The Macroscopic Limit in aStochastic Reaction {
نویسنده
چکیده
Extensive numerical simulation of a reaction{diiusion{system reveal an unusual system size dependence of the uctuation magnitude. If denotes the system size parameter, e. g. particle number, uctua-tions are usually predicted to be of order 0:5 (stable case) or 1 (diiusion{type case). In contrast, a scaling like 0:84 is observed in a combined birth{death and random{walk process, which is described by a multivariate chemical master equation and corresponds to the Fisher equation in the macroscopic limit. The system. In this work we consider the following simple nonlinear reaction{ diiusion{system: Particles are distributed along one spatial coordinate and move by way of diiusion. They react according to the scheme A *) 2A. Thus, the presence of A-particles at some location leads to further production, and at the same time reactions of two A-particles will destroy one of them. Macroscopic Description. Mean eld theory yields a deterministic reaction{ diiusion equation by completely neglecting the uctuations. In the present case, this is the Fisher equation. It is a nonlinear partial diierential equation for the space and time dependent concentration variable c(x; t): @c @t = @ 2 c @x 2 + c ? c 2 : (1) Here, is the (space-and time{independent) diiusion coeecient, and both reaction rates have been absorbed by choosing appropriate units for x and t. Equation (1) was proposed by Fisher to model the spread of advantageous genes 1] and its relevance stems from the fact that it admits stable travelling wave solutions.
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