Abstract
Systems characterized by more than one temperature usually appear in nonequilibrium statistical mechanics. In some cases, e.g., glasses, there is a temperature at which fast variables become thermalized, and another case associated with modes that evolve towards an equilibrium state in a very slow way. Recently, it was shown that a system of vortices interacting repulsively, considered as an appropriate model for type-II superconductors, presents an equilibrium state characterized by two temperatures. The main novelty concerns the fact that apart from the usual temperature T, related to fluctuations in particle velocities, an additional temperature was introduced, associated with fluctuations in particle positions. Since they present physically distinct characteristics, the system may reach an equilibrium state, characterized by finite and different values of these temperatures. In the application of type-II superconductors, it was shown that , so that thermal effects could be neglected, leading to a consistent thermodynamic framework based solely on the temperature . In the present work, a more general situation, concerning a system characterized by two distinct temperatures and , which may be of the same order of magnitude, is discussed. These temperatures appear as coefficients of different diffusion contributions of a nonlinear Fokker-Planck equation. An H-theorem is proven, relating such a Fokker-Planck equation to a sum of two entropic forms, each of them associated with a given diffusion term; as a consequence, the corresponding stationary state may be considered as an equilibrium state, characterized by two temperatures. One of the conditions for such a state to occur is that the different temperature parameters, and , should be thermodynamically conjugated to distinct entropic forms, and , respectively. A functional is introduced, which presents properties characteristic of an entropic form; moreover, a thermodynamically conjugated temperature parameter can be consistently defined, so that an alternative physical description is proposed in terms of these pairs of variables. The physical consequences, and particularly, the fact that the equilibrium-state distribution, obtained from the Fokker-Planck equation, should coincide with the one from entropy extremization, are discussed.
Highlights
The linear Fokker-Planck equation (FPE) represents one of the most important equations of nonequilibrium statistical mechanics; it describes the time evolution of a probability density P(~x, t) for finding a given particle at a position ~x, at time t, diffusing under an external potential [1,2,3,4]
In the absence of external potential, the FPE reduces to the linear diffusion equation, usually associated with the description of the Brownian motion; a confining external potential yields the possibility of a stationary-state solution for a sufficiently long time
A special interest was given to a concrete physical application, namely, a system of interacting vortices, currently used as a suitable model for type-II superconductors, which exhibited such a behavior [38,43,45,46,65]. This nonlinear Fokker-Planck equation (NLFPE), that appears as a particular case of the one derived in [51,52], presents two diffusive terms: (i) A linear contribution, obtained in the usual way, i.e., by applying an additive uncorrelated thermal noise in the system [2,3,4]; (ii) A nonlinear one, characterized by a power in the probability, like in the NLFPE of [8,9], which emerged from a coarse-graining approach in the vortex-vortex interactions
Summary
The linear Fokker-Planck equation (FPE) represents one of the most important equations of nonequilibrium statistical mechanics; it describes the time evolution of a probability density P(~x, t) for finding a given particle at a position ~x, at time t, diffusing under an external potential [1,2,3,4]. A special interest was given to a concrete physical application, namely, a system of interacting vortices, currently used as a suitable model for type-II superconductors, which exhibited such a behavior [38,43,45,46,65] This NLFPE, that appears as a particular case of the one derived in [51,52], presents two diffusive terms: (i) A linear contribution, obtained in the usual way, i.e., by applying an additive uncorrelated thermal noise in the system [2,3,4]; (ii) A nonlinear one, characterized by a power in the probability, like in the NLFPE of [8,9], which emerged from a coarse-graining approach in the vortex-vortex interactions.
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