Energy fluctuation and discontinuity of specific heat

Abstract

Specific heat per particle (cv) of an ideal gas, in many occasions, is interpreted as energy fluctuation per particle (△ǫ 2 ) of the ideal gas through the relation: △ǫ 2 = kT 2 cv, 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 3-D free Bose and Fermi gases. We also have explored the same for harmonically trapped cases. We have obtained a hump of △ǫ 2 /kT 2 c (cl) v around the condensation point for 3-D harmonically trapped Bose gas. We have discussed the possibility of occurring phase transition with discontinuity of heat capacity from existence of such a hump for other Bose and Fermi systems.

Energy fluctuation of ideal Fermi gas trapped under generic power law potential $U=\sum_{i=1} ^d c_i |\frac{x_i}{a_i}|^{n_i}$ in d dimension

“…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 [1].…”

“…al. [1] In this section we summarize the interesting findings relating energy fluctuation of ideal Fermi gas trapped under generic power law potential. It is seen in the studies that [5], all the thermodynamic quantities of free Fermi (Bose) gases can be presented in terms of Fermi function (Bose function) depending on dimensionality d. Now, the thermodynamic quantities of trapped Fermi (Bose) gases can still be written in terms of Fermi function (Bose function), using the concept of effective volume and effective thermal wavelength [13,19].…”

“…Since then, an increasing attraction is noticed in the subject of trapped quantum gases. Although a lot of theoretical studies [1,12,13,14,15,16,17,18,19] are done on quantum gases trapped under generic power law potential, U = d i=1 c i | x i a i | n i , none of these contained detailed discussion on energy fluctuation ∆ǫ 2 , until the recent paper of Biswas et. al.…”

“…al. [1] where they discussed the energy fluctuation for free and harmonically trapped quantum gases in three dimensional space. In their paper they have also conjectured a relation between discontinuity of C V and energy fluctuation ∆ǫ 2 , suggesting the appearance of a hump in ∆ǫ 2 kT 2 over its classical limit does indicate a discontinuity of C V .…”

“…al. [1] by choosing d = 3, n = ∞ (free Fermi gas in three dimension) and d = 3, n = 2 (harmonically trapped Fermi gas in three dimension).…”