Indian Journal of Chemistry Vol. 44A, February 2005, pp. 286-290

Notes

Probabilistic cellular automata model for reaction diffusion systems

K A Khan* & R A Khan

Department of Physics, S.P. College, Sri nagar (J&K) 190001 , India

Received 30 November 2004

Class of cellular automata (CA) for modeling reaction diffusion systems has been presented. The construction of the CA is general enough to be applicable to large class of reaction diffusion equation s. Tne automata are based on running average procedure and on probabilistic table look up to implement diffusion reactions. The evolution of probabilistic CA simulates a spatially distributed process, given by a partial differential equation (POE). The solutions are very isotropic despite the discreteness and the ani sotropic nature of the CA. The adverse effects of discretization are overcome by the use of probabili stic rules. In order to obtain the assurance of the proposed method, the probabilistic CA approach for evolution of reaction diffusion process usually given by POE is proposed.

Cellular automata (CA's) are well known and widely used systems. They are discrete dynamical systems with simple construction but complex self-organizing behavior, which provide a comprehensive study of the complex natural phenomenon l

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• It is made up of cells like the points in a lattice or like the squares . of a checkerboard. It is called automaton because it follows a simple digital (discrete)rule.

Reaction-diffusion (R-O) systems are an important class . of systems to investigate nonlinear behavior. They also represent I1!any problems arising in physics, chemistry, biology, and other disciplines. CA models have been used in many applications to model reactive and diffusive systems4

,5. Most uses of CA can be classified into one of four approaches: (i) Ising­type models of phase transitions; (ii) lattice gas models (the lattice gas method was initially developed to model hydrodynamic flows and has been extended in many directions6

,7; (iii) systematic investigation of the behavior of CA by investigating all rules of a certain class (e.g" all possible rules for one­dimensional automata with two states and nearest neighbor interaction); and (iv) qualitative discrete

*For correspondence Phone: 0194-2490268; Fax: 0194-246828 E-mail: khurshid545@rediffmail.com Khurshid545@yahoomail.com

modeling (including operational use of CA as an alternative to POEs8, These models are generally based on qualitative rather than quantitative information about the system to be modeled. A probabilistic CA model is constructed which preserves the qualitative features deemed most relevant and it is then investigated. Existing CA models for reaction-diffusion systems9

-12 fall into the

category (iv), i,e., they show qualitatively "correct" behavior and are restricted to certain R-O models and certain types of phenomena.

The main idea behind the proposed class of CA is careful discretization. Here we introduce a class of CA that models the POEs also in a quantitatively correct way, The basic idea is not to try to describe a complex system, using difficult equations but simulating this system by interaction of cells following easy rules. In other words: Not to describe a complex system with complex equations but let the complexity emerge by interaction of simple individuals following simple rules I 3

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• They are absolutely stable, have no rounding of errors and are less ti~e consuming. Moreover, border conditions in CA are straight forward, which makes them appropriate to simulate luids in porous media.

In this paper we describe a class of CA which IS

suitable for modeling many RO systems in a quantitatively correct way. The new CA is operationally more efficient than the reactive lattice gas methods, which also achieve quantitative correctness I6

-18

, We first describe the construction of the probabilistic CA; then we present the automata using a chemical reaction described by a second order POE as an example. The procedure of transforming a real spatial function into a Boolean cellular array and

. the method for constructing a probabilistic CA approximation of a POE system along with reaction diffusion process are also presented in this paper.

Probabilistic CA as an Alternative to Partial Differential Equation

The heart of the method is the transformation of a real spatial function into a Boolean array whose averaged form approximates the given function. Two parts of a given POE (a differential operator and a function) are approximated by a combination of their Boolean counter-parts. The resulting CA transition

NOTES 287

function has a basic (standard) part, modeling the differential operator and the updating part modifying it according to the function value. The general form of PDE representation is as follows

n = 1,2, g = 1, ... l, ... (1)

where y7l1 is an n-order differential operator, fg (UI • . .

u/) - an arbitrary function with the domain and the range in the interval (0, 1).

If n = 1, the equation system of the type of Eq. (1) describes, for example, electromagnetic process. If n = 2, i.e. V2 is a Laplacian, then Eq. (1) represents a reaction-diffusion phenomenon. When numerical methods of PDE solution are used, it is precisely these operators which cause the main troubles in providing stability of computation. Nevertheless, for both above types of differential operators there exists well-known CA, whose evolution simulates the corresponding processes. The examples may be found in Simon's model 10, where a CA-model is · proposed for electromagnetic field and in Weimar modeI4

•s, where

CA-diffusion models are studied. This brings up to the idea of replacing the finite difference form of V" u by simple and reliable CA models, which are further referred to as standard CA's. Clearly, this transfers the computation process to the Boolean domain, which requires to perform the addition of fg (u\, ... u/) to a standard CA result in the Boolean form. Such a procedure is further referred to as updating. Its formal representation is as follows .

After time and space discretization with l = tr, t = 0, 1, . . . , and x = hxi, Y = h.j, Z = hzk, the system (1) takes the form

U g (t + 1) = ug (t) + d'V"ug (t) + 'r f g CUI (t), .... ,u l (t)),

g = 1, ... . ,l. " . .. (2)

where d'V"ug is a finite difference representation of

the differential operator, dl depends on -c, d and h, particularly in the case of n = 2, d' = -cd/h2

• In the right hand side of Eq. (2) two first terms are responsible for a process to be simulated by a standard CA, while the function Jg(u\, . . .. ,UI) should be turned out into a Boolean form and added to the standard CA result at each iteration of the simulation process .> Since there are l variables to be simulated, the naming set M is considered to be composed of l parts, M = Ug=1 Mg,

forming a layered structure so that each mg = (i, j. k)g

E Mg has a single name with the same (i. j; k) in each

g-th layer. Let the values of ug (t, x, y. z), g = 1, ... . , l, be the

solutions of Eg. (1), which are taken as reference. Then the problem of constructing a CA, which simulates the same process, is stated as follows. Given a PDE in the form of Eq. (2) and an initial

array " OR (0) = {(ulII (O), . m)}, a CA should be

constructed whose evolution starting from ' tHe

Boolean discretization Q B (0) == {(u/1I (0); 'm)} . of

QR'(O) provides at each t-th iteration ' for any' mEN! that

UIII

(t) - <, (t) < E ' " (3)

where v'is the averaged vafue ~ver a certain averaging neighborhood. The latter may be different in different layers, so, NAv (mg) ={ (Uj (mg), ¢j (mg»: j = 0, .. . ,qg } with ¢j(mg) E Mg .

As it was mentioned above, the transition rule of a reSUlting CA is a combination of two procedures: 1) computation of the next state of a standard part and 2) updating it according to the functions f" ... . ,.Ii values. The first procedure follows the chosen standard 'CA­model, which is not described here, but a representative example is given in detail in the next section. The updating procedure relies upon the same probabilistic rule than that used for Boolean discretization of a real function, but with the account that the function value constitutes only a part of the total averaged cell state.

If fg (V'llli (t), .. .. , V'lli/ (t)) = hili> 0, then the updating should increase the amount of one's in NAv (mg ) .

Hence, a cell (0, mg) may, probably, be changed into (1, mg). Since in any averaging neighborhood NAv (mg)

there are qg (1-w' rug) zeros, then the probability of that change is

1,,, (t) . p . = --,,-' --

(0."" )-'->(1.",,) 1 _ I () w", t , .,. (4)

when flllg < 0, the updating should decrease the resulting averaged value, which is done by changing the cell (1, mg) into (0, mg) with the probability

.. . (5)

288 INDIAN J CHEM, SEC A, FEBRUARY 2005

Denoting the right hand sides of Eqs (4) and (5) as r(jlllg) and T(jmg) , respectively, the updating procedure is as follows.

II'if Will, (t) = O,flll, (t) > 0, T+ (fill, ) > rand (l),

VIII, (t + I) = 0, if Will, (t) = 1, /'''' (t) < 0, r (/'''' ) > rand (1),

VIII (t) otherwise .. .. .(6)

where rand (1) is a random number from the interval (0, 1).

Let the result of standard CA' s tth iteration in each gth layer be Q s (t) = {( Wg (t), mg)} and its averaged form-.Q:. (t) = {(w'g (t), mg)}, then the updating

procedure according to Eq. (3) should meet the following condition. For any cell mg E Mg, g = 1, .. . ,1, and any t = 1,2, .....

<, (t) - (w;',. (t» + Ix «, (t), .. .. , <, (t» < E ' " (7)

where all terms are real values in the interval (-1,1). From above, it follows that a CA, whose evolution

simulates the same process than a given PDE, is a multilayer cellular array, each layer corre~ponding to an equation of the system. Each cell is capable to perform two functions: a Boolean function of a standard CA, and updating the result according to Eq. (6). Both functions are computed by all cells in parallel, the whole array changing its global state iteratively. Since Eq. (6) may be considered .as a function of probabilistic neuron, such a cellular array is also called cellular-neural automaton.

Simulation of reaction diffusion processes by prob.abilistic CA

A number of computer experiments can be canied out in order to obtain the assurance that the proposed method works properly . Our model for simulation of reaction diffusion process is based on the fact that the founding element is the cell, surrounded by neighbouring cells in case of Moore Neighbourhood of 2D CA lattice. .

In 2D CA the cells are arranged in a two­dimensional grid with connections among the neighbouring cells. Consider a 2D CA comprising rn x n cells organized as an m x n array with m rows and n columns. The state of the CA at any time instant can be represented by an m x n binary matrix. The neighbourhood function specifying the next state of a particular cell of the 2D CA is affected by the current state of itself and eight cells in its nearest neighbourhood.

Mathematically, the next state q of the (i, j),h cell of a 2D CA is given by

.[qi_l,j_l (t), qi-l .j (t), qj-I,j+ l (t)]

qi/t + 1) = I qi.j-l (t), qi}t), qi.j+l (t)

qi+l ,j-l (t), qi+I.j (t), qi+l.j+l (t)

Here I is the Boolean function of 9 variables. To express transition rules of such 2D CA model, the specific rule convention along with tranSItIOn behaviour is as shown in Figs ] a, I band 1 c respectively.

The central box repre,ents the current cell (that is, the cell being considered) and all other boxes represent the eight nearest neighbours of that cell. The number within each box represents the rule number associated with that particular neighbour of the current cell-that is, if the next state of a cell is dependent only on its present state, it is referred to as rule 1. If the next state of a cell is dependent on the present state of its right neighbour, it is referred to as rule 2. Similarly, if the next state of a cell is dependent on the present state of its top-left cell it is referred to as rule 64 and so on. These are called primary rules. In case, the next state of a cell depends on the present state of itself and/or its one or more neighbouring cells (including itself), the rule number will be the arithmetic sum of numbers of the relevant cells. For example, rule 3(=1+2). These are called secondary rules.

The simulation of 2D reaction-diffusion process by proposed CA model is usually given by a PDE. In this case, the CA diffusion is called as Block-Rotation (BR-diffusion) model, which is used as a standard CA part.

BR-diffusion is a synchronous two step CA, processing the cellular array OB with the naming set M= {(i,j) : i,j = O, .... }. Two partitions into square 2 x 2 blocks are defined on the array. The first one consists of even blocks, their diagonal cell name component sums (i + j) being even. The second partition is the odd one. In both cases, the cell neighborhood N(i,j) of a cell (v, (i,j» E On is the block, where this cell is in its left top corner i.e.

N(i,j) = {(vo,(i,j », (v" (i+lJ », (V2, (i+lJ +1», (V3, (i,j +l)}. . .. (8)

CA rules are the following probabilistic ones.

NOTES 289

64 128 256

32 1 2

16 8 4

(b) (c) (a)

2DCA grid 2DCA transition dynamics 2DCA rule formation

description

Fig. I-A cell at the centre and its possible transition along different directions in a 2D CA is shown in Figs la, I b & Ic represent the lattice, a cell and the rule description respectively.

lS~ (I) if rand (1) < p,

Nij(t+l)= Si~(t)ifrand(1)?(1:-p),

N ij (t) otherwise

... (9)

where rand (1) is a random number in the interval (0,1), and the sub-arrays in the right-hand side are as follows

S'dt)={ «vJ,(iJ)),( V2,U+ 1 J)),(V3,(i+ 1 J+ 1)), (vo,(i,j+ 1)) }

S"i,iCt)= { « V3,(i,j)),( vo,(i+ 1 J)),( vJ,(i+ 1 J+ 1 )), (v2,(i,j+ 1))}.

At the first step of each iteration, the above rules are applied to the even blocks, rotating them on 1[/2 either clockwise with the probability equal to p, or counter clockwise with the same probability. At the second step, the same rules are applied to the odd blocks. In BR-diffusion, the model with p = 1/2 is proved to simulate diffusion with the value d' from Eq. (2) equal to 3/2 and another diffusion coefficient value may be obtained by appropriate choice of p, 1',

and h.

Chemical reaction A standard CA consists of an array of cells, each of

which can be in one of a finite number of possible states, updated synchronously in discrete time steps, according to a local identical interaction rule. The state of a cell at the next time step is determined by the current states of a surrounding neighborhood of cells 14-16. A great number of second order POE systems modeling a reaction of Belousov-Zhabotinsky

has been proposed and studied. One of them (a simplified one) from Tyson model 19 is chosen as an example for replacing the POE system by a probabilistic CA. Using the notation of the paper the POEs may be expressed as follows:

au)at = d (a 2ul /aX2 + a2ur/al)+ II (u"u2 ),

... (0)

du2 / dl = 12 (u"u2 )·

in Eq. (10) d is a diffusion coefficient,

.t; (u"u 2 ) = 1/ d (u l (1- ul ) - ZU2 (u l - s)/ (u l + s)),

12 (u l ' u2 ) = ul - u2

where s is the initial density of the substances in the medium, i.e. (UI(O) = s, U2 (0) = s), z(x, y) is the spatial non-linear function of both concentrations. With x = 0, y = 0 being chosen to be waves centre, the constraints on z (x, y) are expressed as follows: 0.5<z (O,O)<(1-s). The process begins when in the homogeneous mixture of two substances with the certain gel there appears a spot of saturated U I.

According to the order of a POE, the discrete array is a 2-layered one, i.e.

M =M'UM' M' I 2' g

={(i,j)g :i,j=-n ... ,O, ... ,n,g =1,2n=150}.

The initial arrays QII, (0) = {(s,U, j)g)}, g = 1,2,

s = 0.1, except that in the center of QII1

(0) there is a

square of cells with u, = 1. Boolean discretization of ~2HI (0) is achieved by

setting cell states v = 1 with the probability

290 INDIAN J CHEM, SEC A. FEBRUARY 2005

t=O t=20 t=30 t=40 t=50

Fig. 2-CA evolution at different intervals, displaying the reaction diffusion of a step in the averaging CA.

~I', =I) = s, except the square in the center of QR"

where v = I determinately. Since the second equation in (10) has no diffusion part, there is no need to perform Boolean discretization of Q R (0). ,

Each iteration of the CA approximating the POE system (10) consists of the following computations. (i) An iteration of BR-diffusion is applied to

Q R resulting in Q s (t+I); (ii) the functions!, and/2 , I

are computed for all cells (i, j) E M I forming cellular arrays Q F, and Q F,' The latter is also used as

QR, (t+I); (iii) updating QS,

(t+l) is performed

according to (7) resulting in Q u (t+I); and (iv) I

QUI (t+l) is averaged to obtain QR, (t+l).

The evolution of CA for reaction diffusion phenomenon at different intervals is shown in Fig. 2 . .

Discussion In this study we have reported the construction of

probabilistic CA model which works parallel to the POEs. The main success of the proposed model is the adequate modeling of fluids in porous medium and other complex dynamic systems. The most spatial dynamical processes have a well studied mathematical representation in the form of POEs. Thus, it seems very natural to look for a formal method of transforming a given POE system into a CA, whose evolution approximates its nonstationary solution. The resulting probabilistic CA model is a paradigm of fine grained parallel computational model, intended for, being included in a unified

technology for parallel programming. This new class of CA is suitable for modeling many RO systems in a quantitatively con-ect way. The new CA is operationally more efficient and more versatile than the reactive lattice gas methods, which also achieve quantitative con-ectness. They are also related to coupled map lattices and discrete methods derived from them.

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