Stochastic energetics of non-uniform temperature systems

Abstract

We propose an energetic interpretation of stochastic processes described by Langevin equations with non-uniform temperature. In order to avoid Ito-Stratonovich dilemma, we start with a Kramers equation, and derive a Fokker-Planck equation by the renormalization group method. We give a proper definition of heat for the system. Based on our formulations, we analyze two examples, the Thomson effect and a Brownian motor. The latter realizes the Carnot efficiency.