Direct numerical method for counting statistics in stochastic processes

TL;DR

This paper introduces a direct numerical approach using generating functions to compute transition statistics in stochastic processes, avoiding Monte Carlo simulations and applicable to finite state systems.

Contribution

The paper presents a novel numerical method based on generating functions for calculating moments of transition counts in stochastic processes, bypassing Monte Carlo methods.

Findings

Successfully calculated moments for a two-state model with time-dependent transition rates.

Method applicable to any finite state stochastic process.

Potential to analyze current statistics in nonequilibrium systems.

Abstract

We propose a direct numerical method to calculate the statistics of the number of transitions in stochastic processes, without having to resort to Monte Carlo calculations. The method is based on a generating function method, and arbitrary moments of the probability distribution of the number of transitions are in principle calculated by solving numerically a system of coupled differential equations. As an example, a two state model with a time-dependent transition matrix is considered and the first, second and third moments of the current are calculated. This calculation scheme is applicable for any stochastic process with a finite state space, and it would be helpful to study current statistics in nonequilibrium systems.