Poissonian stationary distribution

A sufficient condition for having Poissonian stationary distribution in a certain class of birth and death processes was given by Whittle (1968). However, his assumptions are slightly different from those of chemical reactions, therefore the search for precisely defined assumptions or for certain classes of reactions is necessary. 

Theorem. The necessary and sufficient condition for a simple birth and death-type process to have a Poissonian stationary distribution (if it has a nondegenerate stationary distribution at all) is that it be a linear one (Erdi Toth, 1979). 

Particular results can be derived for certain classes of the Markov population process (Kingman, 1969). Necessary and sufficient criteria were given for polynomial simple Markov population processes to have Poissonian stationary distributions when the detailed balance condition holds. Special relations among the coefficients are necessary to get Poissonian distribution (Toth Torok, 1980 Toth, 1981 Toth et al., 1983). 

Toth, J. (1981a). Poissonian stationary distribution in a class of detailed balanced reactions. React. Kinet. Catal. Lett., 18, 169-73. 

The master equation has been investigated for a sequence of unimolecu-lar (nonautocatalystic) reactions based on moment generating functions [6] these yield Poissonian stationary distribution for single intermediate systems in terms of the number of particles X of species X, with X that number in the stationary state... 

Exercise. Generally, in any open system with reservoirs A, B, C,... and reactants X, Y, Z,..., if the stationary state does not transport matter among the reservoirs, the stationary distribution is multi-Poissonian. 

In the literature of stochastic reaction kinetics it was often assumed that the stationary distributions of chemical reactions were generally Poissonian (Prigogine, 1978). The statement is really true for systems containing only first-order elementary reactions, even when inflow and outflow are taken into account (i.e. for open compartmental systems see Cans, 1960, p. 692). If the model of open compartmental systems is considered as an approximation of an arbitrary chemical reaction near equilibrium, then in this approximation the statement is true. 

Some historical remarks. The physical assumption adopted by van Kampen (1976) is that the grand-canonical distribution of the particle number of an ideal mixture is Poissonian. Based on this — strongly restrictive — assumption, and utilising the conservation of the total number of atoms the stationary distribution can be obtained. This stationary distribution can be identified with the stationary solution of the master equation, and it is not Poissonian in general, even for large systems. 

Hanusse (1976, pp. 86-8) claimed that in the case when not only monomolecular reactions are in the system, the stationary distribution can be different from the Poissonian. Zurek Schieve (1980) have simulated a particular chemical reaction exhibiting a non-Poisson distribution. 

If the stochastic model of a chemical reaction can be identified with a simple birth and death process, and the process has stationary distribution, then it is precisely Poissonian, if the reaction is an open compartmental system. 

Random-walk models for diffusion on the lattice can be implemented in various ways. Typically, particles randomly hop between a node r and its neighbors r e -T"(r). If there are no restrictions on the number of particles at a node, the stationary distribution for this diffusive stochastic process is Poissonian, ... 

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