Mesoscopic Equation for the Particle Density

Let us introduce the average density of particles p(x, n) at point x at time n. We assume that the number of particles per unit length around jc is large enough that we can neglect the random fluctuations in the number density. In this case, the particle density p x, n) obeys the integral balance equation 

This equation states that the particle density at time n + 1 is the sum of the densities at intermediate points x - z at time n multiplied by the probability of transition from X - z to X. This is a mesoscopic description. Although it only deals with the mean density of particles p(x,n), it involves a detailed description of the movement of particles on the microscopic level. Equation (3.13) is the same as the Kolmogorov forward equation (3.12). The solution to (3.13) can be rewritten as aconvolution 

If the PDF is w x) = j5(x — a) + (x+fl), jumps Z can take only two values, a and —a, with equal probabilities. In this case we have a recurrence equation 

This equation can be recognized as a finite difference approximation of the diffusion equation 

This becomes clearer if we let the time step be of size x instead of size 1. Then the recurrence equation (3.15) can be rewritten as 

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So far we have considered a single mesoscopic equation for the particle density and a corresponding random walk model, a Markov process with continuous states in discrete time. It is natural to extend this analysis to a system of mesoscopic equations for the densities of particles Pi(x,n), i = 1,2,..., m. To describe the microscopic movement of particles we need a vector process (X , S ), where X is the position of the particle at time n and S its state at time n. S is a sequence of random variables taking one of m possible values at time n. One can introduce the probability density Pj(jc, n) = 9P(X < x,S = i)/dx and an imbedded Markov chain with the m x m transition matrix H = (/i ), so that the matrix entry corresponds to the conditional probability of a transition from state i to state j. 

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