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

Abstract

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.

Cardy limits of 6d superconformal theories

“…Although it naively seems subleading in the Cardy limit, it can provide a comparable contribution of order O( 11 2 ) when v D is sufficiently close to the singularity. Such saddle points are known as the saddle point near singularity [50,51]. Among the many possible singularities, the one that is closest to the contour C is located at e −v D = e −m/τ .…”

Heat fluctuations and initial ensembles

“…In this study, we adopt the modified saddle point integral method [21] and search for the modified saddle points λ * (p) by extremizing…”

“…When the modified saddle point λ * is nearby the singularities, the integral along the steepest descent path should be performed with special care, because it becomes a non-Gaussian integral, described in detail in the Appendix of Ref. [21]. Now we present the results for different regions of p as follows.…”

“…where A is usually scaled dimensionless and h(A/τ ) is called a large deviation function (LDF). Many interesting properties on the LDF were found on such as the current fluctuations [18,19], the (non-Gaussian) exponential tail [20], the everlasting initial memory threshold [14,21], and so on. If the detailed FT holds, the Gallavotti-Cohen (GC) symmetry is expressed in terms of the LDFs as…”

“…We also calculated a finite-τ correction for the heat PDF, where the singularity structure becomes more complicated. Using the modified saddle point integral method recently developed by us [21], we exactly obtained the LDF of the heat up to O(τ −1 ) and thus the FT violation is measured up to the same order. Interestingly, the finite-τ correction of the FT violation does not vanish in the infinite initial-temperature limit, which implies the non-commutativity between the two limits of the infinite τ and the infinite initial temperature.…”