A Diffus ion-Limited Reac t ion

Martin A. Burschka

Inst. fiir theor. Physik III, H.-Heine-Universit/it, D-37083 Diisseldorf, Germany

A b s t r a c t . Fluctuations in diffusion-controlled reactions lack the necessary fea- tures for a mesoscopic description. We show how the correlations dominate the dynamics by juxtaposing the macroscopic dynamics of a cellular automaton model for the diffusion-controlled limit to the deterministic diffusion-reaction rate equation for the same reaction. A more detailed N-body master equation is then presented in which explicit diffusion-controlled limits are explained.

1 W h at Is Special About Diffusion-Limited React ions?

Common diffusion-reaction-rate equations are similar in structure and con- cept to the hydrodynamic equations for simple fluids and appear to be a simpler paradigm for general purposes. To the statistical physicist, however, systems with ongoing chemical reactions pose a more fundamental challenge than simple fluids in that two familiar properties of fluctuations cannot be taken for granted there:

a) slow variation in comparison to the fast microscopic (i.e. molecular) dy- namics due to conservation laws on the microscopic level,

b) continuous variation, as it is found if the fluctuating quantities are sums of many small contributions which may vary more or less independently (e.g. all sorts of densities).

Fluctuations with these two properties allow an extension of the determinist ic dynamics of continuously varying macroscopic observables into mesoscopic stochastic dynamic laws without touching on the microscopic discrete details. Fluctuations in the mass-, momentum- , and energy- densities in simple fluids are prime examples amenable to such t reatment .

Diffusion-reaction rate equations, too, are kinetic laws for densities, named concentrations, just like the hydrodynamic kinetic equations. This formal analogy to does not carry very far, however, because the concentrat ions are not conserved under the microscopic dynamics (property (a)) and, in gen- eral, do not vary slowly 1. So these equations apply only in regimes where the effect of the reactions is small compared to t ransport over macroscopic distances. In particular, this excludes the diffusion-controlled regime [1-9] far f rom equilibrium, where the reaction rate is limited by t ranspor t of the sin- gle reactant particles towards each other. There the microscopic correlations

1 There may exist other quantities which are conserved under the microscopic dynamics, which do not necessarily vary continuously.

A Diffusion-Limited Reaction 89

may dominate the dynamics so that working assumptions which are common for fluctuations in fluids may turn out as unphysical.

In the following, we s tar t with two extreme models of a simple diffusion- reaction system with a reversible autocata lyt ic reaction, namely the mean- field diffusion-reaction-rate equation and a stochastic cellular au toma ton mod- el for the diffusion-controlled limit case to show how the macroscopic dynam- ics deviate in both cases. We then explain a slightly more general model, in which we discuss explicitly ways to take diffusion-controlled limits. The re- duced l imit-dynamics in our example depends on a special conservation law, which determines the convenient choice of variables. It corresponds concep- tually to the hydrodynamic level of description in fluids al though there is no formal similarity to the BBGKY-hierarchy of kinetic equations for the mul- tiple point densities (mixed moments) . We conclude with open questions for future research and a "message".

2 Two Descript ions of a Dif fus ion-React ion S y s t e m

We consider a common autocatalyt ic process, like it is found in the famous pat tern-forming Belouzov-Zhabotinsky-reaction:

A ~ 2A (1)

There ~, it is it is implemented as

BrO 3 +IHBr02]+ 2Fe(II) + 3 H + ~- 2[HBr021+ 2Fe(III) + H20

In the regime where - - apar t from H20 -- the substances Br03, Fe(II), H +, Fe(III), are in ample supply with effectively constant concentrations, the dynamics occurs entirely in the concentrations and correlations of bro- mate HBr02. The mean-field rate constants gl , t;2 apply in the absence of correlations between the HBrO~.-molecules, e.g. in thermal equilibrium or for a short t ime following an initial condition with no correlations 3.

Actually, A could stand for other entities like e.g. individuals of a pop- ulation. The model resembles the logistic system, in that it combines the autocatalyt ic process (--~) with a simple inhibitory process (~--). On the mi- croscopic level we will consider the unique version of it which complies with the "detailed balance" condition, i.e. tha t there exists a thermal equilibrium state with reversible microscopic dynamics.

The simplest description of this system is the deterministic mean-field diffusion-reaction rate equation for the concentration p (r; t) of A:

Otp (r; t) = DAp (r; t) + ~;2P (r; t) (Peq--P (r; t)) (2)

in the Oregonator model[10] of this reaction a This could be prepared as a thermal equilibrium state for some other value of an

external parameter (temperature, [H +] , eta.)

90 Martin A. Burschka

--~_x Diffusion-control can only be incorporated via t ime and con- with P~I - , ~ " centrat ion-dependent reaction rate "constants", changing the status of the equation from mean-field approximat ion of a more detailed theoretical model to an interpolation formula with fitted parameters for a special case. Failure of this equation to describe experimental da ta qualitatively is strong evidence for correlations on the molecular scale.

A minimal model for the diffusion-limited reaction must account for mul- t iple-point correlations, like the following stochastic cellular au toma ton on a cubic lattice with the processes:

D ~ 2

t~l/A2d ) t~2/(2__2 tied)

I @ ] I ~2/ (2%'d) t @ 1 @ /'£1/(%d) ) ] @ ) (4) where the squares denote adjacent lattice sites, which may be occupied (-) or vacant. Equation (3) shows the diffusive jumps which make every particle perform a random walk with j u m p rate D~ -2 , where t~ is the lattice con- stant. On large scales this appears as diffusion with diffusion constant D in correspondence with the first rhs. term in (2 ) - - at least in the absence of the reaction process. In (4), the transitions towards the center mimic the prolifer- ation reaction X --~ 2X and the reverse ones from the center the coagulation reaction X ,-- 2X. Again, the mean field values are displayed in leading or- der in e (4). They lead to the reaction terms in (2 ) - - at least for statistically homogeneous initial conditions, i.e. t ranslation invariance of all expectat ion values of the density: For the coagulation, the mean-field probabil i ty for two

adjacent sites to be occupied is (pgd) 2 . Together with the density of pairs of sites d~ -d this gives the mean-field coagulation rate of n2p 2 (5). Similarly, for proliferation, the rate ~ / ( 2 d ) to generate a new particle at an adjacent vacant site together with the density of occupied sites p and the number of adjacent vacant sites 2d + O (gd) gives the mean-field proliferation rate niP •

The diffusion-controlled limit of this model can be identified by two equiv- alent interpretations: a) Equate the probabil i ty rate for two adjacent occupied sites to coagulate with their probabil i ty rate to come into "contact" - (i.e. occupy the same site) by a diffusive jump: 2Dg -2 = n2/(dgd) , b ) S t i p u - late tha t instantaneous partial local equilibrium with respect to the reaction at every site with respect to the reversible 6 reaction, wherever the neces- sary reactants are present. So in our case if a vacant site r 4- ei adjacent to an occupied site r becomes occupied (e.g. processes leading from the right

4 in higher orders, the density of occupied sites has to be distinguished from the density of particles

s In (4), processes from the center to both sides consume one occupied site each. 6 i.e. compatible with the "detailed balance" condition

A Diffusion-Limited Reaction 91

1

0.8

0.6

0.4

0.2

0

. . - o . . . 13" ~ " ,

q

6 0:2 024 0:6 0:8 t '

Fig. 1. Trajectories in the (p (t), ti (t))- plane (-: mean-field approx.)

column to the center in (34), the site r should remain occupied or vacant according to the equilibrium occupation probability peqg d + 0 (g2d) [11], so this probability should equal the quotient of the two rates h i / ( 2 d D g - 2 ) . For both interpretations, the result for the jump rates is Dg -2 for the coagula- tion jumps, i.e. n2 = 2dDg d-2) and p~qDg -2 for the proliferation jumps (i.e. g l ~ 2dpeqDg d-2) (see our concluding section, however).

3 M a c r o s c o p i c S i g n s o f D i f f u s i o n - C o n t r o l

During the course of the cellular automaton dynamics, the particle positions become increasingly correlated - regardless of any special initial condition. How (2) ceases to apply is evident from the actual relation between ~(t) and p (t) displayed in Fig.1. It shows for the cellular automaton in d = 2 dimensions the rate of change t5 (t) in units of Dp~q over the concentration p (t) in units of Peq following various homogeneous initial conditions with no correlations (marked by dots) and relative concentrations p (0)/peq = 2 -8, 2 -13/2, 2 -11/2, 2 -9/2, 2 -7/2, 2 -5/2, 2 -3/2, 2 -1, 2 -1/2 . The drawn lines are averaged trajectories from 2 l° MC-realizations of a system With 29 x 29 sites, periodic boundary conditions, and equilibrium occupancy of peqg 2 = .0694 . The dashed line indicates the mean-field approximation of the rate ~ (t) 4Dp(t) (Peq - p (t)). The initial fast decline in the reaction rate is not resolved.

Evidently, there is no general autonomous kinetic law for the density, un- Far from equilibrium the rate of less the system stays "close" to equilibrium. 7

change ~ (t) depends not only on p (t) but also on their history { p (t ' ) I t ' < t}.

7 p (t = 0) < ~peq or p (t) < 4 - - - - g p e q

1

0.8

0.6

0.4

0.2

0 6 0:2 0:4 0:6 0:8

92 Martin A. Burschka

Fig. 2. Trajectories in the (p(t), So (t))- plane (corresponding to rigA)

This is a sure sign that the dynamics is determined by additional slow vari- ables, even if the set of dependent variables in the macroscopic dynamic law does not include them - - acting like a Procrustean bed for the actual dy- namics.

How the additional slow variables show in the long range fluctuations is displayed in fig.2. The same initial conditions have been used but a differ- ent ordinate, namely the large wavelength correlations. More precisely, the ordinate measures the deviation from binomial independent fluctuations in the number N of occupied sites out of the L d sites in the system: So (t) =

Z-z1 ( ( N (N - 1)) - (1 - L -d) (N) 2 ) . The quanti ty plays the same role here

as the familiar static structure factor for long wavelengths: limk~o S (k, t) , and they coincide in the limit of spatial continuum and infinite system size. Obviously, the p (t),~ (t) -trajectories differ most for the same initial condi- tions and abscissas where the correlations are most pronounced. Due to the initial and final absence of correlations, the trajectories start and end on the abscissa, which overlaps with the mean-field trajectory.

In physical analogy to complex liquids, the mean-field equation for the density can be extended by including additional fields for microscopic order s into a new macroscopic description. This requires a more general model which comprises the mean-field description and the stochastic diffusion-limited cel- lular automaton as special limits. In the following section, we present such a model.

s e.g. the static structure factor or (equivalently) the pair correlation function

A Diffusion-Limited Reaction 93

4 T h e M a s t e r E q u a t i o n

A reliable intuitive starting point for analytic computations is the kinetic law for the probability PN(rl, ..., r g ; t)~ gd (9) that at t ime t the system contains exactly N A-particles, namely at positions r l , ' " , r g . It is a multivariate master equation which allows for multiple occupancy of every site - - unlike the above cellular automaton. After specifying the terms in the master equa- tion, we show how to recover the mean-field rate constants from it and then give a more concise formulation in terms of the generating functional.

The contributions in the master equation describe the processes: 1) The individuals perform random walks with jump rate 2Dr -2, where D is the macroscopic diffusion coefficient:

N d

Z ve- ' Z(E( - 2 + E3'))P e (5) n = l i=1

where [ ~ i ) f ( r n ) : = f(r,~ + £ei) for any function f ( r ) and ei is the ith basis vector ( i e { 1 , . " , d } ). 2) For every particle (say at rm ) there is a (conditional) probability rate

p~qgdR-da~,N+l (with a,~,~ = ( ~ ) to produce a particle at rN+l (A 2A):

[eqeNdR-d E E am,N+1 ~rr++l,r~VN-l(rl,...,rn,...,rN;t)- PN [d rn=l r/v+1 \n#---m

(6) where R measures the (microscopically small) range of interaction 1°. 3) Every ordered pair has a probability rate R-da,~,N+l to coagulate from its positions rm, rN+l into a single individual at rm (A ~- 2A) (11).

N ( N ) R--d~Nd E E am,N+1 gN+l(rl,''',rN+l;t)~ d- E (~ r .+ l , r~PN

rn=l rN+1 nne=m 1

(7) The mean field rate constants ~i,2 can be recovered from this by applying

the mean-field approximation for the microscopic dynamics in two steps: i) Average out a11 correlations by spatial averaging of the initial distribution

"9 We will drop the arguments of PN in our notation where they are obvious. i0 For this scaling and any given function a (r) , the mean field reaction rates ~1,2

become independent of R in the leading order in 11 The total rate of the two particles to coagulate irrespective of the location of the

product is twice that amount.

94 Martin A. Burschka

and all te rms in the master equation (i.e. repeated averaging for all t imes t). This amounts to the substitution

PN(rl,. . . , rN; t ) e N~ -+ N! (gd//2) lv "PN (8)

where T)N is the probabili ty to find exactly N particles in the system and /2 = ~ r gd measures the total size of the system 12. This leads to a mas ter equation in number space as an intermediary result:

Ot~])N = R-dEa(r~R) ~d (Peq ([Z-I --1) N:PN -[- ([: -- 1)'N(N-1)'n t'N] (9) r

where now F f ( N ) = f ( N + 1). 2) Obtain the familiar rate equation for the concentration by expanding in

powers o f / 2 -½ following the substitutions:

7"u : rx (~, t)

o N ~ °-~N ~ (10) ~-~ = +

In leading order (/21/2) this leads to [12]:

at dR( t ) = --g2p(t) (p(t) -- Peq) with t~2 : = peq R-d EaQr~R) ~d (11)

N = p ( t ) /2 + ~/2U2

I: = e x p ( / 2 - 1 / 2 ~ )

More concisely, the kinetic law and the normalizat ion condition are ex- pressed in terms of the generating function [12,13]:

oo 1 XrNPN•Nd c({<;,): E E... E . . .

N : I rx r N

(12)

where the notat ion emphasizes tha t G depends on all values xr not jus t on the value xR for a particular position R [12]. For the general s tate with concentration p(r; t) and two-point factorial cumulant [14] g(r , r ' ; t) we have

i ~e~ ) (13) -~ ~ g ( r , r / ; t ) (Xr - 1 ) ( X r , - 1 ) + " "

i.~r n

1~ In thermal equilibrium - - where all correlations are zero - - this substitution is exact with :PN = (:~)N (exp (pY2) -- 1)--1 g > 1 due to "detailed balance".

N ! ~ - -

A Diffusion-Limited Reaction 95

where the dots denote higher multiple point cumulants. The normalization condition is then G({xr};t)l,-i = 1 and the kinetic law is

= w c + n a (14)

(o o) with: TG = De-: E E (x~ - x~+~e,) Ox~le, ~ G (15)

i=1 r

n c x , ) x , , - ---0-° a (16) - E ----R----- ( 1 - - Peqe8 r , r I (~Xr

From a macroscopic point of view, the range of the interaction R and the function a (z) appear as irrelevant microscopic detail whereas the mean-field rate constants t¢1,2 might be measured macroscopically, provided all corre- lations between particle positions could be avoided e.g. by vigorous stirring. Also, the description provokes questions like (a) how small has ~11 to be for some given value of R before the mean field description fails or (b) can this mode] reproduce some observed discrepancies between mean-field cal- culations and experiment. All of this motivates a reduction of the kinetic description for the diffusion-controlled relaxation to thermal equilibrium.

In preparation for such a reduction in the next section, we expand a (...) into a power series in [r - rl[ and e:

2

T~G = '~2 E n2~Ti(v)a + 0 (~2R4e 4) + O (,~2R 6) (17) b,w~O

T~(~)G=Q2. EA~,(I_x(r))x(r , ) (e_ d O ) ~ G (18) r

where A~f (r) ---- e -2 ~d= l ~ i ( [ (q-i) _ 1) f (r) . All details o f the interac-

tion kernel R-da (1~[) are then described by the parameters Q2~. They have been chosen so that (a) Q0 = 1 , (b) in lowest order g they contain all in- formation about the shape of a (r) apart from its overall magnitude, and (c) they all remain finite (and nonzero in general) in the continuum limit e -+ 0 as well as in the diffusion-controlled limit g~-,~ --+ 0.

5 T h e D i f f u s i o n - C o n t r o l l e d L i m i t

On the molecular scale, diffusion control has to be defined in more detail than for the cellular automaton in section 2, where the relevant length and time scales (e and e2/D) are not clearly related to the interaction and size of the single A-particles. In the following, we therefore measure lengths in units of the typical distance between nearest neighbor particles p-[qUd and times in units of the typical time it takes a diffusing particle to cover a typical volume per particle 1/(Dp~/d). On these scales, the actual reaction rates

96 Martin A. Burschka

should remain of order unity even close to the diffusion-controlled limit, where the mean-field rates may be much larger (see fig. 1) - - except for a short initial transient regime during which the short range correlations adjust themselves.

The simple limit scaling e = k l 1 = k~l with R,/~ = const, leads to

trivial limit dynamics, however, as then the expectat ion t ime (~1 log 2) -1 for an isolated reactant particle to produce another one at a distance it/ > 0 vanishes for ~ --4 0 so that relaxation of a finite system to global equilibrium then occurs instantaneously - - irrespective of any diffusion. So, nontrivial l imit dynamics depend a finite velocity Rnl even in the diffusion-controlled limit, so all small quantities related to the interaction have to become small in a related way in this limit. In the following, we therefbre consider also /~ := Rp~q 1/d as small parameter , in addition to c = ki -1 = k~-a. Later we

will also discuss the case of small [ := fp-~l/d. The reaction operator is then in lowest order

7~G = C17~(°)G + RZTiO)G + 0 (19) T '{?

We will consider the second rhs te rm to be at most of order unity in the following.

The eigenvalues of the dominant rhs operator 7~(°) are all nonpositive and real with a finite gap at zero for sufficiently small f, so for small e, all nonzero eigenvalues of ¢-1T~(°) become strongly negative. All trajectories in phase space are then rapidly a t t racted towards the zero subspace of 7~(°) , and - - after a short initial transient regime with duration of order c - - the dynamics of the system is determined entirely by the dynamics within this subspace, i.e. trajectories stay close to this subspace apar t f rom higher order corrections. 13

It may be helpful to reformulate this in terms of part ial equilibria: Part ial equilibrium with respect to the full reaction process means tha t 7~G = 0, i.e. for any pair of interacting 14 sites r , r ' which occur in the sum in (16) applies the same of the two conditions: (a) °--L--G = 0, so both sites are vacant 0:%(0

probabil i ty one, or (b) ( ~ - Peqg d) G : 0 so G depends on g

with both

xr, and xr only through factors exp (Peq£dX) and the occupation numbers % i ~ f

N~,, N~ are both Poisson distributed with expectation value (Nr , ) - peq~ d. In particular, this excludes tha t both sides fulfill different conditions, say (a) applies to site r and (b) applies to r ' , because then we have in (16) the te rm

with ( 0 ) __ Peq~d 0 b-7:7~ G ¢ 0 which is not zero. Consequently, this part ial

equilibrium is a one-parameter family of spatially homogeneous states:

G ({xr}) : Po2r (1- Po) exp (Peq~dE(xr-- 1)) (~0) rEV

lz for details on the appropriate systematic adiabatic elimination scheme see [15] ,4 (i.e. a ( ! L ~ ) # 0 )

A Diffusion-Limited Reaction 97

For all these states T G = 0 so diffusion is irrelevant then. On the contrary, diffusion can control the rate of the pair reaction if

the relaxation towards a partial equilibrium occurs locally and towards a variety of local equilibrium states. This variety depends on the degeneracy of the zero-eigenvalue of 7~ (°) which, in turn, is equivalent to the existence of microscopic conservation laws. So diffusion can still dominate the long t ime dynamics in local partial equilibrium 7~(°)G = 0 where the sites r need not be all in the same state, but:

G ({xr}) = H (P~(O)gVaC(x~)+(1-p~(O))g°C¢(x~)) (21) r~V

where gVa¢(x)=land g°CC(x)= (e p°qt% - 1 ) ( e peqte - 1) -1 are the single-

site generating functions for sites in part ial equilibrium, i.e. 7~(°)g°¢¢(x) = ~(°)gvac(x) = 0

In order to formally isolate the reduced slow (long-time) dynamics fi'om fast processes in the master equation, one defines two complementary pro- jection operators[5,15]

S G : = lim exp (hT~(°)) G h--+c~ and Y" : = 1 - S (22)

As all eigenvalues of 7~(°) are nonpositive, S projects into the zero subspace of T~(°) where the slow dynamics occurs. One then expands:

= s w) (s + 7 ) a (23) OtSG

.~ (~--17~(0) -~- ]/~) (S + .~')G (24) Ot.YG =

where : : ( - ' ) -7--, e . Crucial observations are: (a) $~(o) = 0

in (23) , so all terms in order e -1 vanish there and (b) ~ ( ° ) S = 0 in (24) , so the dominant terms (i.e. lhs and terms of order e -1 on the r h s ) f o r m a closed kinetic law for the decay of .~'G on the fast ("microscopic") t ime scale of order ~ . Taken together, this means, that during the transient regime t ~ (9 (e) the fast component ~ G decays in general by a factor independent of c while the slow component S G remains almost static. Following this "initial t ime- slip", f G remains of order e and is largely determined ("enslavement") by the inhomogeneous te rm in (24), namely 3CWSG . The dynamic law for SG , (23), then becomes autonomous in the leading orders 15.

2 OtSG({xr}; t) = S T S G + R~STZ(1)SG (25)

(.

1~ In higher order in e, the kinetic law for SG turns out to depend on an increasing detail of the reaction kernel a (r) through the parameters Q2~.

98 Martin A. Burschka

A kinetic law for the complete G (and the common kinetic hierarchies) can be reconstructed from this and (G-in-cumulants), but there is no sys temat ic way to cut them off as all levels s tar t at the same order in e. On the other hand, after the transient regime, the change in G is determined entirely by the projection onto the zero subspace of TO(°), SG, where the state at each site is only distinguished as occupied or vacant. A practical reduced description for the reduced dynamics in this subspace is obtained by t ransforming f rom the continuous independent variables xr to the dichotomous occupation indicator variables s,. C {0, 1} .via

{ * d r,

es({sr};t)=/% g°C¢(z) sa({srzI;t)

Ps({sr} ; t) is the probabil i ty to find the sites {rls~ = 1} occupied and t h e sites {rls,. = 0} vacant. The reduced kinetic law is then expressed terms of the familiar flip operators 16 c~ + and crr as:

d = S'l-SsPs({, , ,}; t) + S • ( 1 ) 8 8 P s ({Sr, }; t ) (27)

The first rhs term in (27) corresponds closely to the cellular au toma ton model in section 2:

d (,gTS)s P s ( { s r , } ; t ) = Dg-=peqgdEE (1-cr:)a+[(+i)a+c~TYs (28)

r,4- i=1

Peqg d d ( +Dg-2exp(peqga)- 1 E E (1 -- ~+) Crr[(: t :Ocr+~rPs +

r,-I- i=1 "4-(1 -- ° ' r [(=ki)cr+~r r ] Crr+ l= ( + i ) crr P$ )

On the rhs, the three products of ar and [rCrr factors acting on Ps on the three lines describe the gain and loss due to the following processes: 1) proliferation of an occupied site r adjacent to an occupied site r + g e i 2) coagulation of two adjacent occupied sites at r and r 4-gei into an occupied site r 4- gei, vacating site r, 3) diffusive jumps of an occupied site from r 4- gei to r, i.e. exchange of oc- cupancy between vacant sites r and occupied neighbor sites r 4- gei (lr) The prefactors agree with the rates given for the cellular au toma ton model in the diffusion limited case in section 2 and can be understood as follows:

16 For any function f ( s~ ) : a + f ( s r ) = s~f (1 - s .) , a T f (s t) : (1 - s . ) f (1 - s . ) . 17 The first operator [(~:i) should be understood to act only on the following (*~.

A Diffusion-Limited Reaction 99

(1) The conditional probabil i ty to find n particles at a site known to be occu- oct d ~ o~ (x)l~=o. The j u m p rate for a every particle is D~ -~ pied is p~ = 1_~ d--~g

, so summed over n > 2, the total rate for one particle to j u m p to a certain neighbor site from a site known to be occupied without vacating tha t site is Dg -2 ~,~°°=2 np,~. This is the prefactor of the first sum in (27). (2) and (3) The conditional probabil i ty to find only one particle at an occu- pied site is p~CC = dgO~C (x)Ix=0" So the rate for an occupied site to become vacated by one particle jumping to a certain neighbor (occupied or not) is D£-2p~ ~, which equals the prefactor in the second sum in (27). Notice, however, that in section 2 the mean-field approximat ion (2) has been taken to refer to occupied sites and are finite, whereas here the mean-field rates refer to particles and have been scaled to diverge in the limit e -+ 0.

The second rhs term in (27) is explicitly:

- - ,97¢(1),9 Ps({sr}; t)=

R2 Peq~ d d r r (Jr r 1-S --~-Q2e-2Peq(]_exp(_Peqed)) E E (1 -- a t-) a + [ ( + i ) - + ~ r - ~

r,4- i=1 d

-~-R2Q2e-2 (Peqed) 2 e - d E E ( 1 - o ' ~ ) o ' r E ( r ' 4 - i ) o ' ~ o ' r g $ (29) c 4 sinh 2 fleqf d r , : t : i=1

The operator sums are the same as the first two rhs sums in (28) so this contributions just modify the rates of the resp. processes. However, the pref- actors grow much faster as g --~ 0 than in the reduced diffusion operator . The physical reason is that the actual reaction process due to a particle at r affects a number sites which is of the orderg -d and the reduced quadrupole contribution above approximates this by an operator acting only on the 2d sites g: r(+i)r (r = ] . . .d) . This does not lead to a divergence of the equilibrium number of particles in the system because the orders in ~ of the rate con- stant for the two processes still differ by the same as in the reduced diffusion operator discussed above. It shows however, tha t quadrupole effects of the

reaction process dominate normal diffusion for small g unless ~-wl]~l in the limit.

6 C o n c l u s i o n s

The main point has been to show how finite j u m p rates for the stochastic cellular au tomaton can be obtained in a limit with diverging mean-field rates from the more detailed master equation given concisely by (14). This has shown tha t the j u m p rates explained for the diffusion-controlled l imit in sec- tion 2, are not unequivocally determined: They depend on how the l imit is taken in the more detMled model which resolves the single particles, i.e. on

100 Mm-tin A. Burschka

hm ~-~-g. How this affects the macroscopic appearance of the dynamics, e.g. the trajectories in figs I. and 2. remains to be checked in a simulation.

Also, there remains the open question, how the higher orders in the e, R, f- expansion scale and under which conditions they may may dominate .

Finally, we would like to stress how spurious the relation of the j u m p rates to the mean-field rate constants is, when considered on a molecular scale, be- cause in the diffusion-controlled limit, the processes of diffusion and reaction are not separate. There appears to be no general and simple cont inuum limit g --~ 0 of the cellular au tomaton itself, and a continuous macroscopic descrip- tion from which the density and the correlations can be computed without simulation still remains to be found. Only in one dimension, this has been achieved for this system so far[15].

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