Physica D 45 (1990) 278-284 North-Holland

R E L A X A T I O N P R O P E R T I E S O F E L E M E N T A R Y R E V E R S I B L E C E L L U L A R A U T O M A T A

Shinji T A K E S U E Faculty of Science, Gakushuin University, 1-5-1 Mejiro, Toshima.ku, Tokyo 171, Japan

Received 24 January 1990 Revised manuscript received 28 February 1990

The relaxation properties of a number of elementary reversible cellular automata are studied. Different rules have different types of relaxation behavior. Possible relationships between the thermodynamic behavior and phase space structure are discussed.

1. I n t r o d u c t i o n

Reversible cellular a u t o m a t a provide a very ef- ficient means to examine the basis of various con- cepts and me thods in s tat is t ical mechanics.

Stat is t ical mechanics, founded by Gibbs, can be applied to the dynamica l sys tems where the vol- ume of phase space is preserved and energy is con- stant . Most successful applicat ions, of course, have been made to Hami l ton ian systems. In Hamil to- nian systems, the former condi t ion is satisfied due to Liouvil le 's theorem and energy is Hami l ton ian itself.

Because cellular a u t o m a t a have discrete dynam- ical variables, the cardinal i ty of phase space is finite in a sys tem with a finite number of sites. Therefore, reversible dynamics leads to the preser- vat ion of phase space volume. Moreover, some rules have addi t ive invariants [1,2]. Unlike Hamil- tonians, these addit ive invariants do not com- pletely govern the t ime evolution of the system. Still, one can calculate t h e r m o d y n a m i c quanti t ies with those quant i t ies by a s t anda rd stat is t ical me- chanics a rgument . Tha t is, a par t i t ion funct ion can be calculated from the invariants. Thus , such sys tems can be used as models of stat ist ical me- chanics.

An example of the above type of cellular au-

1 Present address: Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA.

t o m a t a is Creu tz ' s determinist ic Ising dynamics [3]. Creutz added variables represent ing kinetic energy to the usual two-dimensional Ising Hamil- tonian and a t tached to the model a dynamics which conserved the total energy. This model serves as a microcanonical s imulat ion, where tem- pera ture is calculated via the mean value of the kinetic energy at a site. Good agreement is ob- t a i n e d between simulat ions of this model and ex- act solution of the two-dimensional Ising model. A simple version of determinist ic Ising dynamics called Q2R also agrees with theory, at least in the high t empera tu re region [4]. In addit ion, lattice- gas a u t o m a t a also belong to this ca tegory of cellu- lar au tomata . Due to the parallel na ture of cellular au toma ta , these models can be s imulated more ef- ficiently on parallel machines than some types of Monte Carlo simulation.

All of these cellular a u t o m a t a satisfy symme- tries and conservat ion laws character is t ic of the physical systems modeled. Studies of these models always assume, but do not prove, the sat isfaction of the ergodic-theoret ical condit ions required to guarantee t h e r m o d y n a m i c behavior. Hence, it is worthwhile to ask what rules can realize thermo- dynamic behavior. To answer this question, a sys- temat ic s tudy of a family of simple reversible cel- lular a u t o m a t a was under taken. Since cellular au- t o m a t a are par t icular ly simple in s t ructure , such a s tudy is expected to shed new light on the foun-

0167-2789/90/$ 03.50 (~) 1990 - Elsevier Science Publishers B.V. (North-Holland)

S. Takesue / Relazation of reversible CA 279

dations of statistical mechanics. Elementary reversible cellular au toma ta

(ERCA) are probably the simplest family of non- trivial reversible cellular au tomata . Their t ime evolution rules have the following form:

or t+ l t t t ^ t i : f ( o ' i - - l ' O'i ' gri+ 1 ) XOR a i , ( la)

.t+1 t ( lb) i z O'i,

where a~ and ~ denote the state of site i at t ime t, each of which takes values in the set {0, 1}, XOR is the "exclusive OR" operation, n a m e l y # XOR v : U + v - 2 # v , and f : {0, 1} 3 --~ {0, 1} is a Boolean function of three variables. This func- tion f determines each rule which is represented by the number ~ a , , , ~ 24a+2~'+~f(A, #, v) with an "R" appended to it. For example, if f(A, #, v) : A XOR v, the rule is called 90R. The reversibility of the dynamics is evident when the t ime reversal evolution is explicitly writ ten as

O" i = O" i

~t--1 ^ t ^ t ^ t t O'i ---- f(o ' i -1 ' O'i, O'i+ 1 ) XOR a i . (2b)

This equation has the same form as eq. (1). Tha t is, ERCA are not only reversible but also t ime re- versal invariant as is the case for usual mechanical systems.

In a previous paper [2], the ERCA rules which possessed an additive conserved quanti ty of the type

= Z F( ° ' i ' ° i + 1 ' ~ i , ~ i + 1 ) ( 3 )

i

were determined. In summary, about half of the rules have the above type of additive conserved quantities. In many such rules, however, there are conserved quantit ies that are l o c a l i z e d (i.e. they are bound to a part icular site or group of sites). In these rules, statistical mechanics fails since energy propagat ion is inhibited. There remain seven rules which have one or more additive invariants but no local conservation laws. Only these rules qualify as candidate thermodynamic models. These rules are listed in table 1 along with their additive in- variants. As is seen from table 1, there appear to be only four kinds of additive conserved quanti- ties for these rules. They are named energy (a) through (d) as in the table. A rule can conserve

more than one energy and conversely an energy may be invariant under more than one rule. In fact, the rules 90R, 91R, and 123R each have two kinds of energy functions.

Consider a segment in a chain of an ERCA as a subsystem. According to statistical mechanics, if the subsystem is much smaller than the full sys- tem (but still contains a sufficient number of sites), then the rest of the system can be regarded as a heat bath for the subsystem. In this case, a canon- ical distr ibution of subsystem energies should be realized.

In ref. [6], it was shown numerically that two of these rules, 26R and 90R, do indeed yield a canon- ical equilibrium distribution of subsystem energy. The t empera tu re of the system was measured via the distribution of subsystem energy. The mea- sured t empera tu re agreed with a value obtained theoretically using ensemble theory. In the simu- lation, however, the thermodynamic limit should be suitably taken according to the rule: For rule 26R, the number of sites N does not have to be large as compared with the number of i terations T, while in rule 90R, N must be as large as T. This is because any orbit has a period of at most 2N under rule 90R. Other rules in table 1 also realize the canonical equilibrium distribution of subsys- tem energy [7].

In the above experiment, initial conditions were randomly chosen under a given value of energy. Thus, the system was considered as in an equilib- r ium state from the beginning. However, in order for a state to be truly in equilibrium, the canon- ical distr ibution is insufficient and relaxation to the state must also be observed.

In this paper, the relaxation properties of these rules will be discussed in detail. Numerical simu- lations will be used to show how the type of relax- ation behavior depends on the cellular au tomaton rule. Possible causes for these differences in behav- ior will be discussed.

2. R e l a x a t i o n experiment

These experiments are mot ivated by a tex tbook example of relaxation to equilibrium. Consider a chamber divided into halves by a parti t ion. Ini- tially, only one half of the chamber is filled with

280 S. Takesue / Relaxation of reversible CA

Table 1 Rules with addit ive invariants. All the rules tha t have additive conserved quantit ies but no local conservation laws are shown, together with energy function F(c~,~, ~ ,~ ) defined by eq. (3).

Rules Energy function F(~ , f~, 6, ~)

26R 90R 91R, 123R 77R 94R, 95R

(a): (~ - ~})2 + (a - f~)2 (a): (ol-~)2 +(&_]3)2 (d): otf~- &/3 [or (5 - /3 ) 2 - (ot-f~)2] (b): l + c t & + f ~ - [ 1 - 2 ( 1 - c ~ ) ( 1 - ~ ) ] [ 1 - 2 ( 1 - & ) ( 1 - ~ ) ] , (d): ct~-5/3 [or (&-f~)2-((~-f})2] (c): ~ ( 1 - 2d - 2f~) - 6,~(1 - 2a - 2~) (d): a/3-&f~ [or (& - / 3 )2 - ( a - / ~ )2 ]

air, while the other half is empty. Wha t will hap- pen if, at t ime 0, the part i t ion is removed? One anticipates that the air will spread rapidly and eventually homogeneously fill the entire chamber. This change is irreversible: the air effectively never spontaneously shrinks back into half of the cham- ber. This is what we mean by relaxation.

Analogous relaxation was tested in numerical simulations of four rules from table 1, 26R, 90R, 91R, and 123R. An initial condition was randomly chosen under the constraint that energy of the initial configuration is completely concentrated in one half of a cyclic chain, while the other half con- tains no energy. The t ime evolution of the distri- bution of energy was then observed. The experi- ment was repeated for various values of energy.

Fig. 1 i l lustrates the tempora l variation of the energy values in the half initially filled with en- ergy. Here "energy" is the additive invariant (a) in table 1 for rules 26R and 90R, and (b) for rules 91R and 123R. Other invariants (if they exist) were ignored. For rules with the same type of in- variant, the same initial configuration was used. The number of sites was 1000 and the number of i terations was 20 000 for each rule. Each point in the figure represents the energy averaged over 40 t ime steps.

As is seen from fig. lb , rule 90R does not re- lax. The oscillation is exactly periodic and never damped. In fact, rule 90R behaves as a noninter- acting ideal gas which consists of particles with velocities -4-1 [6]. Energy (a) represents the num- ber of particles and energy (d) represents the total momentum. Thus, when applied to a finite system, rule 90R does not have good ergodic-theoretic properties.

As mentioned in section 1, however, rule 90R realizes the canonical distr ibution of subsys tem's

energy in the limit of large systems. This is not the contradiction it appears to be. The initial condi- tions and the ways limits are taken are different in the two cases. In the first case, the initial con- dition was randomly chosen, though with a fixed total energy. In this case, the system can be con- sidered to be in an equilibrium state from the be- ginning. In the second case, however, the system is initially in a nonequilibrium state. Further, in the first case, the system size and simulation t ime were simultaneously increased, while in the sec- ond case the system size was held constant while t ime was increased. The former corresponds to er- godicity of infinite systems, and the lat ter corre- sponds to ergodicity of finite systems. It has been mathemat ica l ly proved tha t noninteracting ideal gas has Bernoulli property (the top of the hierar- chy of ergodic-theoretical properties) [8].

The other three rules do show relaxation, how- ever there exist certain differences among these rules. These will now be discussed.

Fig. l a illustrates overdamped relaxation in rule 26R. The energy value in the half initially filled with energy rapidly decreases until it approaches the equilibrium value. However, the equilibrium value is not easily reached. In fact, it is known that this rule yields normal thermal conductivity [9]. Therefore, the coarse-grained energy distribution at position x at t ime t, ¢ (x , t ) , should follow the diffusion equation

O¢(x,t) 02 = D ~ x 2 ¢ ( x , t ) , (4)

at least near equilibrium. The observed behavior is consistent with this equation.

Rules 91R and 123R exhibit damping oscilla- tions as displayed in figs. lc and ld. These two figures are very similar. For both cases, the pe-

S . Takesue / Relamation of reversible CA 281

RULE = L::'BR SI7 'F = 1800 T O T ¢ ~ L E N E R G Y =

E n e r ~

(a ) o

Time

Energ9

RULE = £ ~ R S I 2 E = I I a ~ 6 TOTAL ENERGY = 4 0 0

• .. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

• . . . .. . -. . ., -. . . . .. . . . . .

( b ) • , . . . . , . , . . . . , • ~ 8 ~ 1 ~ 1 6 ~

Time,

580

Energ9

250

RULE = 91R $ I ; ~ = 1@@8 TOTAL ENERGY = f

1

i :

! :- L !.

.:.F.

] "-

" C "~'Z"

,; t

(c ) e

Time,

Fig . 1. T e m p o r a l v a r i a t i o n of e n e r g y v a l u e s in t h e h a l f of t h e s y s t e m i n i t i a l l y f i l led w i t h e n e r g y in r u l e s (a ) 26R, (b) 9 0 R , (c) 9 1 R , a n d (d ) 123R. E a c h v a l u e is a n a v e r a g e ove r 40 t i m e s t e p s .

282 S. Takesue / Relaxation of reversible CA

RULE = 12"~

E~ro9

S I 2 E = 100B TOTAL ENERGY 502

~5o

(d) 0

Time

Fig. 1. Continued.

r iods of the osc i l la t ions are s l igh t ly l a rge r t h a n 3 N , and the r e l axa t i on t imes are on the o rde r of 105. However , while the osc i l la t ion in rule 123R r em a i ns u n d i s t o r t e d even for large T, t h a t of rule 91R becomes d i s to r t ed .

C o m p u t a t i o n of the d i s t r i b u t i o n of s u b s y s t e m energy c lea r ly reveals the difference be tween rules 91R and 123R. In the p rev ious p a p e r [6], in i t i a l cond i t ions were chosen at r a n d o m so t h a t the sys- t e m can be t hough t of as a lways in an equ i l i b r i um s ta te . Here n o n - e q u i l i b r i u m in i t i a l cond i t ions are used, and it is d e t e r m i n e d whe the r equ i l i b r i um s t a t e s a re reached , and if so, w h e t h e r these s t a t e s follow a canon ica l d i s t r i b u t i o n .

The energy d i s t r i b u t i o n of a s u b s y s t e m , P(E), is a canon ica l d i s t r i b u t i o n if i t can be w r i t t e n as

P(E) ~ D(E)e -#E, (5)

where D(E) is the dens i ty of s t a tes , n a m e l y the p r o p o r t i o n of such conf igura t ions t h a t have energy E in all the poss ib le conf igura t ions of the subsys- t em, and ~ is the inverse t e m p e r a t u r e . T h e dens i ty of s t a t es was o b t a i n e d exac t l y t h r o u g h ca lcu la t ion of energy values for all the poss ib le conf igura t ions of the s u b s y s t e m . T h a t is, when the s u b s y s t e m con ta ins n s i tes , D(E) is the n u m b e r of t he config- u r a t i o n s wi th energy E d iv ided by 4 n, which is the t o t a l n u m b e r of s u b s y s t e m conf igura t ions . T h e en- e rgy d i s t r i b u t i o n P(E) was numer i ca l l y ca l cu l a t ed by t ak ing t ime average over every 106 t ime s t eps in a long run. Then , the l i nea r i t y of ln[P(E)/D(E)] versus E was checked.

0

-5

(a)

in [P(E)/D(E)]

10.

5

I I I I ~ I

5 10 15 20 25 E

In [P(E)/D(E)]

10

5

0

(b) '~ 5 10 15 20 2'5 E

Fig. 2. Subsystem energy distribution function for rules (a) 91R and (b) 123R. The system size is 1000 and the subsys- tem size is 15. The total number of time steps is 10 7. The initial condition is the same for both rules. Triangles: first 108 iterations for each rule. Crosses: tenth 108 iterations for (a) and second 108 iterations for (b).

U n d e r b o t h rules the energy d i s t r i b u t i o n con- verged wi th in t he s imu la t ion t ime . However , the even tua l d i s t r i b u t i o n s in the two cases a re wide ly

S. Takesue / Relaxation of reversible CA 283

Table 2

Numbers of pe r iod ic orb i t s in pe r iod ic chain wi th size N

N

Rule 5 6 7 8 9 10 11 12 13

26R 54 200 240 898 918 3722 3518 15066 12870

90R 156 700 1758 8230 21853 104968 285978 1.4 × 108 2.6 × 108 91R 90 296 518 1960 2984 12284 15266 73422 76026

123R 68 216 354 1138 2342 6368 8948 40854 44188

different from each other. Figs. 2a and 2b display the result of the simulation with 107 t ime steps for each rule. It is found that rule 91R does not lead to the canonical distribution whereas rule 123R does. However, the shape of the eventual distribu- tion function does not depend on the position of a subsystem. Thus, some relaxation occurs even in rule 91R, but this is not relaxation to the canon- ical distribution. In addition, rule 26R shows the behavior similar to rule 123R and rule 90R similar to rule 91R in the same kind of experiments.

It should be remarked that this behavior is not caused by the existence of energy (d) in table 1. In general, a system with two additive invariants has two kinds of temperature . One might guess that the other t empera tu re affects the thermodynamic behavior. However, in the present case, energy (b) and energy (d) have different symmetr ies with re- spect to the left-right inversion. Thus, if one sets initial conditions according to some distribution of energy (a) only, the t empera tu re corresponding to energy (d) should be infinite. This is assured by the fact that rule 123R shows the relaxation despite the same initial condition as in the case of rule 91R.

One possible origin of the curious behavior of rule 91R is the existence of energy with interac- tion among more than two sites. If this type of invariant exists, it may affect the relaxation be- havior. Actually, it has recently been found that rule 91R has an additive invariant writ ten as

: ~ G(oi, ~ri+l, ~i+2, °'i, °'i+1, ~i+2), i

(6)

while rule 123R does not [10 I. Whether or not this quant i ty really causes the behavior shown in fig. 2 is now under investigation.

3. Summary and discussion

In this paper, the relaxation of an initial prob- ability distribution to an equilibrium distribu- tion under four elementary reversible cellular au- t oma ta (ERCA) was investigated. Numerical ex- per iments revealed that each rule exhibits behav- ior distinct from the others. Rule 90R does not relax. Rule 26R relaxes with overdamping. Rules 91R and 123R show similar damped oscillations but the equilibrium subsystem energy distribu- tions for the two rules are quite different.

Can one develop a unified understanding of these differences? Ergodic theoretical propert ies such as mixing do not seem to be applicable to ERCA. Indeed, these rules are not ergodic [2]. On a cyclic chain, every orbit of an ERCA is a periodic orbit. In a relatively small system, the number of periodic orbits can be calculated. Table 2 displays the numbers in the four rules with system size 5 through 13. The number of periodic orbits in- creases approximate ly exponentially with system size for all of these rules. On the other hand, the number of possible values of energy grows only linearly with the system size. This means that a huge number of orbits coexist on an energy sur- face, that is, ergodicity is broken.

One may ask then, why has thermodynamic behavior been observed at all? This question is equivalent to asking what type of conserved quan- tities influence the thermodynamic behavior of the system. If one labels the periodic orbits in a fi- nite chain, the labels become invariants. I call this type of invariant ultimate, because such in- variants classify orbits completely. The ul t imate invariants necessarily exist for systems on a pe- riodic chain. However, it is almost impossible to write down such invariants unless one knows the

284 s. Takesue / Relazzation of reversible CA

phase space s t r u c t u r e comple te ly . Some of these invar ian t s m a y be w r i t t e n as add i t i ve inva r i an t s or as local conse rva t ion laws. However , mos t of these invar ian t s p r o b a b l y canno t be expressed in a sys- t e m a t i c manne r . I t is ques t ionab le even w h e t h e r a r e l a t i onsh ip exis ts be tween the invar ian t s for dif- ferent N ' s . Such u n s y s t e m a t i c quan t i t i e s m a y not cause p e r c e p t i b l y dev i a t i on f rom t h e r m o d y n a m i c behav io r . Q u a l i t a t i v e and q u a n t i t a t i v e t e s t s of th is hypo t he s i s will be made in the future .

F ina l ly , a c o m m e n t on a poss ib le connec t ion be tween revers ib le ce l lu lar a u t o m a t a and cont in- u u m d y n a m i c a l sys tems . T h e exis tence of add i t i ve conserved quan t i t i e s is s imi la r to the p r o b l e m of the a n a l y t i c i t y of in tegra l s in c o n t i n u u m d y n a m - ical sys tems . W i t h o u t the r e s t r i c t i on o f ana ly t i c - i ty, in tegra l s of m o t i o n necessa r i ly exist . Th i s is equiva len t to l abe l ing orb i t s . The i n t eg rab i l i t y of d y n a m i c a l sy s t ems is d e t e r m i n e d by the presence of ana ly t i c in tegra ls . In add i t i on , t ha t rules wi th- out add i t i ve or loca l ly conserved quan t i t i e s show s imi la r bi t p a t t e r n s for a lmos t all in i t i a l cond i t ions sugges ts a s i m i l a r i t y to d y n a m i c a l chaos. Th i s par - al lel be tween revers ib le ce l lu lar a u t o m a t a and con- t i n u u m d y n a m i c a l sys t ems shou ld be i nves t i ga t ed fur ther .

Acknowledgements

I would like to t h a n k Dr. H o w a r d G u t o w i t z for his k ind help wi th i m p r o v e m e n t of the manusc r ip t .

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