A Diffusion-Limited Reaction

A Diffusion-Limited Reaction Martin A. Burschka Inst. f¨ ur theor. Physik III, H.-Heine-Universit¨ at, D-37083 D¨ usseldorf, Germany

Abstract. Fluctuations in diffusion-controlled reactions lack the necessary features 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

What Is Special About Diffusion-Limited Reactions?

Common diffusion-reaction-rate equations are similar in structure and concept 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) dynamics 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 deterministic 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 treatment. 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 concentrations are not conserved under the microscopic dynamics (property (a)) and, in general, do not vary slowly1 . So these equations apply only in regimes where the effect of the reactions is small compared to transport over macroscopic distances. In particular, this excludes the diffusion-controlled regime [1-9] far from equilibrium, where the reaction rate is limited by transport of the single 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 start with two extreme models of a simple diffusionreaction system with a reversible autocatalytic reaction, namely the meanfield diffusion-reaction-rate equation and a stochastic cellular automaton model for the diffusion-controlled limit case to show how the macroscopic dynamics deviate in both cases. We then explain a slightly more general model, in which we discuss explicitly ways to take diffusion-controlled limits. The reduced limit-dynamics in our example depends on a special conservation law, which determines the convenient choice of variables. It corresponds conceptually to the hydrodynamic level of description in fluids although there is no formal similarity to the BBGKY-hierarchy of kinetic equations for the multiple point densities (mixed moments). We conclude with open questions for future research and a “message”.

2

Two Descriptions of a Diffusion-Reaction System

We consider a common autocatalytic process, like it is found in the famous pattern-forming Belouzov-Zhabotinsky-reaction: κ1 * 2A (1) A ) κ2 There2 , it is it is implemented as BrO3− + HBrO2 + 2F e(II) + 3H + * ) 2 HBrO2 + 2F e(III) + H2 O In the regime where — apart from H2 O — the substances BrO3− , F e(II), H + , F e(III), are in ample supply with effectively constant concentrations, the dynamics occurs entirely in the concentrations and correlations of bromate HBrO2 . The mean-field rate constants κ1 , κ2 apply in the absence of correlations between the HBrO2 -molecules, e.g. in thermal equilibrium or for a short time following an initial condition with no correlations3 . Actually, A could stand for other entities like e.g. individuals of a population. The model resembles the logistic system, in that it combines the autocatalytic process (*) with a simple inhibitory process ()). On the microscopic level we will consider the unique version of it which complies with the “detailed balance” condition, i.e. that 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 ρ (r; t) of A: ∂t ρ (r; t) = D∆ρ (r; t) + κ2 ρ (r; t) (ρeq −ρ (r; t)) 2 3

(2)

in the Oregonator model[10] of this reaction This could be prepared as a thermal state for some other value of an £ equilibrium ¤ external parameter (temperature, H + , etc.)

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Martin A. Burschka

with ρeq = κκ12 . Diffusion-control can only be incorporated via time and concentration-dependent reaction rate “constants”, changing the status of the equation from mean-field approximation of a more detailed theoretical model to an interpolation formula with fitted parameters for a special case. Failure of this equation to describe experimental data qualitatively is strong evidence for correlations on the molecular scale. A minimal model for the diffusion-limited reaction must account for multiple-point correlations, like the following stochastic cellular automaton on a cubic lattice with the processes: D`−2

• κ1 / (2d) •

* ) ¡

κ2 / 2d`

d

¢ •

•

* )−2

•

(3)

* )

•

(4)

D` ¡ ¢ κ2 / 2d`d κ1 / (2d)

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 jump rate D`−2 , where ` is the lattice constant. 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 proliferation reaction X * 2X and the reverse ones from the center the coagulation reaction X ) 2X. Again, the mean field values are displayed in leading order in ` (4 ). They lead to the reaction terms in (2)— at least for statistically homogeneous initial conditions, i.e. translation invariance of all expectation values of the density: For the coagulation, the mean-field probability for two ¡ ¢2 adjacent sites to be occupied is ρ`d . Together with the density of pairs of sites d`−d this gives the mean-field coagulation rate of κ2 ρ2 (5 ). Similarly, for proliferation, the rate κ1 / (2d) to generate a new particle at an adjacent vacant site together with the¡ density of occupied sites ρ and the number of ¢ adjacent vacant sites 2d + O `d gives the mean-field proliferation rate κ1 ρ . The diffusion-controlled limit of this model can be identified by two equivalent interpretations: a) Equate the probability rate for two adjacent occupied sites to coagulate with their probability rate to come into ¡“contact” - (i.e. ¢ occupy the same site) by a diffusive jump: 2D`−2 = κ2 / d`d . b) Stipulate that instantaneous partial local equilibrium with respect to the reaction at every site with respect to the reversible6 reaction, wherever the necessary reactants are present. So in our case if a vacant site r ± ei adjacent to an occupied site r becomes occupied (e.g. processes leading from the right 4

5 6

in higher orders, the density of occupied sites has to be distinguished from the density of particles In (4), processes from the center to both sides consume one occupied site each. i.e. compatible with the “detailed balance” condition


1

91

o o o

0.8 0.6

o

0.4 o

0.2 0

o o oo

0

0.2

0.4

0.6

0.8

1

Fig. 1. Trajectories in the (ρ (t) , ρ˙ (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 ρeq `d + O¡ `2d [11], ¢ so −2 this probability should equal the quotient of the two rates κ1 / 2dD` . For both interpretations, the result for the jump rates is D`−2 for the coagulation jumps, i.e. κ2 = 2dD`d−2 ) and ρeq D`−2 for the proliferation jumps (i.e. κ1 = 2dρeq D`d−2 ) (see our concluding section, however).

3

Macroscopic Signs of Diffusion-Control

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 ρ˙ (t) and ρ (t) displayed in Fig.1. It shows for the cellular automaton in d = 2 dimensions the rate of change ρ˙ (t) in units of Dρ2eq over the concentration ρ (t) in units of ρeq following various homogeneous initial conditions with no correlations (marked by dots) and relative concentrations ρ (0) /ρeq = 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 210 MC-realizations of a system with 29 × 29 sites, periodic boundary conditions, and equilibrium occupancy of ρeq `2 = .0694 . The dashed line indicates the mean-field approximation of the rate ρ˙ (t) ≈ 4Dρ(t) (ρeq − ρ (t)). The initial fast decline in the reaction rate is not resolved. Evidently, there is no general autonomous kinetic law for the density, unless the system stays “close” to equilibrium. Far from equilibrium7 the rate of change ρ˙ (t) depends not only on ρ (t) but also on their history { ρ (t 0 )| t 0 < t}. 7

ρ (t = 0) ≤

1 ρ 10 eq

or ρ (t) ≤ 45 ρeq

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Martin A. Burschka

1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Fig. 2. Trajectories in the (ρ(t) , S0 (t))- plane (corresponding to fig.1)

This is a sure sign that the dynamics is determined by additional slow variables, even if the set of dependent variables in the macroscopic dynamic law does not include them — acting like a Procrustean bed for the actual dynamics. 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 different ordinate, namely the large wavelength correlations. More precisely, the ordinate measures the deviation from binomial independent fluctuations in d the³number N of occupied sites out ´ of the L sites in the system: S0 (t) = ¡ ¢ 2 1 hN (N − 1)i − 1 − L−d hN i . The quantity plays the same role here Ld

as the familiar static structure factor for long wavelengths: limk→0 S (k, t) , and they coincide in the limit of spatial continuum and infinite system size. Obviously, the ρ (t),ρ˙ (t) -trajectories differ most for the same initial conditions 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 8 into a new macroscopic description. This requires a more general model which comprises the mean-field description and the stochastic diffusion-limited cellular automaton as special limits. In the following section, we present such a model. 8

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


4

93

The Master Equation

A reliable intuitive starting point for analytic computations is the kinetic law for the probability PN (r1 , ..., rN ; t)`N d (9 ) that at time t the system contains exactly N A-particles, namely at positions r1 , · · · , rN . 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 equation, 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 2D`−2 , where D is the macroscopic diffusion coefficient: N X

n=1

D`−2

d X i=1

(EI (−i) − 2 + EI (+i) )PN `Nd n n

(5)

f (rn ) : = f (rn ± èi ) for any function f (r) and ei is the ith basis where EI (±i) n vector ( i ∈ {1, · · · , d} ). 2) For every particle (say at r³m ) there ´ is a (conditional) probability rate |rm −rn | d −d ρeq ` R am,N +1 (with am,n = to produce a particle at rN +1 (A * R 2A) :

ρeq `N d R−d

N X

X

m=1 rN +1



 am,N +1

N X

n=1 n6=m



 δrN +1 ,rn PN −1 (r1 ,...,rn ,...,rN ; t) − PN `d 

(6) where R measures the (microscopically small) range of interaction10 . 3) Every ordered pair has a probability rate R−d am,N +1 to coagulate from its positions rm , rN +1 into a single individual at rm (A ) 2A) (11 ).

R−d `N d

N X X

m=1 rN +1



 am,N +1 PN +1 (r1 , · · · , rN +1 ; t)`d −

N X

n=1 n6=m



 δrN +1 ,rn PN 

(7) The mean field rate constants κ1,2 can be recovered from this by applying the mean-field approximation for the microscopic dynamics in two steps: 1) Average out all correlations by spatial averaging of the initial distribution 9 10

11

We will drop the arguments of PN in our notation where they are obvious. For this scaling and any given function a (r) , the mean field reaction rates κ1,2 become independent of R in the leading order in ` The total rate of the two particles to coagulate irrespective of the location of the product is twice that amount.

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Martin A. Burschka

and all terms in the master equation (i.e. repeated averaging for all times t). This amounts to the substitution ¡ ¢N PN (r1 , · · · , rN ; t)`N d → N ! `d /Ω PN

(8)

wherePPN is the probability to find exactly N particles in the system and Ω = r `d measures the total size of the system12 . This leads to a master equation in number space as an intermediary result: ´ X ³|r|´ ³ ¡ ¢ ∂t PN = R−d a R `d ρeq EI −1 − 1 N PN + (EI − 1) N (NΩ−1) PN (9) r

where now EI f (N ) = f (N + 1). 2) Obtain the familiar rate equation for the concentration by expanding in 1 powers of Ω − 2 following the substitutions: PN = Π (ξ, t)

¯ ¯ ¯ ∂ ¯¯ ∂ ¯¯ ∂ξ ¯¯ ∂ = + ∂t ¯N ∂t ¯ξ ∂t ¯N ∂ξ

,

N = ρ (t) Ω + ξΩ 1/2

,

µ

EI = exp Ω

−1/2

∂ ∂ξ

¶

(10)

In leading order (Ω 1/2 ) this leads to [12]: d ρ(t) = −κ2 ρ(t) (ρ(t) − ρeq ) dt

with

κ2 : = ρeq R−d

X ³|r|´ a R `d (11) r

More concisely, the kinetic law and the normalization condition are expressed in terms of the generating function [12,13]: ∞ X X 1 X ··· x r1 · · · x rN P N ` N d G({xr }; t) = N! r r N =1

1

(12)

N

where the notation emphasizes that G depends on all values xr not just on the value xR for a particular position R [12]. For the general state with concentration ρ(r; t) and two-point factorial cumulant [14] g(r, r0 ; t) we have G ({xr }; t) = exp

Ã

X r

`d ρ (r; t) (xr − 1) +  1 X 2d + ` g(r, r0 ; t)(xr − 1)(xr0 − 1) + · · ·(13) 2 0 r,r

12

In thermal equilibrium — where all correlations are zero — this substitution is N exact with PN = (ρΩ) (exp (ρΩ) − 1)−1 , N ≥ 1 due to “detailed balance”. N!


95

where the dots denote higher multiple point cumulants. The normalization condition is then G({xr }; t)|xr ≡1 = 1 and the kinetic law is ∂t G({xr }; t) = T G + RG

d X X

(14) ¶

µ

∂ ∂ G (15) − ∂x ∂x r+è r i i=1 r ¶ µ X µ|r−r 0 |¶ ∂ ∂ d −d a R − ρeq ` G (16) RG = R (1 − xr ) xr 0 ∂xr ∂xr 0 0

with: T G = D`−2

(xr − xr+èi )

r,r

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 κ1,2 might be measured macroscopically, provided all correlations between particle positions could be avoided e.g. by vigorous stirring. Also, the description provokes questions like (a) how small has κ−1 to be 1 for some given value of R before the mean field description fails or (b) can this model reproduce some observed discrepancies between mean-field calculations 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 − r 0 | and `: RG = κ2

2 X

ν=0

R(ν) G = Q2ν

¡ ¢ ¡ ¢ R2ν R(ν) G + O κ2 R4 `4 + O κ2 R6

X r

∆νr0 (1 − x (r)) x (r 0 )

µ

`−d

∂ − ρeq ∂x (r )

(17) ¶

¯ ¯ ∂ ¯ (18) G 0 ∂x (r ) ¯r 0 =r

´ Pd P ³ where ∆r f (r ) = `−2 i=1 ± EI (±i) − 1 f (r ). All details of the interac¡¯ ¯¢ tion kernel R−d a ¯ Rr ¯ are then described by the parameters Q2ν . They have been chosen so that (a) Q0 = 1 , (b) in lowest order ` they contain all information 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 ` → 0 as well as in the diffusion-controlled limit κ−1 1,2 → 0.

5

The Diffusion-Controlled Limit

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 (` and `2 /D) are not clearly related to the interaction and size of the single A-particles. In the following, we therefore measure lengths in units −1/d of the typical distance between nearest neighbor particles ρeq and times in units of the typical time it takes a diffusing particle to cover a typical 2/d volume per particle 1/(Dρeq ). 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 ² = κ ¯ −1 = κ ¯ −1 with R, ` = const, leads to 1 2 −1 trivial limit dynamics, however, as then the expectation time (κ1 log 2) for an isolated reactant particle to produce another one at a distance R > 0 vanishes for ² → 0 so that relaxation of a finite system to global equilibrium then occurs instantaneously — irrespective of any diffusion. So, nontrivial limit dynamics depend a finite velocity Rκ1 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 therefore consider also −1/d ¯ := Rρeq R as small parameter, in addition to ² = κ ¯ −1 =κ ¯ −1 1 2 . Later we −1/d will also discuss the case of small `¯ := `ρeq . The reaction operator is then in lowest order µ 4 ¶ R R2 (1) −1 (0) R G+O ,² . (19) RG = ² R G + ² ²

We will consider the second rhs term to be at most of order unity in the following. The eigenvalues of the dominant rhs operator R(0) are all nonpositive and real with a finite gap at zero for sufficiently small `, so for small ², all nonzero eigenvalues of ²−1 R(0) become strongly negative. All trajectories in phase space are then rapidly attracted towards the zero subspace of R(0) , and — after a short initial transient regime with duration of order ² — the dynamics of the system is determined entirely by the dynamics within this subspace, i.e. trajectories stay close to this subspace apart from higher order corrections.13 It may be helpful to reformulate this in terms of partial equilibria: Partial equilibrium with respect to the full reaction process means that RG = 0, i.e. for any pair of interacting14 sites r,r 0 which occur in the sum in (16) applies the same of the two conditions: (a) ∂x∂(0) G = 0, so both sites are vacant r ´ ³ with probability one, or (b) ∂x∂(0) − ρeq `d G = 0 so G depends on both r ¡ ¢ xr 0 and xr only through factors exp ρeq `d x and the occupation numbers Nr 0 , Nr are both Poisson distributed with expectation value hNr 0 i = ρeq `d . In particular, this excludes that both sides fulfill different conditions, say (a) 0 applies ³ to site r and ´ (b) applies to r , because then we have in (16) the term ∂ with ∂x − ρeq `d ∂x∂ 0 G 6= 0 which is not zero. Consequently, this partial r r equilibrium is a one-parameter family of spatially homogeneous states: Ã ! ³ ´ X d G {xr } = P0 + (1 − P0 ) exp ρeq ` (xr − 1) (20) r∈V

13 14

for details on the appropriate systematic adiabatic elimination scheme see [15] 0 (i.e. a( |r−Rr | ) 6= 0)


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 R(0) which, in turn, is equivalent to the existence of microscopic conservation laws. So diffusion can still dominate the long time dynamics in local partial equilibrium R(0) G = 0 where the sites r need not be all in the same state, but: ³ ´ Y G {xr } = (pr (0) g vac (xr ) + (1 − pr (0)) g occ (xr )) (21) r∈V

³ ´³ ´−1 d d where g vac (x) = 1 and g occ (x) = eρeq ` x − 1 eρeq ` − 1 are the single-

site generating functions for sites in partial equilibrium, i.e. R(0) g occ (x) = R(0) g vac (x) = 0 In order to formally isolate the reduced slow (long-time) dynamics from fast processes in the master equation, one defines two complementary projection operators[5,15] ´ ³ and F := 1 − S (22) SG : = lim exp hR(0) G h→∞

As all eigenvalues of R(0) are nonpositive, S projects into the zero subspace of R(0) where the slow dynamics occurs. One then expands: ³ ´ ∂t SG = S ²−1 R(0) + W (S + F) G (23) ³ ´ ∂t FG = F ²−1 R(0) + W (S + F) G (24) 2

where W : = T + R² R(1) +O

³

R4 ² ,² −1

´

. Crucial observations are: (a) SR(0) = 0

in (23) , so all terms in order ² vanish there and (b) R(0) S = 0 in (24) , so the dominant terms (i.e. lhs and terms of order ²−1 on the rhs) form a closed kinetic law for the decay of FG on the fast (“microscopic”) time scale of order ² . Taken together, this means, that during the transient regime t ∼ O (²) the fast component FG decays in general by a factor independent of ² while the slow component SG remains almost static. Following this “initial timeslip”, FG remains of order ² and is largely determined (“enslavement”) by the inhomogeneous term in (24), namely FWSG . The dynamic law for SG , (23), then becomes autonomous in the leading orders 15 : ∂t SG({xr }; t) = ST SG + 15

R2 SR(1) SG ²

(25)

In higher order in ², the kinetic law for SG turns out to depend on an increasing detail of the reaction kernel a (r) through the parameters Q2n .

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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 systematic way to cut them off as all levels start at the same order in ². On the other hand, after the transient regime, the change in G is determined entirely by the projection onto the zero subspace of R(0) , 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 transforming from the continuous independent variables xr to the dichotomous occupation indicator variables sr ∈ {0, 1} .via ¶ X Yµ (1 − sr0 ) g vac (xr0 ) + sr0 g occ (xr0 ) PS ({sr }; t)(26) SG({xr }; t) ≡ {sr } r0

PS ({sr }; t) = lim

z→∞

Ã

g occ (z)

! − P sr0 r0

SG({sr z}; t)

PS ({sr }; t) is the probability to find the sites {r|sr = 1} occupied and the sites {r|sr = 0} vacant. The reduced kinetic law is then expressed terms of the familiar flip operators16 σr+ and σr− as: d PS ({sr 0 }; t) = ST S S PS ({sr 0 }; t) + SR(1) SS PS ({sr 0 }; t) dt

(27)

The first rhs term in (27) corresponds closely to the cellular automaton model in section 2: (ST S)S PS ({sr 0 }; t) = D`−2 ρeq `d

d XX ¡ r,± i=1

d

+D`−2

XX ρeq `d d exp (ρeq ` ) − 1 r,± i=1

µ

¢ 1 − σr− σr+ EI (±i) σr+ σr− PS r

(28)

(1 − σr+ ) σr− EI (±i) σr+ σr− PS + r ¶ ³ ´ + + (±i) − + 1 − σr− EI (±i) σ σ E I σ P r r r r r S

On the rhs, the three products of σr and EI r σr 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 ± èi 2) coagulation of two adjacent occupied sites at r and r±èi into an occupied site r ± èi , vacating site r, 3) diffusive jumps of an occupied site from r ± èi to r, i.e. exchange of occupancy between vacant sites r and occupied neighbor sites r ± èi (17 ) The prefactors agree with the rates given for the cellular automaton model in the diffusion limited case in section 2 and can be understood as follows: 16 17

For any function f (sr ): σr+ f (sr ) = sr f (1 − sr ) , σr− f (sr ) = (1 − sr ) f (1 − sr ). (±i) The first operator EI r should be understood to act only on the following σr .


99

(1) The conditional probability ¯ to find n particles at a site known to be occu1 dn occ ¯ g (x) . The jump rate for a every particle is D`−2 pied is pocc = n n n! dx x=0 , so summed over n ≥ 2, the total rate for one particle to jump to a certain neighbor P∞site from a site known to be occupied without vacating that site is D`−2 n=2 npn . This is the prefactor of the first sum in (27). (2) and (3) The conditional probability to find only one particle at an occu¯ d occ ¯ g (x) . So the rate for an occupied site to become pied site is pocc = 1 dx x=0 vacated by one particle jumping to a certain neighbor (occupied or not) is D`−2 pocc 1 , which equals the prefactor in the second sum in (27). Notice, however, that in section 2 the mean-field approximation (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 ² → 0. The second rhs term in (27) is explicitly: ´ R2 ³ SR(1) S PS ({sr }; t) = ² S

d XX ¡ ¢ ρeq `d R2 Q2 `−2 ρeq 1 − σr− σr+ EI (±i) σr+ σr− PS r d ² (1 − exp (−ρeq ` )) r,± i=1 ¡ ¢2 d XX ¢ ¡ ρeq `d R2 −2 −d σr+ σr− PS (29) + Q2 ` 1 − σr+ σr− EI (±i) ` r 2 d ² 4 sinh ρeq ` r,± 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 prefactors grow much faster as ` → 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 order`−d and the reduced quadrupole contribution above approximates this by an operator acting only on the 2d sites EI (±i) r (r = 1...d) . This does not lead to a divergence of the equilibrium r number of particles in the system because the orders in ` of the rate constant for the two processes still differ by the same as in the reduced diffusion operator discussed above. It shows however, that quadrupole effects of the ¯2 reaction process dominate normal diffusion for small ` unless `R ¯d ² ß1 in the limit.

6

Conclusions

The main point has been to show how finite jump rates for the stochastic cellular automaton can be obtained in a limit with diverging mean-field rates from the more detailed master equation given concisely by (14). This has shown that the jump rates explained for the diffusion-controlled limit in section 2, are not unequivocally determined: They depend on how the limit is taken in the more detailed model which resolves the single particles, i.e. on

100

Martin A. Burschka ¯2

lim `R ¯d ² . How this affects the macroscopic appearance of the dynamics, e.g. the trajectories in figs 1. and 2. remains to be checked in a simulation. Also, there remains the open question, how the higher orders in the ², R, èxpansion scale and under which conditions they may may dominate. Finally, we would like to stress how spurious the relation of the jump rates to the mean-field rate constants is, when considered on a molecular scale, because in the diffusion-controlled limit, the processes of diffusion and reaction are not separate. There appears to be no general and simple continuum limit ` → 0 of the cellular automaton itself, and a continuous macroscopic description 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].

References 1. von Smoluchowski M., Phys. Z. 17,585 (1916), Z. Phys. Chem. 92 129 (1917) 2. Weiss G. H. and Rubin, R. J., Advan. Chem. Phys. 52 363 (1982) 3. van Kampen N. G., Int. J. Quant. Chem. Symp. 16, 101 (1982) 4. Calef D.and Deutch J. M., Ann. Rev. Phys. Chem. 34 493 (1983) 5. Gardiner C. W., Handbook of Stochastic Methods (Springer, Berlin) (1983) 6. Rice S. A., Comprehensive chemical kinetics, vol. 25. Diffusion-limited reactions (Elsevier, Amsterdam) (1985) 7. Keizer J., Chem. Rev. 87, 167 (1987) 8. Ovchinnikov A. A., Timashev S. F. and Belyy A. A., Kinetics of DiffusionControlled Processes (Nova, New York) (1989) 9. Felderhof B. U., Theory of Diffusion-Controlled Reactions(preprint) (1996) 10. Field R. J., Noyes R. M., Nature237 390 (1972), J. Chem. Phys. 60(5)1877 (1974), Noyes R. M., Field R. J., K¨ or¨ os E., J. Amer. Chem. Soc. 94, 1394 (1972) 11. Burschka M. A., Europhys. Lett. 16, 537 (1991) 12. van Kampen N. G., Stochastic Processes in Physics and Chemistry, (NorthHolland 1981) 13. Burschka M. A., J. Stat. Phys. 45(3/4) 115 (1986) 14. van Kampen N. G., Phys. Reports 124 69 (1985) 15. Burschka M. A., Doering C. R., Horsthemke W. H. (to be published)