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Classical and quantum stochastic thermodynamics

The stochastic thermodynamics provides a framework for the description of systems that are out of thermodynamic equilibrium. It is based on the assumption that the elementary constituents are acted by random forces that generate a stochastic dynamics, which is here represented by a Fokker-Planck-Kramers equation. We emphasize the role of the irreversible probability current, the vanishing of which characterizes the thermodynamic equilibrium and yields a special relation between fluctuation and dissipation. The connection to thermodynamics is obtained by the definition of the energy function and the entropy as well as the rate at which entropy is generated. The extension to quantum systems is provided by a quantum evolution equation which is a canonical quantization of the Fokker-Planck-Kramers equation. An example of an irreversible systems is presented which shows a nonequilibrium stationary state with an unceasing production of entropy. A relationship between the fluxes and the path integral is also presented.