Fluctuation Theorems in Overdamped Theory with Multiple Reservoirs

Abstract

In this chapter, we consider a non-isothermal system simultaneously coupled to multiple heat reservoirs with different uniform temperatures and discuss the fluctuation theorem in the overdamped limit. When we consider the overdamped approximation in this system, velocity degrees of freedom, which are to be eliminated in the approximation, make positively divergent contributions to the entropy production. This is because the velocities relax not to an equilibrium state but to a nonequilibrium steady state in the presence of multiple temperatures. As a result, two naive approaches to the overdamped approximation, which begin with either the Langevin equation or the Fokker-Planck equation, fail to evaluate thermodynamic quantities. To obtain an appropriate overdamped description, we begin with the underdamped description of the system and discuss its stochastic thermodynamics and the fluctuation theorem. Then, as a tryout, we utilize the assumption of the time-scale separation and the technique of the singular expansion to derive overdamped stochastic thermodynamics for a system with a single uniform temperature and show its consistency with the well-established isothermal overdamped stochastic thermodynamics. We generalize this method to a system with multiple heat reservoirs with different uniform temperatures and derive an appropriate overdamped description. Despite the singular behavior of the fast degrees of freedom, we show that the fluctuation theorems are valid for the dynamics of positional degrees of freedom.