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Physical Science International Journal

Application of Fractional Classical and Quantum Mechanics to Statistical Physics System

Zineb Korichi, Mohammed Tayeb Meftah

Published Jan 10, 2015• 0 citations
Publisher

Abstract

In this work, we focus our study on some systems in statistical mechanics by using a recent developpment of the fractional classical and quantum mechanics. At first, we present the thermodynamical properties of N classical and quantum oscillators in 3-dimensional space. In each case, the Hamiltonian, describing the system, exhibits fractional powers. Secondly, in the context of the fractional quantum mechanics, we calculate the thermodynamical properties for the systems: the trapped ideal Bose and Fermi gases. For the trapped ideal Bose system, Bose- Einstein condensation and the thermodynamics properties are discussed and plotted with respect to the fractional powers and the temperature. It is shown that the trapped ideal Bose gas exhibits a phase transition at a temperature TC for which the specific heat presents a discontinuity depending on the fractional powers. All calculations are performed in 3-dimensional space.

DOI
10.9734/psij/2015/20366

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In this work, we focus our study on some systems in statistical mechanics by using a recent developpment of the fractional classical and quantum mechanics. At first, we present the thermodynamical properties of N classical and quantum oscillators in 3-dimensional space. In each case, the Hamiltonian, describing the system, exhibits fractional powers. Secondly, in the context of the fractional quantum mechanics, we calculate the thermodynamical properties for the systems: the trapped ideal Bose and Fermi gases. For the trapped ideal Bose system, Bose- Einstein condensation and the thermodynamics properties are discussed and plotted with respect to the fractional powers and the temperature. It is shown that the trapped ideal Bose gas exhibits a phase transition at a temperature TC for which the specific heat presents a discontinuity depending on the fractional powers. All calculations are performed in 3-dimensional space.
The density of states and thermodynamic properties of an ideal gas system trapped in a generic power-law potential in an n -dimensional space are studied. A unified description for the Bose, Fermi and classical gases is given by using the grand potential of the system. Consequently, not only the results in current textbooks of statistical mechanics but also some new general important conclusions, such as the conditions for the occurence of Bose-Eisntein condensation in a trapped Bose system in any dimensional space, can be derived directly using the results of this paper.
In this work, improvements are introduced to the current models of the ideal Fermi gas and the ideal Bose gas by incorporating the quantum nature of phase space, which is directly linked to the uncertainty principle. These improved models build upon the recently developed concepts of quantum phase space (QPS) and the QPS representation of quantum mechanics. The Hamiltonian operator for a gas particle and its eigenstates are first determined, and quantum statistical mechanics is used to derive the thermodynamic properties of the ideal gas. Analytic expressions for thermodynamic quantities—including the grand canonical potential, particle number, internal energy, von Neumann entropy, and pressure—are derived, along with the corresponding thermodynamic equations of state for both bosons and fermions. These corrections are particularly significant at low temperatures and in confined volumes, where quantum effects related to system geometry, such as shape and size, become significant. The results also establish a direct link between thermodynamic functions and the quantum statistical variances of momentum. Importantly, the improved models recover well-known classical relations of the ideal gas in the high-temperature and large-volume limits, ensuring consistency with classical physics. By addressing quantum corrections and their thermodynamic implications, this work provides a foundation for further applications in nanoscale systems, quantum gases, low-temperature physics, ultracold physics, and astrophysics.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.
The quantum statistical mechanics of an ideal gas with fractional exclusion (i.e., Haldane-Wu) statistics in arbitrary dimensions is discussed. The general formulation for pressure and density of the system is obtained in a closed form in terms of the DDD-dimensional momentum representation, which can be regarded as a natural generalization of the classic results for Fermi and Bose gases. Using this, it is shown that ideal gases with fractional exclusion statistics can be regarded as composites of fermions and bosons, and that no condensation occurs at low temperature except for the pure boson case.
Thermodynamic properties of ideal Bose gas trapped in an external generic power law potential are investigated systematically from the grand thermodynamic potential in ddd dimensional space. The most general conditions for Bose-Einstein condensate and the discontinuous conditions of heat capacity at the critical temperature in presence of generic power law potential are presented in this manuscript. The dependence of the physical quantities on external potential, particle characteristics and space dimensionality are discussed. The more general results obtained in this paper presents an unified illustration of Bose-Einstein condensation of ideal Bose systems as they reduces to the expressions and conclusions available in the literature with appropiate choice of power law exponent.
This paper presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation---namely, the fraction of particles in the ground state and the specific heat---are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a three-dimensional box, the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit---here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.