A probabilistic lattice gas cellular automaton model of a chemically reacting system is constructed. Microdynamical equa
Physica D 47 (1991) North-Holland
132-158
REACTIVE
LATTICE
GAS
AUTOMATA
Anna LAWNICZAK Department
of Mathematics
and Statistics,
University
of Guelph, Guelph, Ontario,
Canada I?1 G 2 Wl
David DAB Faculte’ des Sciences,
Raymond
C.P. 231,
Universite’ Libre de Bruxelles,
1050 Bruzelles,
Belgium
KAPRAL
Chemical Physics Toronto, Ontario,
Theory Group, Department Canada, M5S IAl
of Chemistry,
University
of Toronto,
and Jean-Pierre
BOON
Faculte’ des Sciences, Received
15 Februry
C.P. 231, UnivetsitP
Libre de Bruaelles,
1050 Bruzelles,
Belgium
1990
A probabilistic lattice gas cellular automaton model of a chemically reacting system is constructed. Microdynamical equations for the evolution of the system are given; the continuous and discrete Boltzmann equations are developed and their reduction to a generalized reaction-diffusion equation is discussed. The microscopic reactive dynamics is consistent with any polynomial rate law up to the fourth order in the average particle density. It is shown how several microscopic CA rules are consistent with a given rate law. As most CA systems, the present one has spurious properties whose effects are shown to be unimportant under appropriate conditions. As an explicit example of the general formalism
a CA dynamics is constructed for an autocatalytic reactive scheme known as the Schlijgl model. Simulations show that in spite of the simplicity of the underlying discrete dynamics the model exhibits the phase separation and wave propagation fluctuations
phenomena expected for this system. Because of the microscopic on the evolution process can be investigated.
nature of the dynamics the role of internal
1. Introduction Nonlinear chemically reacting systems exhibit many different types of spatial and temporal behavior, for example, chemical oscillations and waves [1,2]. Typically the analysis of such phenomena is based on a reaction-diffusion equation. This macroscopic description will be adequate if the phenomena of interest occur on sufficiently long distance and time scales and fluctuations do not play an important role. The macroscopic reaction-diffusion equation has its basis in an underlying molecular dynamics which is necessarily quite complex for a chemically reacting system. In this article we explore a class of microscopic models for nonlinear reaction-diffusion systems. We adopt the probabilistic lattice gas automaton approach [3,4] w h ere space, time and particle velocities are discrete. In spite of this simple microscopic description the macroscopic behavior of the automaton is consistent with that obtained from the reaction-diffusion equation and, furthermore, the lattice gas analysis allows one to explore the role of internal fluctuations on the dynamics of this nonlinear system. A number of simplifications of the microscopic dynamics is made in the construction of the cellular 0167-2789/91/$03.50
@ 1991 - Elsevier Science Publishers
B.V. (North-Holland)
Physica D 47 (1991) 132-158 North-Holland
REACTIVE
LATTICE
GAS AUTOMATA
Anna LAWNICZAK Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada NIG 2W1 David DAB Facultd des Sciences, C.P. 231, Universitd Libre de Bru~elles, 1050 Bruzelles, Belgium Raymond KAPRAL Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada, M5S 1A1 and Jean-Pierre BOON Facultd des Sciences, C.P. 231, Universitd Libre de Bruxelles, 1050 Bruzelles, Belgium Received 15 Februry 1990
A probabilistic lattice gas cellular automaton model of a chemically reacting system is constructed. Microdynamical equations for the evolution of the system are given; the continuous and discrete Boltzmann equations are developed and their reduction to a generalized reaction-diffusionequation is discussed. The microscopic reactive dynamics is consistent with any polynomial rate law up to the fourth order in the average particle density. It is shown how several microscopic CA rules are consistent with a given rate law. As most CA systems, the present one has spurious properties whose effects are shown to be unimportant under appropriate conditions. As an explicit example of the general formalism a CA dynamics is constructed for an autocatalytic reactive scheme known as the SchlSgl model. Simulations show that in spite of the simplicity of the underlying discrete dynamics the model exhibits the phase separation and wave propagation phenomena expected for this system. Because of the microscopic nature of the dynamics the role of internal fluctuations on the evolution process can be investigated.
1. I n t r o d u c t i o n N o n l i n e a r c h e m i c a l l y r e a c t i n g s y s t e m s e x h i b i t m a n y different types of s p a t i a l a n d t e m p o r a l behavior, for e x a m p l e , chemical oscillations a n d waves [1,2]. T y p i c a l l y t h e a n a l y s i s of such p h e n o m e n a is based on a r e a c t i o n - d i f f u s i o n e q u a t i o n . T h i s macroscopic d e s c r i p t i o n will be a d e q u a t e if the p h e n o m e n a of i n t e r e s t occur on sufficiently long d i s t a n c e a n d t i m e scales a n d f l u c t u a t i o n s do not play a n i m p o r t a n t role. T h e macroscopic r e a c t i o n - d i f f u s i o n e q u a t i o n has its basis in a n u n d e r l y i n g m o l e c u l a r d y n a m i c s which is necessarily q u i t e c o m p l e x for a c h e m i c a l l y r e a c t i n g system. I n this article we explore a class of microscopic m o d e l s for n o n l i n e a r r e a c t i o n - d i f f u s i o n systems. We a d o p t the p r o b a b i l i s t i c lattice gas a u t o m a t o n a p p r o a c h [3,4] where space, t i m e a n d particle velocities are discrete. I n spite of this simple microscopic d e s c r i p t i o n the macroscopic b e h a v i o r of the a u t o m a t o n is c o n s i s t e n t with t h a t o b t a i n e d from the r e a c t i o n - d i f f u s i o n e q u a t i o n a n d , f u r t h e r m o r e , the lattice gas analysis allows one to explore the role of i n t e r n a l f l u c t u a t i o n s on the d y n a m i c s of this n o n l i n e a r s y s t e m . A n u m b e r of simplifications of the microscopic d y n a m i c s is m a d e in the c o n s t r u c t i o n of the cellular 0167-2789/91/$03.50 (~) 1991 - Elsevier Science Publishers B.V. (North-Holland)
A. Lawniczak et al. / Reactive lattice gas automata
133
a u t o m a t o n model. Normally we shall be interested in reactions taking place in a solvent, so interactions of the chemical species with the solvent must also be taken into account. A complete description of the reactive and nonreactive dynamics would then entail a consideration of several species. However, it is possible to construct a much simpler model t h a t still preserves the features of the full system dynamics. The only aspect of the solvent dynamics that is of interest is the fact that collisions with the reacting molecules can change the velocities of these species. If the solvent is in excess then collisions between solvent molecules will occur frequently and will serve to maintain the velocity distributions of these species close to equilibrium. On the other hand, the system can be constrained to lie far from equilibrium by fixing the concentrations of some species; without loss of generality we consider this to be the case for all species except one, say X. In view of these considerations we m a y ignore the details of the dynamics of all species but X , as well as the solvent and focus solely on the dynamics of X . In qualitative terms the cellular a u t o m a t o n model is constructed in the following way. Space is made discrete by restricting the dynamics of the X species to take place on a square lattice. In addition to the X species we assume there exist "ghost" particles which serve to account for the presence of other species and solvent molecules. The X molecules can undergo two types of collisions: elastic and reactive. Elastic collisions are modeled by assuming that the particles undergo a r a n d o m walk generated by local r a n d o m rotations at a node of the lattice. Reactive collisions consist of random species changes in accord with the kinetics of the system and r a n d o m rotations to simulate the effect of velocity changes that occur as a result of the reaction. Particles move from node to node with unit velocity. This basic model consists of the single X species with four discrete unit velocities. We note t h a t in this ghost particle description (i) the microscopic reactive dynamics does not satisfy mass conservation, and (ii) the fact that the concentration of the ghost species is constant m a y influence the observed fluctuations in the X species. We give a precise m a t h e m a t i c a l formulation of the lattice gas cellular a u t o m a t o n rules and the microdynamical equations for the Boolean fields describing the system state. Space and time scaling transformations are considered in order to investigate the kinetic regime where the a u t o m a t o n can be reduced to a continuous space and time B o l t z m a n n equation. We also present a reduction of the linearized Boltzmann equation to a generalized reaction-diffusion equation. A discrete lattice Boltzmann equation is derived and the spurious invariants inherent in most lattice gas models [5] are discussed. A u t o m a t o n simulations are presented for the application of this class of a u t o m a t a to a specific reaction-diffusion scheme: the SchlSgl model [6].
2. C e l l u l a r a u t o m a t o n
model
T h e cellular a u t o m a t o n model for a chemically reacting system [7,8] can be described in formal terms as follows: Particles of species X move on a square lattice £~ c Z 2, which is a square centered on the origin with sides equal to the integer part of e -1 and periodic b o u n d a r y conditions. At each node, labeled by the discrete vector r , there are four cells labeled by an index i defined modulo four. The cells are associated with the unit velocity vectors c{ connecting the node to its four nearest neighbors. (We assume i increases counterclockwise and 1 corresponds to the positive direction of the x-axis.) An exclusion principle forbids two particles to be at the same node with the same velocity; therefore, each cell (r, {) has only two states coded with a Boolean variable s/(r): s~(r) = 1 = 0
occupied, unoccupied.
(2.1)
A configuration of particles at a node r at time k is described by a random vector r/(k, r ) -- ( y i ( k , r)//4=l with values in a state space S of all (24 -- 16) four bit words; i.e., S = {s = ( s i ) i4= l : si = 0
or 1 for i = 1 , . . . , 4 } .
(2.2)
134
A. L a w n i c z a k et a L /
Reactive lattice gas a u t o m a t a
A configuration of the lattice/:~ at time k is described by a Boolean field n ( k ) -- {n(k, r): r ~ £~},
(2.3)
with values in a phase space F = S & of all possible assignments s(-): s(.)
=
{s(r)
=
(si(r))?_,: r • £ ~ } .
(2.4)
The evolution of the system occurs at discrete time steps and the cellular a u t o m a t o n updating rule can be defined on the Boolean field (2.3). At each node at each time step the updating rule consists of propagation followed by collisions, which m a y be either elastic or reactive. For generality we assume that collisions occur with probability p (thus, no collision with probability (1 - p)). The lattice gas a u t o m a t o n rule consists of a product R o C o P of three basic operators: rotation R, chemical transformation C and propagation P. We first describe these building blocks of the lattice gas dynamics and then express the updating rules in terms of the microdynamical equations for the Boolean fields. 2.1. Propagation During the propagation step each particle moves in the direction of its velocity from its cell to a nearest-neighbor node. The velocity of each particle is conserved during this operation. We denote the propagation operator by P and the configuration in the ith direction after the propagation step by ~P. Thus, P: ~?~(k, r) = y,(k - 1, r - ci).
(2.5)
2.2. Rotation Particles change their velocities as a result of r a n d o m rotations R. The rotation operations are local and thus only involve particles at a given node; clearly rotations do not change the number of particles at a node. At each node, independently of the others, the configuration of the particles is rotated by 7r/2 or - 7 r / 2 with probability 1/2. In more formal terms we let { & / 2 ( k , r): r •
k = 1, 2 , . . . }
(2.6)
be a sequence of independent copies of r a n d o m variables ~ / 2 such that the probabilities P r satisfy Pr(~,/2 = 1) = P r ( ~ / 2 = 0) = 1/2.
(2.7)
If r]~ denotes a configuration in the ith direction after a rotation at a node we have R : l]iR = ~rr/21]i+3 q- (1 - ~r/2)T]/q_l,
i = 1,...,4,
(2.8)
where the subscript addition is modulo four. We m a y also write this equation in the form = ,i + &/2(-,i
+ 7,+3) + (1
-
+
-
+
(2.9)
which defines the collision t e r m A~(r/) (indeed the rotation operator redistributes the velocities as collisions with solvent particles would) t h a t takes on the values { - 1 , 0, 1}. In (2.8) and (2.9) we have dropped the dependence of ~h on k and r. We adopt this convention in the sequel when confusion is unlikely to occur. Rotations can also be described in terms of a probability matrix. A particular rotation at a node can t f f be defined by an input state s = (Sl, s2, s3, s4) and an output state s' = (s~, 82, 8 3 , 8 4 / with an associated probability R ( s , s'). For example, for the rotation s = (1, 0, 1, 1) --~ s' = (1, 1, 0, 1) the associated probability is
135
A. L a w n i c z a k et al. / Reactive lattice gas a u t o m a t a
(2.10)
R ( ( 1 , 0 , 1, 1}; (1, 1,0, 1}) : R(s: s') = 1/2.
It is convenient to define a r o t a t i o n m a t r i x for all s and s', even if the r o t a t i o n rules do not provide for an actual transition between s a n d s~; for such transitions R(s; s') can simply be set to zero. Note also t h a t for an input state t h a t does not change, we have R(s; s t) -- 1 for s -- s'. In this way the entire set of collision rules can be neatly encoded into a single 16 × 16 r o t a t i o n m a t r i x t t with elements R(s; s').
2.3. Chemical transformation In the chemical t r a n s f o r m a t i o n step at each node, i n d e p e n d e n t l y of the others, particles are r a n d o m l y created or annihilated in reactions of the t y p e a X ---, f i X with the net reaction probabilities P ~ = P ~ ( e ) (a,/3 = 0 , . . . , 4) regardless of their velocity state. T h e diagonal elements P~o of the transition probability m a t r i x P = [P~z] correspond to nonreactive events and we m a y write
P~
: 1- ~
P~.
(2.11)
T h e off-diagonal elements of P for which c~ > /3 correspond to reactive transitions where particles are destroyed while if a < / 3 particles are created. Next we describe the chemical t r a n s f o r m a t i o n s in terms of the m i c r o d y n a m i c a l variables ~. Let
Table 1 Rotation probability matrix R. 0000
1000 0100
I00O 0100
1/2
1/2
0010 0001
0010
II00
i010
I001
0101
0011
0110
Iii0
II01
I011
0111
IIII
I//2
1/2 1/2
1/2
0001
1/2 i/2
II00
1/2
1/2
I010 I001
1/2
1/2
0101 0011 0110
1/2 1/2
1/2 1/2 1/2
III0 II01
1111
1/2
1/2
1/2
IOli 0111
1/2
1/2
1/2 1/2
11
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136
{~(k,r):i=l,...,4;reL~,k=l,2,...},
a#~,
0_ / 3 , or 0 if a < / 3 .
(2.22)
As in the case of rotations, chemical t r a n s f o r m a t i o n s can be described by a chemical t r a n s f o r m a t i o n p r o b a b i l i t y m a t r i x . For given input a n d o u t p u t states each reaction has an associated p r o b a b i l i t y which we write as C(s; s'). For e x a m p l e , for the reaction s = (1, 1, 0, 0} ---* 8' = (1, 1, 1, 0), the associated p r o b a b i l i t y is C (
