  Recently, there has been a considerable progress on the issue of the
thermodynamic second law, which is known as the law of entropy increase or
irreversibility. In particular, a novel symmetry known as the Gallavotti-Cohen
symmetry is found in nonequilibrium (NEQ) fluctuations, which leads to
so-called fluctuation theorems. The thermodynamic second law is a simple
corollary of fluctuation theorems, from which one can predict quantitatively
how often NEQ processes violate the law of entropy increase. Violations
disappear in the thermodynamic limit, but can be observed reasonably well in
small systems. In this article, we will briefly introduce the stochastic
thermodynamics and derive various fluctuation theorems, including the total
entropy production (EP), the work-free-energy relation, the excess and
house-keeping EP, and the information entropy.
