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Brazilian Journal of Physics

Thermodynamics of an ideal gas of bosons harmonically trapped: equation of state and susceptibilities

Vı́ctor Romero-Rochı́n, Vanderlei Salvador Bagnato

Published Sep 1, 2005• 44 citations
Publisher

Abstract

We present theoretical aspects concerning the thermodynamics of an ideal bosonic gas trapped by a harmonic potential. Working in the Grand Canonical ensemble we are able to properly identify the extensive thermodynamic variable equivalent to the volume and the intensive thermodynamic variable equivalent to the pressure. These are called the "harmonic volume" and the "harmonic pressure" and their physical meaning is discussed. With these variables, the problem of Bose-Einstein condensation is studied in terms of the behavior of the corresponding equation of state and in terms of measurable susceptibilities such as the heat capacities, the isothermal compressibility and the coefficient of thermal expansion. From the analysis, an interesting analogy with Black-Body radiation emerges, showing that at and below the critical temperature, the non-condensate fraction of atoms behaves thermodynamically like a gas of massless particles.

DOI
10.1590/s0103-97332005000400004

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We present theoretical aspects concerning the thermodynamics of an ideal bosonic gas trapped by a harmonic potential. Working in the Grand Canonical ensemble we are able to properly identify the extensive thermodynamic variable equivalent to the volume and the intensive thermodynamic variable equivalent to the pressure. These are called the "harmonic volume" and the "harmonic pressure" and their physical meaning is discussed. With these variables, the problem of Bose-Einstein condensation is studied in terms of the behavior of the corresponding equation of state and in terms of measurable susceptibilities such as the heat capacities, the isothermal compressibility and the coefficient of thermal expansion. From the analysis, an interesting analogy with Black-Body radiation emerges, showing that at and below the critical temperature, the non-condensate fraction of atoms behaves thermodynamically like a gas of massless particles.
Some thermodynamic features of an assembly of a finite number of bosons trapped in a bounded harmonic potential are investigated. P–V isotherms are drawn for both the degenerate and the non-degenerate phases. At any temperature, the pressure is a decreasing function of volume, unlike the free Bose gas for which the pressure becomes independent of volume in the degenerate phase. At the absolute zero of temperature the quantum pressure for a spherical enclosure of radius r0 equal to aho, aho being the characteristic harmonic oscillator length, is found to be of order 10-10 Torr while for [Formula: see text] it is of order 10-12 Torr. The isothermal compressibility has a sharp drop near the critical point and becomes negligibly small for temperatures above the critical temperature, irrespective of the size of the trap. The coefficient of thermal expansion also shows a sudden drop at the critical temperature. The specific heat at constant pressure shows a peak and is well-defined in the degenerate phase, in contradistinction to the ideal Bose gas. Results recently derived by Singh on the basis of local equilibrium theory are found to be in good agreement with our numerical computations.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.
Abstract The critical properties of the phase transition from a normal gas to a Bose–Einstein condensate (superfluid) of a harmonically confined Bose gas are addressed with the knowledge of an equation of state of the underlying homogeneous Bose fluid. It is shown that while the presence of the confinement trap arrests the usual divergences of the isothermal compressibility and heat capacities, the critical behavior manifests itself now in the divergence of derivatives of the mentioned susceptibilities. This result is illustrated with a mean-field like model of an equation of state for the homogeneous particle density as a function of the chemical potential and temperature of the gas. The model assumes the form of an ideal Bose gas in the normal fluid while in the superfluid state a function is proposed such that, both, asymptotically reaches the Thomas–Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the global thermodynamics of the harmonically confined gas, from which we analyze its critical properties. We discuss how these properties can be experimentally tested.
Thermodynamics of non-uniform systems has always been a challenging topic in the physical sciences.In the field of quantum gases, that subject is mainly studied with ultracold samples trapped in a harmonic potential, the first kind of confinement ever made to atomic fluids. 1 Nascimbène et al. 2 have already described a method to measure locally the equation of state and heat capacity of a harmonically trapped quantum gas.Nonetheless, local measurements cannot provide the volume-dependent susceptibilities of an inhomogeneous system, namely the thermal expansion and compressibility, meaning that the thermodynamic description of harmonically trapped gases is still incomplete.To solve that issue, Romero and Bagnato 3, 4 proposed canonical variables of work for a gas in a harmonic trap, which work like pressure and volume for a gas in a vessel and allow global measurements on the system.Therefore, we describe here our methodology to measure the thermodynamic susceptibilities of a harmonically trapped gas around the Bose-Einstein transition using the Romero-Bagnato formalism.Our experiments were divided in different phases, each having the harmonic trap with unique frequencies, related to the formalism's extensive variable.In each phase, we prepared individual samples of rubidium-87 gas in the trap and imaged them using the time-of-flight technique.Next, we fitted our images with the bimodal model and found the in situ density profiles of the samples with the standard regression procedures.Then, we determined the phase's equation of state as a function of temperature and number of atoms.For the thermodynamic analysis, we used an empirical model that we developed to fit the equation-of-state curves, allowing us to represent our data in terms of mathematical coefficients.In that way, we found the curves of internal energy, heat capacity, thermal expansion and compressibility, thus achieving a full thermodynamic description of a harmonically trapped gas.Our method and its results are unprecedented in the literature and might contribute to the further understanding of non-uniform systems, as well as the future development of quantum thermal engines.
A gas of interacting atoms confined by a three dimensional anisotropic harmonic potential is studied. It is shown that there appear "new" thermodynamic variables instead of the usual pressure and volume: the latter is replaced by (the inverse of) the cube of the geometric average of the oscillator frequencies of the trap, and the former by the harmonic pressure responsible for the mechanical equilibrium of the fluid in the trap. We discuss the origin and physical meaning of these quantities and show that the equation of state of the gas is given in terms of these variables. The equation of state of a cold gas of interacting Bose atoms in the Hartree-Fock approximation is presented. We indicate how the harmonic pressure can be measured in current experiments.
The thermodynamic properties of an ideal q-boson gas trapped in an n-dimensional harmonic potential are studied, based on the distribution function of q-bosons. The critical temperature Tc,q of Bose-Einstein condensation (BEC) and the heat capacity C of the system are derived analytically. It is shown that for the q-boson gas trapped in a harmonic potential, BEC may occur in any dimension when q ≠ 1, the critical temperature is always higher than that of an ordinary Bose gas (q = 1), and the heat capacity is continuous at Tc,q for a one-dimensional system but discontinuous at Tc,q for a two- or multi-dimensional system.