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Recently, there has been a considerable progress on the issue of thethermodynamic second law, which is known as the law of entropy increase orirreversibility. In particular, a novel symmetry known as the Gallavotti-Cohensymmetry is found in nonequilibrium (NEQ) fluctuations, which leads toso-called fluctuation theorems. The thermodynamic second law is a simplecorollary of fluctuation theorems, from which one can predict quantitativelyhow often NEQ processes violate the law of entropy increase. Violationsdisappear in the thermodynamic limit, but can be observed reasonably well insmall systems. In this article, we will briefly introduce the stochasticthermodynamics and derive various fluctuation theorems, including the totalentropy production (EP), the work-free-energy relation, the excess andhouse-keeping EP, and the information entropy.

Entropy and Thermodynamic second laws: New perspective - stochastic thermodynamics and fluctuation theorems