Zon-Cohen singularity and negative inverse temperature in a trapped-particle limit

We study a Brownian particle on a moving periodic potential. We focus on the statistical properties of the work done by the potential and the heat dissipated by the particle. When the period and the depth of the potential are both large, by using a boundary layer analysis, we calculate a cumulant generating function and a biased distribution function. The result allows us to understand a Zon-Cohen singularity for an extended fluctuation theorem from a viewpoint of rare trajectories characterized by a negative inverse temperature of the biased distribution function.