Exact solutions of discrete master equations in terms of continued fractions

Abstract

We present the continued fraction solution for the stationary probability of discrete master equations of one-variable processes. After we elucidate the method for simple birth and death processes we focus the study on processes which introduce at least two-particle jumps. Consequently, these processes do in general not obey a detailed balance condition. The outlined method applies as well to solutions of eigenmodes of the stochastic operator. Further we derive explicit continued fraction solutions for the Laplace transform of conditional probabilities. All the various continued fraction coefficients are given directly in terms of the transition rates and they obey recursion relations. The method is illustrated for the stationary solution of a simple nonlinear chemical reaction scheme originated by Nicolis.

Multi-phonon relaxation and generation of quantum states in a nonlinear mechanical oscillator

“…. Symbols denote numerical results averaged over one period and dashed lines show the behavior according to (35). (c) Time dependence of |ρ 0,1 (τ )| 2 at k B T / = 1/2 for α = 1 (•), α = 3/2 ( ) and α = 3 ( ).…”

“…At finite temperatures, when in addition to the losses, one or two vibron excitations (γ ± > 0 or γ 2± > 0) contribute, the stationary (time independent) probability distribution ρ n,n can be found using the detailed balance condition [33,34] (see Appendix B). If one and two-vibron processes compete (γ ± > 0 and γ 2± > 0), a solution for ρ n,n can be given in terms of a continued fraction [35] or (for γ 2+ = 0) in terms of confluent hypergeometric functions [36].…”

“…It can be seen that in the long-time limit the decay rates do not depend on the initial oscillator displacement α. As discussed above, the off-diagonal (35). (c) Time dependence of |ρ 0,1 (τ )| 2 at k B T /Ω = 1/2 for α = 1 (•), α = 3/2 ( ) and α = 3 ( ).…”

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

“…This is due to the fact that except for the special case of a simple birth and * Supported in part by Deutsche Forschungsgemeinschaft and by National Science Foundation Grant CHE78-21460 death process (nearest neighbor transitions only) satisfying automatically detailed balance, the master equations with multiple transitions do in general not obey a detailed balance relation. In a previous paper [4] we derived for a discrete master equation with one-particle and two-particle jumps a continued fraction representation for the stationary solution P~(n),n=0,1,2 .... , which in addition is very appropriate for a computer evaluation. In this paper we continue this investigation in more detail.…”

“…2 we first briefly review the general results for the two-particle jump master equations obtained in [4]. Using the continued fraction representation for the transition function [4,5] in =P~(n)/P~(n-1), we derive an equivalent reduced difference equation for the stationary probability. This reduced form is of major importance because in many cases it allows for an analytic solution (via the method of Laplace [6]).…”