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arXiv:physics/0210055

Elementary thermodynamics of trapped particles

Martin Ligare

Published Oct 12, 2002• 0 citations
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Abstract

I develop simple thermodynamic relations for a collection of noninteracting classical particles confined in a harmonic trap. The volume of such a trap is not a good thermodynamic variable, so conventional expressions of the first law of thermodynamics and the ideal gas law must be modified. I use the frequency of oscillations about the minimum of the trap as an external parameter characterizing the confinement, and derive elementary relations between particle number, temperature, energy, oscillation frequency, and a generalized pressure, that are analogous to conventional thermodynamic relations for an ideal gas in a rigid volume. I also discuss heat capacities for trapped particles.

DOI
10.48550/arxiv.physics/0210055

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I develop simple thermodynamic relations for a collection of noninteracting classical particles confined in a harmonic trap. The volume of such a trap is not a good thermodynamic variable, so conventional expressions of the first law of thermodynamics and the ideal gas law must be modified. I use the frequency of oscillations about the minimum of the trap as an external parameter characterizing the confinement, and derive elementary relations between particle number, temperature, energy, oscillation frequency, and a generalized pressure, that are analogous to conventional thermodynamic relations for an ideal gas in a rigid volume. I also discuss heat capacities for trapped particles.
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.
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.
We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number NNN of atoms, harmonic oscillator length, deformation of the trap, sss-wave scattering length) fix its thermodynamic behavior in the large NNN limit through two dimensionless scaling parameters. These are the reduced temperature t=T/Tc0t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}t=T/Tc0​ and the ratio $\ensuremath{\eta}$ between the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature Tc0{T}_{c}^{0}Tc0​ of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.
By comparing the thermodynamic properties of ideal atomic gases in a power-law potential and in a rigid box, it is found that the power-law potential is equivalent to the rigid box as far as the macroscopic behaviors of the system are concerned. The dimensionality and volume of the equivalent box are dependent on the parameters charactering the power-law potential. This equivalent relation enables us to treat a trapped ideal gas as a free one, and consequently, several useful conclusions of the trapped-gas system can be easily derived from the corresponding results of the free-gas system.
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.
We present the full thermodynamics of a fluid confined by an arbitrary external potential based on the virial expansion of the grand potential. The fluid may be classical or quantum and it is assumed that interatomic interactions are pairwise additive. We indicate how the appropriate "generalized" volume and pressure variables, that replace the usual volume and hydrostatic pressure, emerge for a given confining potential in the thermodynamic limit. A discussion of the physical meaning and of the measurement of these variables is presented. We emphasize that this treatment yields the correct equation of state of the fluid and we give its virial expansion. We propose an experiment to measure the heat capacity, so that with this quantity and the equation of state, the complete thermodynamics of the system may be extracted. As a corollary, we find that the so-called {\it local density approximation} for these systems follows in the thermodynamic limit, although we also point out that it cannot be used indiscriminately for all local variables. Along the text we discuss the relevance of these findings in the description of the currently confined ultracold gases.