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Communications in Theoretical Physics

Energy Fluctuation of Ideal Fermi Gas Trapped under Generic Power Law Potential U = ∑ i = 1 d c i | x i / a i | n i in d Dimensions

Mir Mehedi Faruk, Md. Muktadir Rahman, Dwaipayan Debnath, Md. Sakhawat Hossain Himel

Published Apr 1, 2016• 0 citations
Publisher

Abstract

Abstract Energy fluctuation of ideal Fermi gas trapped under generic power law potential U = ∑ i = 1 d c i | x i / a i | n i has been calculated in arbitrary dimensions. Energy fluctuation is scrutinized further in the degenerate limit μ ≫ K B T with the help of Sommerfeld expansion. The dependence of energy fluctuation on dimensionality and power law potential is studied in detail. Most importantly our general result can not only exactly reproduce the recently published result regarding free and harmonically trapped ideal Fermi gas in d = 3 but also can describe the outcome for any power law potential in arbitrary dimension.

DOI
10.1088/0253-6102/65/4/465

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Abstract Energy fluctuation of ideal Fermi gas trapped under generic power law potential U = ∑ i = 1 d c i | x i / a i | n i has been calculated in arbitrary dimensions. Energy fluctuation is scrutinized further in the degenerate limit μ ≫ K B T with the help of Sommerfeld expansion. The dependence of energy fluctuation on dimensionality and power law potential is studied in detail. Most importantly our general result can not only exactly reproduce the recently published result regarding free and harmonically trapped ideal Fermi gas in d = 3 but also can describe the outcome for any power law potential in arbitrary dimension.
Energy fluctuation of ideal Fermi gas trapped under generic power law potential U=Σi=1d ci|xi/ai|ni has been calculated in arbitrary dimensions.Energy fluctuation is scrutinized further in the degenerate limit μ>>KB T with the help of Sommerfeld expansion.The dependence of energy fluctuation on dimensionality and power law potential is studied in detail.Most importantly our general result can not only exactly reproduce the recently published result regarding free and harmonically trapped ideal Fermi gas in d =3 but also can describe the outcome for any power law potential in arbitrary dimension.
Energy fluctuation of ideal Fermi gas trapped under 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​ in d dimension
Energy fluctuation of ideal Fermi gas trapped under 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​ have been calculated in arbitrary dimension. Energy fluctuation is scrutinized further in the degenerate limit μ>>KBTμ>> K_BTμ>>KB​T with the help of Sommerfeld expansion. The dependence of energy fluctuation on dimensionality and power law potential is studied in detail. Most importantly our general result can exactly reproduce the recently published result regarding free and harmonically trapped ideal Fermi gas in d=3 (S Biswas, J Mitra, S Bhattacharyya, J. Stat. Mech. P03013, 2015.)
Abstract The average energy per fermion in the case of a Fermi gas with any kinematic characteristic, trapped under the most general power law potential in d -dimension has been calculated at zero temperature. In a previous paper (Acharyya M 2010 Eur. J Phys. 31 L89 ) it was shown, in the case of a free ideal Fermi gas, as the dimension increases the average energy approaches the Fermi energy and in infinite dimension the average energy becomes equal to the Fermi energy at T = 0. In this letter it is shown that, for a trapped system at finite dimension the average energy depends on a power law exponent, but as the dimension tends to infinity the average energy coincides with the Fermi energy for any power law exponent. The result obtained in this manuscript is more general, as we can describe the free system as well as any trapped system with an appropriate choice of power law exponent, and is true for any kinematic parameter.
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.
Abstract From the unified statistical thermodynamics of quantum gases, the virial coefficients of ideal Bose and Fermi gases, trapped under generic power law potential are derived systematically. From the general result of virial coefficients, one can produce the known results in d = 3 and d = 2. But more importantly we found that, the virial coefficients of Bose and Fermi gases become identical (except the second virial coefficient, where the sign is different) when the gases are trapped under harmonic potential in d = 1. This result suggests the equivalence between Bose and Fermi gases established in d = 1 (J. Stat. Phys. DOI 10.1007/s10955-015-1344-4). Also, it is found that the virial coefficients of two-dimensional free Bose (Fermi) gas are equal to the virial coefficients of one-dimensional harmonically trapped Bose (Fermi) gas.