Energy fluctuation and discontinuity of specific heat

The specific heat per particle (c_v) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ϵ²) of the ideal gas through the relation ▵ϵ² = kT²c_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 $△{ϵ}^2/k{T}^2{c}_v^{\left(cl\right)}$ 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. (paper)