System of Two Mesoscopic Equations

We consider the transport of particles A and B with linear reversible reaction A B, One can introduce a two-component system of equations for the densities P x,t) and P2(x,t). We assume that the probability of a transition from A to B during a small time interval of length h is Oih + o(h), and the backward transformation B —A has the probability a2h+o(h). We assume that the reaction is independent of the transport of particles. The probability of a jump during a small time interval h is X h+o h) for particles A and X2h+o(h) for particles B. The balance of particles A and B at the point x can be written as 

These equations are the conservation laws for A and B particles. The first term on the right-hand side of (3.82a) represents the particles A that stay at location x and do not move during the time interval (t, t + h] and do not become particles B. The second term corresponds to the number of particles of type A that arrive at x during (t,t+h] from other points x - z, where the jump length z is distributed according to 

3 Random Walks and Mesoscopic Reaction-Transport Equations 

In the limit 0, we obtain the mesoscopic system of reaction-transport equations 

The main advantage of this Markovian model is that it can be easily generalized to include various nonlinear terms. Later we consider non-Markovian models for which the inclusion of nonlinear effects is a highly nontrivial procedure, see Sect. 3.4. 

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Random Walk with Two States and the System of Two Mesoscopic Equations... 

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