Solution Methods of Master Equations

Abstract

After the derivation of the Master equation, it is logical to introduce and discuss different methods of its solution. One obvious and frequently applied method consists in the approximate transformation of the discrete Master equation in a partial differential equation, namely the Fokker–Planck equation. The not so well-known T-factor method is very efficient in the transformation of the Master equation into difference equations of reduced order, which are easier to handle, and into continued fractions. This method also provides a very elegant way to derive exact and approximate stationary solutions of the Master equation, even in case when detailed balance is not fulfilled. A general graph-theoretical method for the stationary solution developed by Kirchhoff for electrical networks is also presented. In case of detailed balance an exact solution method for the stationary probability distribution is also presented. The chapter closes with exact and approximate solution methods for one-dimensional Master equations with two particle jumps.