The Kolmogorov or Master Equations

In a general context, suppose a given volume V contains a spatially homogeneous mixture of Ni particles from m different populations of initial size noi (i = 1. m). Suppose further that these m populations can interact through to0 specified reaction or diffusion channels Ri (1 = 1. m0). These processes 

The hi are used to express the conditional probability of changes in the population sizes for the Ri reaction from I, to I, I At given the system in n (/,) at t. These probabilities are described by means of the intensity functions IVlA.Vl]m (Ni,.. whereby 

The definition of the hi hazard rates and the model of (9.33) are the only required hypotheses to formulate the stochastic movement or reaction of particles in a spatial homogeneous mixture of m-particle populations interacting through m0 reactions. 

To calculate the stochastic time evolution of the system, the key element is the grand probability function 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation 

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Not only is the master equation more convenient for mathematical operations than the original Chapman-Kolmogorov equation, it also has a more direct physical interpretation. The quantities W(y y ) At or Wnn> At are the probabilities for a transition during a short time At. They can therefore be computed, for a given system, by means of any available approximation method that is valid for short times. The best known one is time-dependent perturbation theory, leading to Fermi s Golden Rule f)... 

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

Using Eq. 13.59 we write the differential form of the Chapman-Kolmogorov equation, or the master equation of the stochastic process X(f), as follows ... 

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