Two-cell stochastic models

The simplest extension of stochastic models of homogeneous kinetic systems for describing spatial effects is the introduction of the two-cell model. According to this approach reactions take place within the homogeneous cells, while transport of matter between the cells is taken into account as a random exchange between cells. 

The two-cell stochastic model of the Schlogl reaction describing first-order phase transition has been carefully studied (Borgis Moreau, 1984b Moreau Borgis, 1984). 

Let P( i, l) denote the absolute distribution function. Accordingly, at a fixed time /, the number of molecules in cell 1 is i, and in cell 2 is 2-Denoting with the operator of the reaction operating on w, which describes the effect of chemical reactions the master equation can be described  

Approximations or simulation experiments are necessary to find even the stationary solution. The behaviour of the system is particularly interesting for small values of D, since large D implies a homogeneous solution. 

The simulation technique used earlier (Frankowicz Gudowska-Nowak, 1982) was adopted by Borgis Moreau (1984b) to calculate the stationary distribution. Its shape is shown by Fig. 6.1. 

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Borgis, D. Moreau, M. (1984b). Two cell stochastic model of Schlogl reaction with small diflfusional coupling, J. Stat. Phys., 34, 631-51. 

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