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 time-dependent 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.