Solution of the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives in statistical mechanics

IF 1 4区物理与天体物理 Q3 PHYSICS, MATHEMATICAL

Theoretical and Mathematical Physics Pub Date : 2024-02-27 DOI:10.1134/s0040577924020107

Z. Korichi, A. Souigat, R. Bekhouche, M. T. Meftah

引用次数: 0

Abstract

We solve the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives for systems exhibiting noninteger power laws in their Hamiltonians. Based on the fractional Liouville equation, we calculate the density function (DF) of a classical ideal gas. If the Riemann–Liouville derivative is used, the DF is a function depending on both the momentum \(p\) and the coordinate \(q\), but if the derivative in the Caputo sense is used, the DF is a constant independent of \(p\) and \(q\). We also study a gas consisting of \(N\) fractional oscillators in one-dimensional space and obtain that the DF of the system depends on the type of the derivative.

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利用统计力学中的黎曼-利乌维尔和卡普托导数求解分数利乌维尔方程

摘要我们利用黎曼-刘维尔导数和卡普托导数求解分数刘维尔方程，以解决哈密顿中呈现非整数幂律的系统问题。基于分数柳维尔方程，我们计算了经典理想气体的密度函数（DF）。如果使用黎曼-柳维尔导数，密度函数是一个取决于动量（p）和坐标（q）的函数，但如果使用卡普托导数，密度函数是一个与（p）和（q）无关的常数。我们还研究了由一维空间中的\(N\)分数振荡器组成的气体，并得出系统的DF取决于导数的类型。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Theoretical and Mathematical Physics 物理-物理：数学物理

CiteScore

1.60

自引率

20.00%

发文量

103

审稿时长

4-8 weeks

期刊介绍： Theoretical and Mathematical Physics covers quantum field theory and theory of elementary particles, fundamental problems of nuclear physics, many-body problems and statistical physics, nonrelativistic quantum mechanics, and basic problems of gravitation theory. Articles report on current developments in theoretical physics as well as related mathematical problems. Theoretical and Mathematical Physics is published in collaboration with the Steklov Mathematical Institute of the Russian Academy of Sciences.