Solutions of the Master Equation

Abstract

Weidlich considers the solutions of the general and the migratory master equation. At first he assumes that the probability transition rates do not depend on time. Furthermore, he assumes that the transition rates satisfy the condition of detailed balance. This condition is fulfilled in the case of his migratory model. Whereas the stationary solution has a very complicated form in general, a simple representation can be given to it in the case of detailed balance; the unique stationary solution of the migratory master equation is derived. Its importance derives from the fact that all time dependent solutions of the master equation must finally evolve into the stationary solution. This property is shown by deriving the master equation version of the famous H-Theorem of L. Boltzmann. Finally, the author goes over to time dependent solutions of the master equation. The general method of solution leads to an eigen value problem which in general can only be solved numerically. For the special case of the migratory master equation with probability transition rates linear in the population numbers, he finds an easily interpretable exact time dependent solution of analytical form. This special case also serves for demonstrating the relation between solutions of the master equation and the mean value equations.