D-Dimensional Statistical Mechanics in Fractional Classical and Quantum Mechanics
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Zineb KORICHI* and Mohammed Tayeb MEFTAH

Univ Ouargla, Fac. des Mathématiques et Sciences de la Matière,

Lab. Rayonnement et Plasma et Physique des Surfaces, Ouargla 30000, Algérie

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ABSTRACT: In this work, we focus our study on some systems in statistical mechanics based on the fractional classical and quantum mechanics in any positive spatial dimension D. At the first stage we present the thermodynamical properties of the classical ideal gas and the system of N classical oscillators. In each case, the Hamiltonian exhibits fractional exponents of the phase space (position and momentum). At the second stage, in the context of the fractional quantum mechanics, we have studied the Bose-Einstein statistics with the related problem of the condensation and the Fermi-Dirac.

KEYWORDS: fractional calculus, fractional quantum mechanics, quantum statistics, partition function.

RÉSUMÉ :Dans ce travail, nous nous intéressons à l'étude de certains systèmes en mécanique statistique basée sur la mécanique classique et quantique fractionnaire dans toute dimension D. Au premier lieu, nous présentons les propriétés thermodynamiques du gaz idéal classique et le système de N oscillateurs classiques. Dans chaque cas, l'hamiltonien présente des puissances fractionnaires de l'espace des phases (de position et l'impulsion). En deuxième étape, dans le contexte de la mécanique quantique fractionnaire, nous avons étudié les statistiques de Bose-Einstein et de Fermi-Dirac avec le problème associé de la condensation.

MOTS-CLÉS : calcul fractionnaire, la mécanique quantique fractionnaire, statistiques quantiques, fonction de partition.

1. Introduction

Since some years, research has intensified and diversified in the calculation of the fractional derivatives. These are applied quickly thereafter, to physics and to engineering. This kind of calculation is due to Riemann and Liouville pioneers. This consists to make derivatives with non-integer order of functions. More precisely, instead to make a derivative of the first order (order 1) or a derivative of second order (order 2), we make a derivative of an intermediate order between 1 and 2. We then speak, for example, of the fractional derivative of order 1/2 or 3/4. This concept was developed recently by I. Podlubny (1999) [1]. In physics, we often deal with differential equations or with partial differential equations. What happens if these equations present the fractional derivative? In quantum mechanics, the results arising from this new concept are discussed by several authors [2-5]. The focus of this paper is to see how the fundamental problems of statistical physics, both classical and quantum, will be affected by using the fractional derivatives. Section 2 start with defining fractional classical Hamiltonian for two classical systems: N particles ideal gas and N independent fractional oscillators in D-dimensional space. Using the canonical ensemble, we have developed the thermodynamical properties of the system (N particles ideal gas and N independent fractional oscillators in D-dimensional space). By performing the limit ,  and D=3, we have recovered the well-known results for the 3-dimensional ideal gas and N independent oscillators. In section 3, we have studied the ideal gases (Bose and Fermi gases) in grand canonical ensemble. We emphasize a subsection to discuss how the critical temperature of the ideal Bose gas is affected by the fraction parameter . By putting  and D=3, we have recovered the standard results relative to the quantum Bose gas. We close this work by a conclusion in section 4.

2. Classical statistical mechanics

2. 1. Ideal gas

The canonical partition function of a classical ideal gas composed of N particles occupying a volume V at a temperature T is given by: 

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