Abstract
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.
Highlights
Thermodynamics was conceived as a discipline based on principles and laws that refer to macroscopic quantities such as the principles of energy conservation and of entropy increase, which are the first and second laws of thermodynamics
This was possible because the energy of a system in thermodynamic equilibrium is related functionally to the entropy, which allows the definition of temperature
The approach to the stochastic thermodynamics that we have developed here is based on the FPK equation which is understood as an equation that governs the time evolution of the probability distribution ρ
Summary
Thermodynamics was conceived as a discipline based on principles and laws that refer to macroscopic quantities such as the principles of energy conservation and of entropy increase, which are the first and second laws of thermodynamics. The crucial property of the equilibrium distribution is that the probability depends on the states of the system only through the energy function. Thermodynamic equilibrium is characterized as the state where a process and its time reversal are probable This characterization of equilibrium, embodied in the stochastic thermodynamics, is a dynamical definition, being more comprehensible than the static definition given above in terms of the Gibbs distribution. If we wish that the system reaches thermal equilibrium for long times, usually called thermalization, the solution of the evolution equation for long times must be a Gibbs equilibrium distribution, which is characterized by depending on (x, p) only through the energy function associated to the system. We remark that without J the FPK equation reduces to the Liouville equation of classical statistical mechanics [37],
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