Bose gas with generalized dispersion relation plus an energy gap

TL;DR

This paper analytically investigates Bose-Einstein condensation in a generalized Bose gas with an energy gap and dispersion relation, deriving formulas for critical temperature, condensed fraction, and thermodynamic properties across dimensions.

Contribution

It provides explicit formulas for critical temperature and thermodynamics of a Bose gas with a generalized dispersion and energy gap, extending previous models to arbitrary dimensions and gap values.

Findings

Critical temperature depends on the energy gap and dimension.

Specific heat exhibits a jump at the critical temperature.

High-temperature classical results are recovered regardless of the gap.

Abstract

Bose-Einstein condensation in a Bose gas is studied analytically, in any positive dimensionality ($d>0$) for identical bosons with any energy-momentum positive-exponent ($s>0$) plus an energy gap $\Delta$ between the ground state energy $\varepsilon_0$ and the first excited state, i.e., $\varepsilon=\varepsilon_0$ for $k=0$ and $\varepsilon=\varepsilon_0 +\Delta+ c_sk^s$, for $k>0$, where $\hbar \mathbf{k}$ is the particle momentum and $c_s$ a constant with dimensions of energy multiplied by a length to the power $s > 0$. Explicit formula with arbitrary $d/s$ and $\Delta$ are obtained and discussed for the critical temperature and the condensed fraction, as well as for the equation of state from where we deduce a generalized $\Delta$ independent thermal de Broglie wavelength. Also the internal energy is calculated from where we obtain the isochoric specific heat and its jump at $T_c$. When $\Delta > 0$, a Bose-Einstein critical temperature $T_c \neq 0$ exists for any $d > 0$ at which the internal energy shows a peak and the specific heat shows a jump. Both the critical temperature and the specific heat jump increase as functions of the gap but they decrease as of $d/s$. At sufficiently high temperatures $\Delta$- independent classical results are recovered. However, for temperatures below the critical one the gap effects are predominant. For $\Delta = 0$ we recover previous reported results.