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arXiv:1509.04809 · Journal of Statistical Mechanics Theory and Experiment

A proof of the Biswas–Mitra–Bhattacharyya conjecture for the ideal quantum gas trapped under the generic power law potential in d dimensions

Mir Mehedi Faruk, Md Muktadir Rahman

Published Mar 1, 2016• 2 citations
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Abstract

Abstract. The well known relation for the ideal classical gas , which does not remain valid for quantum systems, is revisited. A new connection is established between the energy fluctuation and specific heat for quantum gases, which is valid in both the classical limit and the degenerate quantum regime. Most importantly, the proposed Biswas–Mitra–Bhattacharyya conjecture (Biswas et al 2015 J. Stat. Mech . P03013 ) relating the hump in energy fluctuation and the discontinuity of specific heat is essentially proved and made precise in this manuscript.

DOI
10.1088/1742-5468/2016/03/033117

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Abstract. The well known relation for the ideal classical gas , which does not remain valid for quantum systems, is revisited. A new connection is established between the energy fluctuation and specific heat for quantum gases, which is valid in both the classical limit and the degenerate quantum regime. Most importantly, the proposed Biswas–Mitra–Bhattacharyya conjecture (Biswas et al 2015 J. Stat. Mech . P03013 ) relating the hump in energy fluctuation and the discontinuity of specific heat is essentially proved and made precise in this manuscript.
The specific heat per particle ( c v ) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ 2 ) of the ideal gas through the relation ▵ 2 = kT 2 c v , where k is the Boltzmann constant and T is the temperature. This relationship is true only in the classical limit and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3D free Bose and Fermi gases. We have explored the same for harmonically trapped cases. We have obtained a hump of around its condensation point for 3D harmonically trapped Bose gas. We have discussed the possibility of an occurring phase transition with the discontinuity of heat capacity from the existence of such a hump for other Bose and Fermi systems.
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.)
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.
Abstract The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential U = ∑ i = 1 d c i | x i / a i | n i are studied carefully. Detailed calculation of Kim et al. (J. Phys. Condens. Matter 11 (1999) 10269) yielded the hierarchy of condensation transitions with changing fractional dimensionality. In this manuscript, some theorems regarding specific heat at constant volume C V are presented. Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition in trapped systems as well.
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.