Master Equation in Configuration Space

Abstract

In many cases we are not interested in the states x _α of the single elements a of a system but only in the number n ^a _x of subsystems of type a which are in state x (e.g. the number of molecules of sort B which are in a bounded or an unbounded state). Since the states of the subsystems a change stochastically, the occupation numbers n ^a _x change stochastically, too. Consequently, the occupation numbers must be described by a probability distribution. Its temporal evolution is given by a master equation in configuration space (the space of occupation numbers). When considering only spontaneous state changeses we find, as approximate mean value equation of the configurational master equation, the master equation in state space from Chapter 1. Taking into account also pair interactions, as approximate mean value equations we obtain Boltzmann-like equations, instead. A generalization to interactions of higher order (between three or more systems) is possible without any problems. Furthermore, the configurational master equation allows a calculation of corrections of the approximate mean value and covariance equations.