Thermodynamics of ultracold trapped gases. Generalized mechanical variables, equation of state and heat capacity

TL;DR

This paper develops a thermodynamic framework for interacting quantum gases in arbitrary traps, introducing generalized variables and equations of state, validated through virial expansion and applied to Bose gases with experimental comparisons.

Contribution

It introduces generalized volume and pressure variables for trapped gases, enabling a comprehensive thermodynamic description and analysis of their properties.

Findings

Derived virial expansion for the grand potential.

Established equations of state using generalized variables.

Compared theoretical results with experimental data for ultracold sodium gas.

Abstract

The thermodynamics framework of an interacting quantum gas trapped by an arbitrary external potential is reviewed. We show that for each confining potential, in the thermodynamic limit, there emerge "generalized" volume and pressure variables ${\cal V}$ and ${\cal P}$, that replace the usual volume and hydrostatic pressure of a uniform system. This scheme is validated with the derivation of the virial expansion of the grand potential. We show that this approach yields experimentally amenable procedures to find the equation of state of the fluid, ${\cal P} = {\cal P}({\cal V}/N,T)$ with $N$ the number of atoms, as well as its heat capacity at constant generalized volume $C_{\cal V} = C_{\cal V}({\cal V},N,T)$. With these two functions, all the thermodynamics properties of the system may be found. As specific examples we study weakly interacting Bose gases trapped by harmonic and by linear quadrupolar potentials within the Hartree-Fock approximation. Comparisons with experimental results of a $^{23}$Na ultracold gas are also presented. We claim that this route should provide an additional and useful tool to analyze both the thermodynamic variables of a trapped gas as well as its elementary excitations.