On the second-law-like inequalities for Langevin subsystems

In this work, second-law-like inequalities for the compound of two coupled Langevin subsystems are discussed. The change rates of medium entropy and dynamic entropy are used to measure the heat dissipation and the phase space compressibility. For stationary state of the compound, we demonstrate that the change rate of medium entropy for one subsystem is bounded by the minus change rate of the dynamic entropy for the other. Our study clarifies the role of inter-subsystem coupling played in stochastic thermodynamics and implies potential applications in molecular motors and nanomachines.