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

TL;DR

This paper proves the Biswas-Mitra-Bhattacharyya conjecture linking energy fluctuation humps to specific heat discontinuities in ideal quantum gases trapped under generic power law potentials across different regimes.

Contribution

It establishes a new relation between energy fluctuation and specific heat in quantum gases and proves the BMB conjecture connecting fluctuation humps with specific heat discontinuities.

Findings

Proved the BMB conjecture for ideal quantum gases.

Established a new relation between energy fluctuation and specific heat.

Validated the relation in both classical and degenerate quantum regimes.

Abstract

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 connection is established between energy fluctuation and specific heat for quantum gases, valid in the classical limit and the degenerate quantum regime as well. Most importantly the proposed Biswas-Mitra-Bhattacharyya (BMB) conjecture (Biswas $et.$ $al.$, J. Stat. Mech. P03013, 2015.) relating hump in energy fluctuation and discontinuity of specific heat is proved and precised in this manuscript.