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Tr/2 are obtained simply by increasing the radius of the circle. However, a + y < Trl2 cannot be achieved this way, as the equatorial disk would no longer cover the entire domain fl. The difficulty that appears is not an accident of the procedure; it reflects rather a gen eral characteristic of the local behavior of solutions of (3a, b) near comer points. The following result is proved in (6].
We consider a capillary tube with general section n in the absence of gravity. That would not be appropriate for the con figuration of Figure 1, as in that case all fluid would flow ei ther out to infinity if y < Tr/2, or to the bottom of the tube if y > Trl2. I will therefore assume that the tube has been re moved from the bath and closed at the bottom, and that a prescribed (finite) volume of fluid covering the base has been inserted at the bottom. It can be shown [5] that every solu tion surface for (la) bounded by a simple closed curve en circling the side walls projects simply onto the base, and thus admits a representation z u(x, y). We then find from (la) =
div Tu
=
2H
=
const.,
Tu =
Yl
'Vu + 1Vul2
(3a)
b
Figure 2. a) Sessile drop; b) Pendent drop.
22
THE MATHEMATICAL INTELLIGENCER
(3b)
Figure 4. Circular section; surface interface.
Figure 5. Hexagonal section; equatorial circle of lower hemisphere.
Figure 6. Water in wedges formed by acrylic plastic plates; g > 0.
a)
THEOREM 1 . 1 : If a + y < n/2 at any corner point P of open ing angle 2a, then there is no neighborhood of p in n, in which there is a solution of (3a) that assumes the data (3b) at boundary points in a deleted neighborhood of P on '2:. In Theorem 1.1 no growth condition is imposed at P. It can be shown that if a + y 2: n/2, then any solution defined in a neighborhood of P is bounded at P. Further, in partic ular cases (such as the regular polygon discussed above) solutions exist whenever a + y 2: n/2. Thus, there can be a discontinuous change of behavior, as y decreases across the dividing mark a + y = n/2, in which a family of uni formly smooth bounded solutions disappears without dis cernible trace.
a
+ 'Y > 7r/2; b)
a
+ 'Y < 7r/2.
This striking and seemingly strange behavior was put to experimental test by W. Masica in the 132-meter drop tower at the NASA Glenn Laboratory in Cleveland, Ohio. This drop tower provides about five seconds of free fall in vac uum, in effective absence of gravity. Figure 6a,b shows two identical cylindrical containers, having hexagonal sections, after about one second of free fall; the configuration did not noticeably change during the remaining period of fall. The containers were partially filled, with alcohol/water mixtures of different concentrations, leading to data on both sides of critical. In Figure 7a, a + y > 1r/2, and the spherical cap solution is observed. In Figure 7b, a + y < 1r/2. The fluid climbs up in the edges and partly wets the top of the container, yielding a surface interface 9' that folds back over itself and doubly covers a portion of n, while
{h Figure 7. Different fluids in identical hexagonal cylinders during free-fall. a)
a
+ 'Y > 7r/2; b)
a
+ 'Y < 7r/2.
VOLUME 24, NUMBER 3, 2002
23
them at the base. Figure 6 shows the result of closing down the angle about two degrees across the critical opening. On the left, a + y 2: 7T/2; the maximum height is slightly below the predicted upper bound. On the right, a + y < 7T/2. The liquid rises to over ten times that value. The experiment of Coburn establishes the contact angle between water and acrylic plastic to be 80° ± 2°. There is not universal agreement on the physical def inition of contact angle. In view of "hysteresis" phenomena leading to difficulties in its measurement, the concept has been put into some question, and the notions of "advanc ing" and "receding" angles were introduced. Also these quantities are not always easily reproducible experimen tally. The procedure just described gives a very reliable and reproducible measurement for the "advancing" angle, when y is close to 7T/2; but if y is small, the region at the vertex over which the rise height is large also becomes very small, which can lead to experimental error. This difficulty was in large part overcome by the introduction of the "canoni cal proboscis" [8, 9, 10], in which the linear boundary seg ments are replaced by precisely curved arcs, leading to large rise heights over domains whose measure can be made as large as desired. The procedure has the drawback that it can require a zero-gravity space experiment over a large time period. Nevertheless, its accuracy has been suc cessfully demonstrated [ 1 1 ] , and it can yield precise an swers in situations for which conventional methods fail.
'Y
Figure 8. Behavior of interface in corner; a
+
'Y < 7TI2.
leaving neighborhoods of the vertices of n uncovered (see Figure 8). Thus a physical surlace exists under the given con ditions as it must, but it cannot be obtained as solution of (3) over n. The seeming "non-existence" paradox appeared be cause we were looking for the surlace in the wrong place. I emphasize again that the change in behavior is dis continuous in terms of the parameter y. Were the top of the container to be removed when a + y < 7T/2, the fluid would presumably flow out the corners until it disappeared entirely to infmity. For any larger y, the fluid height stays bounded, independent of y. In the presence of a downward-directed gravity field, equation (3a,b) must be replaced by divTu = KU + const.
(4a)
v · Tu = cosy
(4b)
in n,
on an, with K = pg/u. There is again a discontinuous change at the same critical y; although in this case a solu tion continues to exist as y decreases across the critical value. The discontinuous behavior is evidenced in the sense that every solution with a + y < 7T/2 is necessarily un bounded at P, whereas if a + y ::::: 7T/2 then all solutions in a fixed neighborhood of P are bounded, independent of y in that range; see the discussion in [7], Chapter 5. This result was tested experimentally by T. Coburn, who formed an angle with two acrylic plastic plates meeting on a vertical line, and placed a drop of distilled water between
24
THE MATHEMATICAL INTELLIGENCER
Property 2. Uniqueness and Non-uniqueness
Let us consider a fixed volume V of liquid in a vertical cap illary tube closed at the base n, as in Figure 9a. Let I = an be piecewise smooth, that is, I is to consist of a fmite number of smooth curved segments that join with each other in well-defined angles at their end points and do not otherwise intersect. One can prove ([7], Chapter 5): THEOREM 2.1: Let Io be any subset of I, of linear Hausdorff measure zero. Then if K ::::: 0, any solution of (4a) in n, such that (4b) holds at aU smooth points ofi'-2.0, is uniquely de termined by the volume V and the data on I'-2.0• Note that if K > 0, then a solution always exists; see, e.g., [ 12, 13]. If K = 0 then further conditions must be imposed to ensure existence; see Property 5 below. In Theorem 2.1, no growth conditions are imposed; nev ertheless, the data on any boundary subset of Hausdorff measure zero can be neglected in determining the solution. This property distinguishes the behavior of solutions of (3) or of (4) sharply from that of harmonic functions, for which failure to impose the boundary condition at even a single point in the absence of a growth condition leads to non uniqueness. Uniqueness has also been established for the sessile drop of Figure 9b. The known proof [14] proceeds in this case in a very different way. But in view of the uniqueness property in these particular cases, it seemed at first natural to expect that the property would persist during a contin uous convex deformation of the plane into the cylinder, as indicated in Figure 9c.
a
c
�g ""'
""'
""' ""'
""' ""' ""' ""' � ""'
""' ""' ""' " ""' ""' ""' ""' ""'
b
QJ
Figure 9. Support configurations: (a) capillary tube, general section; (b) horizontal plate; (c) convex surface.
Efforts to complete such a program turned out to be fruitless, for good reason. Consider, as a possible interme diate configuration in such a process, a vertical circular cylinder closed at the bottom by a 45° right circular cone (Figure 10). If one fills the cone almost to the joining cir cle, with a fluid whose contact angle with the bounding walls is 45°, a horizontal surface provides a particular so lution of (1) with that contact angle. That is the case in any vertical (or vanishing) gravity field. On the other hand, if a large enough amount of fluid is added, the fluid will cover the cone and the contact curve will lie on the vertical cylin der. In this case, the fluid cannot be horizontal at the bound ing walls in view of the 45° contact angle, and a curved in terface will result, as in the figure. It is known that if g :=:::: 0, there is a symmetric solution interface whose contact line is a horizontal circle, and that the interface lies entirely below that circle. Adding or removing fluid does not change the shape of the interface, as long as the contact line lies above the joining circle with the cone. It is thus clear that
Figure 1 0. Non-uniqueness.
one can remove fluid until the prescribed volume is at tained, and obtain a second solution in the container, as in dicated in the figure. The construction indicated can be extended in a re markable way [ 15, 16] : THEOREM 2.2: There exist rotationally symmetric contain ers admitting entire continua of rotationally symmetric equilibrium interfaces ':!, all with the same mechanical energy and bounding the same fluid volume.
This result holds for any vertical gravitational field g. The case g = 0 is illustrated in Figure 1 1. Some physical con cerns about the construction are indicated in [4]; neverthe less, it is strictly in accord with the Gauss formulation. The question immediately arises, which of the family of interfaces will be observed if the container is actually filled with the prescribed volume V. An answer is suggested by the following further result [ 16, 17, 18]: THEOREM 2.3: All of the interfaces described in Theorem 2.2 are mechanically unstable, in the sense that there ex ist interfaces arbitrarily close to members of the family, bounding the same volume and satisfying the same boundary conditions, but yielding smaller mechanical energy.
These other interfaces are necessarily asymmetric. Be cause it is known [ 19] that a surface of minimizing energy exists, the construction provides an example of "symmetry breaking," in which symmetric conditions lead to asym metric solutions. This prediction was tested computationally by M. Calla han [20], who studied the case g = 0 and found a local min imum (potato chip) and a presumed absolute minimum (spoon); see Figure 12. It was then tested experimentally in a drop tower by M. Weislogel [21], who observed the "spoon" surface within the five-second limit of free fall. In
VOLUME 24, NUMBER 3, 2002
25
- - - - - - ... ... _ - - -
,'
- - - - - - - - - - -
, ',
... _ _ _ _ _ _ _ _ _ '
, ,
,
... _ _ _ _ _ _ _
Figure 1 1 . Continuum of interfaces in exotic container; g = 0. All in
terfaces yield the same sum of surface and interfacial energy, bound the same volume, and meet the container in the same angle y = 80°.
a more extensive experiment on the Mir Space Station, S. Lucid produced both the potato chip and the spoon [22]. Her obsetvation is compared with the computed surfaces in Figure 12. Property 3. Liquid Bridge Instabilities, Zero g; Fixed Parallel Plates
In recent years, a significant literature has appeared on sta bility questions for liquid bridges joining parallel plates with prescribed angles in the absence of gravity, as in Figure 13.
The bulk of this work assumes rigid plates and exaniines the effects of free surface perturbation; see, e.g., [23-29) . I n general terms, it has been shown that stable bridges in this sense are uniquely determined rotationally symmetric surfaces, known as catenoids, nodoids, unduloids, or, as particular cases, cylinders or spheres. There is evidence to suggest that corresponding to the two contact angles 'Yb /'2, and separation distance h of the plates, there is a criti cal volume VcrCy1 , y2 ; h) such that the configuration will be unstable if V < Vcr and stable if V > Vcr· That assertion has not been completely proved. Because stability criteria are invariant under homothety, the above assertion would imply that if the plate separa tion is decreased without changing the volume or contact angles, then an initially stable configuration will remain sta ble. In [29], Finn and Vogel raised the question: suppose that a bridge is initially stable; will every configuration with the same liquid profile, but with plates closer together, also be stable? One would guess a positive answer, because wi:th plates closer together there is less freedom for fluid per turbation. But we note that we will have to change the con tact angles, resulting in changed energy expressions, and the requirement of zero volume change for admissible per turbations has differing consequences for the energy changes resulting from perturbations. In fact, Zhou in [26] showed that the answer can go ei ther way, and even can move back and forth several times during a monotonic change in separation h, so that the sta bility set will be disconnected in terms of the parameter h. Zhou considered bridges whose bounding free surfaces are catenoids, which are the rotationally symmetric minimal
Figure 12. Symmetry breaking in exotic container, g = 0. Below: calculated presumed global minimizer (spoon) and local minimizer (potato chip). Above: experiment on Mir: symmetric insertion of fluid (center); spoon (left); potato chip (right).
26
THE MATHEMAnCAL INTELLIGENCER
each of the planes on its boundary, and whose outer sur face ';! is topologically a disk.
A spherical bridge with tubular topology can exist in a
wedge of opening 2a if and only if y 1 + y2 > 7T + 2a. In contrast to the case of parallel plates, whenever this con dition holds, spherical bridges of arbitrary volume and the same contact angles can be found. McCuan proved [31] that
Figure 13. Liquid bridge joining parallel plates; g
if YI + Y2 s; 7T + 2a, then no embedded tubular bridge ex ists. Wente [32] gave an example of an immersed tubular =
0.
bridge, with 'YI = Y2
= 7T/2.
The unit normal N on the surlace ';! of a drop in a wedge
of opening 2a can be continuous to :£ only if ('Yby2) lies in surlaces. She proved that if the contact angles on both
the closed rectangle m of Figure 14. It is proved in [33] that
plates are equal, and if the plates are moved closer to gether equal distances without changing the profile, then an initial stability will be preserved. However, that need not be so if only one of the plates is moved. Let y1 be the
if (Yl, 'Y2) is interior to m then the interface ';! of every such drop is metrically spherical. It is col\iectured in that refer
Zhou showed that there are critical contact angles y'
surface with :£; in fact, there exist surfaces ';! that exhibit
contact angle with the lower plate, and hold this constant; =
< Y1 < Yo then if the up per plate is sufficiently distant in the range Y1 < Y2 < 14.38°, Yo
=
14.97°, such that if y'
7T y�, the corifiguration will be unstable. On moving that plate downward, it will enter a stability interval; on con tinued downward motion, the configuration will again become unstable, and finally when the plates are close enough, stability will once more ensue.
ence that there exist no drops with unit normal to ';! dis
continuous at :£. In [30] it is shown that the col\iecture can
not be settled by local considerations at the "juncture" of the
such discontinuous behavior locally. The col\iecture asserts
that no such surfaces are drops in the sense indicated above.
-
Property 5. C-singular Solutions
As noted in the discussion of Property 1 above, for capil lary tubes of general piecewise smooth section
0, solutions
of (3a,b) do not always exist. Failure of existence is not oc casioned specifically by the occurrence of sharp comers;
Property 4. Liquid Bridge Instabilities, Zero g;
Tilting of Plates
In the discussion just above, motion of the plates was ex cluded from the class of perturbations introduced in the stability analysis. More recently, the effect of varying the inclination of the plates was examined, with some unex pected results [30].
THEOREM 4.1: Unless the initial configuration is spheri cal, every bridge is unstable with respect to tilting of ei ther plate, in the sense that its shape must change dis continuously on infinitesimal tilting.
existence can fail even for convex analytic domains. The following general existence criterion appears in [7]:
Referring to Figure 15, consider all possible subdomains 0* * 0,0 of 0 that are bounded on k by subarcs k* C k and within 0 by subarcs f* of semicircles of radius IOVCiklcos y), with the properties i) the curvature vector of each f* is directed exterior to 0*, and
It should be noted that a spherical bridge joining paral lel plates is a rare event, occurring only under special cir cumstances. A necessary condition is y1 + Y2 > 7T ; for each such choice of contact angles, there is exactly one volume
1t
that yields a spherical bridge.
A spherical bridge can change continuously on plate tilt ing; however, for general tubular bridges, instability must be expected, in the sense of discontinuous jump to another configuration. With regard to what actually occurs, one has
THEOREM 4.2: If Y1 + Y2 > 7T, a discontinuous jump from a non-spherical bridge to a spherical one is feasible. If y1 + 'Y2 s; 7T, no embedded tubular bridge can result from in finitesimal tilting; further, barring pathological behavior, no drop in the wedge formed by the plates can be formed. In the latter case, presumably the liquid disappears dis continuously to infinity. By a "drop in the wedge" is meant a connected mass of fluid containing a segment of the in tersection line :£ of the planes as well as open subsets of
Figure 14. Domain of data for continuous normal vector to drop in
wedge.
VOLUME 24, NUMBER 3, 2002
27
I.
*
Figure 15. Extremal configuration for the functional . "'
ii) each f* meets �. either at smooth points of � in the angle y measured within fl* or else at re-entrant cor ner points of � at an angle not less than y.
Q:
We then have THEOREM 5. 1: A solution u(x) of (3a,b) exists in fl if and only iffor every such configuration there holds
(fl*; y) = l f* l
-
l�*lcos y +
2H cos y > 0
(5)
with
_ 2H - m cos y. lfll Every such solution is smooth interior to fl, and uniquely determined up to an additive constant. In this result, the circulars arcs f* appear as extremals for the functional , in the sense that they are the bound aries in fl of extremal domains fl* arising from the "sub sidiary variational problem" of minimizing . The following result is proved in [34]: THEOREM 5.2: Whenever a smooth solution of (3a,b) fails to exist, there will always exist a solution U(x,y) over a subdomain flo bounded within fl by circular subarcs r0 of semicircles of radius 112Ho, for some positive H0 :::::; H. The arcs meet � in the angle y or else at re-entrant cor ner points of� in angles not less than y, as in Figure 15. As the arcs fo are approached from within flo, U(x, y) is asymptotic at infinity to the vertical cylinders over those arcs.
u=oo Figure 16. C-singular surface interface.
be shown, and no smooth solution exists. If we consider two such domains with different opening angles, reflect one of them in a vertical axis, expand it homothetically so that the vertical heights of the extremals are the same for both domains, and then superimpose the domains at their tips and discard what is interior to the outer boundary, we ob tain the configuration of Figure 18. In this case two distinct C-singular solutions can appear, for the same y, with re gions of regularity, respectively, to the left of one of the in dicated extremals or to the right of the other one. It has not been determined whether a regular solution exists in this case; however, in the "double bubble" configuration of Figure 19, if the two radii are equal and the opening is small enough, then both regular and C-singular solutions will oc cur, for any prescribed y. Finally it can be shown that in the disk domain of Figure 20, a regular solution exists for every y, but there can be no C-singular solution.
We refer to such surfaces U(x,y) as cylindrically sin gular solutions, or "C-singular solutions". The subarcs are the extremals for the functional, corresponding to H = Ho in (10). Figure 16 illustrates the behavior. Such solu tions have been observed experimentally in low gravity as surfaces going to the top of the container instead of to the vertical bounding walls. THEOREM 5.3: C-singular solutions may be unique or not unique, depending on the geometry. They can co-exist with regular solutions, but can fail to exist in cases for which regular solutions do exist. Figure 17 indicates a case in which a C-singular solution appears for any y < (7T/2) - a In this case uniqueness can .
28
THE MATHEMATICAL INTELLIGENCER
Figure 17. If
a
+ 1' < Trl2, there exists exactly one C-singular solu
tion, up to an additive constant; no regular solution exists.
Figure 18. At least two C-singular solutions exist.
Property 6. Discontinuous Reversal of
I illustrate the possible behavior with a specific exam
Comparison Relations
Consider surface interfaces :J' in a capillary tube as in Fig
ure 1, in a downward gravity field g and without volume constraint. The governing relations become divTu = KU
in n, K > 0;
v . Tu = cos
'Y
on �-
(6)
Here u is the height above the asymptotic surface level
ple. Denote by n 1 a square of side 2, and by n(t) = nt the
domain obtained by smoothing the comers of n 1 by circu lar arcs of radius (1
-
t), 0 :::s t
inscribed disk (Figure 22).
::::;
1. Thus, no becomes the
For y � 7T/4, it can be shown that there exists a solution t of (6) in any of the nt. Denote these solutions by u (x; K).
One can prove:
at infinity in the reservoir. About 25 years ago, M. Miranda raised informally the question whether a tube with section
n0 always raises liquid to a higher level over that section than does a tube with section n l :J :J no (Figure 21). An al most immediate response, indicating a particular configu
ration for which the answer is negative, appears in [35]. A number of conditions for a positive answer were obtained; see [36] and [7], Sec. 5.3. A further particular condition for a negative answer is given in [7], Sec. 5.4. Very recently [37] it was found that negative answers must be expected in many seemingly ordinary situations; further, these negative answers can even occur with height differences that are arbitrarily large. Beyond that, the an swer can change in a discontinuous way from positive to negative, under infmitesimal change of domain. What is perhaps most remarkable is that such discontinuous change in behavior occurs for the circular cylinder, which is the section for which one normally would expect the smoothest and most stable behavior.
Figure 20. In a disk, a regular solution exists for any 'Y; but no C singular solution exists.
r ��/,. I I \
'
'
Figure 19. Double-bubble domain. For a small enough opening, both
a regular and a C-singular solution exist, given any 'Y·
Figure 21 . Does Oo raise fluid higher over its section than does 01
over that same section?
VOLUME 24, NUMBER 3, 2002
29
THEOREM 6.2: For aU K > 0,
0 u (x; K) > u1(x; K)
Q( t)
Figure 22. Configuration for example.
THEOREM 6. 1 : There exists Co > 0 with the property that for each t in 0 < t < 1, there exists C(t) > 0 such that
u 1 (x; K) - ut(x; K) > (C(t)IK) - C0
(7)
(8)
in 00. Thus, no matter how closely one approximates the inscribed disk by making t small, the solution in the square will dominate (by an arbitrarily large amount) the solution in Ot if K is small enough. However, the solution in the disk itself dominates the one in the square, regardless of K. The limiting behavior of u 1 (x; K) - u t(x; K) as K ---" 0 is thus dis continuous at the value t = 0, and in fact with an infinite jump. Paul Concus and Victor Brady tested this unexpected re sult independently by computer calculations. Figure 23 shows u 1 - Ut for 'Y = 7T/3, evaluated at the symmetry point x = (0,0), as function of t for four different values of the (non-dimensional) Bond number B = Ka2 , with a being a representative length. In the present case, a was chosen to be the radius of the inscribed disk, so that B = K. One sees that u 1 - u0 is always negative, as predicted, while for any e > 0, u 1 - U13 becomes arbitrarily large positive with de creasing K. Note that the vertical scale in Figure 23 is log arithmic, so that each unit height change corresponds to a factor of ten. Property 7. An Unusual Consequence of
Boundary Smoothing
The discussion under Property 6 above indicates that the specific cause of failure of existence for solutions of (3a,b)
uniformly over Ot . On the other hand, we have
1� �-r----,-----�--r---r---��--�-�
1cl D ... 0 ... "": 101
B= .oo.J 1 B= .0 1 B= 1 B= 1 00
0 •
q_ :I
8 -;-10°
0
0.1
Figure 23. u1 (0; B) - ut(O; B) as function of t; 'Y
is small.
30
THE MATHEMATICAL INTELLIGENCER
0.2
=
7TI3.
0.3
0.4
0 .5 1
0.6
0.7
Note negative values that minimize when t
0.8
=
0.9
0, and large slopes at end points when B
They conjectured (a) that
U(r) is the unique symmetric so
lution of (9) with a non-removable isolated singularity at the origin, and
(b) that 8
= oo. The latter conjecture was
proved by Bidaut-Veron [41], who then later showed [42] that any singular solution satisfying the specific estimate
I
p
ur(r) l -
1
r2
is uniquely determined. The singular solution
Io(x) = A(x) cf>o(x) with re-
VOLUME 24, NUMBER 3, 2002
69
spect to x and setting x 1: T'(1)4>o + Tcf>b(l) 4>6(1). Taking the scalar product with 1: =
=
3
A ' (l)cf>o +
1tT'(1) o + 1 1 cf>o(1) = A'(1) + 1 14>6(1), so that A'(1)
=
1tT'(1)cf>0. Thus, w = 1tT' (1)cf>o,
(3.10)
whose inteipretation is obvious: 4>o is the asymptotic state vec tor whose components are cf>o,k> k = 1, . . . s; T/i1) = Wij Tij is the gain per move weighted by its probability; and 1 t adds it all up. Hence
l§lijil;iiM
(3. 1 1)
'iA +IB
is the expected gain on making a move from state k, and we can also write (3.10) in the form (3.12)
A game, in the terminology we have been using, is fully specified by the weighted transition matrix T(x), which tells us at the same time the probability Tij of a transition j � i and the gain Wij produced by that move. A random composite of games A and B can then be created by choos ing, prior to each move, which game is to be played; A (and its associated move probability and gain per move), say, with probability a; or B, with probability 1 - a. =
aTA(x) + (1 - a)TB(x).
(4. 1)
What has come to be known as Parrondo's paradox (orig inally, a rough model of the "flashing ratchet" [ 1]), is that domain in which both wA < 0 and WB < 0, but WA,B > 0. Much of the phenomenology is already present in a variant of the simple model we have mentioned as background. Let us see how this goes: In both games, A and B, a move is made from white or black to white or black Game A is now defined by a prob ability p, no longer unity, of moving to black, q = 1 - p to white, with a gain of $3 on a move from white, of -$1 on a move from black Hence (with white : j 1, black :j 2) =
TA =
(! !} ( )
q:i3 qlx TA(x) = ; p;i3 pix
cf>oA =
(!}
THE MATHEMATICAL INTELLIGENCER
(4.4)
It follows, most directly from (3. 10), that Hence, in the bold region of Figure 1, for 3/4 < p ::s: 1, we in deed have WA = WB < 0, together with wtA+tB > 0. (Note however that WA = WB > wtA+tB for p < t.) Game Averaging -Another Example
The game originally quoted in this context is as follows [2]: Each move results in a gain of + 1 or -1 in the player's cap ital. If the current capital is not a multiple of 3, coin I is tossed, with a probability p1 of winning + 1, a probability q1 1 - p 1 of "winning" - 1. If the capital is a multiple of 3, one instead flips coin II with corresponding p 2 and q2. Hence the states can be taken as ( - 1, 0, 1) (mod 3), and the associated transition and gain matrices are =
w=
(
0 1 -1
-1 0 1
)
1 -1 . 0
(5. 1)
and then
(4.2)
For the composite game, we imagine equal probabilities, a i• of choosing one game or the other, and indicate this by iA + iB, and now
70
G D·
=
in game B, the roles of black and white are reversed, so that
=
=
(t n (4.5)
Game Averaging - a Simple Example
TA,B(x)
wtA+tB
=
w = 1tT'(l)cf>o = 3
2 2 PlP2 - q lq2 2 + P1P2 + q lq2 - P 1q 1
(5.3)
Now suppose there are two games, the second specified by parameters pi, qi, pz, q2,. An averaging of the two would then define a move as: (1) choose game No. 1-call it A with probability a, game No. 2, B with probability 1 - a; (2) play the game chosen. Because the gain matrix w is the same for both games, this is completely equivalent to play ing a new game with parameters fil = ap1 + (1 - a) pi, fi2 = ap2 + (1 - a)p2, etc., and so (5.3) applies as well. The "paradox" is most clearly discerned by imagining both games as fair, i.e. , p'fp 2 = qyq2, or equivalently
which we combine to read
)_!.� [E(W�) - (E(WN))2 - N A"(1) - N A'(1)2 - N A'(l)] =
!fi!/ (1)Po - ( t/lbt (1)Po? + #/ (1)po . (6. 3)
We see then that
112
(6.4) In other words,
we have found that the standard devia
tion is given asymptotically in
N by
a(w; N) � N-112[A"(l) + A'(1)2 0
+
A ' (1)] 112,
(6.5)
with a readily computable coefficient. For example, in the "Parrondo" case of
lpldii;ifW
(5.1) , where
(6.6)
(5.4) and similarly for pi, P2, creating the "operating cmve" shown in Figure 2; winning games are above the cmve; losing games, below. For games A and B as marked, all averaged games lie on the dotted line between
A and B, and all are winning
A U T H O R S
games. And by continuity with respect to all parameters, it is
clear that
if A and B were slightly losing, most of the con
necting dotted line would still be in the winning region. How
ever, two slightly winning games, close to D and E, would re sult mainly in a losing game. So much for the paradox! The example most frequently quoted is specialized in
B has only one coin, equivalent to two identical = p2 ( = 1/2 for a fair game, point C); and is mod ified in that A and B are systematically switched, rather
that game coins,
pi
than randomly switched. Qualitatively, this is much the same.
ORA E. PERCUS
Asymptotic Variance
251 Mercer Street
Much of the activity that we have been discussing arose from extensive computer simulations
[3, 4],
one have to go to accomplish this? A standard criterion in volves looking at the variance of the gain per move as a function of the number of moves, N, that have been made:
(6. 1) a2(w; N) proceeds routinely from (3. 7) used previously to compute
the same starting point
w=
limN_.oc E(WNIN).
This time, differentiate
once and twice with respect to
x and set x =
(3. 7) both 1 , again as
suming commutativity of limiting operations. Again using
A(1)
=
1,
o(l) =
