洛伦兹熵与奥尔伯特$κ$分布
Lorentzian Entropies and Olbert's $κ$-distribution
摘要：
本文导出了服从奥尔伯特分布（广义洛仑兹概率分布，称为$κ$分布）的系统的各种形式的熵，这些分布在高温等离子体中经常被观察到。在这种系统中的配分函数的一般表达式，以及在类似的Boltzmann-Gibbs概率分布的形式，包括一个可能的指数高能量截断。我们找到了平均能量作为概率函数的表示，并提供了Olbert（洛伦兹）熵的隐式形式以及它的高温极限。得到了与相空间态密度的关系。然后我们发现熵是概率的函数，这是统计力学的基本表达式，在这里也是它的奥尔伯版本。洛伦兹系统通过内部的集体相互作用产生了增加熵的相关性。费米系统不服从奥尔伯特统计，而玻色系统可能在远离零的温度下。
英文摘要：
This note derives the various forms of entropy of systems subject to Olbert distributions (generalized Lorentzian probability distributions known as $κ$-distributions) which are frequently observed particularly in high temperature plasmas. The general expression of the partition function in such systems is given as well in a form similar to the Boltzmann-Gibbs probability distribution, including a possible exponential high energy truncation. We find the representation of the mean energy as function of probability, and provide the implicit form of Olbert (Lorentzian) entropy as well as its high temperature limit. The relation to phase space density of states is obtained. We then find the entropy as function of probability, an expression which is fundamental to statistical mechanics and here to its Olbertian version. Lorentzian systems through internal collective interactions cause correlations which add to the entropy. Fermi systems do not obey Olbert statistics, while Bose systems might at temperatures sufficiently far from zero.
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