journal of ~OLECULAR

L LIQUIDS I t ELSEVIER Journal of Molecular Liquids 71 (1997) 235-243

Fluctuations in diffusive lattice gas automata.

Alberto Su£rez Center for Nonlinear Phenomena and Complex Systems Universit4 Libre de Bruxelles, Campus Plaine , C.P. 231

1050 Brussels, Belgium

Abstrac t

A class of lattice gas automata giving rise to diffusion at the macroscopic level is investigated both analytically and numerically. We focus on the non-equilibrium properties of the automata and, in particular, on the appearance of algebraically decaying long-range fluctuation correlations when the system is constrained to evolve to a non-equilibrium steady state, arbitrarily far from equilibrium. © 1997 Elsevier Science B.V.

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

In order to characterize the behavior of a physical system at the macroscopic level, only a part of the information necessary to specify its microscopic configuration is required. It is thus advantageous to construct model systems with simplified microdynamics, which nonetheless give rise to realistic behavior at the macroscopic level. Lattice gas automata are examples of such model systems. They have been used with success to investigate the evolution of average macroscopic quantities (hydrodynamic flows [1, 2], phase separation [3], chemical reactions [4], etc.) and their fluctuations [5, 6, 7] in fluid-like systems.

In the present work, we propose to study a lattice gas automaton whose evolution at the macroscopic level is given by a diffusion equation, in order to gain insight into the statistical properties of fluctuations in non-equilibrium systems. Section II contains a description of the automaton and a derivation of the macroscopic diffusion equation. In Section III, we carry out an analysis of the density fluctuations. Finally, in section IV we present the results of simulations in a 2-dimensional diffusive automaton, with special emphasis on the appearance of algebraically decaying long-range correlations in automata subject to non-equilibrium constraints.

II M a c r o s c o p i c e q u a t i o n : Di f fus ion .

A lattice gas automaton consists of a set of particles moving on a regular d-dimensional lat t ice/: at discrete time steps, t = n&*, with n an integer. The lattice is composed of V nodes labeled by the d-dimensional position vectors r E £. Associated to each node there

0167-7322/97/$17.00 © 1997 Elsevier Science B.V All rights reserved PII S0167-7322(97)00014-7

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are b channels (labeled by Latin indices i , j , . . . , running from 1 to b). At a given time, t, channels are either empty (the occupation variable ni(r , t ) = 0) or occupied by one particle (ni(r,t) = 1). If channel i at node r is occupied, then there is a particle at the specified node r, with a velocity ei. The set of allowed velocities is such that the condition r + eiAt E £. is fulfilled. We further require that the set {ei)~=l be invariant under space inversion. The requirement that the maximum occupation be of one particle per channel allows for a representation of the automaton configuration in term of bits {ni(r, t); i = 1 . . . . , b; r E £}. The evolution rules are thus simply operations over sets of bits, which can be implemented in an exact manner on a computer.

The time-evolution of the automaton takes place in two stages: propagation and colli- sion. We reserve the notation {n(r, t)) = {hi(r, t))~=~ for the pre-collisional configuration of node r at time t, and the notation {n ' ( r , t )} - * b = {n i (r, t))i=l, for the configuration after collision. In the propagation step, particles are moved according to their velocity

ni(r + c, At, t + At) = n;(r , t). (1)

The (local) collision step is implemented by redistributing the particles occupying a given node r amongst the channels associated to that node, according to a given prescription (which in our case is probabilistic). This step can be represented symbolically by

,q(r, t) = ~ o, ~ . ( r , ~ . - ~ , (2)

where ~{-(r,0}-{o} is a random variable equal to 1 if, starting form configuration {n(r, t)}, the configuration {a) = {a~}~=~ is the outcome of the collision, and 0 otherwise. The number of particles is conserved by this step g ( r , t ) = ]~in~(r,t) = Eini(r, t) . Our choice for the collision rules is that any configuration with the same number of particles as the pre-collisional one is an equally probable outcome of the collision process. Hence, the average configuration after collision, given that we start from configuration {n(r, t)}, is

(n~(r, t))(,,(r,0} ~{~} ai 5 (g(r,t),]~,~= 1 crk) 1 = = -~N(r, t), (3) ~{o} 5 ( N ( r , t ) , ~ = , ak)

where the quantity 5(N, ~k ak) is a Kroeneker delta, equal to 1 if N = Ek ak and equal to 0 if N -Tt ~k crk- We first average Eq. (3) over the initial conditions

1 (N(r, t )) , (4) (n~(r,t)) =

and then apply the propagation step to obtain

(n,(r,t + 1)) = b ( N ( r - cl, t)) . (5)

Finally, summing over the index labeling the channels and subtracting (N(r, t)) on both sides, we get a closed equation for the evolution of the "macroscopic" variable (N(r, t))

~ (exp { - A t c/- Vr) - 1) (N(r , t)) (6) 1

( N ( r , t + A t ) ) - ( N ( r , t ) ) = ~ . =

237

In the continuous limit, for both space and time, Eq. (6) becomes a diffusion equation

O 0-t (N(r,t)) = D V~ (N(r,t)) (7)

1 a 2 with diffusion constant D = ~ X ~7 - The dimensionless parameter X is defined by the

relation t ~ 2 Ei c ic ,= X (~-7) 1 , where a is a characteristic microscopic lengthscale, of the order of a lattice spacing, and 1 is the unit tensor in the d-dimensional coordinate space (it appears thanks to the invariance of {c~}~=~ under space inversion). Let e be the ratio between the characteristic microscopic lengthscale, a, and a typical macroscopic length. Equation (7) is then valid only in a hydrodynamic regime, for long times (of order ~At) and large distances (of order ¼a).

I I I F l u c t u a t i o n c o r r e l a t i o n s .

In order to obtain the equation that describes the fluctuation dynamics, we consider the behavior of products of channel occupation numbers under collision and propagation. As- sume that at a given time t the configuration of the lattice is {hi(r, t); i = 1 , . . . , b; r E L}. The collisions on two different sites of the lattice r ¢ r' are independent, and therefore

" ' t = ~N(r , t )N(r ' , t ) . (n~(r ' t )n j (r ' )}{n(r.t)} (8)

In the case of products that involve the occupation number of channels associated to the same node, we have to distinguish two different cases: For different channels i ~ j,

1 (N(r,t) 1). (9) (n~(r't)n~(r't)}{,,(r,t)) - b(b- 1) N(r ' t )

For i = j , given the Boolean nature of hi(r, t), we have

1 ((n:(r,t))~)~(r.,) ~ = (n;(r, t))~(r,,)~ = ~N(r, t). (10)

We can now apply the propagation step to Eqs. (8-10), average over initial conditions and collect the results into the expression

(n,(r , t + a t ) n y , t + At)) --

[( 1 (N(r , t )N(r ' , t ) } = e x p { - / ' , t c , . V,. - A t c j . v, . ,} 1 - 5ir, r'))

+ 5 ( r , r ' ) { ( l - 5 , j ) ~ ( ( N 2 ( r , t ) } - ( N ( r , t ) ) ) + 5 , Jb (N( r , t ) )} ] . ( l l )

We introduce the definitions

Gii(r,r ' , t) = (Sni(r,t)Snj(r ' , t)} = (n i ( r , t )n j ( r ' , t ) } - (n i ( r , t ) ) (n j ( r ' , t ) } , (12)

C(r , r ' , t ) = ~-~G,j(r,r' ,t) = ( N ( r , t ) N ( r ' , t ) } - ( N ( r , t ) ) ( N ( r ' , t ) } , (13) i j

238

Using Eq. (11) and the fact that

(g ( r , t + At)) ( g ( r ' , t + At))

1 = b Y E exp{--Atci 'Vr--Atcj 'Vr '} (Y(r , t ) ) (N(r',t)), (14) =3

we can write a closed equation for the evolution of the equal-time correlation function

e / 1 s C ( r , r , t + A t ) - C ( r , r , t ) = ~ . . ( e x p { - A t c , . V ~ - A t c j . V ~ , } - l ) C ( r , r , t )

13

[ 1 - b(i, ] "~- /~ j e x p { - - A t e i " ~ T r - - A t c j ' ~Tr, } [ ~ ( ( N 2 ( r , t ) > - b ( N ( r , t ) ) ) ( i ( r , r ') .(15)

The correlation function C(r, r', t) contains two different contributions

C( r , r ' , t ) = CL~(r,r,t)(i(r,r ') + CU~(r,r',t), (16)

where the local-equilibrium contribution is [8]

C:e(r , r , t ) = ~ (N(r,t)> ( b - (N(r, t)>), (17)

and the long-range contribution satisfies the equation

ta , , 1 C ( r , r , t + A t ) - e t a ( r , r , t ) = v ~ ( e x p { - - A t c i ' V r - - A t e j ' V r ' } - l ) C u ~ ( r , r , t )

12

1 [Cta(r - ci At, r - ci At, t) -- Ct~(r, r, t + At)]. (18) + (i(r, r') ~ ,=,

In the derivation of Eq. (18) we have assumed C( r , r , t ) ~ C~( r , r , t ) . The neglected term CLn(r, r, t) is of the order of the inverse of the system size, relative to C~(r , r, t).

In the continuous limit, equation (18) becomes a diffusion equation in 2d-dimensions with a source term

, , ( O) CL~(r'r't)" (19) ~C0 U~(r,r,t) = D (V~ + V]) CU~(r,r,t) + (i(r- r ') DV~ - -~

III.1 F luc tua t ing Hydrodynamics .

Equation (19), which we have derived from the exact microscopic evolution equa- tions, can be alternatively derived in a phenomenological fashion by means of Fluctuating Hydrodynamics[9, 10]. In this theory, one incorporates fluctuations into the macroscopic description by assuming that the quantity (iN(r,t) = N(r , t ) - (g ( r , t ) ) is a random variable obeying a stochastic equation, which is constructed by adding a noise term to the (linearized) macroscopic diffusion equation that governs the evolution of ((IN(r, t))

O( iN(r , t ) --- D (ig(r , t) + .g( r , t ) . (20) V 2 V

239

The random particle flux, g(r, t ) , is assumed to be Gaussian white noise with a vanishing average (g(r, t)) = 0 . The second moment is assumed to take the local equilibrium value

(g ( r , t ) g(r ' , t ' )> = a C=( r , r , t ) 5 ( r - r') 5(t - t'). (21)

With this specification for the noise term, we derive the equation for the pair correlation function for the density fluctuations, C(r, r'; t) = (SN(r,t) 5 g ( r ' , t ) ) ,

0 ~ C ( r , r ; t ) = D (V2r + V~,) C( r , r ' ; t ) + 2 D V , . V , , [ C " ( r , r , t ) 5 ( r - r ' ) ] . (22)

From Eq. (22) and using (16), we derive the evolution equation for the long-range com- ponent of the equal-time correlation function (compare with Eq. (19))

~-~C ( r , r , t ) = D(V~ + V f ) [ C ~ ( r , r , t ) ] + D(V~+ V¢)2[CLE(r,r,t)5(r-r')]

-stC ( r , r , t )5 (r - r ' )= D(V~ + V ~ ) C ~ ( r , r , t ) + 5 ( r - r ') D V ] - N CL~(r,r,t).

I I I . 2 O r i g i n o f t h e l o n g - r a n g e c o r r e l a t i o n s .

Equation (18) corresponds to a 2d-dimensional random walk in the lattice/2 ® £ with a source term in the manifold r = r'. In order to clarify the physical meaning of this process, consider an ensemble of automata initially characterized by a local equilibrium distribution (i.e. C~(r , r ',t0) = 0) with an average occupation of (N(r, to)) particles per node. After the collision step, the average occupation number per channel is (n~(r, to)) = -~ (N(r, to)) . There are correlations only between fluctuations in the same channel

G~j(r,r, to) = Gil~(r,r, to) = 5i.i 1 (N(r, to ) ) (b- (N(r, to))).

If we now apply the propagation step, we obtain for the average channel occupation (hi(r, to + At)) = -~ (U(r -- cl At, to)) , and for the fluctuation correlations

alj(r, r, to + At) = 5ij G~(r - el At, r - cl At, to) 1

(N(r - c, At, to)) (b - (N(r - ci At, to))) = 5,j ~CL~(r - ei At, r - At ci At, to). = 5i~

After the collision step, the average occupation per channel is given by

. 1 b (n i (r, to + At)) = ~ 5 ~ ( N ( r - c k A t , t ° ) ) ' (23)

and the corresponding fluctuation correlations

G~i(r,r, to+ At) = G~(r,r, to+ At) 1 Ls

= (n~(r, t o + A t ) ) ( 1 - ( n ; ( r , to+At) ) ) = -~C (r ,r , t o+At ) .

24O

In the next step, these correlations, which have their origin in the Boolean nature of the oc- cupation number, remain local (equilibrium like) and do not contribute to the appearance of long-range behavior. Correlations between fluctuations in different channels become

. 1 ~ ( C ~ ( r - ci A t , r - ci At, to) - - C ~ ( r , r, to + A t ) ) . Gij(r'r ' t°+At) - b(b-1) ~=1

(24) This expression is obtained by making use of the fact that after the collision step the b(b- 1) pairs of channels {i, j} with i ~ j are equivalent, and that the number of particles is a collisional invariant, which implies Eii Gii = ~ i j G~j. The terms Gi.i with i • j are the ones responsible for the appearance of long-ranged contributions, given that, after the propagation step, correlations between different channels on the same node become correlations between channels on different nodes. The time evolution of these correlations corresponds to a random walk in a lattice of dimension double of the actual physical lattice. The total strength of the source term for the long-range correlations is

1 b ~-~GTi(r,r, to+ At) = -g y-~(CL"(r-c, zXt, r - e , A t , to)-C~"(r,r, to+ At)), (25) i~j "= i=1

which is precisely the expression appearing on the r.h.s, of (18). T h e contribution neg- lected in (18) corresponds to the fraction of "walkers" that return to the origin, which is negligible compared to the number of walkers created by the source term.

At equilibrium, translational invariance implies that the long-range correlations are either constant (in a closed system [7]), or zero. A more interesting situation is en- countered when the automaton is subjected to non-equilibrium constraints. In particular, at a steady state there exist algebraically decaying correlations, with a characteristic length of the order of the system size. In the next section, we report the results of a steady state simulation in a two-dimensional diffusive automaton with fully stochastic collision rules.

IV Simulation results.

We carry out simulations in a 2-dimensional automaton, composed of random walkers moving on a 11 x 11 square lattice (Ly = 11, L, = 10). There are four channels per node, corresponding to the velocities' pointing to the four nearest neighbors. In lattice units (At = 1, a = 1), the diffusion constant is ¼. The simulations are performed under non- equilibrium steady state conditions: Along the Y direction, we assume periodic boundary conditions (i.e. nodes with y = L~ are identified with nodes L~ = 0). In the X direction, the nodes at the boundaries of the system are kept on average at fixed but different densities. In particular, in the present simulations, nodes with z = 0 are kept fully occupied ( N(z = 0, y, t) = 4) and all channels associated with nodes with z = L~ are kept empty at every time step (N(x = L~,y,t) = 0. The quantities measured in the simulation are the reduced density

1 L~-I Pss(X) = 4Lu ~ (N(x'y't))ss' (26)

y=0

241

and the long-range component of the steady state fluctuations, averaged over the Y direc- tion

L y - I L ~ - I

c'-~(~,~ ') = ~ ~ c '~ (x ,y ,~ ' ,y ' ) , (27) It=O y'=O

where

C u ~ S ( x , y , x ' , y ') = ( $ N ( x , y , t ) $ N ( x ' , y ' , t ) ) ~ - ( ~ N ( x , y , t ) $ N ( x ' , y ' , t ) ) u ~ s s , (28)

with 5 N ( x , y, t) = N ( x , y, t) - ( N ( x , y, t))ss , and

' ' t 1 (N(z,y,t))ss(4_ (N(x,y,t))~) (29) ( ~ N ( x , y , t ) $ N ( x , y , ))L~s = 4

In the simulations, the ensemble average is computed by means of a time average, taking advantage of the stationarity of the steady state.

According to the theory developed in the previous sections, the stationary solution of the diffusion equation (7) under the specified non-equilibrium constraints is a linear density profile. The reduced density at steady state is pss(x) = p(O) + ~13x (see Fig. I), with the gradient given by fl = ~ ( N ( z = L , , y, t) - g ( x = O, y, t)) .

0 .8

.~ 0.6

(D t-~ 0 .4

0 .2

i i ° '%0 2.0 4.0 e.o 8.o lo.o

Node(x) Figure 1: Average density profile (full line) and typical density profile (dashed line) in the steady state.

The full line in Fig. I is a plot the average density of the 11 x 11 automaton (averaged over 10 s time-steps), subject to the non-equilibrium constraints N ( x = O, y, t) = 4, N ( z = L~, y, t) = 0; the dashed line is a typical (i.e. not averaged) density profile. At steady state, using (17), the equation for the fluctuation correlations, Eq. (19), becomes

( V ~ + V , 2]rlcu~tr, ,r',D, - ~fll 2 $ ( r _ r ' ) = 0. (30)

Averaging (30) over the Y direction, we get

( 0 2 0 2 ) 1 u~ ' /32L~ 5(x - x') = O, (31) ~-~+~7 c (~,~,t) -

242

whose solution, subject to the specified constraints, is

B 2 L~ cLRSS(x,x ') -- 4 L= {x' (L= - x) 8(z - z') + x (L= - x') 8(x' - x)}, (32)

where 8(z) is the step function. In Fig. II(a) we have plotted the equal-time correlations C=(z, z ') between fluctuations at node x and fluctuations at x' = 5, averaged over the Y direction. In Fig. II(b) the full line is the long-range component C~SS(x, x'), as obtained from the simulation, and the dashed line corresponds to the approximation given by (32).

12.0 . . . . 0.0 . . . . . .

9.0

~ ~ - 0 . 5 6.0

"x "x x" x

3.o o - 1 . o

0.0 ~- : Simulation (b) ............ Theory

-3.0 , , , l -1.5 ' ' ' ' '

0.0 2.0 4.0 6.0 8.0 10.0 0.0 2.0 4.0 6.0 8.0 10.0 Node(x) Node(x)

Figure 2: (a) Equal-time correlation function at steady state, CSS(x,x' = 5); (b) Long- range contribution Ct~Ss(x, x' = 5) .

V Conclusions.

The present simulations show that the terms neglected in (18) are indeed negligible, even in relatively small automata. Thus, the validity the local equilibrium hypothesis in- volved in the derivation of the equations of fluctuating hydrodynamic [9, 10] is confirmed. Furthermore, Fig. II reveals the presence of algebraically decaying correlations in diffusive systems with a constant gradient [11, 12, 13, 14, 15]. Long-range correlations are a generic feature of non-equilibrium systems which possess conserved quantities and exhibit large- scale anisotropy [16, 17, 18]. For the class of automata studied here, we have elucidated the physical origin origin of these correlations, and further confirmed that in diffusive systems they are proportional to the square of the gradient (which measures the deviation from global equilibrium), and exhibit a power-law decay of the form I r - r ' l (2-d) , except for the critical dimension d = 2, where the decay is logarithmic [14]. Since we average over one of the actual automaton dimensions, the present simulations correspond to an effectively one-dimensional system; whence the linear decay of the long-range correlations.

Thus, we conclude that lattice gas automata, despite their simplified microscopic dy- namics, exhibit realistic behavior both at the macroscopic level (diffusion equation) and the mesoscopic level (fluctuation correlations). Hence, they constitute a useful alternative to more realistic, but also more expensive, simulation techniques.

243

Acknowledgments

This work has benefited from the support of the European Commission under contracts ERBCHBG-CT93-0404 and ERBFMBI-CT95-0087.

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