Boltzmann-type equations for elementary reversible cellular automata

PHYSICA ELSEVIER

PhysicaD 103 (1997) 190-200

Boltzmann-type equations for elementary reversible cellular automata S. Takesue Department of Fundamental Sciences, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-01, Japan

Abstract A Boltzmann-type equation is introduced as an approximation for calculating the time evolution of the probability distribution in elementary reversible cellular automata. A number of properties are discussed from the approximation. Applications to heat conduction problem are exhibited. Simulation results suggest a connection between conserved quantities and diffusive or ballistic behavior.

I. Introduction In reversible cellular automata the phase space volume is preserved due to the reversibility and the discreteness of states. Since statistical mechanics is based on the preservation of phase space volume and energy conservation, if one finds a conserved quantity which can be regarded as energy in a reversible cellular automata, statistical mechanics can be constructed on the system. Deterministic Ising dynamics [1-3] and lattice-gas automata [4] are examples of such systems. Since time evolution of cellular automata is deterministic, the problem of ergodicity readily arises in the same manner as in Hamiltonian dynamics. Under what conditions can statistical mechanics be applied? What is the. origin of irreversibility of macroscopic behavior? These are the difficult problems remaining since the age of Boltzmann. However, use of cellular automata has some advantages over Hamiltonian systems. One is ease of numerical simulations. Since the structure of cellular automata is very simple, simulations of large systems are easier than Hamiltonian systems. The other is exactness. Because states in cellular automata are discrete, they are free from numerical errors. Thus, numerical studies of cellular automata are expected to shed light on the problem of ergodicity. As possibly the simplest deterministic models that allow the construction of statistical mechanics, I introduce a family of one-dimensional reversible cellular automata called elementary reversible cellular automata (ERCA) [7]. So far I studied equilibrium behavior [5] and heat conduction [6,10] exhibited by ERCA and clarified significant differences seen among the models' behavior. In particular, simulations of heat conduction have revealed that some models show diffusive behavior and others show ballistic behavior. However, the origin of the difference has not been clarified. In this paper, I make some modifications on the models so that some properties of the original models are retained and others are discarded and try to understand what property causes the difference. This is done by introducing 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PH S0167-2789(96)00260-6

191

S. T a k e s u e / P h y s i c a D 103 (1997) 1 9 0 - 2 0 0

Boltzmann-type equations, which are applied to heat conduction situations. Simulation results indicate that diffusive behavior is generic if there is a unique additive conserved quantity. In Section 2, the results so far obtained in the investigations on ERCA are briefly reviewed. Details will be reported elsewhere [6]. In Section 3, the Boltzmann-type equations are introduced and some properties of the equations are explained. In Section 4, results of numerical calculations are shown. Section 5 is devoted to summary and discussion.

2. ERCA

ERCA [7] are defined as the family of one-dimensional reversible cellular automata where each site i takes two Boolean values ai', 3-[ ~ {0, 1} at integer time t and they are updated at every time step according to a rule of the form a[+l

= f(ai_

t

t

1, a i , a [ + I ) •

^ o "t,

(1)

(2)

(~[+1 = d '

where .f is a Boolean function of three variables, and • denotes the exclusive OR operation defined as 0 @ 0 = 1 @ 1 = 0 and 0 @ 1 = 1 G 0 = 1. The following alternative notations are also used in place of Eqs. (1) and (2):

x: +' = g(x~_,,x[,x[+,),

(3)

where x: = ( # , &/t), and g(x:_ l , x[, x[+ 1) ---- ( f ( # _ j , ~/t, a/t+1) @ # , at). The reversibility of ERCA is evident from the fact that the time-reversal evolution is explicitly written as [1] a[-I

(4)

~- ~.[,

^1-1 ^t ^t ai = f ( a i _ 1, a i ,

a/+l) ~ a/"

(5)

This couple of equations take just the same form as Eqs. (1) and (2) only with the exchange of variables a ' s and 3-'s. Namely, ERCA are not only reversible in the sense that a unique predecessor is determined for any configuration but also time-reversal invariant. There exist 256 rules in ERCA and each rule is referred to by the number Y~,a.b.,. 24a+2b+"f( a' b, c) followed by an R. The symmetry transformations of reflection (i.e., f ( a , b, c) ++ f ( c , b, a)) and Boolean conjugation (i.e., f ( a , b, c) +-* f ( 1 - a, 1 - b, 1 - c)) classify the 256 rules into 88 equivalence classes, each of which is represented by the rule with the smallest number in it. Without loss of generality I can restrict the extent of considerations to the set of 88 representative rules. For one-dimensional lattice dynamical systems with nearest neighbor interactions of the form (3), additive conserved quantities are defined in the following way. Consider such a dynamical system under the periodic boundary condition of period N. Then, if a quantity of the form N

q~({x/)) = Z

F(x: . . . . . x:+e,),

(6)

i:= I

where ot is a given nonnegative integer, is constant (i.e., q~({x:+l}) = • ({x[ })) for arbitrary configurations {x:}; q~ is called an additive conserved quantity of range t, and F is its density function. A necessary and sufficient condition for an additive quantity to be conserved was obtained in [8]. That is, F is the density function of a conserved quantity if and only if it satisfies the equation G ( X 0 , xI . . . . .

xot+2) - F ( x I . . . . .

X~+l) = d(xo .....

xoe+l) - d ( X l . . . . .

xo~+2)

(7)

S. Takesue/Physica D 103 (1997) 190-200

192

for any x0 . . . . . x~+!, where G(xo, x! . . . . . xa+l) = F(g(xo, xl, x2) . . . . . g(xc~-!, x,~, xc~+l)) and J is defined through F and G as

J(xo, xl

.....

+{(

Xoe+l) = Z

.....

.....

Xi-I

--

G ~ P . . . . . /;, x0, xl

. . . . .

Xi

,

(8)

i=0

where ~P. . . . . /; denotes the n arguments take a same value P which is arbitrarily fixed. Since the left-hand side of Eq. (7) is the time difference of F and the right-hand side is minus the spatial difference of J, it is an equation of continuity and J is interpreted as a flux. If a rule g and integer ~ are given, the above equation becomes a condition for F. Since F is generally represented in an expansion form in ERCA, one can derive possible additive conserved quantities for a given rule by the use of condition (7). The complete list of additive conserved quantities of range ot < 2 for ERCA is found in [8]. In particular, 48 equivalence classes out of the 88 have additive conserved quantities of range ~ = 1. If a rule possesses an additive conserved quantity, the reversibility and the existence of an additive conserved quantity enable one to introduce statistical mechanics to the system. This is done by regarding the additive conserved quantity as an energy or a Hamiltonian. For example, the partition function is written as Z =

Z

e-/~q'(lxi})"

(9)

XO,...,XN-I

By the standard arguments of statistical mechanics, the temperature/3- t is obtained as a function of energy per site. However, there is a point to be taken carefully. It is not uncommon that not only an additive quantity but also values of its density function F(x[ . . . . . x[+c~) are conserved. If it is the case, the set of possible local configurations {(x[ . . . . . x[+a) It c 7/} is restricted by the initial configuration and some configurations are lost from the set. Because statistical mechanics assigns positive probabilities for any configuration, it does not agree with the actual behavior in this case. Therefore, such rules that have this type of localized invariants have to be excluded. As a result, I keep only the seven rules shown in Table 1. The second column of Table 1 shows what quantities are conserved for the rules in the first column. The following four kinds of additive conserved density functions of range u = 1 are: Fa(o" , 6, # , / 2 ) = (o- - / 2 ) 2 ÷ (6 - U) 2,

(10)

Fb(a, 6, U,/2) = 1 + a 6 + #12 -- [1 -- 2(1 -- ~r)(1 --/2)][1 -- 2(1 -- 6)(1 -- #)],

(11)

Fc(cr, 6, U,/2) = ~r/2(1 - 26 - 2/z) - 6/z(1 - 2cr - 2/2),

(12)

Fd(O, 6, # , / 2 ) = (6 -- #)2 _ (o" --/2)2.

(13)

The third and the fourth columns of Table 1 show the numbers of ot = 2 and ot = 3 additive conserved quantities for each rule. To discuss heat conduction, two heat baths with different temperatures are attached to the system. This is carried out as follows. For a given rule I choose an additive conserved quantity of range ot and ignore others if they exist. Let the system be composed of N + 2 sites which are numbered 0 - N + 1. At each time step, values of sites 1-N are updated according to the deterministic rule (3) and afterwards values of sites 0 and N + 1 are chosen with the conditional probabilities

pL(XO[Xl . . . . . Xc~) ----.A/'Lexp[--/3LF(xo . . . . . xa)],

(14)

pR(XN+! IXN+I-~ . . . . . XN) = .A/'R exp[--/3RF(XN+I-~ . . . . . XN+I)],

(15)

193

S. Takesue/Physica D 103 (1997) 190-200 Table I Rules that have ~ = 1 additive conserved quantity and no local invariants Rule

a = 1 conserved density

c~ = 2

ot = 3

Simulation

y = 1

y = 2

26R 77R 90R 91R ')4R 95R 123R

Fa Fc Fa, Fd Fb, F0 Fd Fd Fb, Fd

0 2 2 1 0 1 0

0 0 4 0 0 2 0

D B B B D B B

D D B B D D B

D B B B D B B

In the last three columns D means diffusive behavior, and B means ballistic behavior. where F is the density function o f the selected additive c o n s e r v e d quantity and .ME and A/'R are the normalizations. Consequently, the transition probability from state x = ( xo, x l . . . . .

t

!

x u + 1) to state x ' = ( x o, x I . . . . .

!

X N + I ) is given

by

N ..... P ( x ' I x ) = pL(XolXl ' ' ' " ' " Xot)pR(XN+llXN+l_ct ' ' '

XN) ' H 3 ( x i', g ( x i 1 , X i , X i + l ) ) , i=1

(16)

where 3(a, b) is the K r o n e c k e r ' s delta, i.e., 3(a, b) = 1 w h e n a = b and 3(a, b) = 0 otherwise. Let p~ (x) denote the probability that the state o f the system at time t is x. Then, the time evolution of the probability measure is given by pi+1 (x) = ~

P ( x l x t ) p t (x').

(17)

Xt This equation corresponds to the L i o u v i l l e equation for H a m i l t o n i a n dynamics. In the case where flL = fir -----/3, the Gibbs measure

1-" N+I-c~ Peq(X)~J~eqeXp[--/5 ~

] F(xi

.....

(18)

xi+ot)

is stationary in time, i.e. Z

P ( x l x ' ) P e q ( X ' ) ~- Peq(X).

(19)

X~ This can be proved with use o f the equation o f continuity (7). In fact, use o f Eqs. (14)-(16) rewrites the left-hand side of the above equation as follows• Z

P(x]xt)Peq (x') ~ J~['LA['RJ~'eqe -[3[F(xO'''''x°')+F(xu+l . . . . . . . ~N+I )]

X~ x

~(x~, g ( x s - 1 , xi, x i + j ) ) exp

Eli x'

-fl

i= l

F(x~ . . . . . i =O

xi+ ~)

'

.

(20)

F r o m the equation of continuity I have

N -~'

N-u F(x~

.....

Xi+ot ) =

F(g(xi_ I , X i ,

i=1

Xi+ I )

.....

g(xi+a_

I , Xi+et,

Xi+c~+l))

i=1 -

J (x~),..

' • ,xc~+l) + J ( X N'

a .....

' XN+I)'

(21)

194

S. Takesue / Physica D 103 (1997) 190-200 N-ct

Under the influence of the deltas in Eq. (20) the sum in the right-hand side equals Y]~i=l F(xi . . . . . xi+ot ). Thus P e q is factored out from Eq. (20) and the remaining factor is proved to be unity. Relaxation to the equilibrium state from arbitrary initial conditions is observed numerically for the rules in Table 1. Thus it is confirmed that this boundary condition rightly represents the heat bath. In the case where/3i~ 5~/~r~, energy transport occurs. Assume the convergence pt ~ Pss as t ~ ~ . If the heat flux averaged with respect to the stationary measure pss is f o r m a l l y expanded into a power series of VT (T = / ~ - l ), one obtains Fourier's law and the Green-Kubo formula (Ji)ss = - x V T ,

(22)

1

tc-

Z(J(O)J(t))eq N T 2 t=0

1-

(23)

to O ( V T ) , where J ( t ) = Y]~i J ( x [ . . . . . x[+c~ ) and Ji = J ( x i . . . . . xi+cD. Numerical simulations of heat conduction have been carried out for the seven rules in Table 1 and their additive conserved quantities with el = 1 [6,10]. Two different types of behavior are observed. One is diffusive behavior realized by rules 26R and 94R. In automata with these rules the global temperature gradient is formed, the mean heat flux is proportional to the temperature gradient in large systems, the thermal conductivity computed from temperature gradient and the one from the Green-Kubo formula show a quantitatively excellent agreement, and the stationary state measure is asymptotic to the local equilibrium measure PIe(X) =

J~lle

exp

-

~i F ( x i . . . . .

xi+ct)

(24)

in the limit N --> cx~. The other is ballistic behavior. This behavior means that the global temperature gradient is not formed in large systems, though some gradient can be observed in relatively small systems. Moreover, the mean heat flux is proportional not to the temperature gradient but to temperature difference. The fourth column of Table 1 shows what rules belong to each type.

3. Boltzmann-type equation Now I derive the Boltzmann-type equations. I start from the Liouville equation under the heat-bath boundary condition (17) or equivalently p t + l (X) = pL(XOIXI . . . . .

X c t ) p R ( X N + I IXN+l-ot . . . . .

~(Xi, g(x~_ 1, X i' , X i + l ) )

XN) x~

pt(x').

(25)

i=1

Let y' be a positive integer greater than or equal to or. The (y + 1)-point distribution function on sites n to n + F is defined as the reduced distribution of pt,

pt×,n (Xn, Xn+l, . •., Xn+×) =

p t ( X o ' X I ' •.., XN+I).

Z

(26)

XO,...,Xn I,Xn+y+l ..... XN+I

Performing the summation on Eq. (25), I obtain the following BBGKY-like hierarchy. t+l (Xn, Xn+l . . . . . Py.n

Xn+y) =

Z

n+y I-I ~(xi, t g(xi-l,

, t , t x i , X i + l ) ) P r '+ z , n - I ( x n_ 1' x n . . . . .

t Xn+r+l )

x' ~..... x',+y+l i=, (27)

S. Takesue/Physica D 103 (1997) 190-200

195

for n = 1, 2 . . . . . N - V, and pt+l

~

z,o (xo . . . . . x),)

-=

pt+l zx v.N_~v+I ~ N - V + I ,

x Dr-l- 1

pL(xOlXt . . . . . . u ) r z _ l , 1(Xl . . . . . ....

p ty+_ll , N _ y _ l

XN+I )

(28)

Xy),

tZx N - y + ,

.

. X N. ) P R. ( X.N + I. I X N. - u .+ I ,

(29)

XN)

at the boundaries. Here, I have used the condition V -> u. Like the usual BBGKY hierarchy, the time evolution of (., + 1)-point distribution functions is governed by the (y + 3)-point distribution functions. Thus, this hierarchy is not closed until it reaches p itself. To get a closed-form equation for {Py,n }, ! use the following decoupling approximation ( ~ Boltzmann factorization) pt

.,

y+2.n_l(Xn_l,Xn,

,

,

' ~ . . . . . x,+× ~ t ~ . . . . . x,,+~,) , . . . . X n + v + l ) ~ p t×,n_l(Xn_llX,, I)P~.n(xn X pt , , , y,n+ 1 (Xn+~,+ I ]Xn+ I . . . . ' Xn+y ),

(30)

~ ' h e r e p ty , n - l ( X n -r 1 ] x i , "' . . . . x 'n + y - l )

a n d p tF. n T. l ( X , n~-y~-, - ., IX nt + l ' . . . . Xn4_y ) a r e c o n d i t i o n a l probabilities. Inserting the decoupling formula into the BBGKY hierarchy (27), l arrive at the Boltzmann-type equations,

ynr-I

t+l

P×,n (xn,xn+t . . . . . x,+×) =

,

]

111 8(xi'g(xi-"x;'x;+l))J

Z

P~/,n-1 (x~

I .....

X'n+7-1)

~-'~a-~-;----~z,n-I(a,Xn . . . . .

n+v-l)

x'n I ..... X'n+•+ I Li=n t

t

/

pt tx, .... x r y.n+l ~ n+l' n+y+l )

(31)

× PY'n(Xn . . . . . xn+×) ~ b Ptz,,, + t (xn+l' . . . . . Xn+ , ' × b) fl)r n = 1,2 . . . . .

N - V, and

pt+l

×,o (xo . . . . . x r) -= pL(Xolxj . . . .

•

xu) Z

pt+l ×.l (xl . . . . . x z,

(32)

b)

b

and pt+l rx y.N-y+l t N-y+I,

...,

XN+I) :

pR(XN+llXN+I-u,

.... XN) Z

P y,N t+l

y t"a , x N - v + l

.....

XN)

(33)

a

fi)r the boundaries n =- 0 and n = N + 1 - or. This derivation can be applied to a periodic system and then Eq. (31) is obtained for any n. In that case, if one assumes a translationally invariant stationary state, the equation for the stationary state is reduced to the mean field equation used in the local structure theory by Gutowitz et al. [I 1]. Thus, in this sense, the present approximation scheme can be considered as a generalization of the local structure theory. The Boltzmann-type equations thus derived have some nice properties. The first is that the compatibility of probabilities is inherited. In the Boltzmann-type equations the state of the system is described by a set of (V + I )point distribution functions. Because these distribution functions have overlaps in arguments, a y-point distribution function P×_ 1.n÷l can be made from one of the two different (V + l)-point distribution functions P×,~ and P×.,, +1 by reduction. Of course the results must be identical. This is the compatibility condition• In the Boltzmann-type equations, if

Z

e~,n(xn, x,,+l . . . . . Xn+z)

Xn

=

Z Xn+y+l

P~,'+I (x"+l . . . . . x,+×,Xn+y+l) = P×-I,,+I(x,,+I . . . . . xn+×)

(34)

196

S. Takesue / Physica D 103 (1997) 190-200

holds at time t, the compatibility condition at the next time step "', Xn+y)=

Z p t + l (y,n xn,Xn+l,.

Z

Xn

pt+l , " " , X n + y , Xn+y+l) "y,n+l(Xn+l

(35)

Xn+y+ I

is automatically satisfied. This property is proved by straightforward calculation as follows: Z

~t+l.

r~,~ tx~ . . . . . xn+×)

Xn

-__

~ ~

,

,

t

, t e~,n(Xn .....

,

, , "tn+y)x"'

xt . . . . . X;n+y+I Li=n+l

t

t

Xn ..... X n + y + I

Z

n+y+l j , -.

~-.,b y,n+l ( n+l . . . . . Xn+y, O) t

Z

y , n + l ~, .+1 . . . . . pt ~x I

t

y,n [,Xn . . . . .

6(xi'g(xi-l'xi'x;+l))

Li=n+l

Xn+y)

ZaP~,n(a,x'n+l,---~'

. . . . Xn+y )

t

t

I

P~',n+I(Xn+I . . . . . Xn+y+l)

pt+l y , n + l (Xn+l . . . . , Xx+y+l).

Xn+y+l

In the transformation from the first equality to the second I have used the assumption of the compatibility at time t. The second property is that additive conservation laws of range or' < )/are satisfied in the sense of expectations. Let F be a density function of an additive conserved quantity of range u' and J be the flux associated with ,#. /~ may be or may not be F itself. The average of f'(Xn . . . . . Xn+a') with respect to P~,n is (F(xn .....

X n + u ' ) ) t ----

Z

F(xn .....

Xn+ed)Py.n(Xn,Xn+l .....

(36)

Xn+y).

Xn , . . . , X n + y

Then, utilizing the Boltzmann-type equation (31) and the equation of continuity, it is proved for 1 < n < N - u' that

(F(x, . . . . . x,+~,))t+l = (F(xn . . . . . Xn+a'))t + ( J ( x n - I . . . . . Xn+a'))t - (](xn . . . . . Xn+c~'+l))t,

(37)

where {J(x~_ 1. . . . . x~+~,)),

=

Z Xn

ff(Xn-I . . . . . Xn+u')P~,n_l(Xn-l[Xn . . . . . Xn+y-l)P~,n(Xn . . . . . Xn+y)

(38)

I ..... X n + y

and (J(xn .....

=

xn+ot'+ 1))t

~

](Xn,

""

., xn+,,'+l)P y,n-tt I (Xn+ y

IXn+ l . . . . .

Xn-F Y ) P~,n (Xn . . . .

Xn+ y )"

(39)

Xn,...,Xn--y+l

Thanks to the compatibility of probabilities the following equality holds, pg

~.,_j(xn-lrx. . . . . .

.- - P. ~ , n. - I .( X n. - l ,

x,+×-l)P~..(Xn.

.

.

.. x , + y )

t (x.+y Ix. . . . . . x . + y _ 1). x . + y _ 1) P~,.

(40)

S. T a k e s u e / P h y s i c a

D 103 (1997)

190-200

197

Thus the two J s given by Eqs. (38) and (39) are consistent. At both ends of the system (n = 0 and N + 1 - d ) , the mean energy is given by

(/~(xo .....

x~,)), =

Z

F'(xo . . . . . xo~')pL(xOIXl . . . . . x,~) Z

.ro ...... ~7

(41)

P~.I(Xl . . . . . x z , b ), b

(P(.':N+I-~,', . . . . XN+I)),

=

Z

~'(XN+I-~', . . . . XN+I)PR(XN+IIXN+I-o~ . . . . . X N ) Z d , N _ y ( a .

XN +1 - y ...... ~"N+ I

XN-y+I . . . . . XN).

a

(42) These represent interaction between the system and the heat baths. Thus, if an additive conservation law has range o~ _< y, it survives in the Boltzmann-type equation. On the other hand, if the range a ' > y, the conservation law looses its sense. This property is utilized to control the additive conserved quantities in Section 4. The last property is about equilibrium solutions. As in the case of the Liouville equation (25), if/~L = flU = /4, Boltzmann measure B . . [ n+×-~ Py.,,(x,,, *n+l . . . . . . ~,,+z) : . A / ' B e x p -/4 Z F(xi . . . . . . r i + ~ )

] ,

(43)

i=n

where A/B is the normalization, is stationary. This is also proved by using the equation of continuity (7). Note that the (V + 1)-point distribution reduced from Peq (X) does not necessarily agree with P~,, (x,, . . . . . x,,+×).

4. Numerical calculations

1 have carried out numerical simulations of the Boltzmann-type equations for the seven rules in Table 1. The conserved densities of ot :- 1 are used for F in Eq. (31). To satisfy the compatibility of probabilities, I have chosen as an initial condition the following uniform distribution

P~).,,(x,,, Xn+l . . . . .

xn+y

)=

4 -z-

I

(44)

This clearly satisfies the compatibility. The time evolution of the probability distributions {PC,,,} are simulated according to Eq. (31 ) until they reach the stationary state. Then local temperatures are calculated from mean energy per site ( F ( x i . . . . . xi+~)) via the equilibrium relation between temperature and energy. I have compared the temperature profiles obtained in the simulations of ERCA with the heat-bath boundary and from their Boltzmann equations. For some rules, the agreement of both data is fairly good even for y = 1. Fig. 1 shows the example of rule 26R, w here the original system and the Boltzmann equation yield almost the same temperature profiles. The agreement is much better in larger systems and virtually no differences are observed when N = 400. Instead of n increase or" y also improves the agreement. Rule 94R shows similar behavior. For rules 90R, 91R, and 123R, neither the original automata nor the Boltzmann-type equation forms a global temperature gradient and the qualitative behavior is not changed by varying y or system size. However, in rules 77R and 95R, the two results are quite different. Although the original rules 77R and 95R show ballistic behavior, their Boltzmann equations of y = 1 support the formation of global temperature gradient. This suggests that the diffusive behavior is realized in these systems. For y = 2, however, the diffusive behavior disappears and global temperature gradient is not constructed. Figs. 2 and 3 illustrate the behavior. The simulation results are summarized as shown in the last two columns of Table 1. Table 1 suggests an interesting connection between the behavior of the system and the number of additive conservation laws. In the original system, all conservation laws are of course valid. However, in the Boltzmann

198

S. Takesue / Physica D 103 (1997) 190-200

1.25

i

v

i

i

~0 ;0%% 1.2

26R Simulation o Boltzmann 7=1 •

#o

1.15 1.1 1.05

-.....

1 0.95 0.9 0.85 ,

0.8 0

20

40

60

80

1O0

i Fig. 1. Simulation results for rule 26R and its Boltzmann-type equation with y = 1. In both cases, system size N = 100, the temperatures of the heat baths a r e f l L = 0.8 and/~R = 1.25.

1.3

i

77R Simulation o Boltzmann 7=1 Boltzmann 7=2 ~*

1.25

1.2 1.15

1.1 1.05 1 0.95 0.9 0.85 0.8 0

50

100

150

200 i

250

300

350

400

Fig. 2. Simulation results for rule 77R and its Boltzmann-type equations with Y = 1 and 2. In all cases, system size N = 400, the temperatures of heat baths a r e fiE = 0.8 and f i R = 1.25.

equation only the additive c o n s e r v e d quantities o f range ct _< }, are valid. For example, in rule 77R, only one conservation law exists in the B o l t z m a n n equation with y =

1 but three conservation laws are present in 2/ =

2. Thus it is noticed f r o m Table 1 that diffusive b e h a v i o r e m e r g e s only in the case where the n u m b e r o f effective additive c o n s e r v e d quantities is one. O n the other hand, if there is m o r e than one conserved quantity, the ballistic b e h a v i o r is observed. This m a y be understood as follows. If there are several c o n s e r v e d quantities, one can e x p e c t m o d e couplings a m o n g them. Thus the d y n a m i c s o f the c o n s e r v e d quantities is generally

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S. Takesue/Physica D 103 (1997) 190-200

1.3 95R Simulation o Boltzmann "},=1 •

1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0

I

I

I

I

20

40

60

80

100

i

Fig. 3. Simulation results for rule 95R and its Boltzmann-type equation with y = 1. In both cases, flL = 1.25 and/4R

=

0.8.

modified from the one in which only one quantity is considered. It is feasible that the existence of other conserved quantities prevents the system from realizing the diffusive behavior. This is the case for rule 90R, which is regarded as a noninteracting ideal gas with a fixed magnitude of particle velocities. There, one of the additive conserved quantities of ~ = 1 is the total number of right-going and left-going particles and the other is their difference. Clearly, the conservation of the difference makes the system collisionless and prevents the diffusion of particles. In other rules, things are not so simple and detailed study on effect of the coupling is to be done. However, the origin of ballistic behavior is an easy part and the real problem is why diffusive behavior is generic w hen there is only one additive conserved quantity. In the Green-Kubo formalism, this means that the sum of (.! (0)J (t)) converges. It is generally a very difficult problem to prove. At present I can only say that my simulation results indicate it.

5. Summary and discussion In this paper, I have introduced the Boltzmann-type equations to ERCA, discussed some properties of the approximations, and reported the results of numerical calculations. Because the Boltzmann approximation is rather general, it may be applied to other types of cellular automata. For example, extensions to higher-dimensional systems can be considered. Since the Boltzmann approximation is a generalization of the local structure theory to inhomogeneous cztse, it will be useful when inhomogeneity is important. The present numerical results suggest that the existence of additive conserved quantities other than the selected one prevents the system from exhibiting diffusive behavior. In lattice gas automata, however, it is common that the system has more than one conserved quantity. Usually the system is constructed so as to satisfy the conservation of particle numbers and momentums. This results in the diffusive motion of momentums, namely viscosity. For this method to work, a number of Boolean variables are necessary at each site. For example, the model devised by Frisch, Hasslacher and Pomeau [13] has six variables per site. ERCA have only two variables

S. Takesue / Physica D 103 (1997) 190-200

200

per site. Maybe because of the smallness of the number of variables, only one diffusive mode is permitted in ERCA. Contrary to the additive conserved quantities, the existence of staggered invariants does not seem to affect the behavior of the system. Staggered invariants are defined as conserved quantities of the form [ 12] N-I tP({x[}) : e 27rit/r Z e2rril/ZF(x[' X[+l . . . . . x[+~), l=0

(45)

where r and ~. are nonnegative integers. Equations of continuity are also defined for staggered invariants. Staggered invariants exist in all automata with the rules in Table t. However, they have little influence on the thermodynamic behavior of the system. This can be explained by the difference of symmetries. For example, the staggered invariant density in the automata with rule 26R is Fd defined by Eq. (13), while the additive conserved quantity is Fa by Eq. (10). In automata with rule 94R, the additive invariant is Fd and the staggered invariant is Fa. The two quantities Fa and Fd have the opposite symmetry properties with respect to reflection and time reversal operations. Thus they may not couple with each other in the extent of linear response.

References [1] [2] [3] [4]

G. Vichniac, Simulating physics with cellular automata, Physica D 10 (1984) 96-116. Michael Creutz, Deterministic Ising dynamics, Ann. Phys. 167 (1986) 62-72. H.J. Hermann, Fast algorithm for the simulation of Ising models, J. Statist. Phys. 45 (1986) 145-151. G.D. Doolen, U. Frisch, B. Hasslacher, S. Orszag and S. Wolfram, eds., Lattice gas methods for partial differential equations (Addison-Wesley, New York, 1990). [5] S. Takesue, Reversible cellular automata and statistical mechanics, Phys. Rev. Lett. 59 (1987) 2499-2502. [6] S. Takesue, in preparation. [7] Shinji Takesue, Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. Basic properties, J. Statist. Phys. 56 (1989) 371-402. [8] T. Hattori and S. Takesue, Additive conserved quantities in discrete-time lattice dynamical systems, Physica D 49 (1991) 295-322. [9] S. Takesue, Nonequilibrium statistical mechanics of reversible cellular automata, in: Dynamical Systems and Chaos, Vol. 2 (World Scientific, Singapore, 1995) 118-122. [ 10] S. Takesue, Fourier's law and the Green-Kubo formula in a cellular-automaton model, Phys. Rev. Lett. 64 (1990) 252-255. [ 11] H.A. Gutowitz, J.D. Victor and B.W. Knight, Local structure theory for cellular automata, Physica D 28 (1987) 18-48. [12] S. Takesue, Staggered invariants in cellular automata, Complex Systems 9 (1995) 149-168. [ 13] U. Frisch, B. Hasslacher and Y. Pomeau, Lattice-gas automata for Navier-Stokes equation, Phys. Rev. Lett. 56 (1986) 1505-1508.