Statistics of trajectories in two-state master equations

TL;DR

This paper derives a simple, analytical expression for the probability of trajectories in two-state master equations, enabling efficient calculation of trajectory-based observables, demonstrated with explicit distributions involving Bessel functions.

Contribution

It introduces a straightforward formula for trajectory probabilities in small-state master equations, facilitating analytical computation of trajectory functionals.

Findings

Derived an explicit formula for trajectory probabilities

Calculated distributions of time spent in a state and number of transitions

Expressions involve modified Bessel functions

Abstract

We derive a simple expression for the probability of trajectories of a master equation. The expression is particularly useful when the number of states is small and permits the calculation of observables that can be defined as functionals of whole trajectories. We illustrate the method with a two-state master equation, for which we calculate the distribution of the time spent in one state and the distribution of the number of transitions, each in a given time interval. These two expressions are obtained analytically in terms of modified Bessel functions.