reactive dynamics in a multispecies lattice-gas automaton

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Reactive dynamics in a multispecies latticegas automaton Raymond Kapral, Anna Lawniczak, and Paul Masiar Citation: The Journal of Chemical Physics 96, 2762 (1992); doi: 10.1063/1.462025 View online: http://dx.doi.org/10.1063/1.462025 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)? J. Math. Phys. 54, 092203 (2013); 10.1063/1.4821640 Diffusion in tight confinement: A lattice-gas cellular automaton approach. II. Transport properties J. Chem. Phys. 126, 194710 (2007); 10.1063/1.2721547 Diffusion in tight confinement: A lattice-gas cellular automaton approach. I. Structural equilibrium properties J. Chem. Phys. 126, 194709 (2007); 10.1063/1.2721546 Latticegas model of NO decomposition on transition metals J. Chem. Phys. 104, 2446 (1996); 10.1063/1.470939 Quantization for latticegas models of water J. Chem. Phys. 65, 1345 (1976); 10.1063/1.433230 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.238.33.43 On: Fri, 22 Aug 2014 21:02:19

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Reactive dynamics in a multispecies latticegas automatonRaymond Kapral, Anna Lawniczak, and Paul Masiar

Citation: The Journal of Chemical Physics 96, 2762 (1992); doi: 10.1063/1.462025 View online: http://dx.doi.org/10.1063/1.462025 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)? J. Math. Phys. 54, 092203 (2013); 10.1063/1.4821640 Diffusion in tight confinement: A lattice-gas cellular automaton approach. II. Transport properties J. Chem. Phys. 126, 194710 (2007); 10.1063/1.2721547 Diffusion in tight confinement: A lattice-gas cellular automaton approach. I. Structural equilibrium properties J. Chem. Phys. 126, 194709 (2007); 10.1063/1.2721546 Latticegas model of NO decomposition on transition metals J. Chem. Phys. 104, 2446 (1996); 10.1063/1.470939 Quantization for latticegas models of water J. Chem. Phys. 65, 1345 (1976); 10.1063/1.433230

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Reactive dynamics in a multispecies lattice-gas automaton Raymond Kapral Chemical Physics Theory Group, Department o/Chemistry, University o/Toronto, Toronto, Ontario M5S lAl Canada

Anna Lawniczak Department 0/ Mathematics and Statistics, Univeristy o/Guelph, Guelph, Ontario NIG 2WI Canada and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Paul Masiar Chemical Physics Theory Group, Department o/Chemistry, University o/Toronto, Toronto Ontario M5S IAI Canada and Ontario Centre/or Large Scale Computation, University o/Toronto, Toronto, Ontario M5S IAI Canada

(Received 17 June 1991; accepted 28 October 1991)

A multispecies reactive lattice-gas automaton model is constructed and used to study chemical oscillations and pattern formation processes in a spatially distributed two-dimensional medium. Both steady state and oscillatory dynamics are explored. Nonequilibrium spatial structures are also investigated. The automaton simulations show the formation of rings of chemical excitation, spiral waves, and Turing patterns. Since the automaton model treats the dynamics at a mesoscopic level, fluctuations are included and nonequilibrium spatial structures can be investigated at a deeper level than reaction-diffusion equation descriptions.

I. INTRODUCTION

The full microscopic description of complex chemically reacting systems is a challenging problem that is difficult to solve by direct simulation of the equations of motion. This is especially true of multicomponent reacting systems con­strained to lie far from equilibrium where many interesting phenomena occur on long distance and time scales. It is well known that in such circumstances interesting spatial and temporal structures can arise as a result of the interplay be­tween the nonlinear reaction kinetics and the diffusion pro­cesses in the system.) Examples include bistability, temporal oscillations, chemical waves, and Turing patterns. While these phenomena are usually described by reaction-diffu­sion equations, their existence is affected by molecular fluc­tuations in the system, especially near bifurcation points.2 If the goal is to attempt to obtain a molecular picture of these rather unusual spatiotemporal structures, it is desirable to seek a route other than direct molecular dynamics (MD) simulations of the equations of motion. There have been studies of simple models of far-from-equilibrium systems us­ing full molecular dynamics,3 but such investigations have been necessarily limited in the number of particles and space and time scales which could be explored. The limitations of conventional MD techniques for the study of such systems has prompted our interest in alternative dynamical schemes.

We present in this paper a lattice-gas cellular automaton (LGCA) description of multi component spatially extended reactive systems. While an outline of the method and illus­trations of its utility were given in Ref. 4, we focus here on the full mathematical description and present a variety of simulation results to illustrate some directions for applica­tions. Lattice-gas cellular automaton modelss have been ap­plied with considerable success to the study of fluid flow problems in hydrodynamics.6

,7 The idea behind such mod-

els is the construction of an artificial dynamics that captures many of the essential features of the real collision processes in reactive media, yet is simple enough to permit simulations on large systems composed of many particles. The reactive LGCA scheme described below satisfies these criteria and simulations of the dynamics of millions of particles for long times is a simple task. The automaton model occupies a posi­tion between full MD and macroscopic reaction-diffusion equation descriptions of the dynamics; it is a mesoscopic model of the dynamics. The LGCA is an alternative to other approximate MD methods, for instance, the Bird methodS which has been applied9 to the study of some of the phenom­ena described in this paper. It also permits a deeper level of description than other discrete methods such as "classical" cellular automata and coupled map lattices. \0 Finally, we note that a multispecies LGCA model has been constructed by Dab, Boon, and Li)) and used to study spatiotemporal structures in a two-variable reaction model; their work com­plements this study.

The outline of the paper is as follows: Section II begins with a brief qualitative description of the LGCA, followed by a mathematical formulation of the automaton dynamics, Sections III and IV present the derivation of discrete Boltz­mann equations, continuity equations, and rate equations that follow from the LGCA equations of motion. In these sections the connection between given reaction-diffusion equations and specific LGCA models is established. This allows the construction of a LGCA dynamics that yields in the macroscopic limit a particular nonlinear chemical rate law. A two-variable reaction scheme, the Selkov model, is studied in Sec. V and an automaton corresponding to this reaction is constructed. The results of simulations of the Sel­kov LGCA are presented in Sec. VI. There we show that the automaton can produce chemical oscillations, chemical rings, and spiral waves characteristic of excitable media, and

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Kapral. Lawniczak, and Masiar: Reactive dynamics in a lattice gas 2763

Turing bifurcations. All of these phenomena are found only in multispecies reacting systems and have motivated the ex­tension presented here. The conclusions are given in Sec. VII.

II. LATTICE-GAS CELLULAR AUTOMATON MODEL

A.Overvlew

The mathematical description of the multispecies reac­tive LGCA is rather technical in character and requires a somewhat cumbersome notation in order to insure genera­lity. For this reason it is useful to give an overview of the principal features involved in its construction. We consider spatially extended systems composed of a number of species X" T = 1, ... ,n, which may undergo chemical reactions of the type

alXI + ... + anXn --!3I X I + ... + !3nXn· (2.1)

Basically, the problem is to construct an approximate dy­namics for both the elastic and reactive collision processes that occur in the real system. To simplify the dynamical de­scription space is made discrete by placing the system on some regular lattice with coordination number m; time is also taken to be a discrete variable. To complete the discrete phase space description, a discrete velocity which can point along any of the m lattice directions is associated with each particle. In order to have a computationally tractable scheme an exclusion principle is assumed where a given node of the lattice cannot be occupied by two particles of the same species with the same velocityY

The collision processes are taken to be short ranged or local and occur at the nodes of the lattice. In the form of the model constructed here, only the dynamics of certain reac­tive species are followed and the solvent and possibly other species, which may be in excess or constrained, are taken into account through their effects on the dynamics of the species of interest. (This restriction is not an essential part of the model.) More specifically, elastic collisions of the X r

species with the solvent (and each other) are modeled by random rotations of the velocities of the X T • Reactive colli­sions change the numbers and velocities of the particles of each species and in the automaton such collisions are effect­ed by combining two operations: local reactions (2.1) at a node that occur with preassigned probabilites which are in­dependent of the velocities, and random rotations. The net effect is a reactive event where both particle numbers and velocities are changed, just as in real reactive collisions. Fin­ally, between such collisions the particles propagate freely and move from one lattice site to another. In more pictoral terms one way to visualize this multispecies LGCA model is as a "stack" of n "species lattices" with identical labeling of the nodes. Each species resides on its own lattice and the propagation and rotation steps are taken independently on each lattice but the dynamics on these lattices is coupled by the chemical reactions.

While not a true molecular dynamics, the model cap­tures the essential features of the full reactive system since the reactive and elastic collision processes mimic those in the real system. We now give a precise formulation of the rules that govern the dynamics of the system.

B. Mathematical formulation

The goal of this subsection is the derivation of micro­scopic equations of motion for the system. These microscop­ic equations constitute the starting point for the derivation of chemical rate laws and reaction-diffusion equations. The links established between the microscopic automaton dy­namics and the macroscopic equations provide a means for constructing automaton models that yield a given rate law or reaction-diffusion equation.

The LGCA model for the chemically reacting system (2.1) can be described in formal terms: particles of species XI"",Xn move on a lattice .Sf' with coordination number m and periodic boundary conditions. Each node of the lattice is labeled by the discrete vector r and each particle has asso­ciated with it a discrete velocity Ci, i = 1 , ... ,m pointing in one of the m possible directions on the lattice. We assume that the Ci are unit velocity vectors connecting the node to its nearest neighbors, where i increases counterclockwise and 1 corresponds to the positive direction of the x axis. The exclu­sion principle prevents two particles of the same species to be at the same node with the same velocity. Consequently, there may be 2n different configurations of particles at a node with the same velocity and 2nm different configurations of parti­cles at a node.

The dynamics of the reactive automaton can be neatly described by a set of microdynamical equations that are the analogs of Newton's equations of motion for this system.s

The microdynamical equations are evolution equations for a set of Boolean random variables that describe the species type and discrete phase space coordinates of the particles.

Let S, be a set of all m X n matrices s = [SiT]' U= l, ... ,m, 1'= I, ... ,n) such that

S = {s = [SiT] :SiT = 0 or 1, (2.2)

Vi = I, ... ,m, 0< ± SiT <n} , T=l

and let Si' I<i<m bea set of all row vectors Si = (SiT> T= l •...• n

obtained from matrices S which belong to S. A configuration of particles at a node r at time k can be described by a m X n random rectangular matrix 11(k,r) = [17iT (k,r)], with val­ues in a state space S. This implies that a row vector 11i (k,r) = (17iT (k,r) > T= l ..... n ofa matrix 11 (k,r) takes values in the state space Si defined above and describes a configura­tion of particles of Xl , ••. ,Xn with the same velocity Ci at a node. A configuration of the lattice.Sf' at time k is described by a Boolean field

11(k) = {11(k,r):rE.Sf'}, (2.3 )

with values in a phase space r = S-"" of all possible assign­ments s('),

s(') = {s(r):s(r)ES,rE.Sf'}. (2.4)

The evolution of the system occurs at discrete time steps and the lattice gas automaton updating rule can be defined on the Boolean matrix field (2.3). At each node at each time step the updating rule consists of deterministic propagation fol­lowed by collisions, which may be either elastic or reactive. As in the one variable LGCA \3 the evolution consists of a

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2764 Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas

product RoCoP of three elementary operators: propagation P, chemical transformation C, and rotation R.

In the propagation step each particle moves in the direc­tion of its velocity from its node to a nearest-neighbor node. The velocity of each particle is conserved during this oper­ation. Letting 1); denote the configuration in the ith direction after propagation we have

P:1);(k,r) =1)i(k-l,r-ci ). (2.5)

The rotation operation can be described in the following way. At each node, independently ofthe others, the particle configuration on each species lattice is rotated through one of the possible angles (JI = I 21T/m, 1= 1, ... ,m with probabil­ity P lr; the probabilities may be different for each species and they must be selected to preserve isotropy.14 To explicitly construct the rotation operator we let {SIT (k,r) :rE..?, k = 1,2,"'}, be independent sequences of identically dis­tributed, independent, Bernoulli-type random variables. Us­ing these variables we may then construct new random vari­ables

yf,. = SIT II (1 - SjT)' (l =l=m) (2.6a) j",,1

and m-I

~T = 1 - L yf,., (2.6b) 1=1

with probabilities

Pr(yf,. = 1) = PIT' (2.7)

In Eqs. (2.6) and (2.7) we have dropped the dependence of Sand ron k and r. We adopt this convention in the sequel when confusion is unlikely to occur. If 11i!- denotes a configu­ration of species XT after a rotation in the ith direction at a node then

m

R:l1i!- = L yf,. 7]i + I,T' (2.8) I

where the addition in the index of 11 is modulo m. Since m

L yf,. = 1, (2.9) 1=1

we may also write Eq. (2.8) in the form m

11i!- = l1iT + L yf,.( -l1iT + l1i+l.T) 1=1

=7]iT + ~i!-(1)T)' (2.10)

which defines the rotation operator ~~ (1)T)' that takes on the values { - 1,0, n.

The propagation and rotation operations are intended to describe the free streaming and elastic collisions that oc­cur in the real system; their net effect is to produce diffusive motions of the various species. Since the rotations are carried out independently on each species lattice and since the rota­tion probabilities need not be the same for all species, the different species may have different diffusion coefficients.14 If only propagation and rotation operations are considered the species lattices are uncoupled and the dynamics of each species is independent of that of other species.

The chemical transformation C couples the dynamics

on the different species lattices and forms a central part of the reactive LGCA model. In this step the reactions (2.1) that the chemical species undergo are modeled by probabilis­tic rules for the increase or decrease of the number of parti­cles. The numbers of the different species in the reactant and product states can be specified by the vectors a = (a l , .•• ,an) and [3 = ([31 , ... ,[3n)' respectively, where O<;.aT,[3T<;.m for each 1" = 1, ... ,n. The configuration at the node r is examined and depending on the stochiometric con­figuration a a transformation to configuration [3, i.e., a--[3 is made with the probability P( a[3), regardless of the veloc­ity states of the particles. The diagonal elements P(aa) of the transition probability matrix P = [P(a[3) 1 correspond to nonreactive events and they satisfy

P(aa) = 1- L P(a[3). (2.11) fJ #,a

The off-diagonal elements of P with aT > [3T correspond to reactive transitions for which particles of species X T are de­stroyed while if aT <[3T particles of species XT are created. The number nafJ of all the possible distinct outcomes of a reaction type a --[3 is given by

v (2.12)

where qT = [3T - aT' It is clear that if qT > 0 then particles of species X T are created while if q T < 0 then particles of species X T are destroyed.

The chemical transformation step can be described for­mally in terms of the microdynamical variables l1iT' Let 'If denote the set of all possible chemical transformations a --[3, i.e.,

'If = {Ca,[3):'<i1" = 1, ... ,n, O<;.aT<;.m, O<;.[3T<;.m}.

(2.13 )

For each input stoichiometric vector a let 'If (a) be the set of all possible output stoichiometric vectors [3 different from a, i.e.,

'If (a) = {f3:(a,[3)E'If ,a=l=[3}. (2.14)

The reaction events in any realization of the evolution process are determined by combinations of random variables S rfJ such that

{SrfJ(k,r):i = 1, ... ,m, rE..?,k = 1,2,"} (a,[3)E'If

(2.15 )

are independent sequences, independent of sequences of ran­dom variables defined in Eq. (2.6), of identically distributed independent Bernoulli-type random variables. Denoting by i k E{I, ... ,m} the velocity indices we define random variables yafJUI , ... ,iq ), for distinct sets of velocity indices {il, ... ,iq },

which govern the chemical transformations a--[3

(2.16 )

where the prime on the product indicates that only distinct (distinct from each other and the i l , ••• ,iq ) ik values are con-

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Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas 2765

sidered. The probability distributions associated with these random variables are

P Pr[y<>PUI •...• iq) = 1] =~,

na{J

and obviously

p Pr[y<>P(iI, ... ,iq ) =0] = l-~.

na{J

(2.17a)

(2.17b)

In terms of the random variables -;Pf3(i1 , ... ,iq ) we may write expressions for random variables -;PI ( r,i) which describe all possibili ties in the reaction a -+ {3, where particles of species X .. in velocity state i are produced ( + ) or destroyed ( - ), with particles of other species undergoing number changes specified by the stoichiometric coefficients of the reaction. The derivation of the explicit forms of these random vari­ables is given in Appendix A.

For a given input stoichiometric vector a and species X T

we use r(a,r) to denote a subset of {l, ... ,m} of distinct

velocity indices corresponding to velocity directions occu­pied by particles of XT

r(a,r) = VI , ... ,ia), (2.18)

and by r (a) we denote one of the possible configurations of directions occupied by species X T for an initial stoichiome­tric vector a

rca) = {r(a,r):r= l, ... ,n}. (2.19)

It is clear that the input vector a may be realized by many configurations of particles, i.e., by many rea). We intro­duce the following products of Boolean fields labeled by an input stoichiometric vector a and a set rca) of occupied velocity directions:

n

Qa[1);r(a)] = II II 'fJjT II (1- 'fJkT)' (2.20) T = I jEr(a,T) k<lr(a,T)

If 'fJ~ denotes a configuration for a species r in the ith direc­tion after a chemical transformation then

'fJ~ = L L Qa[1);r(a)]{(I-'fJiT) L -;P!(r,i) +'fJiT[I- L -;P!(r,i)]} a:ar>1 r(a):r(a.T)31 {Je'C - (a,T) {Je'r: - (a,T)

where

~ + (a,r) = {f3:(a,{3)E~,qT >O},

and

~ - (a,r) = {f3:(a,{3)E~,qT <a},

denote the sets of all possible output states /3 in the reaction a -+/3 in which particles of species X T are produced or de­stroyed, respectively. From Eq. (2.21) we can give a phys­ical interpretation of the processes contributing to 'fJ~. The two terms involving sums on a correspond to processes for which the number of particles of a species X T at a node de­crease (expressed by the condition aT>1) or increase (ex­pressed by the condition aT < m). The first term in each curly bracket arises from reactive events while the second term corresponds to nonreactive events. The Q a[ 1);r(a)] factor insures that the configuration at a node corresponds to an input stoichiometric vector a realized by particles lo­cated at r (a) velocity directions.

For a such that aT>l and r(a,r) 3i

Qa[1);r(a) ]'fJ;T = Qa[1);r(a)],

and for a such that aT < m and r(a,r) 1J i

Qa[l1;r(a)l'f/'T =0.

Since

'fJ;T = L L Qa[1);r(a)], a:ar>1 r(a):r(a,T) 3;

formula (2.21) can be simplified to

(2.22)

(2.23)

(2.24)

(2.21 )

x L -;p! ( r,i) (Je'tf - (a.T)

+ L L Qa[1);r(a)] a:ar<m r(a):r(a,r)3i

x L -;P-f! (r,i). (2.25 ) {Je'tf + (a.T)

The last two terms correspond to processes where the parti­cle number of a species X T decreases or increases at a node, respectively. Formula (2.24) follows from Eq. (2.20) in Ref. 13 and the fact for any species u

L L II 'fJ;q II (1 - 'fJjq) = I, (2.26) I J iel jEJ

where I and J are disjoint subsets ofindices of {l, ... ,m} such that IUJ = {I, ... ,m}.

Finally we define a chemical transformation operator for a species X T, a~ (1);a/3), by comparison of

'fJ~ = 'fJIT + L a~(1);atn, (2.27) (a,f3)e'C,a#f3

with Eq. (2.25). This operator has the property that for (a,{3)E~, a=l={3

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2766 Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas

{

o if a.,. =/3.,.

a~(l1;a/3) = - 1 or 0 if a.,. >/3.,. . 1 or 0 if a.,. </3.,.

(2.28)

The cellular automaton rule consists of successive appli­cations of the transformations described above, i.e., propa­gation followed by collision, RoCoP. The microdynamical equations for the automaton follow directly from the micro­dynamical equations for these transformations. From Eqs. (2.8), (2.10), and (2.27) the result of the sequence of trans­formations RoC is

f r:!.11(+ I,.,. = f r:!.l1i + I,.,. 1=1 1=1

m

+ L ') r:!.a(+ I,.,. (l1;a/3) 1= 1 (a,f3)~,a#f3

= l1i.,. + a:!. (11) + a~(l1)' (2.29)

where we have defined m

a~(l1) = L L r:!.a(+I,.,.(l1;a/3). (2.30) 1= 1 (a,f3)E"C,a#f3

Using this result the microdynamical equations for the cellu­lar automaton for a species X.,. take the form

'TJi.,.(k,r) = 'TJ::' + a:!. (l1P) + a~(l1P). (2.31)

This equation is a compact representation of the full micro­scopic dynamics of the automaton and can serve as a basis for further theoretical analyses.

The automaton rule can be cast in a somewhat more general form that is useful in simulations where the magni­tudes of the diffusion coefficients need to be changed by large amounts. This is easily accomplished by performing differ­ent numbers of propagation and rotation steps on the various species lattices. The operator product po RoC can be general­ized to

n II 0 ±IT(P.,.OR.,.)oC, (2.32) T=l

where the operator 0 with a + I superscript means I applica­tions of the operator in its argument for every application of C, while a - I superscript means application of the operator in its argument every I th application of C.

The evolution of the lattice gas can also be described in terms of a probability distribution PI' [k,s(')] at a time k with the initial distribution f-l. Since the random variables 51.,.,5 fP governing the random rotations and chemical trans­formations, respectively, are independent, the entire Boo­lean field {l1(k); k = 0,1,2, ... } defined in Eq. (2.3) is a Mar­kov process. The Chapmann-Kolmogorov equation for P can be found in Ref. 13. I'

The introduction of the probability distribution P and its corresponding expectation EI' enables one to defin: phy­sically interesting average quantities; the mean population of a species X.,. in the ith direction

N i .,. (k,r) = EI' ['TJi.,. (k,r)],

the local density of a species X.,., m

p.,.(k,r) = L Ni.,.(k,r), i=1

(2.33)

(2.34)

the density of a species X.,. at time k

p.,.(k)=N- 1 LP.,.(k,r), (2.35) re.:t'

where N is the number of nodes, and the local mean velocity of a species X.,.

u (k,r) = {[p.,. (k,r)] - I~,!,= 1 ciNi.,. (k,r), if p.,. (k,r) =/=0 .,. 0, otherwise.

(2.36)

Of course NiT' p.,., and u.,. depend on f-l, which we drop to simplify the notation.

III. BOLTZMANN EQUATION

By taking the expectation value of the LGCA microdyn­amical equation (2.31), we can derive a discrete Boltzmann equation. Since the random variables 51.,. (k,r), 5 ff3 (k,r) are independent and also are independent of the evolution of the cellular automaton up to time k - 1, this implies that

m

+ LPI.,. L EI'Il.(+I,.,.(T(;a/3). 1= 1 (a,f3)E'(f ,a#f3

(3.1)

In the above N::' = N i.,. (k - l,r - c i ) and for any matrix

W= [Wi.,.]' m

C:!.(W) = L PI.,. Wi + I • .,. - Wi.,.' (3.2) 1=1

where addition is modulo m. If for all a, P(aa) = 1, i.e., no chemical transition takes place, each particle performs a ran­dom walk on its species lattice. These random walks are not independent since when two or more particles are simulta­neously present at a node, they are rotated by the same angle in order to preserve the exclusion principle. Except for this weak dependence between trajectories due to the exclusion principle, all elastic collisions are independent and a kinetic Boltzmann equation for the average local number of parti­cles can be derived.

In the presence of reactive transformations collisions can no longer be assumed to be independent. For example, particles which have reacted to produce a new particle can collide reactively again with this new particle on nearby nodes. However, if the chemical transformation probabili­ties P(a/3) , a =/=/3 are sufficiently small, reactive transitions are rare events and we can again assume independence be­tween collisions. In this approximation, we can factorize the expected value in Eq. (3.1), and the expected occupancy Ni .,. (k,r) satisfies the lattice Boltzmann equation

Ni.,.(k,r) = N::' + C:!.(NP) + C~(NP), (3.3)

where m

C~(N) = L PI.,. L Cf+I,.,.(N;a/3). 1= 1 (a,f3)E'tC,a#f3

(3.4 )

In the above

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Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas 2767

L C;+ I.T(N;af3) (aJJ)E 'r. .a ¥ fJ

= L L Qa[N;r(a) ]baT a:a,< m r(a):r(a,r);tJi

- L L Qa[N;r(a) ]daT , (3.5) a:a,> I r(a):r(a.T) 3i

where

m m f3 -a L ... L r r PCaf3), (3.6) P, = a, + I fJ. = 0 m - aT

and

m ar-l m a - (3 daT = L ... L ." L T r P(af3). (3.7)

/11 =0 /1,=0 /1.=0 aT

Details concerning the derivation ofEqs. (3.6) and (3.7) are given in Appendix B.

Since factorization in Eq. (3.3) does not hold exactly unless the system is in equilibrium and in a state strictly satisfying the Boltzmann hypothesis, NiT as computed from these equations will not give the true expected occupancies of the process. Nevertheless, we suppose that under some conditions on the initial state analogous to Eq. C 3.7) in Ref. 13, the automaton inherits the limiting behavior ofEq. (3.3) in some space-time scaling regimes. In the kinetic regime (3.3) converges to the continuous space-time Boltzmann equation

aNir(t,x) ----+ ci'VNiT(t,x) at = C~(N) + C~(N) = CiT(N). (3.8)

Details concerning the derivation of this equation are omit­ted since they are similar to those leading to Eq. (3.8) in Ref. 13.

IV. RATE LAW

The results of Sec. III form the starting point for the derivation of the continuity equation and chemical rate law for the total density PT ofspeciesXT • SummationofEq. (3.8) over the velocity index i gives the continuity equation for local particle density. Since for each /, / = 1, ... ,m and (a,/3)E~, ai=f3

m m

L Cf+I,r(N;af3) = L Czr (N;af3), (4.1 ) i= I ;= 1

and

m

L Ct(N;af3) = 0, (4.2) ;= I

then by Eqs. (2.36) and (3.4) the continuity equation for the local particle density Pr is

apr (t,x) -'---+ V' [uT (t,X)PT (t,x)] at m

= L L C~(N;af3). i= I Ca,/1)e'C.a¥/1

(4.3)

If the system is spatially homogeneous and in eqUilibrium in velocity space, so that for each 7 and each i

Pr(t) Nir(t,x) =--,

m the continuity equation (4.3) reduces to

dPT (t)

dt L (f3r -aT)P(af3)

(a./1)e'6' ,a¥/1

(4.4)

(4.5)

The derivation of Eq. (4.5) is given in Appendix C. Note that the highest power of the density of a given species is limited by the coordination number of the lattice. Ifwe iden­tify coefficients of equal powers of the densities in the above equation with those in the phenomenological rate law we can establish relations between the chemical transformation probabilities and the rate coefficients in the chemical rate law. Using these considerations a set ofLGCA rules consis­tent with a given macroscopic rate law can be constructed.

V. SELKOV MODEL

The Selkov model IS is described by the following mech­anism:

kl k2 k3

A ~X, X+2Y~ 3Y, Y~B. (5.1 ) k_1 k_2 k_3

We imagine that the concentrations of the A and B species are fixed by external flows of reagents and consider the dy­namics of the intermediate species X and Y.16 The chemical rate law then takes the form of two coupled equations for the X and Y concentrations

dpx _ k k k 2 k 3 dt - IPa - - IPx - 2PxPy + - 2Py'

dpy _ k k k 2 k 3 dt - - 3Pb - 3Py + 2PxPy - - 2Py. (5.2)

A phase diagram for this model was constructed earlier by Rehmus et a/. 17 and shows some of the interesting behavior that is found for the spatially homogeneous Selkov model. If we employ a scaling of variables similar to that of Ref. 17,

namely, k3t-+t andpT =PT~k3Ik2' the rate law takes the simpler form

(5.3 )

In our simulations we follow Ref. 17 and select X = 0.1 and K = 1 and consider a and b as bifurcation parameters. With the scaling in Eq. (5.3) we can read off the bifurcation struc­ture of the homogeneous system from Fig. 2 of Ref. 17. From this figure it is clear that even in the spatially homogeneous regime the Selkov model has a very rich bifurcation struc­ture: There are regions in parameter space where the system possesses a single steady state (it may be either a node or a focus or exhibit excitability properties), a limit cycle, and bistability between two fixed points or between a limit cycle

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2768 Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas

and a fixed point. We shall now apply the formalism of Sec. IV and construct a LGCA model for the Selkov reaction.

In order to construct the Selkov LGCA we consider a square lattice (m = 4) 18 with periodic boundary conditions and write Eq. (4.5) for two species as

dpr (1) = i i KlkP~ (t)p;(t), (5,4a) dt 1=0 k=O

where

Klk = (_l)l+km -I-k

1 k

X I I PrUj)( -1) -i-j i=Oj=O

(5,4b)

with 4 4

PrUj) = I I (Pr - ar)PUjli'j'). (5.5) ,.=0/=0

The quantities K lk in Eq. (5.4) can be identified with the rate coefficients of any two-variable reaction model, and thus Eq. (5,4) provides the connection between the phenom­enological rate coefficients and the LGCA reaction proba­bilities. From this comparison of coefficients we obtain ex­pressions for the Pr Uj) in terms ofthe rate coefficients in the Selkov model. We find

PxUj) =V(j-I)(j- 2)k_2

- 1ij(j - 1)k2 - ik_l + klPa,

PyUj) = - V(j - l)(j - 2)k_2

+~i(j-l)k2 -jk3 +k_ 3Pb' (5.6)

Equations (5.6) are a set of constraint conditions that the

qO-I- _7-\0-1-+ +-S--2o-l- +-9-:30-1-+ +-10-40 2 2 ~ ~ ~ '" . . . '" t t t t t 3 3 3 3 3 I I I I I a I-I -+ +-7, 1-1-+ +-S-? 1-1 -+ +-9 -31-1 -+ +- 10-41 I I ., I 1 22222 '" • • .J, • t t t t t 1 1 1 4 1

02-1-+ +-7--12-1- +-8-22-1-+ +-9-32-1-++-10-42 I /1 / I / I / I 2 II 2 jll 2 13 2 14 2 • It" 4- 1&. ,/. ,/ '" tJ' t J' t/' tJ' t ~ /12 ~ /12 T )2 ~ )2 T

03-1-+ +-7-13-1-++-s-23-1-+ +-9-:.33-1-++-1O-~3 • /1 / I / I / I 2 13 2 17 2 19 2 21 2 4- ,/ • .t • .t • .t + t /' t,1l t,1l .,. J'l\. t 6 I~ 6 16 6 III 6 20 22 6 1/ 1/ 1/ 1/ 'I

04-1-+ -7-14-1-+ +-11-24-1 .... +-9~-1-+ +-10-44

FIG. I. The nonzero reaction probabilities P(ijli'/l are indicated by arrows from (ij) ..... (q). The number on the arrow in the figure refers to the index k of the quantities p(k) defined in Eq. (5.7).

reaction probabilities PUjIi'/) must satisfy in order for the LGCA to yield the same macroscopic behavior as the Selkov model. Of course, there are an infinite number of LGCA models that are consistent with these constraints. This is to be expected since a set of macroscopic constraints cannot fully determine the microscopic dynamics. Rather one should regard the LGCA model as a means of specifying a microscopic dynamics that is consistent with a desired mac­roscopic rate law. The microscopic model should then be constructed to mimic as many features as possible of the true molecular dynamics of the system of interest. In this spirit, we restrict particle changes to increases or decreases by one particle in accord with the Selkov kinetics and reactive changes to those that occur in the Selkov mechanism. When these considerations are implemented the number of non­zero elements of the 25 X 25 reactive probability matrix is very small. The structure of the reaction probability matrix is given in Fig. I. The reactive transition probabilities that appear in the figure are expressed in terms of the quantities p(k), (k = 1, ... ,22) which are defined below;

p(1) =k.Pa, p(12) = 16k_ 2,

p(2) = k_ 3Pb' p(13) = 8k2,

p(3) = k3, p(14) = (32/3)k2 ,

p(4) = 2k3 ,

p(5) = 3k3 ,

p(6) = 4k3 - k _ 3Pb'

p(7) = k_.,

p(8) = 2k_ l ,

p(15) = 64k _ 2'

p(16) = 64k_ 2 -16k2,

p(17) = 16k2,

p(18) = 64k _ 2 - 32k2,

p(19) = 24k2,

p(9) =3k_ p p(20) =64k_ 2 -48k2,

p(1O=4k_ 1 -k1Pa' p(21) = 32k2 -16k_ 2,

p(1I) =~k2' p(22) =64k2 -64k_ 2.

(5.7)

Figure I and Eqs. (5.7) completely specify the reactive dy­namics of this Selkov LGCA, but other equally valid proba­bility matrices may be constructed. Finally we note that since the reactive probabilities must satisfy O';;;PUjI(j') <; I there are additional restrictions on acceptable values of the rate coefficients that appear in the Selkov model (or any other model).

VI. SELKOV LGCA SIMULATION RESULTS

Below we present the results of simulations carried out on the Selkov LGCA described above in order to demon­strate that the reactive lattice-gas automaton can provide a new method for the study of spatiotemporal dynamics in chemically reacting media. We have selected parameter val­ues in regions of the Selkov phase diagram where the system has very different kinds of behavior.

A. Stable fixed pOint and limit cycle

The simplest kind of attractor is a stable fixed point in a spatially homogeneous system. However, depending on the system's parameters, the attracting fixed point may be a node or a focus. These two characteristic kinds of behavior for the Selkov LGCA are illustrated in Fig. 2 for a model where the diffusion coefficients of X and Yare equal. In Fig.

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Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas 2769

.098

.094

.090

Cy .086

.082

. 078

.074

(a)

.12

.08

o (b)

.132 .136 .140 .144 .148 cx

2.0 4.0 6.0 10.0

FIG. 2. (a) Phase plane plot of monotone decay to the fixed point from several initial conditions. System parameters: kiP. = 0.000 19006, k _ I = O.lk), k) = 0.005, k2 = k _ 2 = 0.01, and k _ )Pb = 0.000 00559. System size: 256 X 256. (b) Plot of oscillatory decay to the fixed point. The concentrations per velocity direction c, (r = x,y) , are plotted vs time. Sys­tem parameters: kiP. =0.0001897, k_ , =0.lk3, k,=O.OOI, k2 = k _ 2 = 0.01, and k _ 3Ph = 0.000 041 11 and the system size is 512 X 512. The time is in units of 10' time steps.

2 (a) we show in a phase plane plot that starting from several different initial conditions the decay is monotone close to the fixed point (apart from fluctuations) indicating the exis­tence of a stable node. In contrast, in Fig. 2(b) we show the oscillatory decay to the fixed point characteristic of a stable focus. The system is close to the Hopfbifurcation point and shows the characteristic slow decay to the fixed point. The

system state is spatially homogeneous on average in both cases.

Spatially homogeneous stable limit cycle dynamics is observed in other regions of parameter space. Figure 3 shows the evolution to an attracting limit cycle starting from two different initial conditions, one inside the limit cycle and the other outside the cycle. This limit cycle has a strong relaxa­tion character; the parameter values have been chosen to lie far from a Hopfbifurcation point.

In all of the above results the effects oflocal fluctuations on the average species concentrations can be observed in the figures; e.g., the irregular trajectories in the montone decay to the fixed point noted above and the fact that the dynamics is perturbed about the limit cycle attractor .

B. Excitable dynamics

Especially interesting dynamics occurs when the fixed point is excitable and perturbations beyond a threshold val­ue cause the system to make a long excursion in phase space before return to the fixed point. During this long excursion the system is not susceptible to further perturbations; i.e., it is refractory. Small perturbations which do not exceed the threshold relax quickly back to the fixed point. If threshold­exceeding perturbations are introduced locally in an excit­able medium, chemical waves of excitation can form and propagate through the reacting medium. The most frequent­ly encountered chemical waves are produced by this mecha­nism; e.g., those in the Belousov-Zhabotinsky reaction are typically of this type. If a perturbation is introduced locally in an undisturbed excitable medium, a ring of excitation will

.13

.11

C .09 Y

.03

,01

0~-,1~6~~~.2~0~~',2~4~~~,~28~-L~.~32~-L~,3~6~

FIG. 3. Limit cycle oscillation. The evolution from two initial conditions, one inside the limit cycle and the other outside indicated by heavy dots, is shown. System parameters: kiP. = 0.000 1897, k_ I = O.lk" k3 = 0.001, k2 = k _ 2 = 0.01, and k _ 3Pb = 0.000 025 30 and the system size is 256X256. The time is in units of lit time steps.

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2770 Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas

propagate outward. The chemical wave has a refractory tail preventing back propagation of the wave. If two such chemi­cal waves meet they annihilate. If the medium is disturbed or is inhomogeneous the rings of excitation may be broken and the free ends will curl to form spiral waves. These phenome­na account for the structures of the most commonly ob­served chemical patterns. 1,19

It is possible to study these chemical waves using the Selkov LGCA by selecting reaction probabilities to corre­spond to homogeneous kinetics within the excitable region. Just as in real excitable media, we can initiate simple ring growth by introducing a local disk-shaped, threshold-ex­ceeding perturbation in the LGCA. Figure 4 shows the evo­lution of a ring from this initial condition. Note that the system relaxes back to the fixed-point concentration after the chemical wave has passed. The wave is circular indicat­ing that the particle fluctuations in conjunction with the

' . . ,.: ....

" " " '

.... ,

.~;' ~', .

FIG. 4. Evolution of a ring of excitation in the excitable region. First panel shows the system state shortly after the introduction of the local disk­shaped perturbation while the next panel shows the ring of excitation that has evolved from this initial condition. The concentration of Yis shown as gray shades in this and the following figures which depict spatial patterns. System parameters: kIP. = 0.000 083 85, k _ I = 0.lk3 , k3 = 0.0005, k2 = k _ 2 = 0.01, and k _ 3P. = 0.000 002326 and the system size is 512X512.

propagation and rotation steps are strong enough to mask any dependence of the wave shape on the underlying square lattice. The chemical waves annihilate when they collide as in the real excitable system.

The structure and velocities of the automaton chemical waves have been studied. Figure 5 shows the concentration profiles of the Y species as a function of the radial distance from the center of the ring for several different times. The propagating front and refractory tail are clearly evident in this figure. The chemical waves propagate with a constant velocity in this simulation so that curvature effects on the wave propagation speed are too small to be detected in the figure.

Spiral wave formation in the LGCA can be induced by shearing the ring of excitation, simulating that process in the real excitable system. In Fig. 6 we observe that the free ends of the ring curl to form a pair of counter rotating spiral waves. Thus the automaton is able to generate the two major types of chemical waves that are typical of two-dimensional excitable media: rings and spirals.

In the above simulations we introduced a disk-shaped perturbation. The radius of the perturbation must exceed a critical value (critical nucleus) for the perturbation to spread in the medium. One interesting problem to investi­gate is whether spontaneous fluctuations in the system can produce nuclei that exceed the critical radius and thus gener­ate spatial structures. One way to increase the probability of forming such critical nuclei is to decrease the diffusion coef­ficients of the two species so that when the autocatalytic step in the reaction produces a local increase in the concentration of a species diffusion is unable to quickly smooth out this local concentration fluctuation. We have verified that this is

y

160 200

FIG. 5. Radial concentration profiles for the ring growth in Fig. 4 for five equally spaced time intervals. The concentration of Y is plotted in arbitrary units vs the radial distance r. The plots are displaced along the concentra­tion and time axes for clarity.

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Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas 2771

FIG. 6. Spiral wave pair formation from "sheared" ring. The free ends curl to form a vortex-antivortex pair of spiral waves. System parameters are the same as Fig. 4 but the lattice size is 1024 X 1024.

the case by carrying out simulations where the diffusion co­efficients of both species were decreased by a factor of 3 by performing the propagation and rotation steps after every three chemical transformation steps. The results are shown in Fig. 7. In addition to the externally induced perturbation,

~.

, ;

: .. : "

:-:" . ~ .

,', ..... .

one can observe the formation and growth of spontaneous nuclei giving rise to a complex dynamic spatial state that consists of annihilating wave fragments and newly formed nuclei and rings of excitation.

C. Turing patterns

Turing bifurcations occur when a stable homogeneous steady state is destabilized by a difference in the diffusion coefficients of the two species giving rise to a steady inhomo­geneous state. The technical conditions for the appearance of such bifurcations are well known20 and pattern selection has been studied for such systems. 21 While these bifurcations were predicted by Turing22 in 1952 and are speculated to be relevant for pattern formation processes in living systems, it is only recently that experimental evidence for Turing struc­tures has been presented for chemical systems.23 Such ex­periments open up the possibility of detailed and controlled studies of these spatial structures24 and their study is a topic of considerable current interest.2s The reactive LGCA method presented here provides a means for carrying out detailed theoretical and simulation investigations of such bi­furcations.

We have selected Selkov parameters to lie in the stable steady state region near a Hopfbifurcation point where the conditions for the onset of a Turing bifurcation are favor­able.26 The analysis of the Selkov reaction-diffusion equa-

'<: ;"" '", ..... .

FIG. 7. Spontaneous nucleation of rings of ex­citation in the excitable region and spiral wave formation. In addition to ring growth from a disk-shaped perturbation, followed by shear­ing of the ring and spiral wave formation, waves of excitation spontaneously nucleate. In the last frame one sees that the spiral wave­fronts have collided and "pinched off' a wave fragment that can serve as the nucleus of an­other spiral wave pair. System parameters and lattice size same as Fig. 6 but the diffusion co­efficients are one third of those in Figs. 4 and 6.

.. ,: ....

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2772 Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas

tion predicts that a Turing bifurcation for these parameters should occur when DxlDy > 24.21 The evolution of the con­centrations of the X and Y species are shown in Fig. 2(b) for equal diffusion coefficients. One can observe the oscillatory decay to the fixed point characteristic of this region, with small fluctuations about the fixed point concentrations. The system is spatially homogeneous under these conditions as noted earlier. Figures 8 and 9 present the results of simula­tions for the same kinetic parameters but with unequal diffu­sion coefficients. We have selected Dx = 25Dy so that in the activator-inhibitor language species X is the inhibitor and Y is the activator (see Murray, Ref. 1 and Ref. 25). The select­ed diffusion coefficient ratio is larger than that predicted from the deterministic reaction-diffusion equation for a Turing bifurcation so one expects pattern formation. One can see from Fig. 8 that the concentrations of X and Y tend to steady state values, but the results in Fig. 9 show that the system is spatially inhomogeneous. The figure shows the in­homogeneous final state. The spatial structure persists over long times and is occasionally perturbed by concentration fluctuations in the system. In these simulations we used rule (2.32) with + Ix = 5 and -Iy = 5. IfweletDbethediffu­sion coefficient for the automaton updating rule FoRoe, then we have Dx = 5D and Dy = D 15. Fluctuations are quite strong with this selection of diffusion coefficients and one can observe perturbations of the basic stationary inho­mogeneous state, but its prominent features persist in spite of these rather strong fluctuations. As the magnitudes of the diffusion coefficients are increased, always keeping the ratio fixed, fluctuations playa smaller role. Of course, the charac­teristic wavelength of the pattern changes since it depends on

0.3 0 r----.,-----,----,---.,-------,

0.18

0.12

0.06 Cy

o~--~~--~--~~--~~-~ o 1M

FIG. 8. Unequal diffusion coefficients. The concentrations of Cx and cy vs time are shown. System parameters are the same as Fig. 2(b) but Dx = 5D and Dy = D /5, where D is the diffusion coefficient for the basic ReCep rule. System size is 512X 512. The time is in units of 10" time steps.

FIG. 9. Turing pattern. System parameters are those of Fig. 8.

Dx and Dy individually. Simulations have been carried out that confirm these effects.

It is also interesting to examine the bifurcations that arise from the spatially homogeneous oscillatory state as a result of diffusion coefficient differences. For parameter val­ues slightly different from those used in the Turing simula­tion, the homogeneous oscillatory state is stable for equal diffusion coefficients. For these parameter values, if the dif­fusion coefficient ratio is changed to Dx = 20Dy a steady inhomogeneous state results. Hence, both steady and oscilla­tory homogeneous states can be destabilized by diffusion co­efficient differences to yield steady inhomogeneous spatial patterns.28

Studies employing reactive lattice-gas automaton mod­els will allow one to probe these pattern formation processes on a molecular level in order to obtain a deeper understand­ing of how molecular fluctuations influence their formation and growth.

VII. CONCLUSION

The multispecies reactive lattice-gas automaton em­ploys a simple set of evolution equations that give rise to a dynamics like that of real reacting systems. By insuring that the essential characteristics of the microscopic reactive and elastic collision processes are correctly treated a near molec­ular-level description of complex chemically reacting fluids can be constructed. There are several advantages to this type of approach. Since the automaton describes particle motions and reactions, specific chemical reaction schemes can be modeled in contrast to "classical" cellular automaton mod­els where the CA rule is a gross abstraction of the dynam­ics.1O With the exception of the generation of random numbers for the rotation and chemical transformation prob­abilities all the dynamics is carried out at the bit level. Hence, problems associated with numerical errors and instabilities, as is the case for solutions of the reaction-diffusion equation,

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Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas 2773

do not arise. There is no increase in complexity if more com­plicated geometries or boundary conditions are included. This is perhaps one of the most important features of the reactive LGCA method since it permits the investigation of systems similar to those commonly studied in the laborato­ry.29 For example, systems with flows imposed at the boun­daries, like those in the recent investigations of Turing bifur­cations,23,24 spiral wave dynamics,30 and turbulence in inhomogeneous systems composed of catalytic bead ar­rays3! can be investigated.

Apart from these features the LGCA model automati­cally incorporates fluctuations in the dynamics. Conse­quently, a variety of problems that cannot be described by the reaction-diffusion equation can be explored using this technique. These include the effects of fluctuations on bifur­cations in far-from-equilibrium systems, the change in char­acter offluctuations near to and far from equilibrium, fluctu­ation effects on chemical wave propagation processes, to name a few. In the brief sketch of possible applications in Sec. VI we noted that fluctuations can induce nucleation of rings of excitation in excitable media and can affect spatial patterns. Commonly, fluctuation effects in systems far from equilibrium are investigated using master equation meth­ods.2 The LGCA method provides another way to study these phenomena. It is also complementary to interacting particle32 and random walk33 methods that have been used to study chemically reacting systems.

The effects and nature of the reactive LGCA fluctu­ations have not been investigated fully in this paper since our main goal was the mathematical formulation of the automa­ton model and a demonstration of its potential utility. How­ever, as stated above, the fact that fluctuations are present is perhaps one of the most important features of such reactive models and a few comments on how they arise in the dynam­ics are in order.

If the LGCA fluctuations are to resemble those of the real system then every effort should be made to build au­tomaton dynamics that is as close as possible to that of the real system. The elastic and reactive collisions that occur at a node should model the real collisions in the system. Of course, the collisions that occur at the nodes of the lattice should not be regarded as identical to those of the real sys­tem. Rather, they should be considered as effective collision processes, much like those that arise in approximate kinetic theory descriptions of fluids, such as the BGK kinetic equa­tion,34 where the collision operator thermalizes particles at each "collision" with a predetermined frequency. Thus some coarse graining of the real dynamics is already implied in the automaton collision processes that occur at a node of the lattice. In our lattice-gas automaton the velocity is rando­mized in one time step; thus from a microscopic point of view it is reasonable to associate one time step with the velocity relaxation time tR in the system. Correspondingly, the dis­tance a particle travels between such thermalizing collisions is the mean free path A R' and the lattice spacing can be asso­ciated with this length.3s

Such physical considerations can aid in the actual con­struction of the reactive probability matrix since the con­straint conditions corresponding to a given macroscopic rate

law do not fully determine the elements of this matrix. The reactive events occurring at a node should mimic those of the actual reaction mechanism given the scaling described above. While rather bizarre choices of reactive probabilities may yield the same macroscopic rate law, there is no reason to expect such models to have fluctuations similar to those of the physical system if the microscopic processes are differ­ent.

In summary, if an automaton dynamics can be con­structed to correspond to this mesoscopic description of the real system one may hope that the fluctuations will be repre­sented accurately. This is a subtle point and one that will be the subject offuture study.36

In addition to these general considerations it worth re­marking that the magnitude of the concentration fluctu­ations can be tuned by varying the number of propagation and rotation steps relative to the number of chemical trans­formation steps. Clearly, in the limit of an infinite number of propagation and rotation steps per reactive transformation the system is completely "stirred" when reaction occurs and no local concentration fluctuation effects arise from the de­pletion or increase of particles due to reaction. It is this com­petition that can lead to the breakdown of the macroscopic rate law for fast reactions. 35 These effects have not been fully explored but some information on their magnitude has been presented earlier for the Schogl model.!3 A complete investi­gation will require a careful consideration of the relative magnitudes of reactive versus diffusive time scales for sys­tems of interest.

The development of the automaton equations presented here form the starting point for the theoretical analysis of LGCA models. Because of the discrete nature of the velocity space, solutions of the discrete Boltzmann equation are pos­sible and can yield insight into the nature of bifurcations in chemically reacting systems. Furthermore, the character of the correlations among the microscopic random fields and the breakdown of the factorization approximation used to obtain the discrete Boltzmann equation and chemical rate law can be investigated in the context of these LGCA mod­els.

ACKNOWLEDGMENTS

This work was supported in part by grants from the Natural Sciences and Engineering Research Council of Can­ada (R.K. and A.L.) and a grant funded by the Network of Centres of Excellence program in association with the Natu­ral Sciences and Engineering Research Council of Canada (R.K.). We would like to thank the members of the CNLS, in particular S. Chen, G. D. Doolen, and Y. C. Lee, for useful discussions. A portion of the computations were carried on the Cray X-MP at the Ontario Centre for Large Scale Com­putation and we thank the members of this Centre for their assistance. We would like to thank J . Weimar for reading the original version of this manuscript and pointing out a num­ber of errors to us. We are also grateful to B. Lawniczak for assistance in preparation of the manuscript.

J. Chem. Phys., Vol. 96, No.4, 15 February 1992

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2774 Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas

APPENDIX A

In this Appendix we give the explicit expressions for the random variables r"'f (r,i). We must introduce notation to denote the velocity state for the final stoichiometric configu­ration. First we consider processes where particles of species X

T are produced in the reaction a -+/3. The new X T particles

can only occupy velocity directions that are not already oc­cupied by existing X T particles. There are qT = /3T - aT > a such directions, and we denote the set of velocity indices for this process by

r + (a/3,r) = {jl , ... Jq), (AI)

which is a subset of the complement r(a,r)C of r(a,r) in {l, ... ,m}. In an exactly parallel fashion we consider pro­cesses where particles of X T are destroyed in the reaction a-+/3. The number of X T particles that are destroyed is IqT I = aT - /3T' and the set of velocity indices where parti­cles are destroyed is denoted by

r - (a/3,r) = {jl , ... J lqTI }, (A2)

which is a subset ofr(a,r). The other species at the node (apart from X T whose

dynamics is of interest) may participate in the reaction a -+ /3 and their velocity states must be specified. We denote by

B(a/3,r) = {O':q.,. > O,O'#r}, (A3)

and

D(a/3,r) = {O':q.,. <O,O'#r}. (A4)

all the species X.,. (O'#r) for which particles are produced and destroyed, respectively. Then clearly

r B(a/3,r) = {r+ (a/3,O'):UElJ(a/3,r)}, (AS)

contains all the velocity indices where particles of species X.,., O'#rwill be produced and

r D(a/3,r) = {r- (a/3,O'):OED(a/3,r}, (A6)

contains all the velocity indices where the particles of species X.,., O'#rwill be destroyed.

For each input stoichiometric vector a with an initial configuration of particles r (a) such that r (a, r) 3 i and for each /3E~ - (a,r) let us denote by r"'! (r,i) the following sum of random variables:

, r"'! (r,i) = L L L L L r"'P

oeB(ap,T) r + (ap,.,.) KED(ap,T) r - (ap,le) {jlqTI}

where the prime on the sum means that we take a sum over all distinct indices j k such that

(A8)

The random variable r"'! (r,i) describes all the possibilities where the particles of species X T will be destroyed, including always a particle in the ith direction, and particles of the other species will be created or destroyed in the chemical transformation a -> /3 with the initial configuration of parti­cles rca) such that r(a,r) 3i.

For each input stoichiometric vector a with an initial configuration of particles r (a) such that r (a, r) ~ i and for

each /3E~ + (a, r) we denote by r"'f (r,i) the following sum of random variables:

, r"'f (r,i) = L L L L )' r"'P

oeB(ap,T) r + (ap,.,.) KED(ap,T) r - (ap,K) d;,:'}

x [rB(a/3,r),rD(a/3,r),{i,jq)], (A9)

where the prime at the sum means that we take a sum over all distinct indicesjk such that

{jq)={j2, ... Jq)cr(a,r)C - {i}. (AW)

The random variable r"'f (r,i) describes all the possibilities where the particles of species X T will be created, including always a particle in the ith velocity direction, and the parti­cles of other species will be created or destroyed in a chemi­cal transformation a -+ /3 with an initial configuration of par­ticles rca) such that rca,r) ~i.

APPENDIX B

The coefficients baT and daT are expected values of the following random variables:

baT = EI' L r"'f (r,i), (Bl ) {3E'tf + (a,T)

for a such that aT < m, and

daT = EI' L r"'! ( r,i), (B2) {3E'tf - (a,T)

fora such that aT> 1. From Eqs. (2.16) and (A9) it follows that

baT = L L L {3Ectff + (a,T) oeB(ap,T) r + (ap,.,.)

x L L ~ P(a/3) , KED(a{3,T) r- (ap,le) dj- n(a/3)

(B3)

where the prime at the sum means that we take a sum over all distinct indices j k such that

{jQ)={j2, ... Jq)Crca,r)C - {i}. (B4)

Since for a such that aT < m, ~ + (a, r) is a set of all output stoichiometric vectors /3 such that in a reaction a -+ /3, /3T > aT' then Eq. (B3) can be written as follows:

i (m=aaT =:)

f3n =O \.pr T

(BS)

m m m P -a baT = L .. , L . .. L T T P( a/3). (B6)

PI = 0 PT = aT + I P. = 0 m - aT

Similarly we can compute daT for a such that aT > 1. By Eqs. (2.16) and (A7)

daT = L L L {3E'tff - (a,T) oeB(ap,T) r + (ap,.,.)

(B7)

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Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas 2775

where the prime at the sum means that we take a sum over all distinct indicesjkT such that

CB8)

Since for a such that aT> 1, ~ - (a,T) is a set of all output stoichiometric vectors {3 such that in a reaction a~{3, {3T <aT' then Eq. (B7) can be written as follows:

daT

= f ... ai I ... f ( aT - 1 ) P, =0 13,=0 13.=0 aT - {3T - 1

II (m - a K)( a K ) P(a{3) X K'f"T OVqK OV - qK n(a{3) ,

(B9)

and by Eq. (2.12) m a,- I m a - {3

daT = L ... L ... L T T PC a{3). ~=o ~=O ~=o aT

CBlO)

APPENDIXC

The continuity equation for the local particle density PT is

BpT(t,X) ~--+ V' [uT(t,X)PTCt,x)]

at m

= L L C~(N;a{3). CCl) i = I (a./3)eYr ,a'f"p

If the system is spatially homogeneous and in equilibrium in

I

velocity space, so that for each T and each i

PT (t) NiT (t,x) = --= CT,

m then Eq. (CI) becomes

dpT(t) = f L C~(N;a{3). dt i = I (a,p)E'6 ,a 'f"fJ

Also, Eq. (C2) implies that Eq. (2.20) becomes n

Qa[N,r(a)] = II c;K(1_cK

)m-aK

K=l

(C2)

(C3)

= IT IT (_l)l-aK(m -aK)m-PK(t)I. K= II=aK m-/

(C4)

Since the number of different configurations r (a), such that r(a,T) "$i, for a such that aT < m is given by the formula

m-a n (m) nT+ (a,i) = T II'

m K= 1 a K '

(C5)

and the number of different configurations rea), such that r (a, T) 3 i for a such that aT;> 1 is given by the formula

a n (m) n; (a,i) = _T II ' m K= 1 a K

(C6)

then by Eqs. (3.5) and (C4) for each i

'I.(a./3)e·(,a#C~ (N;a{3) = ~a:a,<mbaTn/ (a,i)Qa[N;r(a)] - ~a:ar>1 daTnT- (a,i)Qa[N;r(a)]

= ~ {~a:a,<mbaT(m -aT)nZ=I(~)C:K(1-CK)m-aK

- ~a:a,>1 daTaTTIZ= 1 (~)C:K(1 - cK)m - a.J . (C7)

Using Eqs. (B6) and (BlO) for baT and daT , respectively, we obtain

~(a./3)e'C,a'f"pC~(N;a{3) = ~ {~a:a,<m~P:=o"·~.8,=a'+I·"~.8n=O({3T -aT)P(a{3) TI~=1 (~)C:K(1_CK)m-aK

which can be written compactly as

L C~(N;a{3) =.!. L ({3T -aT)P(a{3) IT (m)C;K(1_CK)m-aK. (C9) (a./3)E't' ,a 'f" 13 m (a,p)E'6' ,a "" fJ K = 1 a K

Since the right-hand side of formula (C9) does not depend on i, then by Eq. (C4) we get the LGCA rate law for the concentrationpT ofspeciesXT , i.e., Eq. (4.5)

dp(t) n m(m)(m-a) _T_= L C{3T -aT)P(a{3) II L K (_l)l-aKm-PK(t)/. dt (a./3)e'C.a'f"p K= 1 1= aK a K m - /

(ClO)

I See, for example, G. Nicolis and L Prigogine, SeljOrganization in Nonlin­earSystems (Wiley, New York, 1977); Oscillations and Traveling Waves in Chemical Systems, edited by R. J. Field and M. Burger (Wiley, New York, 1986); P. Berge, Y. Pomeau, and C. Vidal, Order Within Chaos (Wiley, New York, 1984); J. D. Murray, Mathematical Biology (Spring­er, New York, 1989).

2 The effects of fluctuations on such processes are often treated by a master equation approach. See, for instance, Nicolis and Prigogine, Ref. 1 and C. Vidal and H. Lemarchand, La Reaction Creatrice (Hermann, Paris, 1988).

3 J. BoissonadePhys. Lett. A 74,285 (1979); PhysicaA 113, 607 (1982); P. Ortoleva and S. Yip, J. Chern. Phys. 65, 2045 (1976).

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2776 Kapral, Lawniczak, and Masiar: Reactive dynamics in a lattice gas

4R. Kapral, A. Lawniczak, and P. Masiar, Phys. Rev. Lett. 66, 2539 (1991); R. Kapral, A. Lawniczak, and P. Masiar, in Lectures on Thermo­dynamics and Statistical Mechanics, edited by M. L. de Haro and C. Varea (World Scientific, NJ, 1991), p. 44.

sUo Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56, 1505 (1986); D. d'Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J. P. Rivet, Complex Systems 1, 648 (1987).

6 Lattice Gas Methodsfor Partial Differential Equations, edited by G. Doo­len (Addison-Wesley, New York, 1990); D. d'Humieres, Y. H. Quain, and P. Lallemand, in Discrete Kinetic Theory, Lattice Gas Dynamics and the Foundations of Hydrodynamics, edited by I. S. I. and R. Monaco (World Scientific, Singapore, 1989), p. 102; see also articles in Lattice Gas Methods for PDE's, edited by G. Doolen, Physica D 47, (1991).

7 There have also been studies of turbulent reactive flows: P. Calvin, P. Lal­lemand, Y. Pomeau, and G. Searby, J. Fluid Mech. 188,437 (1989); V. Zehnle and G. Searby, J. Phys. (Paris) 50, 1083, (1989); and to heat transfer and chemical reaction: O. E. Sero-Guillaume and D. Bernardin, Eur. J. Mech. 9,177 (1989).

8 G. A. Bird, Molecular Gas Dynamics (Clarendon, Oxford, 1976). 9 F. Baras, J. E. Pearson, and M. Malek Mansour, J. Chern. Phys. 93, 5747

(1990). lOR. Kapral, J. Math. Chern. 6,113 (1991). II D. Dab, J. P. Boon, and Y.-X. Li, Phys. Rev. Lett. 66, 2535 (1991). 12 The exclusion principle can be implemented in other ways, for example,

by allowing only one particle, regardless of species type, to occupy a node with a given velocity. The manner in which the exclusion principle is ap­plied can have consequences for the range of rate laws that can be investi­gated. For further discussion see Ref. II.

l3D. Dab, A. Lawniczak, J.-P. Boon, and R. Kapral, Phys. Rev. Lett. 64, 2462 (1990); A. Lawniczak, D. Dab, R. Kapral, and J. P. Boon, Physica D47, 132 (1991).

14 B. Chopard and M. Droz, J. Stat. Phys. 64, (1991). I'E. E. Selkov, Eur. J. Biochem. 4, 79 (1968). 16 This formulation, which does not explicitly consider the dynamics of the

A and B species, suppresses fluctuations in these species. A more general treatment of this model where the dynamics of all four reacting species is followed will be given elsewhere. In this more general formulation the exclusion principle leads to the restriction of equal forward and reverse rate coefficients.

17p. H. Richter, P. Rehmus, and J. Ross, Prog. Theor. Phys. 66, 385 (1981).

18 With these rules on the square lattice there is a checkerboard invariant. See Ref. 13 for details.

19 A. T. Winfree, SIAM Rev. 32,1 (1990). 20 See, for instance, Murray, Ref. 1. 21D. Walgraef, G. Dewel, and P. Borckmans, Adv. Chern. Phys. 49,311

(1982). 22 A. M. Turing, Philos. Trans. R. Soc. London, Ser. B 237,37 (1952). 23V. Castets, E. Dulos, J. Boissonade, and P. DeKepper, Phys. Rev. Lett.

64, 2953 (1990). 24Q. Ouyang and H. L. Swinney, Nature 352, 610 (1991). 2S I. Lengyel and I. Epstein, Science 251, 650 ( 1991). 26J. Pearson and W. Horsthemke, J. Chern. Phys. 90,1588 (1989); A. Ar­

neodo, J. Elezgaray, J. Pearson, and T. Russo, Physica D49, 141 (1991). 27 The diffusion coefficient ratio at which a Turing bifurcation is observed

can be reduced by considering parameters closer to the Hopfbifurcation point. We have not attempted such simulations where the relaxation times are longer since our intention was simply to demonstrate the feasi­bility of Turing bifurcation studies.

28X. Y. Li, Phys. Lett. A 147, 204 (1990); A. Rovinsky,J. Phys. Chern. 94, 7261 (1990).

29 Experimental situations like those in gel reactors and continuously fed unstirred reactors can be studied. See, W. Y. Tam, W. Horsthemke, Z. Noszticius, and H. Swinney, J. Chern. Phys. 88, 3395 (1988).

30G. Skinner and H. L. Swinney, Physica D 48, 1 (1990). 31 J. Maselko, J. S. Reckley, and K. Showalter, J. Phys. Chern. 93,2774

(1989). 32T. M. Liggett, Interacting Particle Systems (Springer, New York, 1985);

A. De Masi, P. A. Ferrari, and J. L. Lebowitz, J. Stat. Phys. 44, 589 (1986).

33See, for instance, D. Ben Avraham, Science 241, 1620 (1988); M. A. Burschka, C. A. Doering, and D. Ben Avraham, Phys. Rev. Lett. 63, 700 ( 1989); B. J. West, R. Kopelman, and K. Lindenberg, J. Stat. Phys. 54, 1429 (1989).

"P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954). 3S In a dense fluid t R - ps and ,{ R - A. The chemical relaxation time Tchem

for our simulations is Tchem - 104t R' corresponding to a fast reaction. For such fast reactions our results show that macroscopic self-organization which is robust under internal fluctuations occurs on the scale of hun­dreds of angstroms.

360ne may also scale the observed LGCA properties, like the wave veloc­ities and diffusion coefficients to correspond to a particular physical sys­tem and from this infer the length and time scales in the automaton discre­tization. The fluctuations will not necessarily be accurately given in this case without further coarse graining or other modifications in the dynam­ics.

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