quantity of i n t e r e s t is t h e canonical p a r t i t i o n function
2 where the t r a c e is on L (a)
lNI
and 0 * 1/T, T being the temperature i n
u n i t s i n which Boltzmann's constant is unity. The f r e e energy per u n i t volume is
and the problem i s t o show t h a t with
then
l i m F E F(p,B) J j*
exists.
A s i m i l a r problem is t o show t h a t
the ground s t a t e energy per u n i t volume, has a l i m i t e(p) = l i m E j j* where
The proof we w i l l give f o r t h e l i m i t F(p,B) w i l l hold equally well f o r can be s u b s t i t u t e d f o r F i n a l l statements. j j The b a s i c s t r a t e g y c o n s i s t s of two p a r t s . The e a s i e s t p a r t is t o
e ( p ) because E
show t h a t F
3
is bounded below.
W e already know t h i s f o r E
J
by the
r e s u l t s of s e c t i o n IY. sequence F
i s decreasing.
j
Theorem 12. p =
The second s t e p is t o show t h a t i n some sense t h e
Given N,Q
IN~/IQ~
This w i l l then imply t h e e x i s t e n c e of a l i m i t .
&B
t h e r e e x i s t s a constant C depending only on
0 such t h a t 'F(N,a,B)
Proof. -
Write H
2c
HA+%, where
i s h a l f the k i n e t i c energy.
Then HB
2 b IN I , with b depending only on
z,
by t h e r e s u l t s of Section I V (increasing t h e mass by a f a c t o r of 2 i n HB only changes t h e constant b)
.
Hence Z (N,R, 0)
5 emBblNITr exp (-BHA).
However, Tr exp(-@HA) i s t h e p a r t i t i o n function of an i d e a l gas and i t
is known by e x p l i c i t computation t h a t i t is bounded above by eedlNlwith
1
d depending only on p = I N / 1 Q
1
and 0.
Thus
For t h e second s t e p , two elementary b u t b a s i c i n e q u a l i t i e s used i n t h e general theory of t h e thermodynamic l i m i t a r e needed and they w i l l be described next. A. Domain p a r t i t i o n inequality:
Given t h e domain R and t h e p a r t i c l e
numbers N = (n,k), l e t n b e a p a r t i t i o n of R i n t o
n
a Q
,..., .
1
d i s j o i n t domains
Likewise N i s p a r t i t i o n e d i n t o a i n t e g r a l p a r t s (some of
which may b e zero):
1 N=N+...+N
e
Then f o r any such p a r t i t i o n , r, of 0 and N
.
Here ~r~ means t r a c e over
and
$ is
for the N
defined as i n (58) but with D i r i c h l e t ($I= 0) boundary conditions i
p a r t i c l e s on the boundary of Q
Simply s t a t e d , t h e f i r s t N N
2
2
t o fi
, etc.
1
i
( f o r i=1,...,&).
1 p a r t i c l e s a r e confined t o S1 , the second
The i n t e r a c t i o n among t h e p a r t i c l e s i n d i f f e r e n t domains
is s t i l l present i n
$.
(69) can be proved by t h e Peierls-Bogoliubov
v a r i a t i o n a l p r i n c i p l e f o r Tr ex.
Alternatively, (69) can b e viewed simply
a s t h e statement t h a t t h e i n s e r t i o n of a hard wall, i n f i n i t e p o t e n t i a l on i the boundaries of the fi only decreases 2; the f u r t h e r r e s t r i c t i o n of a i
d e f i n i t e p a r t i c l e number t o each fi
t h a t the t r a c e is then over only the
f u r t h e r reduces 2 because i t means
.
Since eA = e"o'
i s a simple tensor
lNil,
product of operators on each L ~ ( Q ~ ) W i s merely the average i n t e r domain Coulomb energy i n a canonical ensemble i n which the Coulomb interaction i s present i n each subdomain but the II domains a r e independent of each other.
This b a s i c idea i s due t o Grif f i t h s (Grif f i t h s , 1969). i
other words, l e t q (x), xER
i
, denote
the average charge density i n R
In
i
f o r t h i s ensemble of independent domains, namely
with the notation: A
i n fli, dx
1
Xi stands f o r the coordinates of the
I N iI
particles
i means integration over a l l these coordinates ( i n D ) with the
exception of x
J'
and x
is s e t equal t o x; qj i s the charge (-1 o r + z )
of the j t h p a r t i c l e ; e q ( - f 3 H i
i i ) ( X ,Y ) is a kernel (x-space representation)
i
f o r exp ( - B H ~ ) . q (x) vanishes i f x
4 ni.
With the definitions (75) one has t h a t
(70), together w i t h (76) and (74) is t h e desired inequality f o r t h e interIt is q u i t e general i n t h a t an analogous inequality
domain interaction.
holds f o r a r b i t r a r y two-body potentials.
Neither s p e c i f i c properties of
the Coulomb p o t e n t i a l nor n e u t r a l i t y was used. Now we come t o t h e c r u c i a l point a t which screening is brought in. The following venerable r e s u l t from the Principia Mathematics is e s s e n t i a l . Theorem 13 (Newton)
.
L t e p (x) b e an integrable function on 3-space such
t h a t p(x) = p(y) If_ 1x1 = 1y
1
-
-
(isotropy) a 2 p(x) = 0 i f 1x1 > R f o r
be the Coulomb p o t e n t i a l generated by p .
Then i f 1x1 > R
-
The important point is t h a t an isotropic, n e u t r a l charge d i s t r i b u t i o n generates zero p o t e n t i a l outside its support, i r r e s p e c t i v e of how the charge i s d i s t r i b u t e d r a d i a l l y . Suppose t h a t N
i
i s neutral, i.e.
the electron number = z times the
nucleon number f o r each subdomain i n Sl.
Ci
i
is a
ball of i
radius R
i
centered a t a
i
invariant, q (x) = q (y) i f Ix-a i
and q (x) = 0 i f /=-ail involving q
i
> R
i
.
i
I
Suppose a l s o t h a t the subdomain i
.
= Iy-a
i
Then since H~ i s r o t a t i o n
1,
jqi (x)dx
= 0 (by n e u t r a l i t y )
Then, by Theorem 13, every term i n (76)
vanishes, because when
jZi,
q j(y) = 0 i f
1 y-ail
< R
i
since
Q1 i s d i s j o i n t from 0i
.
Consequently t h e average interdomain i n t e r a c t i o n ,
<W>, vanishes. I n the decomposition, n, of
.
a i n t g nl, ..,n II and
.,N'
N i n t o N1,..
we
w i l l arrange matters such t h a t (i) (ii) (iii)
..,nL-' N', ...,N'-' N~ = 0 . nl,.
are balls are neutral
Then <W> = 0 and, using (69) and (70)
I n addition t o ( i ) , ( i i ) , ( i i i ) i t w i l l a l s o be necessary t o arrange matters such t h a t when Q i s a b a l l BK i n the chosen sequence of domains, then t h e sub-domains
2,...,~'-li n
the p a r t i t i o n of BK a r e
smaller
b a l l s i n the same sequence., With these requirements i n mind the standard sequence, which depends on the l i m i t i n g density, p , is defined a s follows: (1) Choose p > 0. (2)
Choose any No s a t i s f y i n g t h e n e u t r a l i t y condition (59).
(3)
Choose Ro such t h a t 3 28(4n/3)pRo
(4)
lNOl
.
For j > 1 l e t
-
be the r a d i u s of the b a l l B and the p a r t i c l e number i n t h a t b a l l . j
It w i l l be noted t h a t t h e density i n a l l the b a l l s except the f i r s t
pjPps
j ) 1 ,
(82)
while t h e density i n the smallest b a l l i s much bigger:
This has been done s o t h a t when a b a l l BK, K 2 1 i s packed with smaller b a l l s i n t h e manner t o be described below, the density i n each b a l l w i l l come out r i g h t ; the higher density i n Bo compensates f o r t h e portion of BK not covered by smaller b a l l s .
The r a d i i increase geometrically,
namely by a f a c t o r of 28. The number 28 may be s u r p r i s i n g u n t i l i t is r e a l i z e d t h a t t h e objective is t o be a b l e t o pack BK with b a l l s of type BK-l,
BK-2,
i n such a way t h a t as much as p o s s i b l e of BK i s covered and
also t h a t
very l i t t l e of BK i s covered by very small b a l l s .
etc.
I f t h e r a t i o of r a d i i
were too c l o s e t o unity then the packing of BK would be i n e f f i c i e n t from t h i s point of view.
I n s h o r t , i f t h e number 28 i s replaced by a much
smaller number the analogue of t h e following b a s i c geometric theorem
w i l l not be true. Theorem 14 (Cheese theorem).
Xj a p o s i t i v e i n t e g e r define the i n t e g e r
Then f o r each p o s i t i v e i n t e g e r K 2 1 i t i s possible K-1 t o pack t h e b a l l BK of radius % (given by 81) with .U (t-j b a l l s of 1=0 radius R ) "Pack" means t h a t a l l t h e b a l l s i n the union a r e d i s j o i n t .
"1
' (27)1-1(28)2j. 1
.
W e w i l l not give a proof of Theorem 14 here, but note t h a t it
e n t a i l s showing t h a t ml b a l l s of radius
can be packed i n BK i n a
cubic array, then t h a t m2 b a l l s of radius %-2 a r r a y i n t h e i n t e r s t i t i a l region, e t c .
can b e packed i n a cubic
Theorem 14 s t a t e s t h a t BK can be packed with (28) 2 b a l l s of type
3
b a l l s of type BK-2,
is the f r a c t i o n of the K- j volume of BK occupied by a l l the b a l l s of radius R i n the packing, then (27) (28) B~-l'
etc.
If f
1
with
The packing i s asymptotically complete i n the sense t h a t lim K*
K-1
1
OD
j=o
f
K-j
= (1127)
1
yj
5
j=1
1
.
It is a l s o "geometrically rapid" because the f r a c t i o n of
IB K I
that is
uncovered i s
The necessary ingredients having been assembled, we can now prove Theorem 15.
Given p and
B > 0, the thermodynamic l i m i t s F(p,B)
e(p)
(65,67) e x i s t f o r t h e sequence of b a l l s and p a r t i c l e numbers s p e c i f i e d
b~ (80) Proof. -
(81). Let FK given by (64) b e the f r e e energy per u n i t volume f o r t h e
b a l l B with N p a r t i c l e s i n it. For K 2 1 , p a r t i t i o n BK i n t o d i s j o i n t K K 1 i domains D , ,nk , where t h e D f o r i=1,. ,I1-1 designate t h e smaller
..
...
b a l l s referred t o i n Theorem 14, and
n 9.
(which i s the "cheese" a f t e r the
holes have been removed) is t h e remainder of BK. copies of B
0
t o (81). 'N
= 0.
J'
2
j (K-1;
The smaller b a l l s a r e
i n each of these place N
The t o t a l p a r t i c l e number i n
%
p a r t i c l e s according j i s then
a s i t should be.
U s e t h e b a s i c i n e q u a l i t y (79) ; <W> = 0 s i n c e a l l the smaller b a l l s a r e n e u t r a l and dividing by
with f
with
j
I. contains no p a r t i c l e s .
IB ~ ,I we
have f o r K
= y j/27 and y = 27/28.
% -> 0.
,
Thus, taking logarithms and
1 that
This i n e q u a l i t y can be r e w r i t t e n as
(89) i s a renewal equation which can be solved e x p l i c i t l y
by inspection:
We now use the f i r s t s t e p , Theorem 13, on t h e boundedness of FK. 0
Since FK F -+ K
-.
1
d must be f i n i t e , f o r otherwise (90) would say t h a t j j=1 The convergence of the sum implies t h a t -+ 0 a s K + m. Hence
2
C,
%
the l i m i t e x i s t s ; s p e c i f i c a l l y
Theorem 15 is t h e desired goal, namely the existence of the thermodynamic l i m i t f o r the f r e e energy (or ground s t a t e energy) per u n i t volume.
There a r e , however, some a d d i t i o n a l p o i n t s t h a t deserve comment.
,A
For each given l i m i t i n g density P , a p a r t i c u l a r sequence of
domains, namely b a l l s , and p a r t i c l e numbers was used.
It can be shown
t h a t the same l i m i t is reached f o r general domains, with some mild conditions on t h e i r shape including, of course, b a l l s of d i f f e r e n t r a d i i than t h a t used here.
The argument involves packing the given domains
with b a l l s of the standard sequence and v i c e versa. but standard, and can b e found i n (Lieb-Lebowitz,
B. -
The proof i s tedious,
1972);
Here we have considered the thermodynamic l i m i t f o r r e a l matter,
i n which a l l the p a r t i c l e s a r e mobile. of some physical i n t e r e s t .
There a r e , however, other models
One i s Jellium i n which the p o s i t i v e n u c l e i
a r e replaced by a fixed, uniform background of p o s i t i v e charge.
With the
a i d of an additional t r i c k t h e thermodynamic limit can a l s o be proved f o r t h i s model (Lieb-Narnhofer,
1975).
Another, more important model is one
i n which the nuclei a r e fixed point charges arranged periodically i n a This i s the model of s o l i d s t a t e physics.
lattice.
Unfortunately, l o c a l
r o t a t i o n invariance is l o s t and Newton's Theorem 13 cannot be used.
This
problem i s s t i l l open and i t s s o l u t i o r i w i l l require a deeper i n s i g h t i n t o screening.
8
-
C. -
A n absolute physical requirement f o r BF(p,B), a s a function of
1/T, is t h a t i t b e concave.
This is equivalent t o t h e f a c t t h a t the
s p e c i f i c heat is non-negative since ( s p e c i f i c heat) Fortunately i t i s true.
-8
2 2 a 8F(p ,B)/a8 2
.
From the d e f i n i t i o n s (57), (58) we s e e t h a t
I n Z(N,Q,$) is convex i n 8 f o r every f i n i t e system and hence BF(N,O,B)i s concave.
Since t h e l i m i t of a sequence of concave functions is always
concave, the l i m i t BF(p,B) is concave i n 8.
Another a b s o l u t e requirement i s t h a t F(p, B) be convex a s a function of p.
This i s c a l l e d thermodynamic s t a b i l i t y a s d i s t i n c t from
t h e lower bound H - s t a b i l i t y of t h e previous s e c t i o n s . t o t h e f a c t t h a t t h e compressibility i s non-negative, (compressibility)-' t h e o r i e s (e.g.
=
PI ap
a2~
= p
( pfi)/ap2. ,
I t i s equivalent
since
Frequently, i n approximate
Van d e r Waals' theory of the vapor-liquid t r a n s i t i o n , some
f i e l d t h e o r i e s , o r some t h e o r i e s of magnetic systems i n which the magnetization per u n i t volume plays t h e r o l e of p ) , one introduces an F with a double bmp.
Such a n F is non-physical and never should a r i s e i n
an e x a c t theory. For a f i n i t e system, F i s defined only f o r i n t e g r a l N , and hence not for a l l r e a l p.
It can be defined f o r a l l p by l i n e a r i n t e r p o l a t i o n , f o r
example, but even s o i t can n e i t h e r b e expected, nor i s i t generally, convex, except i n the limit.
The i d e a behind t h e following proof i s
standard. Theorem 16.
E ( p ) i s a l s o a convex function of p .
f i x e d 8. Proof:
The l i m i t f u n c t i o n F(p,fi) i s a convex function of p f o r each
mis
means t h a t f o r p
and s i m i l a r l y f o r E(p).
a
Ap 1+(I-h)p2,
0
5
5
1,
A s F is bounded above on bounded p i n t e r v a l s
( t h i s can b e proved by a simple v a r i a t i o n a l c a l c u l a t i o n ) , i t i s s u f f i c i e n t t o prove (92) when A = 112.
To avoid t e c h n i c a l i t i e s (which can be
supplied) and concentrate on t h e main i d e a , w e s h a l l here prove (92) when p 2 and p1
are r a t i o n a l l y r e l a t e d :
ap
1
= bp
2
, a,
b, positive integers.
Choose any n e u t r a l p a r t i c l e number M and d e f i n e a sequence of b a l l s B
1
3 with r a d i i a s given i n (81) and with 28(4a/3)pR0 = (atb) system take N
0
s y s r m take N:
= (a+b)M, N j = Zb M, N'
j
= (28)
3j-1
No, j
= ( 2 8 1 ~ j - 4 : (resp. N:
-
j
even.
In
other h a l f place N
2 j
half
i n an obvious notation.
Since l i m
of t h e s e b a l l s p l a c e N'
particles, 0
5 j 5 K-1.
For t h e
2aM, ~:=(28)~'-11:).
b a l l s (Theorem 14) note t h a t the number of b a l l s B number is
.
2 1. For t h e p1 (resp. p2)
I n t h e canonical p a r t i t i o n ,
Consider t h e p system.
/MI
5
T,
is
of BK i n t o smaller j
and t h i s
p a r t i c l e s and i n t h e
Then i n place of (88) we g e t
I n s e r t i n g (89) on t h e r i g h t s i d e of (931,
-
Gp2
= 0, we can take t h e l i m i t K + i n (94) and o b t a i n (92). KE. The convexity i n p1 and concavity i n 8 of F(p ,B) has another -
Since F i s bounded below (Theorem 13) and
important consequence.
bounded above (by a simple v a r i a t i o n a l argument) on bounded sets i n t h e (p ,B) plane, the convexity /concavity implies t h a t i t i s j o i n t l y continuous i n (p,f3). FK + y
4(
This, together with t h e monotonicity i n K of
(see (go)), implies by a standard argument using Dini's theorem
t h a t the thermodynamic l i m i t i s uniform on bounded (p , B ) sets.
This
uniformity i s sometimes overlooked a s a b a s i c desideratum of t h e thermoWithout it one would have t o f i x p and 6 p r e c i s e l y i n
dynamic l i m i t . taking t h e l i m i t
- an
impossible t a s k experimentally.
With it, it is
s u f f i c i e n t t o have merely an increasing sequence of systems such t h a t
J
-t
p and B
+
.I
B.
The same r e s u l t holds f o r e (p)
.
F. -
An application of t h e uniformity of the l i m i t f o r e (p) is the Instead of confining the p a r t i c l e s t o a box (Dirichlet
following.
) one could consider H boundary condition f o r H defined on a l l of n,k n,k 'L (x3) , i.e. no confinement a t a l l . I n this case
IN(
is j u s t the ground s t a t e energy of a n e u t r a l molecule and i t is expected that
41I N ]has a l i m i t .
Indeed, t h i s l i m i t exists and i t is simply
l i m #/IN] = 1 i m p - l e(p)
P*
IW-
.
There i s no analogue of t h i s f o r F(p,B) because removing the box would cause the p a r t i t i o n function t o be i n f i n i t e even f o r a f i n i t e system. G. -
The ensemble used here i s the canonical ensemble.
It is
possible t o define and prove t h e existence of the thermodynamic l i m i t f o r the microcanonical and grand canonical ensembles and t o show t h a t a l l three ensembles a r e equivalent (i.e.
t h a t they y i e l d the same values f o r
a l l thermodynamic q u a n t i t i e s , such as the pressure).
H. -
(See Lieb-Lebowitz,
Charge n e u t r a l i t y was e s s e n t i a l f o r taming the long range
Coulomb force. this l e t N
What happens if the system i s not neutral?
To answer
il be a sequence of p a i r s of p a r t i c l e numbers and domains, 3 but without (59) being s a t i s f i e d . Let Q = zk -n be the n e t charge, 3 j j p j = INjl/liljl a s before, and p j p. One expects t h a t i f (i)
Qj = 0.
3'
-
~ ~ l ~ ~ 1 0- then ~ ' ~the same l i m i t +
On the other hand, i f
F(P ,$)
is achieved as i f
qj I 5 I
(it)
- ~ ~ then~ +
t h e r e is no l i m i t f o r F(N
j
,flj ,B) .
More
fl ,B) + because t h e minimum e l e c t r o s t a t i c energy is too j' j Both of these expectations can be proved t o b e c o r r e c t .
p r e c i s e l y F(N great.
The i n t e r e s t i n g case i s i f
l i m Qj
(iii)
j-
1 n j 1-213
=
0
exists
.
.
Then one expects a shape
A s s u m e t h a t the fl are geometrically j = Ano with Iflo[ = 1 and = p j with p j + p . Let C
dependent lidt t o e x i s t a s follows. s i m i l a r , i.e.
fl
j
b e t h e e l e c t r o s t a t i c capacity of capacity of fl is then C = C A. j .I t h e expectation is t h a t
no;
I N 1li3
it depends upon t h e shape of flO.
The
From elementary e l e c t r o s t a t i c s theory
.
l i m F(Nj,flj,i3) = F(p,6)+crL/2~ j-
I-'02/2c.
Note t h a t ( ~ ~ 11Ifl 2 ~ j
j
j
+
(95) can be proved f o r e l l i p s o i d s and b a l l s . complicated a s t h e r e s u l t is simple.
The proof i s a s
With work, t h e proof could probably
b e pushed through f o r o t h e r domains Ro with smooth boundaries. The r e s u l t (95) is amazing and shows how s p e c i a l t h e Coulomb f o r c e
is.
I t says t h a t t h e s u r p l u s charge Q. goes t o a t h i n l a y e r near t h e J
surface.
There, only i t s e l e c t r b s t a t i c energy, which overwhelms i t s
k i n e t i c energy, is s i g n i f i c a n t .
The bulk of Q
1
i s n e u t r a l and uninflu-
enced by the s u r f a c e l a y e r because t h e l a t t e r generates a constant p o t e n t i a l i n s i d e t h e bulk.
I t is seldom t h a t one has two s t r o n g l y i n t e r -
a c t i n g subsystems and t h a t t h e f i n a l r e s u l t has no cross terms, as i n (95).
I. -
There might be a temptation, which should be avoided, t o suppose
t h a t the thermodynamic l i m i t describes a s i n g l e phase system of uniform
density.
The temptation a r i s e s from the construction i n the proof of
Theorem 15 i n which a large domain BK i s partitioned i n t o smaller domains having e s s e n t i a l l y constant density. a large domain.
Several phases can be present i n s i d e
Indeed, i f B i s very large a s o l i d i s expected t o form,
and i f the average density, p , is smaller than the equilibrium density,
Ps* of t h e s o l i d a d i l u t e gas phase w i l l a l s o be present.
The location of
the s o l i d inside the l a r g e r domain w i l l be indeterminate.
From t h i s point of view, there is an amusing, although expected, aspect t o the theorem given i n (95). that p
C ps.
Suppose t h a t B is very large and
Suppose, also, t h a t a surplus charge Q = 0y2'3
where V is the volume of the container.
is present,
I n equilibrium, the surplus
charge w i l l never be bound t o the surface of the s o l i d , f o r t h a t would give r i s e t o a larger f r e e energy than i n (95). A s a f i n a l remark, the existence of the thermodynamic l i m i t (and hence the existence of intensive thermodynamic variables such as the pressure) does not e s t a b l i s h the existence of a thermodynamic s t a t e .
In
other words, i t has not been shown t h a t correlation functions, which always e x i s t f o r f i n i t e systems, have l i m i t s as the volume goes t o infinity. present.
Indeed, unique l i m i t s might not e x i s t i f s e v e r a l phases a r e For well behaved p o t e n t i a l s there a r e techniques available f o r
proving t h a t a s t a t e exists when the density is small, but these techniques do not work f o r t h e long-range Coulomb p o t e n t i a l .
Probably the
next chapter to be written i n t h i s subject w i l l consist of a proof that' correlation functions a r e well defined i n the thermodynamic l i m i t .
VI
.
Har tree-Fock Theory
Q (see (23) and (25)), As a p r a c t i c a l matter, a good estimate f o r EN even with fixed nuclei, i s d i f f i c u l t t o obtain.
An old method (Hartree,
1927, Fock, 1930, S l a t e r , 1930) is s t i l l much employed.
Indeed, chemists
r e f e r t o i t a s an ab i n i t i o calculation. Without taking any position on the usefulness of a HF calculation, i t might be worthwhile t o present the r e s u l t s of recent work (Lieb-Simon
1973) t o the e f f e c t t h a t HF theory is a t l e a s t well defined, i.e. equations have solutions.
the HF
Unlike the s i t u a t i o n f o r Thomas-Fed theory,
the solutions a r e not unique i n general. To define HF theory l e t 4
{O1,*.',ON)
denote a s e t of N s i n g l e p a r t i c l e functions of space and spin, +i, i n
L2 (P3 ;O 2). Two s p i n s t a t e s a r e assumed here.
Form the S l a t e r determinant
N
D,,,(xl,. ..,%;al,.
(N!)-~''
-.,flN)
detl Oi(xj,aj)
2 3 2N The L (R ;O ) norm of
which i s an antisymrnetric function of space-spin. = det IM' 4' 4 i j i,jP1
1
where M4 i s the overlap (Gram) matrix: ,
The HF energy is
f
= inf( : 0; Z need not be
i s less obvious.
It
may o r may not occur, depending on the d e t a i l s of t h e nuclear configurWe s h a l l have nothing t o say about t h i s l a t t e r case.
ation.
Theorem 17.
I f N < Z+1 = 1+Cz then, f o r any nuclear configuration, there j is s minimizing ( tor Furthermore, the (i & ( can be chosen t o be
g.
orthonormal, i.e. 'M
ij
-
6
id'
The proof of Theorem 1 7 involves a t r i c k which i n retrospect is obvious, but which took some time t o notice. (1)
Eere i s an o u t l i n e
Consider F N ( $ ) as defined by (103), (104) and (105).
This i s
a q u a r t i c expression i n the (i. Both GN(J,) and a r e i n v a r i a n t
JI 4'
under any unitary transformation of t h e form
with R being an N
x
N unitary matrix.
I f R is chosen t o diagonalize M$' ,
we can r e s t r i c t our a t t e n t i o n t o J, such t h a t the gi a r e orthogonal. The minimizing J, w i l l be constructed by taking a weak l i m i t of a sequence
such t h a t
The major d i f f i c u l t y is t h a t a weak l i m i t of orthonormal functions need not be orthogonal.
It could even happen t h a t l i m) ' :4
= ( (independent
IT-
of i ) .
The t r i c k t o overcome the d i f f i c u l t y is t h i s :
minimizing
with
gN(()
+
Instead of
subject t o = 1 consider instead $'
is the N x N i d e n t i t y matrix and the inequality i n (110) is t h a t IN-M JI i s positive semidefinite.
The obvious, but c r u c i a l , f a c t i s t h a t
a weak l i m i t of functions i n SN remains i n SN. I f there is a minimizing JI f o r
%,
the +i can be chosen t o be ortho-
gonal, possibly a f t e r a unitary transformation (107).
2 1. Assume
= 6i
6
i
> 0, a l l i.
Then, since *ESN,
Then t o see t h a t t h e 6i can be
chosen t o be unity, note t h a t g N ( $ ) i s quadratic i n each
+i, with
i f +i ,is replaced by (yi/6i)1'2 yi.
Clearly
aE N/ayi
5 0 (otherwise
yi > 0,
gNi s
Therefore
l i n e a r i n each
can be decreased i n taking yi=O,
which contradicts the assumption t h a t thus
gNis
Oi.
> 0 a t the minimum), and
not increased i f yi i s taken t o be 1.
The problem, then, i s t o show two things: (2a)
there is a minimizing JI f o r eN;
(2b)
M$ does not have a zero eigenvalue. (2a).
This is an application of functional analysis.
one can find a sequence i n t h e Sobolev spaceR('W
Given (108)
converges weakly t o
such t h a t each
3
) , i.e.
.
Vi N
1
i s weakly laser semicontinuous, essenti$ J I i is1 a l l y because W -K is a p o s i t i v e operator and is bounded on ~ ' ( 1 ~ ) . The J, JI
The functional
p o s i t i v i t y of the function
I X - ~ I-'
6
on 1 i s used.
Finally,
because V is a r e l a t i v e l y compact perturbation of -A quadratic f o m s .
Thus J, minimizes EN($) on SN.
i n the sense of
(2b).
I f MQ has. a zero eigenvalue then
%
= eN-l,
$i vanishes ( a f t e r a unitary transformation (107)).
i . e . one of t h e
This is impossible
i f N < Z+l because one can always f i n d a $ orthogonal t o such t h a t 6N(41~""$N-199)
< ~N-l(+l,.-.,~N-l).
4N-1
The property of t h e
hydrogenic Ramiltonian (1) t h a t i t has i n f i n i t e l y many negative eigenvalues i s used i n an e s s e n t i a l way. By a standard argument i n the calculus of v a r i a t i o n s , t h e minimizing
Q s a t i s f i e s t h e Euler-Lagrange equation f o r P N ( Q ) a s follows. Theorem 18.
Lot
such t h a t M'
a
...,(N) be any minimizing Q tor $ arranged
I( = ( I $ ~ ,
5.
It is not necessary t o assume t h a t N < Z+1.
2 3 2 be t h e operator on L (1 ,P ):
a s defined i n (104) and (105).
f o r some hi < 0 .
(Al,.
(ii)
..,%I
..
Then f o r i = 1,. ,N
This is the HF equation. a r e t h e lowest N eigenvalues of H
%
N
$.
1
Xi because t h e r e a r e no f a c t o r s of 1/2 i n i-1 The only s l i g h t l y unusual point is ( i i ) which follows from t h e
I t is not t r u e t h a t
(112).
Let HQ
=
f a c t that g N ( Q ) is quadratic i n each Oi. t h e lowest N i s missing,
%
I f some eigenvalue of H
4'
among
can be lowered by using t h e missing eigen-
function i n s t e a d of t h e (N+j) t h eigenfunction. I n summary, j u s t a s i n the analogous case of TF theory, i t has been shown t h a t t h e nonlinear HF equation (113) not only has a s o l u t i o n , b u t
HF
t h a t among these s o l u t i o n s there i s one t h a t .minimizes t h e EF energy EN It i s not easy t o prove d i r e c t l y t h a t (113) has solutions.
.
I n general, i t is d i f f i c u l t t o say much about a minimizing $. One can-
Despite the deceptive notation, (113) is not a l i n e a r equation.
not say, as one could f o r the l i n e a r Schroedinger equation, t h a t the (Pi can be assumed t o be r e a l o r t h a t Oi(x,o) is a product fi(x)gi(a). assumptions a r e often made i n practice.
These
What can be done i s t o r e s t r i c t
the (Pi from the beginning t o be r e a l and/or product functions such t h a t f o r any i $ j gi = g o r g j
i
i s othogonal t o g
1'
Then the whole analysis
can be done afresh and Theorems 17 and 18 w i l l hold. i n t h i s r e s t r i c t e d class might be greater than
ENg",
manner other r e s t r i c t i o n s can be placed on the bi invariance, f o r example) with the same conclusion. requirement is that f o r any 4 and J, =
The minimum,
-a, EN
In the same
however.
(such as rotation The only e s s e n t i a l
i n the r e s t r i c t e d class,
H$"
is i n the same class.
The overriding question is, of course, h a close i s
%HF
to
I#? It
is d i f f i c u l t t o give a precise answer, but i n two limiting cases HF theory is exact.
One i s the hydrogen atom; the other is the 2
+
-
limit.
It was
i n f a c t a determinantal wave function (971, not the best one t o be sure, t h a t was used i n the variational upper bound leading t o Theorem 5.
i n the sense of Theorem 5.
Thus
References ~ a l i z s ,N.,
1967, Formation of s t a b l e molecules within t h e s t a t i s t i -
c a l theory of atoms, Phys. Rev. 5 6 , 42-47. Birman, M.S.,
1961, Mat. Sb.
2
(97), 125-174; The spectrum of
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Proc. Camb. P h i l . Soc. 26, 376-385. Dyson, F.J.,
1967, Ground-state energy of a f i n i t e system of charged
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F e d , E.,
1927, Un metodo s t a t i s t i c 0 per l a determinazione d i
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5,
602-607.
1930, NHherungsmethode zur Liisung des quantenmechanischen
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2
(1930), 795-805. Gombgs, P.,
1949, "Die s t a t i s t i s c h e n Theorie des Atomes und i h r e
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1969, Free energy of i n t e r a c t i n g magnetic dipoles,
Phys. Rev. 172, 655-659. /Y
Hartree, D.R.,
1927-28, The wave mechanics of an atom with a non-
Coulomb c e n t r a l f i e l d . SOC. 24, 89-110. r
P a r t I. Theory and methods, Proc. Camb. Phil.
Heisenberg, W.,
1927, Uber den anschaulichen I n h a l t der quanten-
theoretischen Kinematik und Mechanik, Z e i t s . Phys., 43, 172-198. Jeans, J.H.,
1915, The mathematical theory of e l e c t r i c i t y and
magnetism, Cambridge University Press, t h i r d e d i t i o n , page 168. Kirzhnits, D.A., 123.
1957, J. Exptl. Theoret. Phys. (U.S.S.R.)
2,115-
Engl. t r a n s l . Quantum corrections t o t h e Thomas-Fermi equation,
Sov. Phys
. JETP, 2 (1957),
Kompaneets, A.S. (U.S.S.R.)
and E.S.
3l, 427-438.
64-71. Pavlovskii, 1956, J. Exptl. Theoret. Phys.
Engl. transl. The self-consistent f i e l d
equations i n a n atom, Sov. Phys. JEW,
5 (1957),
328-336.
Lenard, A. and F.J. Dyson, 1968, S t a b i l i t y of matter. 11, J. Math. Phys
.2 , 698-711.
Lenz, W.,
1932, ijber d i e Anwendbarkeit der s t a t i s t i s c h e n Methode auf
Ionengitter, Zeit. Phys. 77, 713-721. Lieb, E.H.,
1976, Bounds on t h e eigenvalues of t h e Laplace and
Schroedinger operators, Bull. Amer. Math. Soc., i n press. Lieb, E.H. and J.L. Lebowitz, 1972, The c o n s t i t u t i o n of matter: existence of thermodynamics f o r systems composed of e l e c t r o n s and n u c l e i , Adv. i n Math.2,
316-398.
See a l s o J.L.
Lebowitz, and E.H.
Lieb , Existence of thermodynamics f o r r e a l matter with Coulomb forces, Phys. Rev. L e t t . Lieb, E.H.
2 (19691,
631-634.
and H. Narnhofer, 1975, The thermodynamic l i m i t f o r
jellium, J. S t a t . Phys. h 2 , 291-310.
Erratum:
J. S t a t . Phys.
3
(19761, No. 5. Lieb, E. H. and B. Simon, 1973, On s o l u t i o n s t o t h e Hartree-Fock problem f o r atoms and molecules, J. Chem. Phys. a longer paper i n preparation.
5,735-736.
Also
21.
.
Lieb ,E .H. and B Simon, 1975, The Thomas-Permi theory of atonp, molecules and s o l i d s , Adv. i n Math., i n press.
See a l s o E.H. Lieb
and B. Simon, Thomas-Fed theory r e v i s i t e d , Phys. Rev. Lett.33
w
(1973), 681-683. 22.
Lieb, E.H.
and W.E. Thirring, 1975, A bound f o r the k i n e t i c energy
of fermions which proves t h e s t a b i l i t y of matter, Phys. Rev. Lett.35
N'
687-689, Errata:
Phys. Rev. L e t t . (1975),
2,1116.
~or'more
d e t a i l s on k i n e t i c energy i n e q u a l i t i e s and t h e i r application, see a l s o E.B.
Lieb and W.E. Thirring, I n e q u a l i t i e s f o r the moments of
the Eigenvalues of the Schrijdinger Hamiltonian and t h e i r r e l a t i o n t o Sobolev i n e q u a l i t i e s , i n Studies i n Mathematical Physics:
Essays i n
Honor of Valentine Bargmann, E.H. Lieb, B. Simon and A.S. Wightman e d i t o r s , Princeton University Press, 1976. 23.
Rosen, G.,
1971, Minimum value f o r c i n t h e Sobolev i n e q u a l i t y
lid3 5 C I I V ( ( ~ ~SIAM , J. 24.
Appl. Math. N 21, 30-32.
Schwinger, J., 1961, On the bound s t a t e s of a given p o t e n t i a l , Proc. Nat. Acad. Sci. (U.S.) 57, 122-129.
25.
26.
Scott, J.M.C.,
1952, The binding energy of the Thomas Fermi atom,
Phil. Mag. ,3.4
859-867.
Sheldon, J.W.,
1955, Use of t h e s t a t i s t i c a l f i e l d approximation i n
molecular physics, Phys. Rev. 27.
S l a t e r , J.C.,
2,1291-1301.
1930, The theory of complex spectra, Phys. Rev.
3,
1293-1322. 28.
Sobolev, S.L.,
1938, Mat. Sb. 56, 471 (1938).
See a l s o S.L. Sobolev,
Applications of functional analysis i n mathematical physics, Leningrad (1950), Amer. Math. Soc. Transl. of Monographs,
(1963).
29.
Sonrmerfeld, A.,
1932, Aspmptotische Integration der Differential-
gleichung des Thomas-Femischen Atoms, Z e i t . Phys, 78, 283-308. 30.
Teller, E . , 1962, On the s t a b i l i t y of molecules i n the Thomas-Fermi theory, Rev. Mod. Phys. 34, 627-631.
31.
Thomas, L.H., Phil. Soc.
32.
1927, The calculation of atomic f i e l d s , Proc. Camb.
5,542-548.
Von Weizsacker, C.F.,
1935, Zur Theorie der Kernmassen, 96, 431-458. /u
CENTRO INTERNAZIONALE M A T W T I C O ESTIVO (c.I.M.E.
)
REPORT ON RENORMALIZATION GROUP
Be TIROZZI
Istituto d i Matematica, U n i v e r s i t i d i Camerino
C o r s o tenuto a B r e s s a n o n e d a l 21 giugno a1 24 $iugno 1976
REPORT ON RENORMALIZATION GROUP
P r o f . Benedetto T i r o z z i I s t i t u t o d i Matematica U n i v e r s i t s d i Camerino
Introduction
1. I n t e g r a l and l o c a l c e n t r a l l i m i t theorems o f p r o b a b i l i t y t h e o r y and t h e
r e n o r m a l i z a t i o n group method. In t h i s l e c t u r e we want t o p r e s e n t t h e problem o f t h e r e s e a r c h o f automodel p r o b a b i l i t y d i s t r i b u t i o n s i n comparison w i t h u s u a l i n t e g r a l and l o c a l c e n t r a l l i m i t theorems. We t h i n k t h a t t h i s approach is i n s t r u c t i v e f o r u n d e r s t a n d i n g t h e main mathematical i d e a underlyng t h i s kind o f problems. Consider a s t a t i o n a r y d i s c r e t e random f i e l d
-=i' k
S=I
E Zi
and suppose t h a t '0 J-I. f i e s t h e i n t e g r a l c e n t r a l l i m i t theorem i f
jj
, zL
. Then
J6
and t h e sequence
t h e random f i e l d w s a t i s -
G-
1.(1)
T h i s theorem was proven i n t h e p i o n e e r works o f Gnedenko (1),(2),(3) mogorov i n t h e c a s e i n which
5
a r e independent and e q u a l l y d i s t r i b u t e d . For
u s it i s more i n t e r e s t i n g t h e c a s e i n which t h e random v a r i a b l e s dependent and more p r e c i s e l y when
and Kol-
lfL,i e2
5,
a r e not in-
form a Gibbs random f i e l d , ( 4 ) ,
(51, corresponding t o a c e r t a i n p o t e n t i a l
+B 5 (
~ , t b ~IUI>A, )
-A36
(Kk&
K ~ E I
1. 2 )
where we suppose t h a t
$6
X,Xbeing
t h e s p a c e o f r e a l i z a t i o n o f t h e Gibbs
random f i e l d . It is w e l l known, from v e r y g e n e r a l arguments, t h a t a n e c e s s a r y and s u f f i i c i e n t c o n d i t i o h f a r 1.1 t o b e t r u e is t h a t t h e f i e l d $ . , ~ b t must b e s t r o n g
mixing ( 2 ) w i t h mixing c o e f f i c i e n t
-1
and t h a t t h e d i s p e r s i o n
O & * C ~ where
C i s some p o s i t i v e c o n s t a n t , f u r t h e r it is r e q u i r e d a c o n d i t i o n of t h e t y p e of Lindeberg ( 1 ) , ( 2 ) , analogous t o t h e one used f o r t h e c a s e o f independent variables:
where
@(@= ~ & , + - - + $ L is ~ ~some
p o s i t i v e c o n s t a n t and
The c o e f f i c i e n t of s t r o n g mixing i s d e f i n e d i n t h e f o l l o w i n g way: l e t q q p ) be a p r o b a b i l i t y s p a c e where t h e random v a r i a b l e s
4
a r e d e f i n e d and l e t u s
c a l l e t h e minimal r - a l g e b r a g e n e r a t e d by t h e e v e n t s
where
A-CA3and
Then we can d e f i n e t h e c o e f f i c i e n t o f mixing by t h e f o l l o w i n g , q u a n t i t y ' : 1. 4)
ac~)w = A€
1 pC~nB)-Pcn) PCB)\
mO_=
86Mz and we w i l l s a y t h a t t h e f i e l d (
*
se,t6b Z)
is s t r o n g mixing if we have t h a t
A'CK)-8
1. 5 )
K 4 0
Now we can show hcw i r i s p o s s i b l e from t h e knowledge o f 1. 5
t h e behaviour of
O.&
.
t o deduce
I n f a c t from a v e r y g e n e r a l theorem ( 2 ) f o r s t a t i o n a r y p r o c e s s e s we have that 1. 6 )
/ ESk54J -c c ~ ( ( i d h )
c,o
And s o we can g i v e t h e f o l l o w i n g e s t i m a t e f o r t h e d i s p e r s i o n of
E~&=o
g
Suppose now t h a t
Z o(Oj4)43
f o r a l l 9 t h e n we o b t a i n suddenly
,B =
6~23
DS--8-4,
1. 8 )
t h a t is t h e r e q u i r e d c o n d i t i o n on
0.c,
The proof o f t h e c e n t r a l l i m i t theorem under t h e h y p o t h e s i s s a i d above proc e e d s i n t h e f o l l o w i n g way. Divide t h e i n t e r v a l
Cd(*]
i n segments
Gl wl
such t h a t
/=P, I
wd/=q= O>
and
a l a r g e c l a s s o f system o f i n t e r e s t o f S t a t i s t i c a l P h y s i c s .
Now we want t o begin t o examine t h e s i t u a t i o n when
P=rpgC. The
first
o b s e r v a t i c r . z o n s i s t s o f t h e f a c t t h a t it i s no more t r u e t h a t
because of t h e s m a l l e r r a t e o f d e c r e a s i n g o f c o r r e l a t i o n s and s o we e x p e c t a n o r m a l i z a t i o n f a c t o r g i v e n by
a s , i r 4 d . Furthermore,
g i v e n a c e r t a i n po-
t e n t i a l we a r e i n r s r e s t e d n o t o n l y i n t h e e x i s t e n c e o f t h e a s y m p t o t i c j o i n t
p r o b a b i l i t y d i s t r i b u t i o n s f o r t h e normed sums o f s p i n s b u t a l s o i n t h e d e t e r -
&. f o r
mination o f
p a r t i c u l a r h a m i l t o n i a n and a l s o it i s i m p o r t a n t t o
a
know how s t a b l e i s
w i t h r e s p e c t t o '!small1' changes of t h e p o t e n t i a l .
A s an example o f t h e s i t u a t i o n o f t h e known r e s u l t s a b o u t t h e s e problems we
f i n i s h t h i s i n t r o d u c t i o n showing an open problem f o r a onedimensional s p i n system w i t h a p a i r p o t e n t i a l g i v e n by
when
1344
Pa.
The q u e s t i o n is t h e f o l l o w i n g : i s , t h e l o c a l c e n t r a l l i m i t theorem v e r i f i e d const i n t h i s c a s e ? we know b u t we have no i n f o r m a t i o m ltL-u" about t h e c o e f f i c i e n t o f mixing.
E[T.J'J-
-
2 , Kadanoff r e n o r m a l i z a t i o-n group and g a u s s i a n automodel d i s t r i b u t i o n s .
We a r e going t o g i v e a more p r e c i s e f o r m u l a t i o n o f t h e above problems and t o d e s c r i b e some examples of l i m i t p r o b a b i l i t y d i s t r i b u t i o n s . Le?
R
be t h e s p a c e of a l l r e a l numbers. We s h a l l c o n s i d e r a random f i e l d
sb,&6zL.' which t a k e s v a l u e s i n R. -
Def i-a ii i -o n 2 . 1 i
d
,
fc
+e
Let u s c o n s i d e r a r e a l i z a t i o n
nzd
. We
3 /fiL,,-
w i l l d e f i n e an endomorphism on
R
H
of t h e random i n such a way:
K ~ Z ' '2.1.
kLJt5)-i
+
= (Aa(d1313= Y
The i r z n s f o r n a t i o n
b i l i t y measures d e f i n e d on
where
C 6 & R~ 4
i.e.
c
i d j o i n t of
&'
L & i4*cL F~~~~ 4
Ad P
is
when
3.2
e
t h i s argument d o e s n o t hold and wq w i l l t a k e t h e i d e a
of s t a b i l i t y from t h e c l a s s i c a l b i f u r c a t i o n t h e o r y .
~ e T: t MAN b e a d i f f e o m o r f i s m o f a n-dimensional manifold ( 1 6 ) i n i r s e l f
and l e t
x&M
be a f i x e d p o i n t o f T . ( f i g . 3 ) . Let f ( @ ) m e t r i z e d by
be a c u r v e i n
and l e t
* H
para-
be a n-1 submarr
n i f o l d of
such t h a t if
&6fd -
T'''&+x
then
ry
if~f=~'@f"l
and
d+e
a(@)belongs t o M ' i n t e r s e c t i o n between fig. 3
then t h e r e
g(P) and
is o n l y one M which w i l l
be o b t a i n e d f o r a c e r t a i n v a l u e o f
P.
T h i s s i t u a t i o n i s achieved when, and o n l y when, t h e d i f f e r e n t i a l o f T i n )(has o n l y one e i g e n v a l u e b i g g e r t h a n one, t h e n M ' is a d i r e c t i o n such t h a t T r e s t r i /r/ c t e d on M ' is expanding and T r e s t r i c t e d on M is c o n t r a c t i n g . In t h e c a s e when M i s t h e "manifold" o f p r o b a b i l i t y d i s t r i b u t i o n s , T is t h e r e n o r m a l i z a t i o n group
* [ d ) ,is~ & AK
t h a t is a a . p . d .
"manifold" o f p r o b a b i l i t y d i s t r i b u t i o n s such t h a t
4
We have t h a t M is a
3t AK Id)
a c t i n g on it is
c o n t r a c t i n g and M' i s t h e s e t o f i n i t i a l p r . d i s t r . and c l e a r l y t h e v a l u e o f
P
shown b e f o r e w i l l b e t h e c r i t i c a l t e m p e r a t u r e . I f such a s i t u a t i o n is v e r i f i e d we s h a l l s a y t h a t
6
is s t a b l e , t h u s
&
w i l l l o o s e i t s s t a b i l i t y when t h e r e
w i l l be two e i g e n v a l u e s b i g g e r o r e q u a l t h a n one and t h e n w e e x p e c t t h e appear i n g o f a new branch o f automodel p r o b a b i l i t y d i s t r i b u t i o n s . Now we w i l l e n t e r more i n t o t h e d e t a i l s and g i v e an e x p l i c i t c o n s t r u c t i o n of
We w i l l write f o r m a l expres-
t h e tangent space t o a gaussian aut'.prob.distr.
s i o n s f o r sake of s e m p l i c i t y b u t it i s p o s s i b l e t o g i v e t o them an e x a c t and r i g o r o u s meaning using t h e same procedure as in (17). Let
&
be a g a u s s i a n s t a t i o n a r y a u t . p r , d i s t r .
=
E
~
~
5
!2 ~ i &
5e +
=0
)
on
zs
k
and l e t ~ C C ) ;
J ~ A d~ I
, d
2
be t h e c o r r e l a t i o n f u n c t i o n , where
Thm
~(6)% g i v e n
by 2,18, Define t h e m a t r i x
3' can be w r i t t e n f o r m a l l y a s = e
- 9.6ZS Z a~,ks;'
For d e f i n i n g t h e t a n g e n t space i n
Gd
l e t us consider t h e s e t of s t n t i o i n t h e s e n s e t h a t t h e y a r e ab-
nary p r o b a b i l i t y d i s t r i b u t i o n s "near" t o s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o it
5:
3.6.
where (Li,jlk,
;
-2 e
it162
aiJj,4e aidj$$-6.L G% ~62'
C a r e r e a l numbers such t h a t : &hljhrr,k+y
The d e n s i t y of
6:
with respect t o
b4 w i l l
34 - J3 3K )e
G).m=a<jtk,
be c o n s i d e r e d a s a v e c t o r of
t h e t a n g e n t s p a c e , s o we w i l l g i v e t h e f o l l o w i n g d e f i n i t i o n s . D e finition -
3.1.
A form o f d e g r e e *
of t h e process
[~;f-Till
be t h e random
v a r i a b l e given by t h e f o l l o w i n g formal sum
where
aQC1., em a r e r e a l
numbers and symmetric, i n
eA,.- ., em
'
The forms o f d e g r e e can i n t r o d u c e i n Definition 3.2. that
form a v e c t o r space which we w i l l i n d i c a t e
3' . We
3* t h e t o p o l o g y g i v e n by t h e convergence o f t h e c o o r d i n a t e s . A s t a t i o n a r y form o f degree*
faC5+*.-,&+
3~
6%) w i l l be a form Q
d o e s n o t depend on ~6
e*m
Z'
The s p a c e o f s t a t i o n a r y forms o f d e g r e e 4 w i l l b e i n d i c a t e d a s c4tl I t is e v i d e n t t h a t i s a c l o s e d subspace o f
such
i&J
44
Y4
A l a r g e c l a s s of s t a t i o n a r y forms o f d e g r e e 4
way. L e t
d(&,,.. dA) b e
can be o b t a i n e d i n t h e f o l l o w i n g
a c o n t i n u o u s symmetric f u n c t i o n d e f i n e d on t h e 4 d i -
mensional t o r u s T o r n s u c h t h a t i t s F o u r i e r s e r i e s is a b s o l u t e l y convergent f o r
.
4
6
%here
Let u s c o n s i d e r t h e form
- 5 ~Z"ii'A'ocMJ
atA,..., e,,., -
f~rm
. Then
it i s e v i d e n t t h a t t h e
T~
belongs t o
t4tJ
3,
. From
t h e c o e f f i c i e n t s o f t h e form
3.9
w:
obzaifi t h s f o l l o w i n g r e p r e s e n t a t i o n f o r
gcw
T h i s means t h a t , i f t h e f u n c t i o n s
&''
c o i n c i d e on t h e diaq,iial of&+:
(4)
M '
l A . cZb l c ' 0 ( d l ) $ Let
414'
ment Q b S of d e g r e e
c&
t h e n t h e y g e n e r a t e t h s same form
be t h e a l g e b r a i c sum o f t h e l i n e a r s p a c e s
& c ~ L ~ d .
i& gm . Every
may be r e p r e s e n t e d a s a sum o f s i m i l a r s t a t i o n a r y forms
M,a:raCw' where Nl
f e r e n t from z e r o .
(461 -
eie-
c d &
i n t h i s sum o n l y a f i n i t e number of terms i s d i f -
0.3.3.
The s p a c e
3(461
w i l l be t h e t a n g e n t space of t h e g a u s s i a n s t a t i o n a r y
distribution. D i f f e r e n t i a l o f t h e r e n o r m a l i z a t i o n group.
*(4,.
Let u s now g i v e some f o r m a l arguments t o i n t r o d u c e t h e d i f f e r e n t i a l o f t h e renormgroup which we w i l l i n d i c a t e a s >K r e a d y given f o r m a l l y t h e r e l a t i o n between b a b i l i t y measure on
@"
A
?4
AKM&
In s e c t i o n 2 we have a l and
. For d e f i n i n g t h e d i f f e r e n t i a l
5
where G is a pro-
o f t h e a d j o i n t opera-
t o r o f t h e r e n o r m a l i z a t i o n group a c t i n g on t h e space o f p r o b a b i l i t y measures l e t u s c o n s i d e r two p r o b a b i l i t y d i s t r i b u t i o n s which a r e "near", introduced i n
3.5.
,
, then
3.6.
t h a t is
&
we have t h a t t h e y t r a n s f o r m i n t h e
f o l l o w i n g way:
where B i s some s e t b e l o n g i s t o Then t h e t a n g e n t v e c t o r i n
where
$=fAd4)'~
aR&rR1l,
can be f o r m a l l y d e f i n e d a s
Thus
A
d e f i n e d a s t h e t r a n s f o r m a t i o n on't'$ 0.3.4.
3.10.
3 K ~ * ~ d ) 4 1 *E =
w i l l be
g i v e n by
C Q ~ ~ ' ~ ~ )
Now we w i l l g i v e more d e t a i l s about t h i s d e f i n i t i o n . For t h i s aim we s h a l l u s e t h e n - s t o c h a s t i c i n t e g r a l d e f i n e d by I t o and a l s o some c o n c e p t s from g a u s s i a n dynamical systems (18), ( 1 9 ) . Let u s c o n s i d e r a g a u s s i a n dynamical system
,
where
zd 8 is t h e s p a c e
o f r e a l i z a t i o n s o f t h e g a u s s i r n random f i e l d
is t h e g a u s s i a n s t a t i o n a r y a u t . p r o b . d i s t r i b u t i o n
d e f i n e d i n 3.4;
SOD
lfi3 4 T is t h e
s h i f t operator:
Then, from t h e t h e o r y o f s t a t i o n a r y s t o c h a s t i c p r o c e s s e s we have t h a t it i s p o s s i b l e t o r e p r e s e n t t h e automodel g a u s s i a n random f i e l d w i t h t h e h e l p o f i t s s p e c t r a l random measure:
2cdLt) ( 1 8 ) ~ = i t ( d ) , & &f
where t h e random measure
-
measure
forms a c o n t i n u o u s complex normal random wherea
is t h e a l g e b r a o f b o r e 1 s e t s b e l o n -
C-il+g-
ging t o
The c o n t i n u o u s complex normal random measure
th), A&
is a system o f
At,..- ,dm6fi v a r i a b l e s Z['i),...r£(Al*)
random v a r i a b l e s such t h a t f o r e v e r y c o l l e c t i o n o f s e t s t h e j o i n t p r o b a b i l i t y d i s t r i b u t i o n o f t h e random is .qaussian w i t h t h e f o l l o w i n g moments:
Let
T*
3.13.
b e t h e a d j o i n t o f t h e o p e r a t o r T, t h a t is I. a c t s on
f (&)
: fro.
it is e a s y t o s e e t h a t
J* ZcdW = 4 "zc&)
3.15. Let(43,p)
b e t h e b a s i c p r o b a b i l i t y s p a c e where a l l t h e complex random vh-
.
riables
2 Cb)
D.3.5.
We w i l l s a y t h a t a complex random v a r i a b l e i s a 2 a i r e f u n c t i o n ( 1 8 )
of
a r e defined
it(CI-i
with e i g e n v a l u e
-u+*-a++ K
a
2) t h e r e i s a l s o a n o t h e r e i g e n f u n c t i o n
fm[tA,..,t4)=T&
Remark. Note t h a t t h e g l o b a l t r a n s f o r m a t i o n
*
&A*(@)
is t ~ i a n g u l cs o
t h a t i t s g l o b a l p r o p e r t i e s a r e determined by t h e e i g e n v a l u e s and t h e e i g e n functions of t h e transformation
3!A COO.
we can a l s o n o t e t h a t it is pos-
s i b l e t o f i n d a l a r g e r c l a s s o f e i g e n f u n c t i o n s i f we a l l o w t h e & ( t ~ , - - , % ) t o be g e n e r a l i z e d f u n c t i o n s . Now we make a s e l e c t i o n w i t h i n t h e s e t o f t h e e i g e n f u n c t i o n o f p r o p o s i p i o n ( 4 . 1 . 1 , because we want t h a t t h e e f f e c t i v e harniltonian o f d e g r e e
t o s a t i s f y some p h y s i c a l p r o p e r t i e s , where it is e a s y t h a t
with
&
o f prop. ( 4 . 1 . )
by t h e f o l l o w i n g e q u a t i o n :
em
/M
i s connected
.
The r e q u i r e d p r o p e r t i e s a r e 1 ) The e f f e c t i v e h a m i l t o n i a n must be t r a n s l a t i o n i n v a r i a n t
Hm must
2)
be p o s i t i v e d e f i n i t e i n o r d e r t o g e n e r a t e a Gibbs measure
3 ) t h e p o t e n t i a l c o r r e s p o n d i n g t o t h e h a m i l t o n i a n s w i t h more t h a n two p a r t i c l e s
must d e c r e a s e f a s t e r t h a n t h e p o t e n t i a l coming from t h e g a u s s i a n term. Thus
+CP
From (4.21. ) it i s e v i d e n t t h a t t h e s i n g u l a r i t y o f must c a n c e l w i t h t h e b e h a v i o u r o f Let u s e v a l u a t e
Thus
((6)
$(t) CC
in the origin
]*-it
i n the origin
~5(~~'i(c4(T -f)-' iti (r
1 t l -A -q
and
t h u s we f i n d t h e e x p r e s s i o n f o r t h e e i g e n f u n c t i o n o f
ad~ ' ( 4 )
which have a l r e a d y been determined i n ( 1 2 ) . 4.4.
Compact formulae f o r t h e f u l l d i f f e r e n t i a l .
Now we a r e a b l e t o g i v e a r e p r e s e n t a t i o n o f t h e a c t i o n of t h e f u l l d i f f e r e n -
rial
& A*(H~.
We s h a l l Ejee how it a c t s on t h e s p a c e o f forms of t h e t y p e
where
e,,&~#.-~%)i s g i v e n by
(4.24. )
. It
i s u s e f u l t o u s e now o t h e r random
v a r i a b l e s d e f i n e d by
w (t)=
s f
-I e-it&
The t r a n s f o r m a t i o n induced on
0,
w[ej
by t h e p r e c e e d i n g f o r m u l a s ( 4 . 4 , ) ,
where
XCt)
by t h e a c t i o n o f
is a complex g a u s s i a n random p r o c e s s whose c o r r e l a t i o n f u n c t i o n is
~ ( twe)have
and is independent from
H(i)
a n o t h e r form f o r
...,ta)
where
isgivenby
r
4.29.
%,Ct.,...l
t4 1,
r,
Tf3t,+tnjK, K r i
j~,--.,j~
Thus
4.30.
is determined
(4.5.)
determined by t h a t o f W C * ) ~ U ) ' C k t ) Using
AK(Q()
+@ tff +lr f-l ( w )= (t rrl" (dr [dt, - [&d&.I--tuC&) em
-7f
-?i
wt(*f).
Let u s d e f i n e t h e f u n c t i o n
*(O()
So t h a t t h e e i g e n v e c t o r o f
We can
where
determine now how
>Ic gK)KA(4ja c t s
ex)is some polynomial.
We c a l c u l a t e ( 4 . 3 4 . )
w i l l have t h e form
on t h e more g e n e r a l form:
We have
i n t h i s way. S e t
, from
-7T -.
well known theorems ( 1 8 ) X w i l l be a complex g a u s s i a n random p r o c e s s whose var i a t i o n i s determined by t h a t of
Thus we can s e t :
~ ( t ,)and
wl[f) :
Let u s make t h e s u b s t i t u t i o n kt-7
I(
P, ('1%
since
+
Fj)=e&(~.,~t) ,
t/J(+?& and s e t t i n g ~
formuh f o r
4.39.
F
we h a v e :
-w
-Q)
changing
t
:
( % ] = ~ d [ ~ we ) o b t a i n a t r a s f o r m a t i 2 n
--xz
FJ~~,zI=
Choosing
F(x)
pending on
f
t o be a polynomial i n t h e v a r i a b l e
3
, with
c o e f f i c i e n t s de-
we c a n r e w r i t e (4.39. ) i n t h e f i n a l form:
.
w h e r e ~ ~ h ~In ' t~h i s way we have reduced t h e problem o f t h e a c f i o n o f t h e d i f f e r e n t i a l of t h e renormalization group
&A*&)
t o t h e problem o f s t u -
dying t h e p r o p e r t i e s o f t h e i n t e g r a l operator (4.40.). 5. E i g e n f u n c t i o n o f t h e d i f f e r e n t i a l of t h e r e n o r m a l i z a t i o n arouD.
It is ~ o s s i b l et o s t u d v t h e ~ r o ~ e r t i eo fs t h e o D e r a t o r ( 4 . 4 0 . ) . evaluate
x[c)
:
F i r s t we s h a l l
-
From t h e d e f i n i t i o n (4.27. ) of
P u t t i n g (4.42.)
Setting
X[t)
we have t h a t
-
ir.to ( 4 . 4 1 . ) we o b t a i n :
where
r way i f is p o s s i b l e t o show t h a t 4.46.
4 = P-'~(IZ)
and s o we o b t a i n
( Y [ ~ ) = ~ [ K Z ) ~ K ~ - ' ~ ( ~ )
Now it is p o s s i b l e t o f i n d t h e e x p l i c i t e x p r e s s i o n s o f t h e e i g e n f u n c t i o n s o f
pK4+(d) . Let
u s d e f i n e b e f o r e t h e h e r m i t e polynomials:
Then : Proposition 4.2.
The
functions
~(42)'-&&;qf~)) a r e
t h e transformat ion (4.40.) with eigenvalue equal t o
eigenfunctions of
&
Proof. The g a u s s o p e r a t o r
4
.
I -(D
t r a n s f o r m s a n h e r m i t e polynomial
c'? & (~;c(u-B))if T ~ U St r a n s f o r m a t i o n
ction
&, ( 2 , ~ )
i n a h e r m i t e polynomial
a7 6
( 4 . 40. ) t r a n s f o r m s t h e f u n c t i o n
R [ z i 7(a))
i n t h e fun-
KC-O&C~;C(~~K~)-X~ILI~= - K K- mf24?. * fZR/-Kd-tpq~'t))=~
-
The p r o p o s i t i o n is proven.
Remark. P r o p o s i t i o n ( 4 . 2 . ) i m p l i e s t h a t t h e form
is e i g e n f u n c t i o n of
& A (k' )
with e i g e n v a l u e
%-w+i 6%
. F u r t h e r ff$,
form an o r t h o g o n a l system o f random v a r i a b l e s w i t h r e s p e c t t o t h e g a u s s i a n p r o b a b i l i t y measure corresponding t o t h e automodel g a u s s i a n s t a t i o n a r y random field.
5. The
&
-expansion.
I t i s p o s s i b l e now t o g i v e a n a l g o r i t h m which a l l o w s t o f i n d t h e e f f e c t i v e h a m i l t o n i a n c o r r e s p o n d i n g t o a non g a u s s i a n automodel p r o b a b i l i t y d i s t r i b u t i o n . Let u s w r i t e t h e e f f e c t i v e h a m i l t o n i a n i n t h i s way / # ~ ) = ~ ~ i ~ ( U ) a nexpand d t h e corresponding measure i n power o f
where
where
N O is a
&
:
g a u s s i a n a u t . p r . d i s . . Let u s a p p l y
&A'@)
we have t a k e n o n l y t h e t e r m s up t o t h e second o r d e r i n
to
6
because we
a r e going t o f i n d an e x p l i c i t form o f t h e e f f e c t i v e h a m i l t o n i a n o n l y t o t h i s degree of a c c u r a c y . Thus we have t o f i n d
$(g
i n such a way
, BdW)
that:
We a r e s t u d y i n g t h e e f f e c t i v e h a m i l t o n i a n i n a neighbourhood of expected t o be a b i f u r c a t i o n p o i n t and s o If
d= 3
w i l l be d e f i n e d by
which i s
E=d- 2 t
.
t h e s o l u t i o n o f ( 5 . 3 a . ) should be w i t h e i g e n v a l u e 1 but s i n c e
N C Es t €=a-23
we can s t a t e
and s o we r e s o l v e t h e e q u a t i o n ( 5 . 3 a .
up t o t h e accurac.y
and p u t t i n g i n t o t h e e q u a t i o n (5.3b.1 t h e term
OCE)
OCg)
, setting
it w i l l a p p e a r a s :
where L is d e f i n e d by
The
aim o f t h i s s e c t i o n is t o f i n d a s o l u t i o n o f ( 5 . 6 . ) .
The space o f forms where we s h a l l l o o k f o r a s o l u t i o n of ( 5 . 6 . ) i s determined by t h e s t r u c t u r e o f t h e o p e r a t o r
LC&&])
. Let
u s s t u d y it i n some d e t a i l :
we u s e t h e r e p r e s e n t a t i o n o f t h e p r e c e e d i n g s e c t i o n :
tr
( t i t C ( 4 q ~and /%$a) -It c o e f f i c i e n t s depend on 2 . We have: where T[%)
3K~%40() HC ' WJ = E
+
w
whose
X
d
( $ dhldca , , F ( -rf[ ~ * s~,ftt,~~)K-'w't~k)+ -09
+-nI+" d f c t ~ d l * d , h - ) ~ ( t l ) ) .~
5.8-
is some polynomial i n
.(JT
-9 ( r ~ d t ~ & l t ~ , t ~ ~ -n
We can proceed i n an analogous way a s have done f o r f i n d i n g t h e compact form of t h e e i g e n f u n c t i o n s of t h e d i f f e r e n t i a l , t h e o n l y d i f f e r e n c e is t h a t now we w i l l make t h e c o n d i t i o n a l e x p e c t a t i o n w i t h r e s p e c t t o t h e by
and
-lr
we o b t a i n
where
and
-algebra generated
&=I d?h/&(t&k)X[t%) %=~t~(,&a&(*',W x&) -IT +JT
1%
,+a is a m a t r i x g i v e n by
and
making a n a l o g o u s c a l c u l a t i o n s a s b e f o r e w e o b t a i n
where
ltfq-fi)po(e) p,(zrrq-t) 8
yl%,q)
6w
making t h e i n t e g r a l s w i t h r e s p e c t t o
Xa 1 &
dt
and u s i n g t h e theorem
o f moments of g a u s s i a n s y s t e m o f v a r i a b l e s ( 2 0 ) ( w i c k t h e o r e m ) we o b t a i n a sltm
o n l y on t h e c o n n e c t e d g r a p h s b e c a u s e t h e d i a g o n a l s t e r m s o f
elpal t o
YCK~Z,)
.
are
From t h i s r e p r e s e n t a t i o n we car, a l s o d e d u c e t h a t we need
t o c o n s i d e r a s p a c e of forms o f t h e type:
R e p e a t i n g t h e c a i c u l a t i o n s made b e f o r e we o b t a i n t h a t t h e e i g e n f u n c t i o n s o f t h e d i f f e r e n t i a l i n t h e s p c e o f forms o f t h e t y p e ( 5 . 1 2 . ) a r e t h e multidimens i o n a l h e r m i t e polynomial;.
We g i v e t h i s s t a t e m e n t i n more p r e c i s e terms: P r o p o s i t i o n 5 .l. L e t
q(~, Y ) be d e f i n e d
a s i n (5.11. ) and l e t
c o r r e s p o n d i n g m a t r i x . Consider t h e g a u s s i a n random f i e l d sion matrix
ft : i (LCJ t~(ti) + -.+fin) nted
)=e
~ ( e Qr : :net w . . VC&)
5.13.
Let
6 ij
~(3 with
-2 uc& e,m=1
be t h e Wick polynomial o b t a i n e d
by
be t h e disper-
m,L) %(kJ ~'h,,..
making t h e o r t o g o n a l i z a t i o n w i t h r e s p e c t t o t h e g a u s s i a n d i s t r i b u t i o n (5.13.). Then
where
191 zG+-**+4*
The o r t o g o n a l i z a t i o n is j u s t t h e same a s used i n e u c l i d e a n quantum f i e l d theory:
t h e g r a f i c s designed above must be i n t e r p r e t e d w i t h t h e h e l p o f t h e f o l l o w i n g explanations:
-
t o every vertex corresponds a p o i r i t a
- t o every l i n e going -
i n t o the vertex
Jti
t h e l i n e j o i n i n g two v e r t e x e s c o r r e s p o n d s t o
Then t h e d e f i n i f ion of:
nq'&)
-
c o r r e s p o n d s a random v a r i a b l e r(ti)
-..flm (q*):
~ ~ I z ~ ) T ( ~ > = ~ i s g i v e n by
where t h e sums go o v e r a l l t h e p o s s i b l e g r a p h e s o b t a i n e d j o i n i n g p a i r o f l i n e s of t h e p i c t u r e above. We can make u s e o f P r o p o s i t i o n ( 5 . 1 . ) tion (5.6.).
In f a c t we can w r i t e :
where we have s u p p r e s s e d t h e i n d e x
f o r s o l v i n g equa-
.
b u t t h e Wick polynomials must be
understood i n t h e s e n s e of P r o p o s i t i o n ( 5 . 1 . ) .
Thus we c a n w r i t e
We can a l s o expand t h e q u a d r a t i c o p e r a t o r i n ( 5 . 6 . )
with t h e help of hermite
polynomials. I n f a c t we have
we c a n expand
9:
i n t e r m s o f ;he
eigenfunctions
of
gK~IC@)
It is p o s s i b l e t o s e e t h a t t h i s sum i s e q u i v a l e n t t o make t h e g r a f i c expansion
o f b e f o r e and keeping o n l y t h e connected termd. Thus it f o l l o w s from p r o p o s i t i o n ( 5 . 1 . ) t h a t :
and
The integralJY4(~h,kca)dbidta
tv
Of
will
‘r
d i v e r g e s because o f t h e p e r i o d i c proper-
=
: '#cu~+~,&+L) -l'(Cs%&)
, but
it g i v e s a c o n s t a n t which
g i v e no c o n t r i b u t i o n t o t h e p r o b a b i l i t y d i s t r i b u t i o n . T h i s d i v e r g e n c e
a r i s e s because we c a l c u l a t e d i r e c t l y t h e e f f e c t i v e h a m i l t o n i a n which is a sum o f some p o t e n t i a l o v e r a l l t h e l a t t i c e p o i n t s : t h e same c a l c u l a t i o n f o r t h e pot e n t i a l would have g i v e n a f i n i t e term. T u t t i n g (5.18.1,
(5.19.),
(5.15.)
i n t o eq. ( 5 . 6 . )
and n e g l e c t i n g t h e term
we o b t a i n t h e f o l l o w i n g e q u a t i o n s :
6
CY,~)
h t , w K
z-5[44)
(53)
-5
=
&,fit)
w I)K
i6af (YCKCA,K~E)-LYC& z*ca-Nq(,t =isa:[~
putting
OC&) i n t h e h i g h e r F,'3,3'
-
444,
-q(%~2))+ OCbI
o r d e r e q u a t i o n s we o b t a i n t h a t
[2a,24)
(TA,O*)=
=
16 (Y(%,h.t)
G~
7 2 ~ ' C ~ * , T Za)
'
P r o p o s i t i o n 5.2. The f u n c t i o n
(kte&)
belongs t o
flfl')
is r e a l , p e r i o d i c and
symmetric and h a s t h e f o l l o w i n g a s y m p t o t i c
4-t Proof.
The p r o p o s i t i o n f o l l o w s immediately from t h e d e f i n i t i o n (5.11. ) o t
Ute. P, \
. In
f a c t t h e Fourier transform o f
I t is s u f f i c i e n t t o look a t t h e b e h a v i o u r o f
From which it f o l l o w s t h a t :
y.
c"%t
C.4'l&,~~ ) is
bC.4)
g i v e n by
in the origin
From t h e d e f i n i t i o n of t o be i n t e g r a b l e on
R'
fa
( 5 . l Q b . ), it i s c l e a r t h a t we need
484
L%et)
and t o have s i n g u l a r i t i e s which a r e i n t e g r a b l e on t h e
plane.
i t s behaviour
F i r s t w k n o t e t h a t we caw s u b t r a c t from a t large
1%;
)
and t h e e q u a t i o n s o f t h e t y p e ( 5.20. ) w i l l b e s a t i s f i e d
j u s t t h e same. I n f a c t t h e r i g h t hand s i d e of (5.20.) w i l l n o t change.
K
~16 (ktod ~ -(#C,KQ~, ~ K O ~ -QC%i )
1%)
)=
5.23.
we need t o s u b t r a c t t h e behaviour a t l a r g e because
where Reg
only f o r
-
(33) , -
(4~)
q -
h a s a s u f f i c i e n t l y good d e c r e a s e p r o p e r t y a t i n f i n i t e .
But we have t h a t
5 ,z1
/%A-&)
~C"l'
now has, a non-integrable s i n g u l a r i t y
for
a s a consequence of t h e s u b t r a c t i o n . So we s h a l l s e t
is a g e n e r a l i z e d f u n c t i o n d e f i n e d by
-00 Now it i s p o s s i b l e t o show t h e following i d e n t i t y between g e n e r a l i z e d f u n c t i o n s
Which f o l l o w s from t h e d e f i n i t i o n :
Thus
cx.)
i
'
:k4"(fi8{
/ ~ 1 4 ' 2 dftv(*))+Z~"[qC0I k-5
5.27.
=('-@K
/*/MA
b8/L--
/~+-u
+-?
p - 4 -
( i
2d-3
- k3-a ) (7~x4,LPC*))
From (5.24.) and t h e f a c t t h a t neglected i n equation (5.20.)
-w2&w
i-K
3-44
Zol-3
6
we o b t a i n t h a t t h e t e r m
i s e q u a l t o t h e c o n t r i b u t e g i v e n by t h e
c t i o n t o t h e l e f t hand s i d e o f (5.6.)
and s o t h e
-
if
-expansion is c o m p l e t e l y r e s o l v e d up t o t h e second o r d e r .
Now we a r e g o i n g t o i n t e r p r e t t h e s e r e s u l t s i n term o f t h e e f f e c t i v e hamiltonian f o r t h e f i e l d
fa0 1 tlf-* .
We have found t h a t t h e non g a u s s i a n term o f t h e h a m i l t o n i a n h a s t h e f o l l o w i n g
form
Let us.-write
(5.28.)
We o b s e r v e t h a t
i n terms of
y'b) -
5.29.
where
S ( ~ I #. S~i n c e we have a l r e a d y s u b t r a c t e d
o f (5.28.)
from t h e f i r s t two terms
t h e g a u s s i a n b e h a v i o u r we compare o n l y t h e t h i r d term of ( 5 . 2 8 . )
w i t h the g a u s s i a n term. W e w i l l w r i t e t h e l a s t term o f (5.28.)
i n t h e f o l l o w i n a u s e f u l form:
we want t o compare t h e c o n t r i b u t i o n t o t h e e f f e c t i v e i n t e r a c t i o n between two spins
jplfPW
due t o , t h e ' n o n . gaussian:term
w i t h - t h e one-due t o - t h e
g a u s s i a n term. We w r i t e t h e l a s t one i n t h e f o l l o w i n g way:
For t h i s aim we have t o s t u d y t h e a n a l i t i c i t y p r o p e r t i e s o f f o r a l l t h e values of
. Thus we
/qS-
+(&,&)
have t o s t u d y t h e a n a l i t i c i -
t y p r o p e r t i e s o f t h e s e r i e s (5.22.). Let u s s t u d y b e f o r e t h e term i n (5.22.) w i t h
5.32.
q[ua,u*] = -3L- (luL1 -d-s +&, 1c(,/2Hz d+d
where
$o
~UI)
loped t h e f a c t o r
f15f Ut=0 M 4 ) ) -%
go(ar)dhIUt2
is an a n a l y t i c a l f u n c t i o n o f
(L+
IUlldtfg (llr))-id) *
UA
= $yug) , and
-- 1 *'
we have deve-
i n Taylor s e r i e s in t h e nei-
ghbourhood o f t h e o r i g i n , t h u s we can w r i t e
The second term g i v e s n o s i n g u l a r i t y i n t h e o r i g i n because it is a n a l y t i c a l while t h e f i r s t and t h e t h i r d w i l l g i v e
The s i n g u l a r i t y . a r i s i n g by t e r m s where
ffafu+=2~ k
gives a contribu-
t i o n no l a r g e r t h a n
a s it i s p o s s i b l e t o s e s from e l e m e n t a r y c o n s i d e r a t i o n s . Thus t h e main c o n t r i b u t i o n t o t h e e f f e c t i v e i n t e r a c t i o n p o t e n t i a l comes from
. ' e f i r s t term i n (5.33.
) 'and we have t h a t
which must be compared w i t h t h e behaviour o f t h e g a u s s i a n term
-' .
( 5 . 3 6 . ) g i v e n a p o t e n t i a l which d e c a y s a s which is slower t h a n
4-4
, while
(5.35.)
M-6
g i v e s +l
F u r t h e r i n v e s t i g a t i o n can be made i n two pos-
s i b l e directions: A ) To s t u d y t h e h i g h e r o r d e r e q u a t i o n s o f t h e
B) r e g u l a r i z e a l s o t h e t h i r d term o f ( 5 . 2 8 . ) .
E
-expansion
I n t h i s c n s e we would o b t a i n t h e
c o r r e c t behaviour o f t h e i n t e r a c t i o n p o t e n t i a l and a l s o t h e u n i c i t y of t h e s o l u t i o n o f eq. ( 5 . 6 . ) and s o B seems t o be t h e most f r u i t f u l way t o i n v e s t i gate.
6.
Some new r e s u l t s f o r s p i n systems
*1
Let u s c o n s i d e r a d-dimensional l a t t i c e system, l e t
g&)
be t h e c o r r e l a t i o n
f u n c t i o n of a Gaussian automodel random f i e l d
where Then t h e Gaussian a u t . p r . d i s t r .
can be w r i t t e n a s a Gibbs d i s t r i b u t i o n i n t h e
" ~ e c e n t l ~E. 1.Dinaburg and Ja.G. S i n a i proved t h a t for i 4dcg ' t h e gauss i a n automodel d i s t r i b u t i o n a p p e a r s a s t h e l i m i t p r o b a b i l i t y d i s t r i b u t i o n a t
p=w
t
f o r some one-dimensional t r a n s l a t i o n a l l y i n v a r i a n t system e with t h e long range p o t e n t i a l & I % )
-,,
form :
d>t
I t is ~ o s s i b l et o find,
for
l y t i c a l f u n c t i d n and s o
C(C%)
a c a s e when
l i m i t distribution
c(
p
I ,
p
'>
p implies p
%
p ' , so t h a t
> is a pre-
Lattice structure of density matrices. One can show t h a t , with respect t o 2 , the equivalence classes of density matrices form a l a t t i c e . There exists always a "smallestn element, namely, the pure s t a t e s , but only i f the Hilbert space
is finite-dimensional, a "biggest one", namely ( d i m
Convex and concave functions. p
> p',
1.
i f , and only i f , for every convex (or
concave) function f , 2 0 , with f (0) = 0, T r f ( p ) 2 T r f ( p ' ) (or 2 0, resp. In particular, p
Coarse-grainins.
1 Pi
p
'
+ S(p )
)
.
2 S ( p ' ) , but the converse is not true.
Let Pi be a family of pair-wise orthogonal projections with
= 1. Then p (
1 Pi
p Pi.
(In theories about the measurement process, t h i s
is sometimes called "reduction of a s t a t e " ) . I f , i n addition, there e x i s t s a "coarse-grained" density matrix pc = then
2
pi p pi
1 A.
Pi such t h a t T r
< pc.
A i Pi =
T r p Pi,
Among the measures t h a t are compatible with Uhlmann's order relation, the quantum analogues of Renyi's entropies2) play a distinguished role. Let
for a > 0, # 1; So ( p ) = In d i m Ran p (the quantum analogue of the Hartley entropy), Sl( p ) = S ( p ) , Sm(p) =
- In
1 l p 11.
Then S ( p ) i s decreasing i n a6 ) , a
f i n i t e for a > 1, continuous w i t h respect t o the trace norm for 6 > 1 since
I(
~P ar 'Ia
-
(Tr p ' a ) l / a l
2 (Tr
Ip
- p ' l ) i /a
( t h i s is a consequence of the t r i a n g l e inequality7) f o r the v.Neumann-Schatten classes)
. Also
a-entropies look very much l i k e t h e r i g h t entropy, i n p a r t i c u l a r , they are a d d i t i v e (see s e c t i o n " I n e q u a l i t i e s (Two Spaces)") and have been used on s e v e r a l occasions, e.g.
i n non-equilibrium s t a t i s t i c a l mechanics.
Continuity P r o p e r t i e s o f Entropy
I f t h e H i l b e r t space is finite-dimensional,
entropy is c l e a r l y continuous. I n
t h e infinite-dimensional case, entropy is discontinuous with r e s p e c t t o the t r a c e norm, because every " b a l l " {p ' : Tr
Ip - p '1
0)) contains density
matrices with i n f i n i t e entropy. This can be shown e x p l i c i t l y a s follows: l e t p(') < p ( 2 ) be thd eigenvalues of p. Choose N such t h a t
...
Let p
'
for i
2 N,
have t h e same eigenvectors as p , b u t eigenvalues p (i)
= p (i) f o r i < N,
provided t h a t
otherwise one can assume t h a t p (N-l) p ' ( N - l ) = P IN-''
-
C*
(c'
0 , then l e t p-*(i)
p ~ ( f~o r) i , N
= p(i) for i < N
- 1,
a s above. f n both cases, c is
to be chosen such t h a t
hen, T r
Ip - p'l
< E , b u t ~ ( p *=)
=.
Lower semi-continuity f o r entropy. Since S o ( @ )is continuous f o r a > l y a n d S(p)
= l i m Sa (P ) = sup Sa (p) , S (p ) is lower semi-continuous. a+l
Therefore, t h e
a >1
s e t s {p: S ( p ) ( n) a r e closed, t h e i r complements a r e dense, hence they a r e nowhere dense and
i s of f i r s t category. Besides lower semi-continuity,
some other r e s t r i c t e d continuity properties a r e
valid. The most t r i v i a l one is
Convergence of canonical approximations3). ~f p =
1 p ( i )pi,
arranged i n decreasing order, t h e Pi being one-dimensional,
( "canonical
approximation" )
. Then,
S ( pN ) + S (p )
the p(i) being let
.
Much l e s s t r i v i a l i s t h e
Dominated convergence theorem f o r entropyg). I f pn i s a sequence of densify matrices converging weakly t o p , and i f there e x i s t s a compact operator A 2 0 (not necessarily a density matrix) such t h a t pn
