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A general nonequilibrium thermodynamic theory is developed for time-dependentLangevin dynamics, starting from the common definition of nonequilibrium Gibbsentropy. It is shown that the notations appearing in the First and the SecondLaw of thermodynamics could be consistently applied to transient nonequilibriumprocesses. We find out a general equality representing the energy balancerelation for the system, and further an explicit mathematical interpretationfor the extended form of the Second Law is proposed, which only comes intoexistence in the time-dependent case. More applications to several classicthermodynamic processes are finally included.

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