The large deviation function for entropy production: the optimal trajectory and the role of fluctuations

Abstract

We study the large deviation function for the entropy production rate in two driven one-dimensional systems: the asymmetric random walk on a discrete lattice and Brownian motion in a continuous periodic potential. We compare two approaches: using the Donsker–Varadhan theory and using the Freidlin–Wentzell theory. We show that the wings of the large deviation function are dominated by a single optimal trajectory: either in the forward direction (positive rate) or in the backward direction (negative rate). The joining of the two branches at zero entropy production implies a non-differentiability and thus the appearance of a ‘kink’. However, around zero entropy production, many trajectories contribute and thus the ‘kink’ is smeared out.