Equilibrium States in Two-Temperature Systems
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 w...
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MDPI AG
2018-03-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/20/3/183 |
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| author | Evaldo M. F. Curado Fernando D. Nobre |
| author_facet | Evaldo M. F. Curado Fernando D. Nobre |
| author_sort | Evaldo M. F. Curado |
| collection | DOAJ |
| description | 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 θ 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 θ ≫ T , so that thermal effects could be neglected, leading to a consistent thermodynamic framework based solely on the temperature θ . In the present work, a more general situation, concerning a system characterized by two distinct temperatures θ 1 and θ 2 , 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, θ 1 and θ 2 , should be thermodynamically conjugated to distinct entropic forms, S 1 and S 2 , respectively. A functional Λ [ P ] ≡ Λ ( S 1 [ P ] , S 2 [ P ] ) is introduced, which presents properties characteristic of an entropic form; moreover, a thermodynamically conjugated temperature parameter γ ( θ 1 , θ 2 ) 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. |
| first_indexed | 2024-04-11T13:59:47Z |
| format | Article |
| id | doaj.art-e5b5ad3852f44719b415eed695970919 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T13:59:47Z |
| publishDate | 2018-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-e5b5ad3852f44719b415eed6959709192022-12-22T04:20:09ZengMDPI AGEntropy1099-43002018-03-0120318310.3390/e20030183e20030183Equilibrium States in Two-Temperature SystemsEvaldo M. F. Curado0Fernando D. Nobre1Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Urca, Rio de Janeiro 22290-180, BrazilCentro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Urca, Rio de Janeiro 22290-180, BrazilSystems 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 θ 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 θ ≫ T , so that thermal effects could be neglected, leading to a consistent thermodynamic framework based solely on the temperature θ . In the present work, a more general situation, concerning a system characterized by two distinct temperatures θ 1 and θ 2 , 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, θ 1 and θ 2 , should be thermodynamically conjugated to distinct entropic forms, S 1 and S 2 , respectively. A functional Λ [ P ] ≡ Λ ( S 1 [ P ] , S 2 [ P ] ) is introduced, which presents properties characteristic of an entropic form; moreover, a thermodynamically conjugated temperature parameter γ ( θ 1 , θ 2 ) 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.http://www.mdpi.com/1099-4300/20/3/183nonlinear Fokker-Planck equationsgeneralized entropiesnonextensive thermostatistics |
| spellingShingle | Evaldo M. F. Curado Fernando D. Nobre Equilibrium States in Two-Temperature Systems Entropy nonlinear Fokker-Planck equations generalized entropies nonextensive thermostatistics |
| title | Equilibrium States in Two-Temperature Systems |
| title_full | Equilibrium States in Two-Temperature Systems |
| title_fullStr | Equilibrium States in Two-Temperature Systems |
| title_full_unstemmed | Equilibrium States in Two-Temperature Systems |
| title_short | Equilibrium States in Two-Temperature Systems |
| title_sort | equilibrium states in two temperature systems |
| topic | nonlinear Fokker-Planck equations generalized entropies nonextensive thermostatistics |
| url | http://www.mdpi.com/1099-4300/20/3/183 |
| work_keys_str_mv | AT evaldomfcurado equilibriumstatesintwotemperaturesystems AT fernandodnobre equilibriumstatesintwotemperaturesystems |