Plastino Angel R, Wedemann Roseli S
CeBio y Departamento de Ciencias Básicas, Universidad Nacional del Noroeste de la Província de Buenos Aires, UNNOBA, Conicet, Roque Saenz Peña 456, Junin 6000, Argentina.
Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, Rio de Janeiro 20550-900, RJ, Brazil.
Entropy (Basel). 2020 Jan 31;22(2):163. doi: 10.3390/e22020163.
Nonlinear Fokker-Planck equations (NLFPEs) constitute useful effective descriptions of some interacting many-body systems. Important instances of these nonlinear evolution equations are closely related to the thermostatistics based on the S q power-law entropic functionals. Most applications of the connection between the NLFPE and the S q entropies have focused on systems interacting through short-range forces. In the present contribution we re-visit the NLFPE approach to interacting systems in order to clarify the role played by the range of the interactions, and to explore the possibility of developing similar treatments for systems with long-range interactions, such as those corresponding to Newtonian gravitation. In particular, we consider a system of particles interacting via forces following the inverse square law and performing overdamped motion, that is described by a density obeying an integro-differential evolution equation that admits exact time-dependent solutions of the -Gaussian form. These -Gaussian solutions, which constitute a signature of S q -thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker-Planck equation.
非线性福克-普朗克方程（NLFPEs）构成了对一些相互作用多体系统的有用有效描述。这些非线性演化方程的重要实例与基于S_q幂律熵泛函的热统计学密切相关。NLFPE与S_q熵之间联系的大多数应用都集中在通过短程力相互作用的系统上。在本论文中，我们重新审视相互作用系统的NLFPE方法，以阐明相互作用范围所起的作用，并探索为具有长程相互作用的系统（如对应于牛顿引力的系统）开发类似处理方法的可能性。特别地，我们考虑一个通过遵循平方反比定律的力相互作用并进行过阻尼运动的粒子系统，该系统由一个密度描述，该密度服从一个积分-微分演化方程，该方程允许具有-Gaussian形式的精确时间相关解。这些-Gaussian解构成了S_q热统计学的一个特征，它们的演化方式与适当的非线性幂律福克-普朗克方程的解相似但不完全相同。
