Reaction-diffusion master equation

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

Baras, F. Mansour, M. M. Reaction-diffusion master equation a comparison with microscopic simulations. Phys Rev E 1996, 54 6139-6148. 

As earlier in the case of diffusion-controlled concentration decay, one-species reactions, A +A -> A and A + A —> 0, in one-dimension, could be used as a proving ground for the testing theory of particle accumulation, when random particle creation is added. For the former reaction the master equation could be easily derived in the form [95]... 

The master equation approach considers the state of a spur at a given time to be composed of N. particles of species i. While N is a random variable with given upper and lower limits, transitions between states are mediated by binary reaction rates, which may be obtained from bimolecular diffusion theory (Clifford et al, 1987a,b Green et al., 1989a,b, 1991 Pimblott et al., 1991). For a 1-radi-cal spur initially with Ng radicals, the probability PN that it will contain N radicals at time t satisfies the master equation (Clifford et al., 1982a)... 

The reason why the boundaries in physical problems are often natural becomes obvious by looking at the simple example of radioactive decay in IV.6. The probability for an emission to take place is proportional to the number n of radioactive nuclei, and therefore automatically vanishes at n = 0. The same consideration applies when n is the number of molecules of a certain species in a chemical reaction, or the number of individuals in a population. Whenever by its nature n cannot be negative any reasonable master equation should have r(0) = 0. However, this does not exclude the possibility that something special happens at low n by which the analytic character of r(n) is broken, as in the example of diffusion-controlled reactions. A boundary that is not natural will be called artificial in section 7. 

So far we only considered transport of particles by diffusion. As mentioned in 1 the continuous description was not strictly necessary, because diffusion can be described as jumps between cells and therefore incorporated in the multivariate master equation. Now consider particles that move freely and should therefore be described by their velocity v as well as by their position r. The cells A are six-dimensional cells in the one-particle phase space. As long as no reaction occurs v is constant but r changes continuously. As a result the probability distribution varies in a way which cannot be described as a succession of jumps but only in terms of a differential operator. Hence the continuous description is indispensable, but the method of compounding moments can again be used. 

In this Chapter we introduce a stochastic ansatz which can be used to model systems with surface reactions. These systems may include mono-and bimolecular steps, like particle adsorption, desorption, reaction and diffusion. We take advantage of the Markovian behaviour of these systems using master equations for their description. The resulting infinite set of equations is truncated at a certain level in a small lattice region we solve the exact lattice equations and connect their solution to continuous functions which represent the behaviour of the system for large distances from a reference point. The stochastic ansatz is used to model different surface reaction systems, such as the oxidation of CO molecules on a metal (Pt) surface, or the formation of NH3. 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

The results obtained for the stochastic model show that surface reactions are well-suited for a description in terms of the master equations. Since this infinite set of equations cannot be solved analytically, numerical methods must be used for solving it. In previous Sections we have studied the catalytic oxidation of CO over a metal surface with the help of a similar stochastic model. The results are in good agreement with MC and CA simulations. In this Section we have introduced a much more complex system which takes into account the state of catalyst sites and the diffusion of H atoms. Due to this complicated model, MC and in some respect CA simulations cannot be used to study this system in detail because of the tremendous amount of required computer time. However, the stochastic ansatz permits to study very complex systems including the distribution of special surface sites and correlated initial conditions for the surface and the coverages of particles. This model can be easily extended to more realistic models by introducing more aspects of the reaction mechanism. Moreover, other systems can be represented by this ansatz. Therefore, this stochastic model represents an elegant alternative to the simulation of surface reaction systems via MC or CA simulations. 

Let us study now a stochastic model for the particular a+ib2 -> 0 reaction with energetic interactions between the particles. The system includes adsorption, desorption, reaction and diffusion steps which depend on energetic interactions. The temporal evolution of the system is described by master equations using the Markovian behaviour of the system. We study the system behaviour at different values for the energetic parameters and at varying diffusion and desorption rates. The location and the character of the phase transition points will be discussed in detail. 

Fluctuations in nonequilibrium systems have been studied mainly through two approaches the master equation approach16 and more recently the Ginsburg-Landau functional approach.17 In the master equation approach, the microscopic transition probabilities for chemical reactions and diffusion are taken to be given, and a master equation for the spatiotemporal variation of the probability distribution is obtained. Though the explicit solution of the master equation is difficult to obtain, some important general features could be deduced from it. One can show... 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

MC is also successful in far from equilibrium processes encountered in the areas of diffusion and reaction. It is precisely this class of non-equilibrium reaction/diffusion problems that is of interest here. Chemical engineering applications of MC include crystal growth (this is probably one of the first areas where physicists applied MC), catalysis, reaction networks, biology, etc. MC simulations provide the stochastic solution to a time-dependent master equation... 

Here, j) (z = A, B, j = S, H) indicates the average coverage of the surface with species i on surface phase j. The rate constants are indicated with where p indicates the reaction type and q indicates the reactant, if needeci. Note that these rate constants are often not equal to the transition probability W, as used in the master equation (1). In equation (63), the first two terms on the right hand side describe adsorption and desorption of A. These reactions involve only one site and are not influenced by lattice symmetry, hence in this case Two diffusion terms are inserted. 

Vibrational relaxation and excitation and usually the rate-limiting processes for molecular reaction in the gas phase, and their importance has led to many theoretical approaches. The use of an FPE such as described in Section II leads to a diffusion model in energy space, and only applies if the collision kernel P(E, E ) of the master equation is strongly peaked about the initial energy E. This is the weak collision limit in which the energy transfer is small, or comparable to kT. Other approaches, such as the model of Bhatnager, Gross and Krook propose that impulsive collisions randomize... 

In Section III the temporal behavior of diffusion-reaction processes occurring in or on compartmentalized systems of various geometries, as determined via solution of the stochastic master equation (4.3), is studied. Also, in Sections III-V, results are presented for the mean walklength (n). From the relation (4.7), and the structure of the solutions (4.6) to Eq. (4.3), the reciprocal of (n) may be understood as an effective first-order rate constant k for the process (4.2) or (n) itself as a measure of the characteristic relaxation time of the system it is, in effect, a signature of the long-time behavior of the system. 

Regarding the dependence of the reaction efficiency on the dimensionality of the compartmentalized system, the studies reported in Sections III.B.3 and III.B.4 on processes taking place on sets of fractal dimension showed that, consistent with the results found for spaces of integer dimension, the higher the dimensionality of the lattice, the more efficient the trapping process, ceteris paribus. Processes within layered diffusion spaces, which can be characterized using an approach based on the stochastic master equation (4.3), show a gradual transition in reaction efficiency from the behavior expected in c( = 2 to that in = 3 as the number k of layers increases from fe = 1 to k = 11. 

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

The stochastic approach to reaction-diffusion systems is not mathematically well-established. Though spatio-temporal stochastic phenomena ought to be associated with random fields, and not with stochastic processes, the usual investigation of such kinds of physicochemical problems starts from the master equation, and then it is extended by some heuristic procedure. From the physical point of view the role of spatial fluctuations is obviously important. It is well known that the density fluctuations are spatially correlated, and according to the modern theory of critical phenomena (e.g. Fisher, 1974 Wilson Kogut, 1974) small fluctuations are amplified owing to spatial interactions causing drastic macroscopic effects. 

As in the case of homogeneous systems, there are two kinds of stochastic descriptions for reaction-diffusion systems as well the master equation approach and the stochastic differential equation method. Until now we have dealt with the first approach however, stochastic partial differential equations are also used extensively. Most often partial differential equations are supplemented with a term describing fluctuations. In particular, time-dependent Ginzburg-Landau equations describe the behaviour of the system in the vicinity of critical points (Haken, 1977 Nitzan, 1978 Suzuki, 1984). A usual formulation of the equation is ... 

Nicolis, G. Malek-Mansour, M. (1980). Systematic analysis of the multivariate master equation for a reaction-diffusion system. J. Stat. Phys., 22, 495-512. 

In Chap. 5 we discussed reaction diffusion systems, obtained necessary and sufficient conditions for the existence and stability of stationary states, derived criteria of relative stability of multiple stationary states, all on the basis of deterministic kinetic equations. We began this analysis in Chap. 2 for homogeneous one-variable systems, and followed it in Chap. 3 for homogeneous multi-variable systems, but now on the basis of consideration of fluctuations. In a parallel way, we now follow the discussion of the thermod3mamics of reaction diffusion equations with deterministic kinetic equations, Chap. 5, but now based on the master equation for consideration of fluctuations. 

---