Stochastic master equation connectivity

Suppose the geometry of a compartmentalized system is described by a lattice of integral or fractal dimension of given size and shape, and characterized by N discrete lattice points (sites) embedded in a Euclidean space of dimension d = de and local connectivity or valency v. At time t = 0, assume that the diffusing coreactant A is positioned at a certain site j with unit probability. For f > 0 the probability distribution function p(f) governing the fate of the diffusing particle is determined by the stochastic master equation... 

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. 

It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... 

Fox, R. 0. and L. T. Fan. A Master Equation Formulation for Stochastic Modeling of Mixing and Chemical Reactions in Inter-Connected Continuous Stirred Tank Reactors. Instn. Chem. Engrs. Symp. Series 87 (1984) 561. 

It has been seen that the transition from one investors configuration n , n to another can be a single unit motion connected with a product innovation (5.13 a) or a process innovation (5.13 b) of one investor, or a multiple unit motion (5.14a and b) which is most often an imitation process. As innovation always and imitation sometimes is investment under uncertainty, it can be suspected that a deterministic modelling of the motion of the investors configuration will not succeed. But since all investment is risky, the stochastic approach is appropriate. Therefore the well founded master equation formulation will be adopted. 

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