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). 2023 Jul 27;25(8):1132. doi: 10.3390/e25081132.
Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker-Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker-Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker-Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed.
近年来，人们对非加性熵形式的主题表现出越来越浓厚的兴趣，这主要是由于它们在复杂系统领域的潜在应用。基于给定的熵形式应仅取决于一组概率这一事实，其时间演化与这些概率的演化直接相关。在本工作中，我们讨论与非加性熵相关的一些基本方面，考虑它们在连续和离散概率情况下的时间演化，对于这两种情况，分别考虑了福克 - 普朗克方程和主方程的非线性形式。对于连续概率，我们讨论了一个H定理，通过将非线性福克 - 普朗克方程中出现的泛函与一般熵形式联系起来进行证明。该定理确保福克 - 普朗克方程的稳态解与从熵形式的极值中出现的平衡解一致。在平衡状态下，我们表明在标准热力学要求下，卡诺循环对于一般熵形式成立。在离散概率的情况下，我们也考虑主方程描述的概率的时间演化证明了一个H定理。来自主方程的稳态解被证明与从熵形式的极值中出现的平衡解一致。对于这种情况，我们还讨论了热力学第三定律一般如何适用于平衡非加性熵形式。讨论了与以下事实相关的物理后果：从相应演化方程（对于连续和离散概率）获得的平衡态分布与从熵形式的极值中获得的平衡态分布一致、卡诺循环有效性的限制以及一般熵形式的热力学第三定律的适当表述。
