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The well known relation for ideal classical gas $\Delta \epsilon^2=kT^2 C_V$which does not remain valid for quantum system is revisited. A new connectionis established between energy fluctuation and specific heat for quantum gases,valid in the classical limit and the degenerate quantum regime as well. Mostimportantly the proposed Biswas-Mitra-Bhattacharyya (BMB) conjecture (Biswas$et.$ $al.$, J. Stat. Mech. P03013, 2015.) relating hump in energy fluctuationand discontinuity of specific heat is proved and precised in this manuscript.

journal of statistical mechanics

A proof to Biswas-Mitra-Bhattacharyya conjecture for ideal quantum gas  trapped under generic power law potential $U=\sum_{i=1} ^d c_i |\frac  {x_i}{a_i}|^{n_i}$ in $d$ dimension