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

Abstract

We present the full thermodynamics of an interacting fluid confined by an arbitrary external potential. We show that for each confining potential, there emerge "generalized" volume and pressure variables V and 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 discuss how this approach yields experimentally amenable procedures to find the equation of state of the fluid, P=P(VN,T) with N the number of atoms, as well as its heat capacity at constant generalized volume C_{V}=C_{V}(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. We claim that this route provides an additional and useful tool to analyze both the thermodynamic variables of an ultracold trapped gas as well as its elementary excitations.

Critical properties of weakly interacting Bose gases as modified by a harmonic confinement

“…That is, if we replace the harmonic potential by one of a vessel of rigid walls, then P and V are replaced by p and V in Eq. (20) and this expression becomes the usual one for the virial expansion of the pressure [8]. Nevertheless, because the variables P and V are still unfamiliar, it may be useful to grasp their order of magnitude in a typical experiment with 87 Rb at T = 100×10 9 K, with N ≃ 10 5 atoms in a trap of mean frequencyω = 2π(100) Hz.…”

“…LDA provides the bridge between the homogeneous and non-uniform versions of the same system, subject to the corresponding external fields. It can be shown to be exact in the thermodynamic limit [8,43,44]. Its recipe is quite simple: one first obtains the homogeneous equation of state of the particle density as a function of chemical potential and temperature, n = n(µ, T ); then, LDA proceeds replacing the chemical potential by a "local" one, µ → µ − V ext ( r) and, as a result, n becomes the density profile ρ( r; µ, T ) of the inhomogeneous system ρ( r; µ, T ) = n(µ − V ext ( r), T ).…”

The equation of state of one-dimensional Gross-Pitaevskii equation

“…Nevertheless, in fact, in the case of BEC we have two options in order to deal with thermodynamic case by focusing on the particles or the condensates. In the previous papers, some authors proposed the grand canonical partition function to achieve some thermodynamic properties associated with N bosons confined by three-dimensional anisotropic harmonic trap [18,19,20,21], for other similar discussions see also [22,23]. They concerned with N weakly interacting particles by proposing a new variable thermodynamic quantity, namely harmonic volume, which replaces the role of real volume.…”

Semiclassical Hartree-Fock theory of a rotating Bose-Einstein condensation

“…(16) shows that the condensate density is drastically altered from the ideal case, reflecting that the shape of the confining potential has a three-dimensional 'Mexicanhat' shape [24]. Moreover, µ(α) is the relevant energy scale parameterizing the effects of interactions, up to the point in the trap where µ(α) = V rot (r ⊥ , z).…”

“…Performing the integral over this phase space required to calculate many system pa- * Electronic address: ahmedhassan117@yahoo.com rameters, such as the condensate density, the effective potential as well as the chemical potential. However, all of them may be self-consistently parametrized using the Hartree-Fock approximation [15][16][17]. Using the thermodynamical potential, the condensed fraction, the transition temperature, entropy and the heat capacity are calculated.…”