Nonequilibrium Thermodynamics of Time-dependent Langevin Systems: Energy Balance Relation and the Extended Form of the Second Law

A general nonequilibrium thermodynamic theory is developed for timedependent Langevin dynamics, starting from the common definition of nonequilibrium Gibbs entropy. It is shown that the notations appearing in the First and the Second Law of thermodynamics could be consistently applied to transient nonequilibrium processes. We find out a general equality representing the energy balance relation for the system, and further an explicit mathematical interpretation for the extended form of the Second Law is proposed, which only comes into existence in the time-dependent case. More applications to several classic thermodynamic processes are finally included. Many different kinds of approach have been put forward in the last several decades [16], but in contrast to equilibrium systems, with their elegant theoretical framework, the understanding of nonequilibrium steady-state systems is still ∗Mathematical Department and Center for Computational Systems Biology, Fudan University, Shanghai 200433, P.R. China; Email: [email protected]