Stochastic theory reaction diffusion system

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. 

Thermodynamic and Stochastic Theory of Reaction—Diffusion Systems Relative Stability of Multiple Stationary States... 

So far we have considered only homogeneous reaction systems in which concentrations are functions of time only. Now we turn to inhomogeneous reaction systems in which concentrations are functions of time and space. There may be concentration gradients in space and therefore diffusion will occur. We shall formulate a thermodynamic and stochastic theory for such systems [1] first we analyze one-variable systems and then two- and multi-variable systems, with two or more stable stationary states, and then apply the theory to study relative stabihty of such multiple stable stationary states. The thermodynamic and stochastic theory of diffusion and other transport processes is given in Chap. 8. 

Thermodynamic and Stochastic Theory of Reaction-Diffusion Systems I.OOr... 

The idea of homogenous spatial distribution of the particles is based on the concept of well-stirred reactor. However, even microscopic reactions produce local nonhomogenities, which can not be always eliminated by diffusion. There are many contraversions about the stochastic formulation of reaction - diffusion systems. Three directions in the theory of random fields seem to be able to cope with such complexity the theory of random measures, the theory of stochastic partial differential equations relating to trajectories, and the theory of Hilbert space valued stochastic processes. The details are beyond of our scope. 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

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