Exact substitute processes for diffusion–reaction systems with local complete exclusion rules

Lattice systems with one species diffusion–reaction processes under local complete exclusion rules are studied analytically starting from the usual master equations with discrete variables and their corresponding representation in a Fock space. On this basis, a formulation of the transition probability as a Grassmann path integral is derived in a straightforward manner. It will be demonstrated that this Grassmann path integral is equivalent to a set of Ito stochastic differential equations. Averages of arbitrary variables and correlation functions of the underlying diffusion–reaction system can be expressed as weighted averages over all solutions of the system of stochastic differential equations. Furthermore, these differential equations are equivalent to a Fokker–Planck equation describing the probability distribution of the actual Ito solutions. This probability distribution depends on continuous variables in contrast to the original master equation, and their stochastic dynamics may be interpreted as a substitute process which is completely equivalent to the original lattice dynamics. Especially, averages and correlation functions of the continuous variables are connected to the corresponding lattice quantities by simple relations. Although the substitute process for diffusion–reaction systems with exclusion rules has some similarities to the well-known substitute process for the same system without exclusion rules, there exists a set of remarkable differences. The given approach is not only valid for the discussed single-species processes. We give sufficient arguments to show that arbitrary combinations of unimolecular and bimolecular lattice reactions under complete local exclusions may be described in terms of our approach.


Introduction
In recent years, many studies have been undertaken to understand the dynamical behaviour of classical diffusive systems under exclusive restrictions [1]- [13]. This natural problem has been associated with an extraordinarily large variety of phenomena related to various chemical and diffusion processes. Usually, systems in which reactants are transported by diffusion [14,15] are denoted as reaction-diffusion systems. Two fundamental timescales characterize these systems: (i) the diffusion time as a typical time between collisions of reacting particles and (ii) the reaction time defining the inverse reaction rate of neighbouring particles. When the reaction time is much larger than the diffusion time, the process follows approximately the classical kinetic equations. Such a process is reaction-limited. It is characterized by the validity of the law of mass action and by neglecting possible concentration fluctuations. The opposite case-diffusion-limited processes-is much more in the focus of recent investigations [16]- [21]. In this limit, local concentration fluctuations dominate the kinetics [22]- [25]. Although the effect of fluctuations is most pronounced in low dimensions or in confined geometries, there are also non-negligible effects in the traditional three-dimensional space.
The diffusion time is usually defined by the mean-square distance between reacting particles and the corresponding diffusion coefficient. Since the particle diffusion coefficient D depends on the normalized total concentration c via D ∼ (1 − c), a process may also be diffusion-limited in dense systems, although the mean distance between the particles is relatively short. For instance, heterogeneous reaction systems form increasing product layers between the educts regions, so that a particle has to overcome this barrier by diffusion processes in order to find an appropriate reaction partner. It is well known that the educt-product surfaces are strongly controlled by local diffusion-induced fluctuations [26]. On the other hand, several autocatalytic reaction processes generate a transport of particle properties by chemical reactions, so that these effects also contribute to the diffusion. Consequently, the collective diffusion coefficients in dense autocatalytic systems are often controlled by the reaction time and not by the original Brownian motion.
The aim of this paper is to present a general formalism in order to describe a large class of reaction-diffusion systems obeying the exclusion principle. Although we focus our investigation on various single-species reactions, the formalism can be extended to all reaction-diffusion systems considering spontaneous changes of particles with the schematic representation A → B as well as several pair reactions of the general type A + B → C + D.

Classical lattice reaction-diffusion systems
One possible way to approach diffusion and reaction processes under exclusive restrictions is to map the real dynamical processes onto a given lattice structure in the sense of a coarsegraining approximation. Usually, it means that the considered system is divided into small cells. Every cell is occupied by particles or states specified by kinetic rules depending on the situation in mind. Application of the exclusion principle means that the occupation numbers and the number of states, respectively, are restricted. The problem is, therefore, to formulate the dynamics in such a way that these restrictions are taken into account. In particular, such a situation can be described in a rather compact form starting from a master equation on a lattice [5,27,28].
A natural way to describe lattice-diffusion processes is given by Kawasaki exchange processes, which may be written schematically as where A represent a particle, while 0 indicates a vacancy. The elementary rule of the Kawasaki process means that the elements of a pair (A, 0) exchange their positions with an empirically defined transition rate. At sufficiently large scales, the simple rule (1) leads to a diffusion equation for the continuous particle density ρ ∂ρ(r, t) ∂t = D∇ 2 ρ(r, t), which can be solved straightforwardly using well-known standard methods considering initial and boundary conditions. It should be remarked that the diffusion coefficient D describes the collective diffusion of the system, while the particle (or tracer-) diffusion coefficient depends on the particle concentration [29]. Additionally, we introduce lattice reactions. In this paper, we focus on single-species reactions. An important class of single-species lattice reaction processes considering the excluded volume principle are the so-called contact processes [19], defined as evolutionary rules on a set of elementary objects. Since we have only one species of particles, each site of the lattice is either occupied by a particle or is vacant. The transition rules define that a vacant site with n occupied nearest neighbours becomes also occupied with the rate λ, while particles disappear spontaneously, i.e. independent of their surroundings, with the rate β. These processes may be described by the reaction scheme nA + 0 → (n + 1)A and where A represents the particles. The classical kinetic equation describing the evolution of the particle density ρ on the level of a mean-field theory at sufficiently large scales is then given by This equation has two stationary solutions, the vacuum state ρ = 0 and an active phase ρ = ρ 0 (λ/β) > 0. Obviously, the vacuum state is stable in the sense that this configuration remains unchanged after the application of the evolutionary rules. Furthermore, there exist a set of initial configurations (e.g. configurations with isolated particles) belonging to the basin of attraction of the vacuum state. The active phase, characterized by a nonzero density of particles, exists only for a sufficiently large creation rate λ > λ c , otherwise the corresponding stationary solution is unstable and each initial configuration converges to the vacuum state. It should be remarked that a contact process in a d-dimensional space is associated with the directed percolation model in d + 1 dimensions since the dimension along the preferred direction of this model can be interpreted as a time axis. In this sense, the critical value λ c corresponds to the directed percolation threshold. Another important class of lattice reaction processes is given by autocatalytic processes. Here, we have a reversible reaction scheme The forward reaction is again defined by the above-mentioned evolutionary rule, while the backward reaction means that a particle surrounded by n occupied nearest neighbours disappears with the rate β. The corresponding kinetic equation now reads from which we find again two stationary solutions, the vacuum state ρ = 0 and an active phase ρ = ρ 0 (λ/β) = λ/(λ + β) > 0. The vacuum state is again stable in the above-mentioned sense, but its basin of attraction contains only the vacuum state itself. There are other stable configurations (e.g. in case of n 2 configurations with only isolated particles) that are not considered in the classical solution, but the probability that such an initial configuration can be obtained by a certain random process generating all allowed configurations converges to zero for an infinitely large lattice. Thus, the mean field solution ρ = 0 is unstable, while the corresponding solution ρ 0 for the active phase is always stable. Both contact and autocatalytic processes may be extended by the Kawasaki exchange process (1) in order to introduce diffusion rules. Together with the chemical reaction equations, we arrive now at the continuous mean-field equation for the particle concentration The coefficients a and b k depend on the underlying chemical process and the kinetic order n. It should be remarked that, in the case of a spatially homogeneous system ρ(r, t) = ρ(t), we find again the kinetic equations (4) and (6). As mentioned above, we will restrict our investigation to the analysis of pair reactions and spontaneous decays. This means that we have to deal with autocatalytic and contact processes of order n = 1. Thus, we get in case of the contact process, while the autocatalytic process requires Also in the presence of diffusion processes, we find an active phase for a, b > 0. However, the kinetic equations (4), (6) and (7) consider only the mean evolution from an initial state far from the stationary state into the stationary state. Close to the stationary state, fluctuations due to the random realization of the given rules become dominant. In order to keep the influence of these fluctuations, (7) has to be modified by introducing a noise term [30] ∂ρ(r, t) ∂t It seems surprising that the noise is multiplicatively coupled, but a detailed mathematical analysis [31,32] shows that this square root is the proper term under the customary assumption of Poissonian statistics for the reaction part. It should be remarked that there is an additional noise due to the random realization of the diffusive process. But this noise does not affect the critical behaviour of (10) so that we can neglect it for the moment. The noise η(r, t) has a Gaussian character, i.e. we obtain When these fluctuations are taken into account, the critical point separating the vacuum state from the active phase shifts from a = 0 to a c > 0 and the critical behaviour is nonclassical. That is, the autocatalytic process now also has a stable vacuum state corresponding to λ < a c , while the contact process requires λ < a c + β for a stable vacuum state. Furthermore, the nonclassical critical behaviour is characterized by new scaling exponents. For example, the mean density scales now as ρ ∼ (a − a c ) ς , with ς = 0.277 in d = 1, while the mean field exponent is given by ς = 1.
Obviously, the critical behaviour will not be changed if we try to consider the excluded volume effect properly in the analytical description of these diffusion-reaction processes. We remark that the stochastic equation (10) is a coarse-grained evolution equation considering lattice rules only in the collective diffusion coefficient and reaction rates and in the coupling with the noise term. Especially, the exclusion principle itself was not considered by the derivation of (10). Therefore, it may be expected that the evolution of the density at short and moderate times and length scales of a real lattice model is significantly affected by the presence of this principle, so that, in this case, (10) must be replaced by a more general equation.
The aim of the present paper is to obtain a microscopic representation of autocatalytic or contact single-species diffusion-reaction processes of kinetic order n = 1 under a proper consideration of the mutual exclusions in terms of a generalized Fokker-Planck equation. We expect that the microscopic representation with exclusion principle approaches the briefly discussed classical kinetic equation (4), (6), (7) and (10) at large distances and for long times, while significant differences should remain for short and moderate scales.

Master equations and the Fock-space approach
3.1.1. General remarks. We start our investigation from the usual master equations and the corresponding representation in Fock space, respectively. The original master equation contains discrete observables describing the actual occupation of the lattice sites. For the processes in mind, the occupation of a lattice site i is determined by a number n i with n i = 1 if a particle A occupies the lattice site and n i = 0 if the site is vacant. The set n = {n j } of all occupation numbers forms a configuration. From this point of view, the evolution of the statistical probability distribution function P(n, t) can be described by a master equation: where the dynamical matrix L(n, n ) contains all information about the kinetic rules of the lattice system. Following Doi [27] (compare also [5]), the probability distribution P(n, t) can be related to a state vector |F(t) in a Fock space according to P(n, t) = n|F(t) and therefore |F(t) = n P(n, t)|n , respectively, with basis vectors |n . Using this representation, the master equation (12) can be transformed to an equivalent equation in the Fock space The dynamical matrix L(n, n ) of (12) is now mapped onto the evolution operator L = n,n |n L(n, n ) n |.
Usually, this operator is expressed in terms of creation and annihilation operators with Bose properties [27,28,33] or Pauli properties [2,5,8,28]. In order to derive a path integral formulation considering the exclusion principle, we have to construct the evolution operator on the basis of Fermi operators.

Evolution operator for local spontaneous reactions.
First of all, we investigate the spontaneous creation 0 → A and annihilation A → 0 of a particle at a certain lattice site with the creation rate λ and the annihilation rate β. The master equations of these processes can be written as In principle, two basis states are sufficient for the representation of this problem using the Fockspace method. In order to transfer the exclusion principle (the lattice site is either occupied by only one particle or it is vacant) onto the algebraic properties of some basis operators, it seems reasonable to use a Clifford (or Fermi-) algebra, which is characterized by characteristic anticommutation relations. But this approach contains an intrinsic problem. Later we are interested in the extension of the simple one-site problem to a complete lattice problem. This extension implies serious difficulties related to the commutation rules of operators and states. Especially, such a method requires unwanted and uncomfortable manipulations of the sign of the lattice state. Thus, the standard procedure to is apply a Pauli algebra, which is Fermi-like for operators acting at the same lattice site, but Bose-like for operators at different sites [34,35]. Unfortunately, this algebra is not suitable for the generation of clear path integrals. Therefore, we will choose an alternative approach. To achieve this aim, we introduce a Fock space F defined by the four basis states: |n + = 1 (|n + = 0 ) describes the situation that at the lattice site particle A is present (absent). Analogously, |n − = 1 (|n − = 0 ) is the state with the vacancy present (absent) at this lattice site. This construction, although complicated at the first sight, is necessary for the reasons given above. The algebra is completed with the introduction of the usual Fermi operatorsâ 1 , a † 1 ,â 2 andâ † 2 . These operators form a Clifford algebra with the well-known anticommutation relations: and The application of these operators onto the basis vectors |n leads tô and Of course, the dimension of the chosen Fock space is too large for the description of the two allowed states. In order to represent the presence or absence of the particle A (and therefore the absence or presence of a vacancy), we need only two of the four available Fock-space basis states. It seems reasonable to choose the identification The two states form the subspace F s of the total Fock space F(F s ⊂ F). Obviously, the two basis states |n = |0 , |1 of this subspace satisfy the condition while the other two basis states of the complete Fock space violate this relation. After fixing the basis states, we can represent the master equation (15) in terms of this reduced basis. It is clear that the application of the evolution operatorL should not lead to a state outside the subspace F s . In other words, each possible term of the evolution operator is a combination of creation and annihilation operators such that its application onto an arbitrary basis state (21) |n = |(n + , n − ) ∈ F s guarantees the conservation n + + n − = 1. It can be checked that each operator of the form fulfils this condition. Furthermore, the configuration state |F(t) is defined by The master equations (15) can be obtained directly from the general evolution equation ∂ t |F(t) =L|F(t) due to the orthogonality of the basis states n |n = δ n n when we identify K 11 = −β, K 12 = λ, K 21 = β and K 22 = −λ. Thus, (13) together with (23) and (24) can be interpreted as the Fock space representation of the master equations (15).

Spontaneous lattice reactions.
The Fock space F (and therefore also the reduced Fock space F s ) of a spontaneous single-species reaction at a certain site can be extended to a Fock space (and therefore to the corresponding reduced Fock space) of independent spontaneous single-species reactions on N lattice sites. The basis states of the reduced Fock space F s are now defined by the ordered product of local basis states The N factors of the vector n are the local occupation numbers n i ∈ (0, 1) corresponding to the states (21) of the reduced Fock space. The state |F(t) is again a linear combination of all basis vectors and the evolution operator becomeŝ because of the identity and the physical independence of the local spontaneous reactions. Here, we may see the consequence of our choice of the basis states (21). The action of an arbitrary local term of the evolution operator related to the position i changes the state |n at this position, i.e. |n 1 · · · |n i · · · |n N → |n 1 |n i · · · |n N , but the sign of the state remains unchanged due to the bilinear structure of all terms and the special structure of the basis states of the reduced Fock space. In the case of the traditional Fermionic Fock space, the evolution operator contains also linear terms describing the creation and annihilation of particles, so that now we have to deal with position-dependent signs of the total basis states after each applicationL. Finally, it should be remarked again that the anticommutation relations are completely defined by the well-known Fermionic algebra: 3.1.4. Kinetic coupling: autocatalytic reactions. The kinetic restrictions of an elementary autocatalytic step allow the realization of an elementary change A 0 at the lattice site i only if at least one neighbour site is also occupied. This restriction can be considered in the evolution operatorL by the following extension: ( ij is a lattice indicator with ij = 1 if i and j are neighbouring sites, otherwise ij = 0). The additional factorâ † 1,jâ 1,j allows a change of the lattice configuration by a simple unimolecular reaction A 0 at the lattice site i if the neighbouring site j is occupied, i.e. if |n j = |1 . Otherwise, the application ofâ † 1,jâ 1,j onto the state |n j = |0 leads toâ † 1,jâ 1,j |0 = 0. Thus the evolution operator of the autocatalytic reactions may be explicitly written aŝ with the rate λ for the forward reaction A + 0 → 2A and the rate β for the backward reaction 2A → A + 0. Furthermore, the contact process may be interpreted as the superposition of autocatalytic forward reactions with the rate λ and a spontaneous decay A → 0 with the rate β . We getL The evolution operators (30) and (31) may be combined to a general operator describing both autocatalytic and contact processeŝ with λ = λ + λ and z the coordination number of the lattice.

Diffusion processes.
Following the above-discussed construction principle for the evolution operator, Kawasaki exchange processes A + 0 0 + A are described bŷ where we have introduced the diffusion coefficient D controlling the jump rate of a particle A between neighbouring sites. The superposition of the evolution operators for the diffusion steps (33) and for the chemical reaction processes (32) leads to the total evolution operator of the diffusion-reaction system with exclusion,L =L diff +L chem . It should be remarked that the presented Fock-space formalism has the decisive advantage of a simple construction principle for each evolution operatorL on the basis of creation and annihilation operators. Therefore, this method allows investigations of master equations for various evolution processes, e.g. aggregation, chemical reactions [34,35], nonthermal kinetic Ising models [36], nonlinear diffusion [37] and just reaction-diffusion models.

Normalization condition.
Using the representation (26) and the orthogonality of the basis states, one obtains the normalization condition which must hold for all master equations This important property can be simply written as s|F(t) ≡ 1 by using the so-called reference state [36,34]: From this point of view, one obtains from (13) a second possibility to describe the normalization condition Hence, an evolution operatorL conserves the total probability if and only if s|L = 0.

Averages and operators.
The reference state (35) can be used also for the representation of averages. The average of a given observable C(n) as a function of the actual occupation n is usually defined by C = n P(n, t)C(n). On the other hand, the observable C(n) can be associated with an operatorĈ = n |n C(n) n|. Thus, using (26), the average C can be simply written as The average C(t) is time-dependent except that |F(t) represents the equilibrium or another stationary state. Furthermore, a formal equation of motion for this average follows from (37) and (13): For the investigations below, we need the following two operators: These operators indicate the occupation of a given lattice site i, i.e.Â + i corresponds to an occupied site, whileÂ − i is related to a vacancy.Additionally,Â + i andÂ − i allow the projection of the reference state s| onto an arbitrary basis state representing the configuration n via Thus, the probability P(n, t) = n|F(t) can be interpreted as the average of the is equivalent to the occupation probability of the lattice site i, while A − i is the probability that the lattice site i is vacant. From this point of view, A + i can be identified as the averaged concentration c i of particles at site i, i.e. A + i = c i , and therefore A − i = 1 − c i . The operatorsÂ + i andÂ − i fulfil the following identities in the reduced Fock space F s : These identities can be used for a change of operator structures, e.g. the operatorsĈ andĈ obtained from the mappinĝ are identical with respect to the reduced Fock space.

Transition probability.
The formal solution of the evolution equation (13) is given by Using (26), the probability distribution P(n, t) can be written as On the other hand, the evolution of the probability function P(n, t) is given by the well-known Chapman-Kolmogorov equation [40], i.e.
with the transition probability K(n, t; m, 0) describing the transition from the configuration m at the initial time t = 0 to the configuration n at time t. The comparison between (44) and (45) leads to the following expression for the transition probability:

Correlation functions. The correlation between two observables C(n) and D(n) is usually defined by
A simple calculation usingĈ = n |n C(n) n| andD = n |n D(n) n| and the orthogonality of the basis states, i.e. k|n = δ k,n , yields As mentioned above, the Fock space representation is an elegant method especially for describing relatively complex master equations in terms of a quantum mechanical language. Within this framework, further calculations, e.g. different kinds of mean-field approaches or perturbation theory, are always possible [5,34,35,37].

Grassmann path integral
The transition from a Clifford algebra (defined by the anticommutation rules of the operatorsâ † a,i andâ a,i ) to a Grassmann algebra (given by the Grassmann variables α a,i ) is possible because of the isomorphism [38] a The knowledge of the evolution operatorL in (13), see for example (29), allows the derivation of the transition probability K(n, t; m, 0) (46) as Grassmann path integral. We use a representation with a discrete time lattice with the sites t m = m(t/M) and the elementary time interval t = t/M. m is an integer with 0 m M. The transition probability K(n, t; m, 0) follows after carrying out the limit M → ∞. Using the Grassmann variables α = (α 1,1 , α 2,1 , α 1,2 , α 2,2 , . . . , α 1,N , α 2,N ) and where N is the number of lattice sites, one obtains [39] K(n, t; The basis states |n and n| defining the initial and final configurations n and n, respectively, are now simple ordered products of the completely anticommutating Grassmann variables: The functions L ij and L i are obtained from the original evolution operatorL =L diff +L chem , with (33) and (32). Using the usual rules [38,39] for the transition from Fermi operators to Grassmann variables, one obtains and The first exponential factor in the 2nd line of (50) can be separated into a product: Now we consider the first factor of (54). Taking into account the complete anticommutation rules of the Grassmann variables, one obtains for i = j: The introduction of ghost variables ϕ ij leads to a quadratic representation The normalization factor N is irrelevant for the further treatment, therefore we continue without an explicit representation of this quantity. The second factor of (54) is rewritten by analogous steps and one obtains Introducing the ghost variables ψ ij , we arrive at Finally, the third term becomes and therefore and the last term At this point, it should be remarked that only the apparently complicated Fermionic structure of the problem with operators for particles and vacancies allows the manipulations above. The fact that a reaction or diffusion step requires the annihilation of the old state and the creation of the new state generates, for all allowed elementary steps, an evolution operator that consists of the same number of creation and annihilation operators. Especially, this remarkable property allows us to use operators with a Fermionic algebra for a consistent formulation of the master equation in the Fock space representation and, furthermore, to introduce a Grasmann path integral. Another important remark is that an analogous transformation for three-particle processes (and also higher processes) leads to quadratic Grassmann terms containing nonlinear contributions of the ghost variables. These nonlinearities prevent, for three-particle and higher processes, a successful execution of the concept suggested below. Only two-particle processes lead always to a linear coupling between the quadratic Grassmann terms and the ghost variables. Fortunately, many higher-order reactions can be decomposed in a sequence of successive unimolecular and bimolecular reactions, so that the present technique is also available for these chemical processes.
The exchange i ↔ j in the first terms of the exponents containing the coupling between Grassmann variables and ghost variables and taking into account the symmetry ij = ji results in the following expression for the path integral (for M → ∞): with Dϕ Dψ Dω D ω = lim M→∞ M m=1 and The quantities k i (n i , t; n i , 0; [ϕ, ψ, ω, ω]) are Grassmann path integrals with respect to a single lattice site i: Note that the kinetic interaction between the lattice sites is now transferred from the nonlinear Grassmann terms (55), (57), (59) and (61) to the terms in the exponent containing ghost variables that couple pairs of lattice sites in a linear manner. We interrupt our calculation for a short moment and focus our attention on a seemingly independent question, which becomes important for the further analysis of (65).

Ordinary linear differential equation and Fock space representation
A linear differential equation of second order can always be transformed into a system of two linear differential equations of first order. We use the representation: with arbitrary time-dependent coefficients λ(t), β(t), µ(t) and η(t). Note that, with respect to the following investigations, the independent variable t is interpreted as time. We use the following initial conditions: It will be shown that this system can also be transformed into a representation in the reduced Fock space F s with the above-introduced basis states (21). We introduce an evolution operator E associated with the system of differential equations (66): Using the state (66) can be transformed into the evolution equation ∂ t |F(t) =Ê|F(t) with the initial condition |F(0) = S 0 |1 + Q 0 |0 . Note that an evolution operator corresponding to the usual master equation, e.g. (29), fulfils always the identity s|F(t) ≡ 1, see (34) and (35), whereas this conservation becomes invalid for the more general operator (68). Here, one obtains s|F(t) = S(t) + Q(t). For the following investigations, we use the notation and whereÂ + andÂ − are defined in (39). Obviously, the solution of (66) can be represented by the formal expression Using the general properties (40), one obtains the averages with respect to the reduced Fock and These expressions may be interpreted in a twofold manner. Firstly, they are the solutions of the differential equation (66) under well-defined initial conditions contained in |F(0) . This can be verified from the equation of motion (38) for the averages A + (t) and A − (t). After some simple manipulations we arrive at with , i.e. the quantities A + (t) and A − (t) satisfy the same system of differential equations (66) as S(t) and Q(t) and they have the same initial conditions due to |F(0) = S 0 |1 + Q 0 |0 . The identity between A(t) and the solutions [S(t), Q(t)] can also be checked considering the property (40). We obtain and Secondly, from (73), (74) and (40), it is obvious that the transition matrix can be interpreted as the time-dependent averages A ± (t) with respect to the initial conditions In other words, the transition matrix is a complete basis system of solutions with respect to the differential equations (66).

Fock space representation for Ito stochastic differential equations
In principle, the system of ordinary differential equations (66) can be generalized to Ito stochastic differential equations. Therefore, we replace the matrix M(t) by a deterministic term M 0 (t) and a stochastic contribution N W α=1 Ξ α (t) dW α (t)/dt, where dW α (t) are the differential components α of an N W -dimensional Wiener process W(t), i.e.

M(t) dt
Both M 0 and Ξ α may be arbitrary matrices. The Wiener process is characterized by the welldefined properties [40]: With M 0 (t) from (80), the equation ∂ t |F(t) =Ê(t)|F(t) now becomes an Ito stochastic differential equation for the Fock state |F(t) and the corresponding evolution of the averages (75) must be replaced by which is the corresponding Ito stochastic differential equation for the time-dependent average A(t). For the further analysis, it is an important remark at this point that the introduced averages are only performed with respect to the states of the reduced Fock space, while the Wiener processes are not concerned by this averaging procedure. Furthermore, the state |F(t) can be explicitly written as as follows from the comparison with (69), (76) and (77).

Grassmann path integrals for Ito stochastic differential equations 4.3.1. Infinitesimal time step: Fermionic algebra.
The aim of the following investigation is the representation of the solution of an Ito stochastic differential equation in terms of a Grassmann path integral. The first step is the determination of a time-dependent operatorÊ (t) describing the infinitesimal evolution: The operatorÊ (t) can be obtained from a comparison of (82) and (85). Hence we get The introduction of the additional term γâ † 1â 1â † 2â 2 dt is always possible because of the identity (41).After some algebraic transformations (see appendixA), we arrive at the following expression forÊ (t):Ê The difference betweenÊ (t) andÊ(t) stems from the fact that a Wiener process scales as dW α (t) ∼ √ dt.

Infinitesimal time step:
Grassmann algebra. The next step is the transformation of the infinitesimal evolution (85) into a Grassmann path integral. For that purpose, we replace the creation and annihilation operators by Grassmann variables and the corresponding differentiation operators, see (49). Therefore, (87) becomeŝ Furthermore, the representation (84) of the evolution state|F(t) is given in the explicit form The substitution of (88) into (85) leads to the Grassmann representation of the solution of an Ito stochastic differential equation (see appendix B for details). One obtains Note that we have here used the representation |F(dt), α instead of |F(dt) in order to show explicitly the dependence on the Grassmann variables. The function E is given by Note that the matrices M 0 and Ξ α are identical to those in (83). It should be remarked that E (t, α, α ) is not a completely unique expression. Without changing the physical content, this quantity may be modified by the addition of an arbitrary term proportional to α 1 α 1 α 2 α 2 . This possible gauge is a result of the original representation in the reduced Fock space 1 F s , see (41).

Grassmann path integral.
The successive continuation of the above-discussed infinitesimal steps leads straightforwardly to the following result: The discrete timescale is chosen as t m = m dt and we have used again the convention α m = α(t m ). The time dependence of E (t) stems from the stochastic Wiener processes. The discrete version of a Wiener process can be written as dW(t) → W(t m+1 ) = W(t m+1 ) − W(t m ), thus E (t) in (92) must be replaced by E (t m+1 ). 1 Obviously, the set of zero operators in F s (i.e. operators with the propertyĈ|n = 0 for any state |n ∈ F s ) contains more elements than the set of zero operators related to the complete Fock space F (e.g.Â + iÂ − i |n ≡ 0 for |n ∈ F s but there exists at least one state |n ∈ F withÂ + iÂ − i |n = 0). In principle, the complete Fock space has only the trivial zero operator0, with the exception of such combinations of operators that can be identically transformed intô 0 by using the Fermionic anticommutation rules. Now we are able to determine the transition matrix (78) as a Grassmann path integral. Considering that the scalar product using Grassmann variables is defined by where A t|n indicates that this solution belongs to the initial condition corresponding to the map (95). We remind that the quantities A ± (t) are still stochastic observables depending on the actual realization of the underlying Wiener processes. Thus the transition matrix k(n, t; n , 0; W) depends also on this special Wiener process W. The notation A indicates only that these quantities are averages with respect to the states of the original reduced Fock space F s .

Averages with respect to Wiener processes
Usually, the solution of an Ito stochastic differential equation for a special realization W(t) of the Wiener process is not of interest. However, averages over all realizations play an important role. Therefore, we give here the formal average of the transition matrix with respect to the Wiener processes which we need for our further discussions. Using (96), the averaged matrix elements of (97) correspond to the mean quantities A ± (t) with respect to the multi-dimensional Wiener process W (with dimension N W ). The introduction of normalized noise terms χ α with dW α (t) = χ α (t) dt leads to Thus the multi-dimensional Wiener process W with the components W α (α = 1, . . . , N W ) corresponds to a normalized Gaussian measure with the probability distribution with the normalization prefactor N .

Stochastic evolution equation
Now we come back to the original problem and continue the discussion of the path integral (65). Obviously, this path integral can be interpreted as a product N i=1 k i (n i , t; n i , 0; [ϕ, ψ, ω, ω]) averaged over a set of normalized Wiener processes (given by the quantities ϕ ij , ψ ij , ω ij and ω ij or, more explicitly, dW

respectively). Thus (65) can be rewritten as
The quantities k i n i , t; n i , 0; [ϕ, ψ, ω, ω] are path integrals depending on the actual Wiener processes. The comparison between (65) and the general mathematical structure of (91) shows that k i can be represented by (96), i.e.
where A + i (t|n i ) and A − i (t|n i ) are solutions of the corresponding system of Ito stochastic differential equations: with the initial conditions (95). The comparison of (65), (91) and (94) leads to the matrices: and and controlling the strength of the coupling between the Wiener processes and the dynamic variables A i (t). The structure of these matrices is mainly determined by the underlying elementary processes describing diffusion, creation and annihilation processes. It seems to be simple to construct the corresponding matrices also for other elementary unimolecular and bimolecular reactions and diffusion processes because the general concept is the same.

Averages and correlation functions
The knowledge of the transition matrix K(n, t; n , 0) (see equation 100) allows the determination of arbitrary averages, moments and correlation functions. In the following investigations, we confine ourselves to averages and correlation functions of the local occupation numbersÂ + i , see (39). All other quantities can be determined by an analogous procedure. Using (37), we obtain Because of the identity m|Â + i = m|δ m i ,1 , we have with (44) where the quantities A + j (t|n j ) and A − j (t|n j ) are the above-introduced solutions of the stochastic differential equations (102) subject to the special initial conditions (95). An important aspect becomes clear in the last equation. The left-hand side A + i (t) is the local occupation density of the lattice site i for the diffusion-reaction system with exclusion restrictions. Obviously, this quantity contains implicitly the interaction with neighboured lattice sites due to the evolutionary rules. The right-hand side of (109) is a product of quantities, which are apparently independent solutions of stochastic differential equations (102) for each lattice site. The coupling between these quantities is organized by the averaging procedure with respect to the Wiener processes W ϕ ij (t), W ψ ij (t), W ω ij (t) and W ω ij (t), because the stochastic processes which drive the evolution of the Ito differential equations of neighboured lattice points are not completely independent. Hence, the average procedure connects now the lattice sites of the system. The summation over all configurations can be carried out and we get A similar calculation leads to higher moments, e.g.
Correlation functions can be represented by similar expressions, e.g.
Considering that the initial conditions may also be written as (111) can be identically transformed into Note that the initial conditions require A + k (0|n k ) + A − k (0|n k ) ≡ 1, whereas the solution of the system of stochastic differential equations (104) is usually characterized by the property A + k (t|n k ) + A − k (t|n k ) = 1 for t > 0.

Ito stochastic differential equations and Fokker-Planck equation
The determination of averages and correlation functions using (107), (110) and (113) and the solution of the corresponding stochastic differential equations (e.g. by Monte Carlo simulations) is very hard, because of the complicated mathematical structure of (107), (110) and (113). The difficulties stem, on the one hand, from the Wiener processes acting simultaneously at pairs of lattice points. On the other, the local averages A + i (t) and the two site correlation functions A + i (t)A + j (0) depend on the time evolution of A + k (t) and A − k (t) at all lattice sites, see (107), (110) and (113).
There exists always a probability P(A, t; A 0 , 0) determining the transition from the initial values A 0 (at time t = 0) to the final values A (at time t) under the influence of multi-dimensional Wiener processes W ϕ , W ψ , W ω and W ω via the Ito stochastic differential equation (102). The generating equation for this probability can be obtained from the well-known connection between Ito stochastic differential equations and Fokker-Planck equations [40]. One obtains the following generalized Fokker-Planck equation for our diffusion-reaction system: where z is the coordination number of the underlying lattice.

Probability and pseudo-probability
6.2.1. Pseudo-probability. Equation (115) allows the determination of the probability distribution P(A, t; A 0 , 0). Thus, averages (107) and (110) and correlation functions (113) can be obtained directly from this probability. In terms of this probability, the relevant averages are now defined by A real disadvantage is the remaining product on the right-hand side, because it requires the calculation of high moments in order to obtain the simple local occupation rate A + i (t) of the underlying diffusion reaction lattice system, see also (109). A similar situation occurs also for the determination of other moments or correlation functions. All these quantities contain similar product terms, see (109), (110) and (113). Thus, it seems to be reasonable to introduce which leads to the more favourable representation: The new quantity G(A, t; A 0 , 0) is no longer a real probability distribution function, although it plays the role of a probability in the formulae for moments and correlation functions, see for example (118). Therefore, let us speak about a pseudo-probability (117). An important cause why G(A, t; A 0 , 0) is not a probability follows from the fact that this quantity is not positive definite, see (117). However, this pseudo-probability can be generated also by a partial differential equation similar to a Fokker-Planck equation. The insertion of P(A, t; (115) and simple algebraic transformations lead to with the functional dependence G = G(A, t; A 0 , 0). This linear partial differential equation has a structure similar to the usual Fokker-Planck equation.

Transformation of variables.
The pseudo-probability G(A, t; A 0 , 0) simplifies considerably the determination of averages and correlation functions, but the averaging procedure must be performed now over a fraction, see (118). Therefore, a mapping seems to be reasonable. This transformation leads to a new representation of (119) for the pseudo- From (120), the initial conditions are now defined by N i,0 ≡ 1 and φ i,0 = n i .

New probability distribution.
The transformation of the variables (120) changes also the integral measure on the right-hand side of (118). Because of one obtains instead of (118) Equation (118) suggests the introduction of a new distribution function: In contrast to G (N, φ, t; N 0 , φ 0 , 0), the function (N, φ, t; N (117)). Furthermore, the function is a solution of the following partial differential equation: It can be checked immediately that (125) leads to N k=1 (dN k dφ k ) = const. Now we use (117), (124) and (122) and obtain the relation Because of the fact that the integral N k=1 (dN k dφ k ) is independent of time, the average is also conserved. On the other hand, the initial state is always characterized by A + i + A − i ≡ 1, see also (112). Therefore, is characterized by the normalization condition From this point of view, has all properties of a probability distribution function or, because of its dependence on the initial conditions, of a conditional probability distribution function.

Reduced probability distribution. Equation (123) can be written as
Obviously, the knowledge of the total probability distribution (N, φ, t; N 0 , φ 0 , 0) is not necessary for the determination of averages and correlation functions. In principle, the reduced probability is sufficient for the calculation of the above-mentioned statistical quantities. Note that each initial configuration n 0 is equivalent to N i,0 ≡ 1. The integration (129) is performed in each term of (125) under consideration of the usual boundary conditions (N, φ, t; N 0 , φ 0 , 0) → 0 and ∂ N i (N, φ, t; N 0 , φ 0 , 0) → 0 for N → ∞. This yields a Fokker-Planck equation for the reduced probability distribution function: Obviously, it seems more correct to say that this final equation is a Fokker-Planck equation equivalent to the original master equations on a lattice with local complete exclusion rules. The new set of variables φ i replaces the original discrete occupation numbers n i . A first important remark concerns the mathematical structure of the new variables. While the occupation numbers are discrete quantities with only two values, n i = 0, 1, the new variables φ i may be continuous quantities, φ i ∈ (−∞, ∞). However, all moments and correlation functions of the original variables have an analogous representation in terms of the new variables.

Averages and correlation functions
A knowledge of the probability distribution function (129) allows the determination of all relevant averages and correlation functions by simple relations. Thus, the average n i (t) = A + i (t) is given by (107) and (128), i.e.
The connection between the initial probabilities P(φ 0 , 0) and P(n , 0) is given by the simple relation φ i,0 = n i . Thus the average A + i (t) (determined from the Fock space representation and the original master equation) and the average of φ i using the statistical weight are equivalent observables. Higher moments are given by similar relations, e.g.
Furthermore, correlation functions can be determined by using (115) and (134): A further test of these relations is demonstrated in appendix C. On the other hand, it is simple to demonstrate that φ 2 i = n 2 i = n i . This is a consequence of the anticommutation algebra, which was used for the derivation of (130). This algebra excludes multinomials containing variables with a power higher than one. Obviously, this discussion together with some general considerations, suggest that all averages over multilinear combinations φ i φ j φ k · · · φ m are equivalent to the corresponding averages A + i A + j A + k · · · A + m = n i n j n k · · · n m , while all averages of multinomials with a higher degree per variable, e.g. φ 2 i φ j φ k · · · φ m , have no direct physical meaning. A similar statement follows for the two-time correlation function A + i (t)A + j (t) · · · A + k (0)A + l (0) · · · if the multinomials corresponding to the time t and to the initial time are linear in their variables.

Discussion
The derivation of a Fokker-Planck equation describing a substitute process instead of the original process is a particularly elegant technique for the analysis of chemical master equations. Without the exclusion principle, a well-established method is the so-called Poisson representation [41,42]. Here, it was assumed that the probability distribution function P(n, t) of a lattice is a superposition of multivariate uncorrelated Poisson processes In contrast to processes under exclusion rules, the occupation numbers are non-negative integers, n i = 0, 1, 2, . . . . Elementary diffusion steps are defined between neighbour lattice sites, whereas chemical reactions occur between the particles of one site. Thus, chemical reactions are totally local processes, while the elementary diffusion processes require two lattice sites. For our reaction-diffusion system, the function f(θ, t) defined in (134) satisfies the equation which is of a Fokker-Planck form provided λθ i > βθ 2 i . There exists a simple relationship between the moments of the occupation numbers and moments formed by application of f(θ, t) n i n j · · · n m = θ i θ j · · · θ m = dθ[θ i θ j · · · θ m ]f(θ, t).
This relation follows from the factorial moments of the Poisson distribution. However, f(θ, t) is not a real probability distribution function because it cannot be guaranteed that this pseudoprobability distribution is a positive-definite function. But it can demonstrated that (141) is equivalent to a stochastic differential equation with A more detailed analysis shows that the motion takes place over the range (0, λ/β) and both boundaries satisfy the criteria for entrance boundaries. This means, if the initial conditions are located on this range, the system cannot leave this range. The time-dependent functions θ i (t) define an underlying substitute process. The corresponding substitute process for the diffusion reaction system with exclusion rules reads with the noise ξ i (t) defined by Both substitute processes with and without exclusion rules become equivalent for purely unimolecular reactions. In our case, this means D = λ = β = 0, and we get the deterministic equations describing the reaction A → 0. Surprisingly, these processes are not controlled by noise terms, although the corresponding real evolutions of the local occupation number are random processes. This is a typical indication that (137) and (139) describe substitute processes. However, there are essential differences between the substitute processes with and without exclusion rules. The first difference concerns the diffusion process. While the Poisson representation describes diffusion by a deterministic equation the substitute process with exclusion requires an additional noise term The origin of this surprising difference comes from the vacancies. The exclusion principle requires that an arbitrary site is either occupied or vacant. Without the exclusion principle, this conservation law is violated since the basin of vacancies per lattice site is infinitely large. A second difference between both approaches occurs in the deterministic bimolecular reaction part. The exclusion principle allows only reaction terms connecting neighboured lattice sites. In contrast to this behaviour, the bimolecular reaction terms may be local terms if the exclusion principle is neglected. Therefore, the corresponding noise without the exclusion principle may also be a local expression, while the noise term for bimolecular reactions with exclusion rules are necessarily nonlocal.
Another important difference between both representations is the range of allowed values. As mentioned above, the variables of (137) have to be localized over the range 0 < θ i < λ/β. On the other hand, each well-defined initial value θ i (0) = θ 0 i corresponding to an initial distribution f(θ i , 0) = δ θ i − θ 0 i is related to a reasonable probability distribution of the occupation number Thus, an initial value θ 0 i > λ/β should also be allowed. This means that the correlation of the noise terms becomes negative and, consequently, (135) is no longer a Fokker-Planck equation. On the other hand, the process (139) allows the full range −∞ < φ i < ∞ for all variables. Especially, the corresponding equation (130) is a true Fokker-Planck equation in the sense that the kernel of the second-order derivatives is positive definite. This may be checked by the fact that (114) is always a true Fokker-Planck equation with a positive-definite second-order kernel. Thus, during all the following transformations of the probability distribution function and the variables, this characteristic property of (114) remains unchanged. In particular, we get the result that the noise matrix (140) is also a positive-definite object, so that (130) for the substitute process (139) is defined for all possible realizations of the variables φ.
We remark that another approach leading to a field theoretical description [43] of reactiondiffusion systems considering diffusive, reactive and branching processes of order higher than n = 2 also use the Fock space representation and the path-integral technique. But the main difference from our approach is that the original master equation is transformed into a Fock space representation by the application of a bosonic algebra with the consequence that the exclusion principle is completely neglected. In contrast, the fermionic algebra used in the present approach guarantees the exclusion principle, but it becomes very difficult to construct a treatable path integral for reactive processes of orders n > 2.
Finally, we compare the classical evolution equation (10) for a reaction-diffusion system with the coarse-grained versions of the substitute processes (137) and (139). A simple method is a kind of random-phase approximation. Here, we neglect all local fluctuations. Thus we may interpret φ i as a lattice representation of the local particle density ρ(r, t). If the spatial fluctuations of ρ(r, t) are sufficiently small, we arrive at dρ dt = +D ρ + z λ − β ρ − z( λ + β)ρ 2 + ζ(r, t) with the noise ζ(r, t)ζ(r , t ) ∼ ρ(r, t)δ(r − r )δ(t − t ).
Note that the diffusion coefficient D = D 2 contains now also information about the characteristic length scale of the lattice. Thus, the kinetic equation (10) may be interpreted as the coarse-grained version of the stochastic evolution equation (139).

Conclusions
It was shown that master equations of discrete variables describing diffusion processes as well as unimolecular and bimolecular reaction processes on a lattice can be mapped onto a system of Ito stochastic differential equations. These stochastic equations are characterized by the following special properties: (i) All equations appear to be local processes (i.e. processes with respect to one lattice cell or lattice point). Thus, each lattice site i corresponds to a closed system of differential equations consisting of dynamical variables (A + i and A − i ), which are related only to this lattice site. In other words, the differential equations are separated with respect to the dynamical variables of each lattice site, see (102). (ii) The dynamic observables of different lattice sites only seems to be independent. The coupling between different lattice points is a result of the simultaneous action of Wiener processes W α (t) ↔ ϕ ij (t), ψ ij (t), ω ij (t), . . . at pairs of lattice sites.
Thus, only the stochastic processes enforce the kinetic interaction between neighbouring sites in this representation. Since the Ito stochastic differential equations are equivalent to a Fokker-Planck equation, one obtains after some reasonable transformations a probability distribution, which allows the determination of averages, moments and correlation functions by formulae similar to the determination of these statistical variables by the original master equation, see (131)-(133). The decisive difference between the original probability distribution function (corresponding to the master equation) and the new probability distribution related to a Fokker-Planck equation is given by the variables. The master equation of the original diffusion reaction process on the lattice is based on a finite number of discrete variables, while the probability distribution function of the obtained substitute process depends on continuous variables (in the present case φ i ). This mapping from a discrete representation to a continuous one is the main result of the present work.
Partial integration considering the boundary conditions (φ, t; φ 0 , 0) → 0 for φ → ∞ leads to the equation An analogous procedure using (138) and (136) yields The comparison of (C.4) with (C.8) and (C.5) with (C.9) shows a complete correspondence. This agreement confirms the equivalence between the original master equation of the autocatalytic reaction processes and the Fokker-Planck equation (130).