Modified saddle-point integral near singularity for the large deviation function

TL;DR

This paper introduces a modified saddle-point method to accurately compute large deviation functions in stochastic processes, especially near singularities, demonstrated through heat production examples with finite-time corrections validated numerically.

Contribution

A novel saddle-point integration technique for handling singularities in large deviation function calculations, improving accuracy near critical points.

Findings

Effective handling of branch-cut and power-law singularities

Finite-time corrections to large deviation functions derived

Numerical results confirm the theoretical predictions

Abstract

Long-time-integrated quantities in stochastic processes, in or out of equilibrium, usually exhibit rare but huge fluctuations. Work or heat production is such a quantity, of which the probability distribution function displays an exponential decay characterized by the large deviation function (LDF). The LDF is often deduced from the cumulant generating function through the inverse Fourier transformation. The saddle-point integration method is a powerful technique to obtain the asymptotic results in the Fourier integral, but a special care should be taken when the saddle point is located near a singularity of the integrand. In this paper, we present a modified saddle-point method to handle such a difficulty efficiently. We investigate the dissipated and injected heat production in equilibration processes with various initial conditions, as an example, where the generating functions contain branch-cut singularities as well as power-law ones. Exploiting the new modified saddle-point integrations, we obtain the leading finite-time corrections for the LDF's, which are confirmed by numerical results.