Continued fraction solutions of discrete master equations not obeying detailed balance II

Abstract

In this paper we continue to extend our previous investigation of continued fraction (CF) solutions for the stationary probability of discrete one-variable master equations which generally do not satisfy detailed balance. We derive explicit expressions, directly in terms of the elementary transition rates, for the continued fraction recursion coefficients. Further, we derive several approximate CF-solutions, i.e., we deduce non-systematic and systematic truncation error estimates. The method is applied to two master equations with two-particle jumps for which we derive the exact probability solution and make a comparison with approximate solutions. The investigation is also extended to the case of master equations with multiple birth and death transitions of maximal order R.

Non-Markov processes: The problem of the mean first passage time

“…We introduce j<i<N-1 (3.22) yielding in virtue of (3.20) the result in (3.7). It should be noted however that v i =0, for all i implies always detailed balance whereas the two-step process in (3.8) generally does not obey detailed balance [12]" The result in (3.20) holds independent of any specific form of the f2-matrix elements {21,/~i, vi} and independent of a detailed balance relation.…”