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Journal of Dezhou University

The Universal Description of Arbitrary Dimensional Ideal Fermi Gas Under the Conditions of Complete Degeneration and Strong Degeneration of Low Temperature

Zhan Li-bo

Published Jan 1, 2005• 0 citations
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Abstract

we derive the quantum state density, the relation of internal energy and the pressure of arbitrary dimensional ideal gases that have energy spectrum of e = ap5 , and then offer the universal descriptions of chemistry potential, internal energy ,degenerate pressure and relative fluctuation of arbitrary dimensional ideal Fermi gases, through analysis and comparison, we draw a conclusion that when temperature approaches zero, thermodynamic quantities of high degenerate Fermi gases under low temperature trend to that of the complete degenerate Fermi gases.

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Acta Physica Sinica•Jan 1, 2015•2 citations
we derive the quantum state density, the relation of internal energy and the pressure of arbitrary dimensional ideal gases that have energy spectrum of e = ap5 , and then offer the universal descriptions of chemistry potential, internal energy ,degenerate pressure and relative fluctuation of arbitrary dimensional ideal Fermi gases, through analysis and comparison, we draw a conclusion that when temperature approaches zero, thermodynamic quantities of high degenerate Fermi gases under low temperature trend to that of the complete degenerate Fermi gases.
Thermodynamics of ideal Fermi gas trapped in an external generic power law potential U=∑i=1dci∣xiai∣niU=\sum_{i=1} ^d c_i |\frac{x_i}{a_i}|^{n_i}U=∑i=1d​ci​∣ai​xi​​∣ni​ are investigated systematically from the grand thermodynamic potential in ddd dimensional space. These properties are explored deeply in the degenerate limit (μ>>KBTμ>> K_BTμ>>KB​T), where the thermodynamic properties are greatly dominated by Pauli exclusion principle. Pressure and energy along with the isothermal compressibilty is non zero at T=0KT=0KT=0K, denoting trapped Fermi system is quite live even at absolute zero temperature. The nonzero value of compressibilty denotes zero point pressure is not just a constant but depends on volume.
For the ideal Fermi gas that fills a quantum well confined by two parallel planes, there are calculated the thermodynamic characteristics in general form for arbitrary temperatures, namely: the thermodynamic potential, energy, entropy, equations of state, heat capacities and compressibilities. The distance between planes is considered as an additional thermodynamic variable. Owing to the anisotropy, the pressure of the Fermi gas along and transverse to the planes is different, so that the system is characterized by two equations of state and a set of different heat capacities. Limiting cases of low and high temperatures are considered. The temperature dependencies of the entropy and heat capacities at low temperatures remain linear, just as in the volume case, and their dependencies on the chemical potential and density undergo jumps at the beginning of the filling of new discrete levels. It is shown that the behavior of thermodynamic quantities with the distance between plates can be either oscillating or monotonic, depending on what quantity is assumed to be fixed: the volume or surface density. For high temperatures the corrections to thermodynamic quantities are obtained, which are proportional to the ratio of the thermal de Broglie wavelength to the distance between planes.
The thermodynamic properties of a nonrelativistic free-electron Fermi gas is of fundamental interest in condensed matter physics. Properties previously studied in three-dimensions (3D) in the low- and high-temperature limits include the internal energy, heat capacity, zero-field magnetic spin susceptibility, and pressure. Here we report solutions for the temperature dependence spanning these two temperature regimes of the chemical potential, internal energy, magnetic susceptibility, and the heat capacity at constant volume in 1D, 2D, and 3D. Also calculated are the pressure, enthalpy, heat capacity at constant pressure, isothermal compressibility, and thermal expansion coefficient versus temperature in 2D and 3D. Of primary interest here are the detailed dimension-dependent crossovers of these properties between the degenerate and nondegenerate temperature regime, which are graphically illustrated for each of the above properties.
When taking into account the generalized uncertainty principle in statistical physics, the density of states must make a correction, which causes all the results of traditional statistical physics to have different degrees of correction. In high-energy or high-temperature conditions, this amendment can subvert the traditional concept and there are also some certain amendments at low temperatures. In this paper we study the thermodynamic properties of the ideal and weakly interacting Fermi gas in low temperature conditions when the generalized uncertainty principle is taken into account. Firstly, analytical expressions of chemical potential, internal energy and heat capacity at constant volume of ideal or weakly interacting Fermi gas are given. Then the properties of copper electron gas are computed as an example, showing that when the generalized uncertainty principle is taken into account the chemical potential, Fermi energy and the ground state energy increase with the increase of temperature, while the heat capacity decreases. When the temperature is lower than 0.3 times TF0, the internal energy increases with the increase of temperature, but becomes decreased when temperature is high than 0.3 times TF0. These amendments are mostly dependent on particle density, which becomes bigger and bigger with particle density increasing.