Stationary and Time Dependent Solutions of Master Equations in Several Variables

Abstract

It is a well known fact that exact solutions of the Master Equation

$$ \frac{{dP\left( {\left\{ {{X_i}} \right\},t} \right)}}{{dt}} = \sum\limits_\rho {W\left( {\left\{ {{X_i} - {r_{i\rho }}} \right\} \to \left\{ {{X_i}} \right\}} \right)} P\left( {\left\{ {{X_i} - {r_{i\rho }}} \right\},t} \right) - \sum\limits_\rho {W\left( {\left\{ {{X_i}} \right\} \to \left\{ {{X_i} + {r_{i\rho }}} \right\}} \right)} P\left( {\left\{ {{X_i}} \right\},t} \right) $$

which describes the time evolution of the probability function P({X_i},t) in terms of the various gain and loss processes, are few and far between. Already in the case of one variable, the absence of detailed balance precludes any straightforward general solution, even for the stationary solution. (1,2)