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arXiv:1504.06050 · Acta Physica Polonica B

Thermodynamics of Ideal Fermi Gas Under Generic Power Law Potential in ddd-dimensions

Mir Mehedi Faruk, G.M. Bhuiyan

Published Jan 1, 2015• 3 citations
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Abstract

Thermodynamics of ideal Fermi gas trapped in an external generic power law potential $U=\sum_{i=1} ^d c_i |\frac{x_i}{a_i}|^{n_i}$ are investigated systematically from the grand thermodynamic potential in $d$ dimensional space. These properties are explored deeply in the degenerate limit ($μ>> K_BT$), where the thermodynamic properties are greatly dominated by Pauli exclusion principle. Pressure and energy along with the isothermal compressibilty is non zero at $T=0K$, 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.

DOI
10.5506/aphyspolb.46.2419

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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.
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.
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.
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.
The effects of low dimensionality on the thermodynamics of a Fermi gas trapped by isotropic power-law potentials are analyzed. Particular attention is given to different characteristic temperatures that emerge, at low dimensionality, in the thermodynamic functions of state and in the thermodynamic susceptibilities (isothermal compressibility and specific heat). An energy-entropy argument that physically favors the relevance of one of these characteristic temperatures, namely, the nonvanishing temperature at which the chemical potential reaches the Fermi energy value, is presented. Such an argument allows interpreting the nonmonotonic dependence of the chemical potential on temperature, as an indicator of the appearance of a thermodynamic regime, where the equilibrium states of a trapped Fermi gas are characterized by larger fluctuations in energy and particle density as is revealed in the corresponding thermodynamics susceptibilities.