Curado Evaldo M F, Nobre Fernando D
Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Urca, Rio de Janeiro 22290-180, Brazil.
Entropy (Basel). 2018 Mar 9;20(3):183. doi: 10.3390/e20030183.
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 , 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.
具有多个温度特征的系统通常出现在非平衡统计力学中。在某些情况下，例如玻璃，存在一个温度，在该温度下快速变量达到热平衡，还有另一种情况与以非常缓慢的方式向平衡态演化的模式相关。最近，有人表明，一个相互排斥作用的涡旋系统，被视为II型超导体的合适模型，呈现出一种以两个温度为特征的平衡态。主要的新奇之处在于，除了与粒子速度波动相关的通常温度外，还引入了一个额外的温度θ，它与粒子位置的波动相关。由于它们呈现出物理上不同的特征，该系统可能达到一种平衡态，其特征是这些温度具有有限且不同的值。在II型超导体的应用中，已表明θ≫T，因此热效应可以忽略不计，从而导致一个仅基于温度θ的一致热力学框架。在本工作中，讨论了一种更一般的情况，涉及一个以两个不同温度θ1和θ2为特征的系统，它们可能具有相同的数量级。这些温度作为非线性福克 - 普朗克方程不同扩散贡献的系数出现。证明了一个H定理，将这样的福克 - 普朗克方程与两种熵形式的和联系起来，每种熵形式都与给定的扩散项相关；因此，相应的稳态可以被视为一种平衡态，其特征是两个温度。这种状态出现的条件之一是不同的温度参数θ1和θ2应分别与不同的熵形式S1和S2热力学共轭。引入了一个泛函Λ[P]≡Λ(S1[P],S2[P])，它呈现出熵形式的特征性质；此外，可以一致地定义一个热力学共轭温度参数γ(θ1,θ2)，从而根据这些变量对提出一种替代的物理描述。讨论了物理后果，特别是从福克 - 普朗克方程得到的平衡态分布应与熵极值化得到的分布一致这一事实。
