Selected Papers of Morikazu Toda
SERIES IN PURE MATHEMATICS Editor: C C Hsiung Associate Editors: S S Chern, S Kobayashi, I Satake, Y-T Siu, W-T Wu and M Yamaguti
Part I. Monographs and Textbooks Volume 1:
Total Mean Curvature and Submanifolds on Finite Type B Y Chen
Volume 3:
Structures on Manifolds
Volume 4:
Goldbach Conjecture
Volume 6:
Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds
K Yano S M Kon Wang Yuan (editor) Ngaiming Mok
Volume 7:
The Geometry of Spherical Space Form Groups
Volume 9:
Complex Analysis
Peter
BGilkey
TO Moore & EH Hadlock
Volume 10: Compact Ftiemann Surfaces and Algebraic Curves Kichoon Yang
Volume 13: Introduction to Compact Lie Groups Howard D Fegan
Part II. Lecture Notes Volume 2:
A Survey of Trace Forms of Algebraic Number Fields
Volume 5:
Measures on Infinite Dimensional Spaces
Volume 8:
Class Number Parity
P E Conner & R Perils Y Yamasaki P E Conner & J Hurreibrink
Volume 11: Topics in Mathematical Analysis Th M Rassias (editor)
Volume 12: A Concise Introduction to the Theory of Integration Daniel W Stroock
Part III. Collected Works Volume 14: Selected Papers of Wilhelm P. A. Klingenberg Volume 15: Collected Papers of Y. Matsushima
Series in Pure Mathematics - Volume 18
Selected Papers of Morikazu Toda
Edited by Miki Wadati Department of Physics University of Tokyo
lfcBj World Scientific wh
Singapore • New Jersey • London - Hong Kong
Published by World Scientific Publishing Co. Pte. Lid. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, Rivei Edge, NJ 07661 UK office: 73 Lynlon Mead, Totteridge, London N20 BDH
SELECTED PAPERS OF MORIKAZU TODA Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may nol be reproduced in any f or by any means, electronic or mechanical, including photocopying, recording ora information storage and retrieval system now known or to be invented, witho written permission from the Publisher. For photocopying of inaletial in this volume, please pay a copying fee through Ihe Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA. ISBN 981-02-1469-3
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
Foreword
This volume contains selected papers of Dr. Morikazu Toda. The arrangement of papers is in chronological order of publishing dates. His contributions to theoretical physics are legion. There is little need to tell the significance of these papers which speak for themselves. I think, however, it may be interesting and convenient to the reader to mention briefly his scientific career. Morikazu Toda was born in Tokyo on the 20th of October 1917. He graduated from the Department of Physics, University of Tokyo, in 1940. After serving as an associate professor first at Keijo University and then Tokyo University of Education (Kyoiku University), he was promoted to professor in 1952 at the age of 34. He was a professor at Chiba University, Yokohama National University and University of the A i r . Meanwhile, he was a visiting professor at Sao Paulo University and University of Trondheim. He is a professor emeritus of Tokyo University of Education and a member of the Royal Norwegian Academy of Science. Morikazu Toda's main interests are statistical mechanics and condensed matter physics. Among his many contributions, we should mention his works on liquids and nonlinear lattice dynamics. The one-dimensional lattice where nearest neighboring particles interact through an exponential potential is called the Toda lattice. The Toda lattice is a miracle and indeed a jewel in theoretical physics. He received the Mainichi Shuppan-Bunka prize for his contribution to the theory of liquids in 1947 and the Fujiwara prize for discovery of the Toda lattice in 1981. I am grateful to Professor Akinobu Shimizu and Dr. Noriko Saitoh, Yokohama National University, for their efforts in preparing this work. I am glad to acknowledge the cooperation of World Scientific Publishing and especially of Ms. H. M . Ho who took charge of compiling this volume.
Miki Wadati, editor
This page is intentionally left blank
vii
Preface S o m e of m y scientific papers are collected i n this book. M y scientific career s t a r t e d just before the Second W o r l d W a r , and some of my works of t h a t t i m e were w r i t t e n in Japanese. A m o n g these, a paper on the v i r i a l theorem, translated for this issue, presents a m e t h o d of statistical t h e r m o d y n a m i c s . F o r a p e r i o d of time, I was then interested i n the theory of l i q u i d s , a n d low temperature physics. F r o m about 1957, we h a d a g r o u p of physisists w h o were interested i n exact solutions rather t h a n a p p r o x i m a t e methods, since computer experiments were just reveali n g t h a t c o n v e n t i o n a l p e r t u r b a t i o n methods sometimes failed i n g r a s p i n g specific features of p h e n o m e n a such as the l o c a l i z a t i o n of wave functions a n d v i b r a t i o n a l modes a r o u n d i m p u r i t i e s , enhancement of heat flow by n o n l i n e a r i t y of i n t e r a c t i o n between lattice particles, and so on. M y s t u d y of nonlinear s y s t e m was advanced when I found papers of J . Ford(1961,64), w h o examined the v i b r a t i o n of lattices w i t h s m a l l number of particles a n d clarified that certain one-dimensional nonlinear l a t t i c e s m a r v e l l o u s l y s u s t a i n the character of l i n e a r m o d e s (his s t u d y was i n i t i a t e d b y the famous c o m p u t e r e x p e r i m e n t by E . F e r m i et a l . , b u t I c o u l d not see it at that t i m e ) . I thought there would be a l a t t i c e m o d e l w i t h p a r t i c u l a r i n t e r a c t i o n between particles, w h i c h a d m i t s exact p e r i o d i c waves, a n d finally found the integrable nonlinear lattice system. It is m y great pleasure to t h a n k m a n y friends w h o have shared interest i n the l a t t i c e p r o b l e m s and nonlinear problems, i n c l u d i n g E . T e r a m o t o , H . M a t s u d a , S. Takeno, F . Y o n e z a w a , N . Saito, T . K o t e r a , Y . Ichikawa, R. H i r o t a , M . W a d a t i , J . S a t s u m a and S. W a t a n a b e . It was found desirable to have a collection of some of my papers to d i s t r i b u t e a m o n g seminars i n several laboratories, and first I i n t e n d e d o n l y to b i n d copies a n d reprints. However, one of my colleagues k i n d l y suggested to m a k e a b o o k , and this is the o u t c o m e . M o s t of m y papers on n o n l i n e a r lattices are presented.
It also includes some of m y earlier
works w h i c h seem to have some o r i g i n a l i t y a c c o r d i n g to m y own d o g m a and prejudice.
It is m y sincere hope t h a t this b o o k may give basis for
discussion and development for furture study. It is a pleasant d u t y to thank M . W a d a t i for e d i t i n g a n d w r i t i n g the F o r e w o r d . I a m grateful to the staffs of the D e p a r t m e n t of A p p l i e d
viii
M a t h e m a t i c s , F a c u l t y of E n g i n e e r i n g , Y o k o h a m a N a t i o n a l U n i v e r s i t y for k i n d efforts w h i c h e n a b l e d the p u b l i c a t i o n of t h i s b o o k , e s p e c i a l l y to A . S h i m i z u a n d N . S a i t o h . T h a n k s are also due to the staff of W o r l d Scientific P u b l i s h i n g C o . for k i n d assistance.
A p r i l , 1993
Morikazu Toda
Contents
Foreword ( M . Wadati) Preface ( M . Toda) Reprinted papers T h e S o l i d States of H? a n d D2 Proc. Phys-Matk. Soc. Japan 22 (1940) 503-507 S e c o n d a r y E l e c t r o n E m i s s i o n from P u r e M e t a l s Proc. Phys-Math. Soc. Japan 25 (1943) 207 O n the V i r i a l T h e o r e m Recent Problems in Physics (1948) 93-112 N o t e s o n the T h e o r y o f H i g h P o l y m e r Solutions (with A. Iskihara) J. Polymer Science 7 (1951) 277-287 O n the R e l a t i o n between F e r m i o n s a n d Bosons J. Phys. Soc. Japan 7 (1952) 230 N o t e s o n F e r m i a n d Bose Statistics (with F. Tahano) J. Phys. Soc. Japan 9 (1954) 14-18 O n the T h e o r y of Q u a n t u m L i q u i d s . I. Surface T e n s i o n a n d Stress J. Phys. Soc. Japan 10 (1955) 512-517. Diffusion i n V e l o c i t y Space a n d T r a n s p o r t P h e n o m e n a Transport Processes in Statist. Meek. (1958) 148-154 L o c a l i z e d V i b r a t i o n a n d R a n d o m W a l k (with T. I\otera and Y. Kogvre) J. Phys. Soc. Japan 17 (1962) 426-433 S t a t i s t i c a l D y n a m i c s of Systems of I n t e r a c t i n g Oscillators (with Y. Kogure) Progr. Theoret. Pkys. Suppl. 23 (1962) 157-171 S o m e P r o p e r t i e s of the P a i r D i s t r i b u t i o n F u n c t i o n J. Pkys. Soc. Japan 19 (1964) 1550-1554. One-Dimensional Dual Transformation J- Phys. Soc- Japan 20 (1965) 2095
z
One-Dimensional Dual Transformation Progr. Tkeorei. Phys. Suppl. 36 (1966) 113-119
90
V i b r a t i o n of a C h a i n w i t h Nonlinear Interaction J. Phys. Soc. Japan 22 (1967) 431-436
97
Wave Propagation in Anharmonic Lattices J. Phys. Soc. Japan 23 (1967) 501-506
103
M e c h a n i c s a n d S t a t i s t i c a l M e c h a n i c s of N o n l i n e a r C h a i n s J. Pkys. Soc. Japan, Suppl. 26 (1969) 235-237
109
Waves i n N o n l i n e a r L a t t i c e Progr. Theoret. Pkys. Suppl. 45 (1970) 174-200
112
T h e C r i t e r i o n for the E x i s t e n c e o f a G a p i n the O p t i c a l B a n d of D i s o r d e r e d M i x e d C r y s t a l (unpublished, 1970) I n t e r a c t i o n of S o l i t o n s w i t h E l e c t r o m a g n e t i c Waves Physica Norvegica 5 (1971) 203-207
139 146
A n E v i d e n c e for the E x i s t e n c e of K i r k w o o d - A l d e r T r a n s i t i o n (with M. Wadati) J. Pkys. Soc. Japan 32 (1972) 1147
151
T h e E x a c t J V - S o l i t o n S o l u t i o n o f the Korteweg—de V r i e s E q u a t i o n {with M. Wadati) J. Pkys. Soc. Japan 32 (1972) 1403-1411
152
A S o l i t o n a n d T w o Solitons i n a n E x p o n e n t i a l L a t t i c e a n d R e l a t e d E q u a t i o n s (with M. Wadati) J. Pkys. Soc. Japan 34 (1973) 18-25
161
B a c k l u n d T r a n s f o r m a t i o n for the E x p o n e n t i a l L a t t i c e (with M. Wadati) /. Pkys. Soc. Japan 39 (1975) 1196-1203
169
A C a n o n i c a l T r a n s f o r m a t i o n for the E x p o n e n t i a l L a t t i c e (with M. Wadati) J. Pkys. Soc. Japan 39 (1975) 1204-1211
177
D e v e l o p m e n t of the T h e o r y of a N o n l i n e a r L a t t i c e Progr. Theoret. Phys. Suppl. 59 (1976) 1-35
185
xi
Chopping Phenomenon of a Nonlinear System (with R. Hirota and J. Satsuma) Progr. Theoret. Phys. Suppl. 59 (1976) 148-161
220
Problems in Nonlinear Dynamics Rocky Mountain J. Math. 8 (1978) 197-209
234
Solitons and Heat C o n d u c t i o n Physica Scnpta 20 (1979) 424-430
247
I n t e r a c t i o n o f S o l i ton w i t h a n I m p u r i t y i n N o n l i n e a r L a t t i c e (with S. Watanabe) Jf. Phys. Soc. Japan 50 (1981) 3436-3442
254
E x p e r i m e n t on S o l i t o n - I m p u r i t y I n t e r a c t i o n i n N o n l i n e a r L a t t i c e U s i n g L C C i r c u i t (with S. Watanabe) /. Phys. Soc. Japan 50 (1981) 3443-3450
261
T h e C l a s s i c a l Specific H e a t of the E x p o n e n t i a l L a t t i c e (with N. Saitoh) /. Phys. Soc. Japan 52 (1983) 3703-3705
269
Interest i n F o r m i n J a p a n a n d the W e s t Science on Form (1986) 1-8
272
Coupled Nonlinear Waves Physica D 33 (1988) 317-322
280
N o n l i n e a r D u a l L a t t i c e {with Y. Okada and S. Watanabe) /. Phys. Soc. Japan 59 (1990) 4279-4285
286
P a r t i t i o n F u n c t i o n of N o n l i n e a r L a t t i c e
293
Nonlinear Dispersive Wave Systems (1992) 435-443 Academic Career of Morikazu T O D A
303
Bibliography of Morikazu T O D A
305
List of Misprints
313
1
1940]
The Solid
States of H and
503
D. 2
Proc. Phys. Math. Soc. Japan 22 (1940).
The
Solid
States
of JT mid
B y Morikazu
Z
D. z
TODA
(Kead May 11, 1940}
§1.
Introduction
C r y s t a l lattice of h e l i u m and hydrogen are k n o w n to have singular properties due to their large zero point vibrations. C l u s r u s ' attributed largo difference of lattice spacing and heat of sublimation of solid TI~ and D to the difference of their zero-point vibrations, and H o b b s calculated quantitatively these properties of solid hydrogen, assuming intermolecular force of Lennard-Jones type between pair of molecules. Since H o b b s ' expression for k i n e t i c energy is rather arbitrary interpolation and contains an adjustable parameter, i n this paper the problem of solid hydrogen is reinvestigated u s i n g a different method i n the hope to drive some informations concerning to the intermolecular force from crystal data.' 0
(Z>
2
31
A s the zero-point v i b r a t i o n has fairly large amplitude, attack from the usual lattice theory of h a r m o n i c v i b r a t i o n (Debye) is inadequate, and (1) K. Clusius unci E. Bartlio Ionic, Zeits. Phys. Chem. B 30 (1935), 937. 12) Hobbs, Journ. Ghcm. Phys. 7 (1830). 31S. (3) At the annual meeting, on April 1940. of tlie Physical-Mathoniatiral Society we were informed that the similar metlioii was applied to the study of solid helium by T. Nagamiya, Proc. Phys.-Math. Soc. Japan, this issue.
2
504
Morikazu TODA.
[Vol. 22
i t seems neccessary to lake i nt o account the deviations from h a r m o n i c vibration i n the zero order a p p r o x i m a t i o n . These circumstances suggest " cage m o d e l , " w h i c h is. successfully used i n some of modern theories of l i q u i d state. §2.
Energy
"We assume that the interaction energy d>(r) between two hydrogen molecules is approximately represented as a function of distance f between their centers of mass, a n d neglect the dependence of 4>( ) tho orientation of the molecular axes. T h i s approximation seems sufficient since the distance of the nearest neighbour molecules is 4 ~ 5 times larger than the nuclear separation i n single molecule, a u d is supported by the fact that the hindrance of free rotation of molecules of fortho) hydrogen lattice occurs at far lower temperature than fusion. r
o
n
A s s u m i n g tp(r) of the form T
T
we take for the f o l l o w i n g calculations the values of A aud B, w h i c h were determined by Lencard-Jones from gas data:
• W - y ^ - T T j W
erg.
A l t h o u g h tho determination of these constants was made by classical method, q u a n t u m correction of these constants seems small, since the gas data at extremely low temperatures were not used. N o w consider any molecule A i n the lattice, i n w h i c h the distance between nearest neighbours is a. W e classify a]] molecules of the lattice into shells according to the distances a from A i n the e q u i l i b r i u m position, zi nearest neighbours belong to the first shell, z next nearest neighbours to the second shell and so on. I n e q u i l i b r i u m positions molecules belonging to a shell i are arranged on the sphere 7z,, whose center is A. Potential energy for A is 2*sf»(juo)i where B denotes a l l other molecules i n the lattice. I n the cage model, this potential is replaced by a n expresion derived from i t by averaging w i t h respect to the direction of A displaced r from its rest position: t
a
1
Vit) = T A f U(VoJ + r -2a,rcos6) &n6d3dj(g )^j(q)e~ '' ,
op-
erating K.E. or V.iJ. operators from the left and taking the trace. Thus, using thermodynamic K.E. and V.R., we get P=-^(K.E.-V.R.)
(3.4)
which holds irrespective of the statistics of the particles, and of the type of the forces such as nuclear forces. In classical statistics K.E.
= ^NhT
holds, and (3.4) reduces to (1.7).
Further, if the interaction is absent, it reduces to the equation of state of an ideal gas. When the interaction is absent, the equation pV — | A ' . £ . applies to all statistics including the Fermi and Bose particles.
13
In classical statistics, we have the sum-over-states
where n(T,y)= / Jo /a
u
••• f Jo Ja
e- '
V = L
h T
d
q i
.:dq
9 N
,
S
and the pressure is given as
^_
,
_d
f c T
,
n
l o g n =
_ kT _ _L^f^
Writing r/; = Lxi, we have f • Jo
t fat-•• Jo
dx
5
N
e-
u
^
k
T
and therefore
where the second term on the r.h.s. is nothing but 2/3 times the V.R., l v -
dU
l v -
and (3.5) is the statistical mechanical expression for (1.1).
§4. T h e Variation Principle and the V . R . Theorem Assuming a certain approximate wave function with a parameter, we may obtain the best energy eigenvalue by using the variation principle
E =
Jj>(q)H{q)i>{q)dq /
iji(q)^(q)dq
14
To solve this problem by assuming an expansion-contraction variation of the wave function, we introduce a parameter A and put V"(?>
x
A
= H 9)
= #pSi> ^ > • • ')•
If we change A from A to A j , the extension of the wave function changes by t
the factor A i / A . P u t t i n g 2
%
= xi
(4.1)
we nave
ii>(\q)i>(\q)dq j$(x)H(xlX)i>(x)dx
(4.2)
/ i>(x)4>(x)dx As before, we assume
H
M
= - \ l L
V
l
+
J
Y,Y, (r)P
(4-8)
where P denotes exchange of coordinates and spins. Since
with Xr, — p,t, t
we have
£
(
A
)
^
—
L
—
J
—
M
-
and 2 5 i ? ( A )
=
/ ^ ) [ - ¥ £ v i - i E E f ^ ^ A ) / ' ] ^ ) ^
X** 2
" A "
/0(x)0(*)d* /
r
[-1 £ V ? - | E D | r J ( r ) F ] ^ ( A ) d g g
/0(A?) (Ag)d? v
(
4
6
)
15
Therefore the minimum condition 6E(\)/5X
— 0 gives the V . R . theorem (with-
out external force) K.E.
-
V.R.
(4.7)
where Jy(Ag)[-|£v;] (A )rf v
K.E.
=
g
g
/ yy{Xo)i>(Xq)dq ^fi>{x)[-\Evl}i>{*)dz j i/>(x)t/)(x)(ia: JflA(Xq)iP(Xa)d L
q
Jr=p/>.
(4.8)
both include A. The virial theorem (4.7) determines the parameter A of the approximate wave function. For this purpose, the denominators /
l
4i( x)il (x)dx i
of the above K.E. and V.R. may be omitted.
§5. Supplementary Remarks The virial theorem can take alternative forms according to the type of the force. Since
. are of the form A— exp(i(p-
where jf and r a r e the momentum and the position. We take the volume V of the nucleus i n place of the contract!on-expansion parameter \ . T h e K.E. is the Fermi energy, which is 4x g f V
4rr P%V
3
5 Mh '
5 Mh
3
for protons and neutrons respectively, with the maximum Fermi energies P
p
and P . n
Thus
= | J'"J
f
f
r
r
Pp{ i> 2)[ ^A )]pn-(r2,ri)dr dr 1
2
24
where ? =\ t> -~- r~i |. The density matrix p, with the factor 2 for the spin, is
p
r
o 1
P„r
/ .
Pr
P r\
p
p
For the potential well J(r) = -J(0)
(v < a),
0{r>a)
we have V.R. = KF
— ^sin^j, - x c o s z „ J ^sin x p
1
^
/
n
W
- x coaa: ^ J ( 0 ) n
n
A
with
The volume V of the nucleus is determined by the virial theorem V.R. = Since x ,x x»
~
20
2
Ma 1
97"ft "
The energy is minimized when Z — iV" = A / 2 . In this case we have / 110 0
2
MMa a"
A - i
and therefore 10 l
2
/
^
2\2/3
J(0)AfV
'
or, for the nuclear radius „
_
^
.1/3
K.E..
25
1 3
P u t t i n g a = 1 . 3 x l 0 - c m , J(0) ~ 0.1 M . U - , we get r
1 3
1
3
~ 1.5 x l O " , ! / , and in
0
1
3
tKe same approximation the potential energy turns out to be — ^ 3 - V J ( 0 ) x / a , to give E = A".£. + P . £ . ~ -0.010.4 ( M . U . ) . These coincide with the known properties of nuclei.
[IX] Vaporization
Heat and Melting
Heat of Metals
Following the method
of Wigner-Seitz, let e^f, the K.E. belonging to the wave function ipa with the wave vector k — 0, and approximate the total energy as a sum of en* and the Fermi energy Ep i n such a way that
K.E. where (K.E.)j
=(e fc
+ {K.E.)i
0
= -E
(virial theorem),
is the sum of kinetic energies of the electron and heat motion
of metallic ions; but we can neglect the latter.
Ne k is nearly equal to the 0
energy of ionization L of the neutral atoms. Therefore, the vaporization heat to decompose a metal into constituent neutral atoms can be approximated by
The melting heat may be approximated by the change in E, or — Nsp,
due to
the volume change A V of melting. Thus for the melting heat we have 2
We calculated L
v
from ep, and L
AV
m
2,
AV
from the measurement of L
v
and
AV/V.
We obtain the following reasonable agreement with experimnt. The unit used is kcal/mole.
£
L
« { calc. rexp. \ calc.
m
Li 46 66 0.23 0.46
Na 30 44 0.70 0.50
K 26 28 0.50 0.46
Rb 25 24 0.52 0.57
Cs 24 21 0.50 0.40
Cu 76 93 3.2 2.6
Ag 65 73 2.7 2.0
Au 83 73 3.1 2.3
26
[X] Metallic
Hydrogen
The force in an assembly of electrons and protons is
the pure Coulomb force. Let r, be the half of distance between atoms, and
2
the expansion-contraction parameter. The Fermi energy is written as A A , or K.E.
-
2
1.105e*a A . H
The potential energy comes from the pro ton-electron interaction —(3/2)e A J
(electrons with uniform density is roughly assumed), electron-electron interaction +(3/5)e*A and the contribution of the exchange force which amounts to 2
—0.453e A. Thus we have the potential energy as B\, or P.E.
= -l.SCe'A.
In the equilibrium state without pressure, the virial theorem, K.E.
=
-\P.E.,
gives r, = T- = 1.63ajf , E - -11.29 e.V. = - 0 . 8 3 4 J V E is thus higher than the energy of atomic hydrogen. If we treat the electron wave function more rigorously, E will be lowered. But still it will be higher than the molecular state of hydrogen.
[XlJ/Vesswre due to Light
A s an example of relativistic case, we consider
electro-magnetic wave, or photons. T h i s is the case of limit where the rest mass m —* 0, and the speed is c (/5 —t 1). We need some modifications. B y (1.8) we have
27
1
where p is the external pressure. As rac /\J\ — ft is the total energy, i n the 2
2
limit 0 —* l j E Tac
—0
2
goes to the energy E of the electro-magnetic
field. Therefore the pressure due to light is given as -
\E-
It must be noticed that there is a difference of facter 2 compared with the limit
§7. Extension of the Theorem A s a mechanical problem, further extension to relativity and quantum mechanics, for example to the second quantization, is desirable. It seems also reasonable to think of certain analogous theorems in thermodynamical pairs such as n
a f l
d A' in p.N (/.i = chemical potential), besids the
pair p and V in pV. Kirkwood discussed nN in this context to some extent, but without any imputant success. There will be futher possibibty of application of the V . R . theorem, or the like, i n treating the many-particle systems.
Addenda* § 1 . Q u a n t u m Mechanics For a system with the Hamiltonian
lm we have i = \(xH in
- Hx) = - , rn
* - j S j W r - l « ~ g . * Added in 1992.
28
so that
ady state, we get the virial thei
which can be easily extended to systems with many degrees of freedom.
52. Pressure Equation It seems worthwhile to show that pressure of a thermodynamic system can be deduced from the force between the wall of the container and the molecule of the system. For this purpose we consider a system in a cylinder with a piston, and calculated the force on the piston. Let x — L be the position of the piston, xj the position of the j-the molecule, w(L — Xj) be the potential of the force between the piston and the j - t h molecule (j = 1,2, • • -, N), and U the remaining potential energy including interaction between the molecules. If the cross-section of the cylinder is A, the pressure on the piston can be written as 1 / • • • / ^ • • • ^ ( - S / •••/ d
Xl
J
g
-
• • • dz
N
?
^
1
) e x { - ( t 7 + E i ML ~ * , ) ) / * T - } P
e x { - (ll + E j w(L - Xj P
where V denotes the volume of the system, we have
where S3 is the configurational partition function
}
))/kT
29
We can assume the potential w(L — Xj) as steep as we wish, so that IT i f° ^ - * ' > = (oo £
x
1
i < x)>L.
Then we have n(iF,Fj= j
-j
dx --dz e p(-U/kT) 1
N
X
;
where the integration is performed over the volume V of the system.
§3. Statistical Thermodynamics When the contribution from the momenta pj, • •• ,pj is taken into account, the partition function takes the form Z(T,V)=
J ... J =
d ---dp dx ---dx exp(-H/kT) Pl
f
1
i
J(T)Q(T,V),
where H is the Hamiltonian H = K + U with the kinetic energy K and J(T) is the integral
J { T )
=
h j'" j
d
K
kT
^••• Pf^{- i )-
The internal energy E of the system is given as /•••/dpi E
•••dpfdxi
•••dx Hexp^-ff/kTj }
=
/•'••••/ dpi • • - dpfdxi • - -dxj d 6(1/HT)
log Z,
and the pressure as P =
kT—\o Z. %
expy—H/kTJ
30
The heat dQ given to the system is calculated by using the energy conservation law which asserts that dQ = d.E + PdV. However since the above relations lead to kdlogZ
= -Ed(±)
+
£\ = -(2R,l; n) - 2 f
n
HO sin Bde
(10)
The actual calculation of (10) is very complicated even in the case of a polymer whose degree of polymerization is 2—the dimer. In this case / of equation (4) is .expressed by the following equations defined in respective domains: /. f
= 2 ( =
1
_
- L + 5 5X 2 1
^ ) +
(11)
:
A i_iS£ -82 40' x
2- < x < 4 " V
(12)
34
280
A. ISIHARA AND M . TODA
2 < i < 2 Av = 2 / m / ( l + \x
- g}j
V l
0 < x < 2
(13) (14)
x = l/R
(15)
The curve n = 2 i n Figure 1 corresponds to the above equations.
20
/
2
3
4
f
6 X
Fig. 1. The coefficient/in the expression of second virial coefficient us. the axial ratio i = l/R or L/R of solute molecules, n represents the degree of polymerization, n = => corresponds to the case in — 1)1 — £ is constant.
While equation (11) is simple, equation (13) is somewhat complicated. If x is very small, (14) can be expressed as a power series of x:
It is interesting to calculate the limit when n becomes infinite at a fixed value of L = (rt — 1)/. Then the polymer will take a shape such as shown in Figure 2. Since this is an ovoid shape we can utilize the general formula that has already been given by one of ua: 1
35
THEORY OF HIGH POLYMER SOLUTIONS
281
(17)
4 \v
where v and s are the volume and the surface of the molecule respectively, and p is the mean radius, or the mean of the mean radius, of curvature; and these quantities are given by the following integrals: v = fHRxRi da s « (1/2) fH(R, + R )d P - (l/4w) fHdv 2
ul
(18)
Here H is the length of perpendicular from an origin of the ovoid to the contact plane having the direction (0, ) C 2 6 )
|R-r.|'
where p is the hydrodynamic pressure, « is the friction constant, 17 is the viscosity of the solvent, V , is the relative velocity at the sth segment situated at r. Consider the case in which the distribution v(r). of segments about the center of gravity is given. Then equation (25) becomes:
J
(PR _ r ]-V(r)) grad
'(*)
R
s
Accordingly, by operating V on both sides of this equation and using equation (26) we obtain: V V'(R) = - ^ | - 8 T V ( R ) , ( R ) + 2 g r a d j S
(
[
R
~ ^ ' J ^ ,fr) # « |
or in case of a spherical particle in a laminar flow: 1
KV
V "V(R) = - V(R) + - grad p Vn
(27)
Vf
This is just the equation used by Debye and Bueche. Conversely, starting from equation (27) we can arrive at (25). remember the condition: div V = 0
If we (28)
Equation (27) becomes: l
div (XV) + (lA»)v- p - 0
(29)
X= W W
(30)
where:
Integrating (29), we obtain: m f
div XV
On the other hand it holds the relation:
„„ C (V, grad X) .
39
THEORY OF HIGH POLYMER SOLUTIONS
y^!^dr
y"diY(XV/|R-r|)A.
+ j(xV,grad-~^)
285
dr (32)
and because of the integral of the left-hand side of this equation vanishes we can rewrite (31) as:
which is equivalent to (26). Also, because p given by equation (33) is: K
.
f([R J
- r]V(r)) | R - r |
•. W
and the integral of (27) is:
we can easily re-form equation (34) into (25)—the form used by Kirkwood and Riseman. Thus, the relation between the fundamental equations is confirmed. U n fortunately i t is very difficult to solve these equations rigorously. Kirkwood and Riseman replaced the matrix M appearing in equation (25) by a mean value, whereas Debye-Bueche made an almost equivalent approximation in the probability density—they took i-constant inside certain sphere. In the simplest case of dumbbell model, we can avoid replacing M in equation (25) b y its mean value, but in turn the motion of the dumbbell in laminar flow must be approximated in some way. If the dumbbell rotates with a constant angular velocity of q/2 (q is the velocity gradient), we arrive at a different k value from that of S i m b a i n the expression of specific viscosity. Again, i f the distribution v(r) is replaced by an expression of the Gaussian type, i t must also be recognized that this is not applicable at a large distance from the center of the macromolecule. 13
14
Apart from these situations we must pay attention to the experiment and the theory of Fox and F l o r y , according to which we must first consider the volume effect of polymer segments in the modified Staudinger equation (24). 16
In conclusion, we wish to express our thanks to Professors P. J. Flory and B. H . Zimm for their kind suggestions in preparing this paper.
References 1. J. G. Kirkwood and J. Riseman, J. Chan. Phys., 18, 512 (1950). 2. W. G. McMillan and J. E . Mayer, J. Chem. Phys., 13, 276 (19*5). 3. B. H . Zimm, J. Chem. Phys., 14, 164 (1946). 4. A. Isifaara, J. Chem. Phys., 18,1446 (1950). Because this paper is too condensed, the reader should refer to the more elaborate articles: A. Isihara and T. Hayashida, J. Phyt. Soc. Japan, 6, 40, 46 (1951).
40
286
A. ISIHARA A N D M . TODA
5. P. J. Ffory and W. R. Krigbaura, J. Chem. PAy»_ 18, 1086 (1950). 6. A. Isihara, J. Chan. Phyt., in press. 7. P. Outer, C. J. Carr, and B. H. Zimm, J. Chem. Phyt.. 18, 830 (1950). 8. P. Debye and A. M . Bueche, J. Chem. Phyt., 16, 573 (1948). 9. J . G. Kirkwood and J. Riseman, J. Chem. Phyt., 16, 565 (1948). 10. R. Houwink, J. prakl. Chem., 157, 15 (1940). P. J. Flory, J. Am. Chem. Sue., 65. 1901 (1943). W. C. Carter. R. L. Scott, and M . Magat, ibid., 68, 1480 (1946). 11. W. Haller, KolloidZ., 56,256 (1931); 61,26 (1932). V/. Kuhn. Z. phytik. Chem.. A161, 1 (1932). " M . L. Huggins, J. Phyt. Chem., 42, 911 (1938); 43, 439 (1939). P. Debye, J. Chem. Phyt.. 14,636 (1946). H. A. Kramers, J. Chem. Phyt., 14,415 (1946). J. M . Burgers, Second Report on Viscosity and Plasticity, Elsevier, Houston-Amsterdam, 1938, p. 113. A. Peterlin, Z. Phyt., I l l , 232 (1938). 12. See J. M . Burgers in the above reference. 13. R. Simha, / . Research Noll Bar. Standard*, 48i +09 (1950). 14. A. Isihara, J. Phyt. Soc. Japan, 5, 201 (1950). 15. T. G. Fox and P. J . Flory, J. Phyt. Colloid Chem., 53, 197 (1949).
Synopsis We have noted some correspondences in the theories of high polymer solutions. This article is divided in two independent sections: (/) Osmotic pressure. The second virial coefficients for polymers of the pearl-necklace type are given and discussed in two limiting cases—the dumbbell and the infinitely polymerized molecule having a definite length. The variation of the coefficients against the polymerization is predictable from these limiting cases (Fig. 1). Attention must be paid to the manner of taking the abscissa). The case of infinite polymerization can be considered as corresponding to polymers of a compact single-body type such as globular proteins. Thus, an interesting correspondence in the second virial coefficients can be judged for the pearl-necklace and the compact single-body type of polymers. The general feature of the coefiicient is also discussed. {II) Intrinsic viscosity. The relation between the two methods (KirkwoodRiseman and Debye-Bueche) for intrinsic viscosity of high polymer solutions is discussed. Because therespectivefundamental equations are equivalent to each other, we may choose whichever of these two methods we like.
Resum6 Nous remarquons qu'il y a deux relations de correspondence entre les theories des solutions de hauts polymeres. Ce memoire est divisf: en deux parties independantes comme suit: (/) Pression osmctiqtte. Les deuxieme coefficients du viriel pour le polym6re d'un type du collier de perles sont donnas et discuUs dans deux cas exhemes, e'eata-dire, celui d'un type de la cloche suspendue et celui d'une molecule infiniment polymerisee ay ant une longueur definie. On pent prevoir la variation de ces quantites quand la polym6rization s'avance, en deduisant de ces deux cas (Fig. 1. Remarquez la facon de choisir I'abscisse). On peut aussi cousiderer le cas de polymerisation infinie comme correspond ant a celui d'un type d'un seul corps compact tel qu'une molecule de proteine globulaire. Alors, on peut deduire une correspondance interessaute d'un type du collier de perles et pour celui d'un seul corps compact. Aussi discutons-nous la mode generate de la variation de ces coefficients. {11} ViscotiU inlrintfque. Nous discutons la relation entre la methode de Kirk wood-Riseman et celle de Debye-Bueche, toutes les deux traitant la viscosity intrinseque des solutions des hauts polymeres. Et nous trouvons qu'on peut choisir une de ces deux methodea, comme on le desire, pare* que les Equations fondnmentales sont equivaientes entre eux.
Z u sa mmen fassung Wir haben einige Gleichwertigkeiten in der Theorien iiber Hoch-Polyraerere Zulosungen. Diese Abhandlung ist in zwei unabbaogige Abechnitten geteilt: {I) Oimotischer
THEORY OF HIGH POLYMER SOLUTIONS
287
Druek. Die zweite Virialkoeffizienten for rierletuialsschmuckformige Polymeren sind in zwei Grenzfallen gegebeu und diskuriert—eineraeits der Haute], und anderseita der unbegrenzt polymerizierta Molekur von bestimmteu Langeu. Die Verapderung der Koefuzienten gegeniiber der Polymerization iflt aua diesen Grenzfalien klar abgeseheu (Fig. 1. Eine Beachtung soil zu den Methoden der Whal von der Abazisse gerichtel werden). Der Fall von luibegreiizten Polymerization kann demjenige von kompakten einfachen Polymeren angesehen werden. Auf dieaer Weise kann ein interessantes korreapondunz zwischen der zweiten Virialkoeffizienten von PerlhalsacbmuckgestaJte Polymeren und derjenigen von kompakten einfachen Molekulen festgestetlt. Die allgemeine Gestaltung der zweiten oamotischen Eoeffizienten iat auch diakuriert. (//) Inlrirwticht VUkoiilai. Beziehungen zwischen der zwei Methoden, die eine von Kirk wood-Rise man und endere von Debye-Bueche iiber die intrinaische Viskositat der Hoch Polymere Zulosungen gegebene, aind diakuriert. Es ist beweist daaa die fundamentale Gleicbungeo der beiden Arbeiten gleichwertig sind, so es ist mogticb ein beltebiges von beiden zu uehmen.
Received February 15, 1951
42
Sho t Notes
£30
(Vol. 7
r
J . PHYS. SOC. JAPAN 7 (1952) 230
On the Relation between Fermions and Bosons By Morikazu
1-0
TODA
Institute of Physics, Tokyo Bunrika University (Received February 1, 1952)
r.-d
Indeed these expansions can be readily verified with* the partition function of N Boae-oscillators *N = 1/(1 -£*>--• (1 -1*); thus tlie partition function for FermE-oeeillators approaches, with increasing N, to that of Bose-osciFlators, i.e. to U [l — x y. This is an extension of a theorem of the 0-functions: n
The method of Bound developed by Einstein'), ]•:!,,:h. Tomonaga) and others aims to approximate B system by an aggregate of harmonic oscillators. If we restrict ourselves to the problem of a FermionBystem aod ita corresponding Boson-system, the fundamentals of the relation between them will be as follows: Consider a system of non-interacting particles in a harmonica field. In terras of the Hermite polynomials the normalized wave function for a particle may be written in suitable units as 3
JSQ-q-jiTtt-T-q*}*
1
'—V
1
Now, turning to thefieldtheory, wo construct thenuantiHed wave functions which are where OA and b , the annihilation operators, together with tlieir conjugates a * and b„+, the creation operators, fulfil the commutation relations n
n
=(2"?i IVlTj-'Aj-'^ffXr) its eigenvalue being n+—.
We construct antisym-
metric wave functions, or determinants
and symmetric wave functions, or permanent
The correspondence between the Fcrrniou and Bofionnelda may be sununerized as follows: The waveeqTjations
witli the commutation relations
Can be unified, ¥ and 0 are counterparts of each other. in which we place tlie X's and as 0 >! and N>1 . For a system of free particles in a two dimensional box of length L, we have b = hV8mL>. It is trivial to justify the classical limit where 2¥*S|J. Osborne ) pointed out that the conventional method failed when the temperature was sufficiently low as the summation could be no longer replaced by the integration and that the lowest level must be separated from the integral. In such a way he obtained an apparent Bose condensation, or a beginning of the abrupt accumulation in the ground state. In our treatment setting A~=2b, we obtain for the condensation temperature, 3
In the notation of Osborne, 4if3iV/it= 7 V / T , so the left hand side becomes a'sexp(—Tm'/T). Thus the Osborne's conditions for an abrupt accumulation a' = 5fi0/2 is compared with the above equation « ' = l - e - " * « 2 f r / 3 if bfi-b{3, which means that the socalled Bose condensation cannot take place when the width d of the box is finite. An abrupt transition of phases occurs only in the limit d--> oo. We calculate the value of a and then the specific heat as a function of the temperature. To do so we use the following equations.
+ Un ~ -4bSY (e.._ ~ {( {3 ' ~)P.. aNn N n ~_ 2_ " _ i xdx } NfJ{3--Nk- n cn+ ezn _ 1 NfJ{3 +N eZn-l 4b{3NJ. e'-1 '
Table 1. TIT.
TIT.
d=l4.2A
(J' aE -N&ji
0. 35 3.16
0.87 2.04
1.15 0.68
1.73 -0. 52
3.46 -1.215
8.615 -1.85
17.3 -6.00
0.61
0.96
1.08
1.31
1.54
1.68
1.60
1.16 -0.28
1.73 -0.48
3.46 -1.25
1. 36
1.91
1.33
0 .87 0.026 1.11
_ _--L-_--"---'-_~=====__ where
xn=4b{3Nn/" . The ratio A of 4b{3N/" to c{3 determines the width d of the box, and the value of c{3
_
_
-._
determines the temperature. We take A=l and B, which correspond to d=7.11A and 14.2A, respectively. The results are shown in Table I and Fig. 1.
46 17
Notes on Fermi and Bose Statistics
1954)
Fermi system can be regarded as the excitation of Bosons at the Fermi surface. § 4. Sound 1) In the first instance, let us consider free Fermions in a one-dimensional box of length L. The energy levels are given by £ =h'nVSmL', n=l. 2, 3, ••- . If there are JV particles, the levels, are occupied up to the Fermi surface jj = A/; the zero point energy of the system is %
m
Fig. 1. The specific heat of free Bosons in a narrow box. Its width d is taken 7.11 A (curve I) and 14.2A (curve II). The dotted curve corresponds to d=~. To is the condensation temperature in the case of d= «.
E.-
8mL*
Since the level density at the Fermi surface is given by
, . t/r,-i
)
1
=_X rJ-* 2 t» rtt dr,jj (2) We impose on 0 the boundary condition ^=0 xp(x, x'j/TrpCx, xT), (7) at the wall. Rewriting the variables as and the hydrostatic pressure is given by i - i A l - f & O , z-tzKl + 81), z-»«/(l + « ) , P=l{P„+P +P„) • (8) (3) We must note that the pressure obtained and neglecting the higher powers of SX we above does not express the stress in the get the wave equation system, but it does express the force exerted on each side of the container. P
T
(0<E, y, a • (4) with
To get the stress in the system we shall calculate the equation of motion. The current density at r is, with normalized p, M=% Tr 2 f j j - S(x-r) + S( 2i j.i\dxi
x p(x. x'}. (9) The rate of change of this quantity is easily (0<st, y, z(l+Bi)z.
(17)
This corresponds to elongation in the a-direc(14) tion and at the same time contraction in the s-direction in the same amount SX. AccordThis is the coarse-grained average of a and re- ingly the liquid film is expanded in the xpresents macroscopic value of the stress tensor direction, but the volume of the system and at x. are areas of the vapor-wall and the liquidwall contact are not changed if we neglect § 3 . Surface Tension the terms higher than the second power with The free energy of the system of liquid respect to the deformation SX and the ratio and its vapor in thermal equilibrium can be d/l. written in the form We know physically and as a general rule F^F^Fv+F^+F^+F,. , (15) that any macroscopic restriction imposed on where Fi and F„ denote the free energies of a quantum mechanical system does not violate the liquid and vapor in bulk, and F™, Fiu, the thermodynamic nature of the system. and F „ are the free energies of vapor-wall, Therefore we select the set of eigen-states liquidwall and liquid-vapor interfaces respec- which correspond to the system of liquid and its saturaged vapor mentioned above. Let
(29') normal) we can proceed analogously, and obtain The constant t can be determined using the . n 2m !32') expression for surface tension we have just ~ 4 k obtained. In fact from Eq. (29) the partition We may combine these results in the form function for one particle is seen to be
•HH^ 1
Z=\^-»i"g{E)dE (33)
{-l£flihil where the summation is carried out over all paths from 0 to R along the lines of steps, each connecting adjacent nearest neighbours and passing the same line as many times as we want, 5 is the number of steps contained in a path, and 3£ is a certain normalization constant. Counting the number of paths by a combinatory method we may easily write down the explicit form of u(R) as follows: *»=** 1/6. Farther, as
c(lil + Ijl + Ikl ;P) c(lil+ /jl+ Ikl-1 ;P)
lil+IiI+lkl+2P 6 [(lil+lj l+lkl+P+2)/3] < ,
we can see
I V(lil+ljl+lkl)1< 16gV(lil +IiI+ lkl -1) 1 . Consequently, for Igl ^
—
ap-^-"
(3.9)
and the divergent character of the series 2 (1/p) we can conclude that «{0, 0)/3t increases infinitly as g approaches —1/4 from 0. This fact means that for lighter impurity localized vibration is always possible in 2-dimensional case. By the aid of the expansion formula for the modified Bessel function Jj{xj, u(j, k) in Eq. (3.7) is easily seen to be (3.10) with
u(; fc) = ( - l ) ' M j"drexp{-r(^-4r)>/i(2TrtM2rr), +
pl
A
m±M
v
/ w + w + 2
!
(ljl+lmH-l),(lj|+l),(!*l+l): 16iT ),
1/1+1*1+2
\i\+\k\+i
IJI+IAI+1 (3.11)
66
1962)
431
Localised Vibration and Random Walk
where «F ( , , , ; , , ; ) is the Pochhammer's generalized hypergeometric function. Eq. (3.11) agrees with Eq. (3.7). Then .F,( 16g*) converges inside of the unit circle in 16g* plane except at the origin; |16#*| Si Kiven by Eq. (4.17), ' i f. Therefore if we take irW-gJ (4.25) WOW**.. ' which coincides with Eq. (4.18). then, Eq. (4.16) becomes identical with Eq. (2.10). References The solution of Eq. (4.12) with the initial 1) E. W. Montroll and R. B. Potts: Phys. Rev. condition (4.17) is easily seen to be" 100 (1955) 525. J
l
+1
1
y
!
E
4
V
16
v e s
f
o
r
4
( 4 i l 7 )
6M=AeX( )
Pit)
X(t)
(18)
+p(_0)X(t)
£
SPJ> J
2
2^X(T)^-X (r) -f^XC*)^] X(t)
jl-X(t)-^X(
f f
)rf
f f
1
)
-^X(rf)rf)
X(t) | l - X ( r ) 2
ft
2
l-X (r)-^X (i) (19)
Equations (18) and (19) are identical with Rubin's results. These equations are formally valid also in quantum mechanical case. The only difference in this case is that an is to be replaced by tfn^-^-ftfiaCOth
and accordingly Xit)
J'
(20)
by
• and
\ 2kT
Sn
ki
(21)
the initial value X ( 0 ) is to be changed accordingly. In
the following, however, we shall confine ourselves to the classical
case. It has been assumed that the initial displacements and momenta are normally distributed. T h i s leads to the consequence that the displacement and momentum x(t), pit) of particle 0 constitute a stationary Gaussian process. Next problem is to ask whether this process is Markoffian. The
73
Statistical
Dynamics
of Syste?ns of Interacting
Oscillators
condition that the stationary Gaussian process {xit), pit)) that the correlation matrix R(t — ti) satisfies the relation
161
be Markoffian is
}
Rit -t,)=R(t -t,)R(t -t ) a
i
where r j>i2^>ri.
'
(22)
1
The correlation matrix is defined as
a
1
i
Vpit^xit^/wxpty*
where, from S , , or terms of X(t), we have
< K O K * < ) >/*>
2
;
1
<x > = <x (t = v' kT,
Xit)
i
(p*} = = kT.
In
^X't) (23)
U2
-u X(t)
\-^X{a)da
Further, since X(0) = 1, X(0) =0, we have R(At)
(0%x . + A x = [) (s= -N, !
a
l
t
x + «>(! x + S A,x, = 0. Thus x and x
t
- 1 , 1,
AO. (37}
constitute the model S. Frequencies and interaction constants
76
164
M . Toda and Y. Kogure
in this model are
W e have therefore
The motion of the isotope is determined by the function X{t), E q . (16) of the preceding section, in which the direction cosines cpn and the frequency i3 of the «-th normal mode are given by the simultaneous equations n
We first note that
2
2
to - J2
M'
2 ( A M 1)
2
» | - J3
i(AM-l) /here 1
yj
-
1
S
S m
•
-
-
-
—
(42)
2(/Y-H)
T o calculate the sum we note the identity KS . ., 1 sin 2(AM-1 ).r n [sinS 2(AM-1) - s i n " x J = —2 ». s i n- 2,,x — ,'
(4,i)
from which we get ,v
Y\ ,-i
1 — . .
its 2(AM-1)
5 l n
1 (N+1) - — — -— . , • a sinxcosxL ~
S
m
cos^(AM-1)^sin 2 (AM- l ) x
cos2x "j sin2xj
X
(44)
In our case 2
sin x = G*/4r, sin x cosx = (4r) " ' f l y ^ r - f l
2
.
(45)
77
Statistical
Dynamics
of Systems of Inter-acting
Oscillators
165
W e have therefore, for N^>1, 2
{(2r - fi*) - Q V A T - Q * c o t 2 ( N + l ) x ) .
= M r
(46)
Since a > i H 2 r M / M \ we have the relation
2
On the other hand, differentiating E q . (46) with respect to fi , we obtain, for A > 1
i£i ( - f - f i * )
8
M '
r
X c o t 2 ( A + \)
X
+~
2fi
-2fi 2 i/47
V 2
(N+1) (1 + c o t 2 ( i V + l)ar).
2
(48)
W e have therefore, for N^>1
Actually, uij forms quasi-continuum from u> = 0 to o> = 4r and the same holds for fi . T h e number of normal modes with frequency between fi and Q + dQ is f
f
n
^(£)rffi=
, 2(iV+l) " I 0
__^fi_ »/4r"~fi
(o +«>? ^r4r+# 2
^
_
!
(AT—°°).
(57)
a
Thus, we have
which gives the same result as E q . (53) (cf. reference 5 ) ) . Before closing this section, we shall present a slightly modified system for which the calculation can be carried out rigorously. T h i s is a system of a chain of atoms subject to the same equation of motion as above except for the central isotope to which another spring is attached. Equations (33) to (46) hold for this model except that _g=-||-+*>*
(59)
79
Statistical
Dynamics
of Systems of Interacting
Oscillators
where to is a constant standing for the strength of the new spring. (47) is then modified as 4_3 - - ~ a >
(M-s
with s i n x = j2S/4r.
2
167 Equation
(60)
/
= _ . v 4 r - Q l cot 2 ( N + 1 ) x
Further, we find
= T r W 4£- (1 + -ot 2<JV+ JV+1 M ,
1 W
+
J3S(4r-_«)
M ' 1
M
_ J ( 4 r - _ 3 )
+
_.
(61)
[ ( ^ - - l ) _ 2 - ^
a
So that for Q ^ l , provided that a>,;!>ai , X ( r ) can be approximated by 9 X
(
0
=
;
* l
m
i"(j3 -_" j " 2
3
2
_
^
(62)
=—e""siniaf ,
when
is very large and b very small.
§4.
M o t i o n of the c e n t e r of mass
It is of some interest to compare the with the motion of the center of mass of uniform continuum. The argument of §2 T h e calculation is easily modified to examine of a portion of a one-dimensional lattice.
result of the preceding section a portion of a one-dimensional can be applied to this system. the motion of the center of mass
The displacement of the position x in the continuum at time t can be written as u(x, t) = ff-
S ? . ( 0 s i n ^
(0<x
