Oylukan Alp Doga, Shizgal Bernard
Department of Mathematics University of British Columbia Vancouver, British Columbia V6T 1Z4, Canada.
Institute of Applied Mathematics University of British Columbia Vancouver, British Columbia V6T 1Z4, Canada.
Phys Rev E. 2023 Jul;108(1-1):014111. doi: 10.1103/PhysRevE.108.014111.
Nonequilibrium systems in chemistry and physics are generally modeled with the Boltzmann, Fokker-Planck, and Master equations. There has been a considerable interest in the nonequilibrium distributions of electrons and ions in space physics in different environments as well as in other systems. An often-used empirical model to characterize these distributions, especially in space physics, is the Kappa distribution. There have been numerous efforts to provide a theoretical basis for the Kappa distribution that include the Fokker-Planck equation with specific drift and diffusion coefficients. Alternatively, the maximization of the Tsallis nonextensive entropy provides the desired Kappa distribution. This paper examines three families of Fokker-Planck equations that provide a steady-state Kappa distribution as well as a myriad of other nonequilibrium distributions. The relationship of these works with analogous studies of distributions with asymptotic high-energy tails is also considered. It is clear that the many different nonequilibrium distribution functions that can occur cannot all be rationalized with Gibbs-Boltzmann statistical mechanics, which uniquely gives equilibrium distributions, or with the Tsallis nonextensive entropy, which gives uniquely the Kappa distribution. The current research is directed towards an improved understanding of the origin of nonequilibrium distributions in several specific systems.
化学和物理学中的非平衡系统通常用玻尔兹曼方程、福克 - 普朗克方程和主方程来建模。在不同环境下的空间物理学以及其他系统中，电子和离子的非平衡分布一直备受关注。一种常用于描述这些分布（尤其是在空间物理学中）的经验模型是卡帕分布。人们进行了大量努力为卡帕分布提供理论基础，其中包括具有特定漂移和扩散系数的福克 - 普朗克方程。另外，通过最大化Tsallis非广延熵也能得到所需的卡帕分布。本文研究了三类福克 - 普朗克方程，它们能给出稳态卡帕分布以及大量其他非平衡分布。同时也考虑了这些工作与具有渐近高能尾部的分布的类似研究之间的关系。显然，出现的许多不同非平衡分布函数不能全部用吉布斯 - 玻尔兹曼统计力学（它唯一地给出平衡分布）或用Tsallis非广延熵（它唯一地给出卡帕分布）来合理化解释。当前的研究旨在更好地理解几个特定系统中非平衡分布的起源。
