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The Journal of Physical Chemistry B

Thermodynamics and Phase Transitions in a Fluid Confined by a Harmonic Trap

Vı́ctor Romero-Rochı́n

Published Apr 21, 2005• 7 citations
Publisher

Abstract

We study a fluid of interacting atoms confined by a three-dimensional anisotropic harmonic potential, similar to those produced by the magnetic traps used to confine cold atoms. We show that instead of the usual thermodynamic variables pressure and volume, no longer existing in this case, there appear "new" variables: the volume is replaced by (the inverse cube of) the geometric average of the oscillator frequencies of the trap, and the hydrostatic pressure is replaced by an intensive variable, conjugate to the previous one, and responsible for the mechanical equilibrium of the fluid in the trap. We discuss the origin and physical meaning of these new variables. With the aid of molecular dynamics simulations we show the emergence of novel liquid, vapor and solid-like phases in a classical fluid. In particular, we calculate the liquid-vapor-like coexistence curve and show evidence for the appearance of a critical point. These phase transitions should be observable in fluids of not-so-cold alkaline atoms.

DOI
10.1021/jp0443052

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We study a fluid of interacting atoms confined by a three-dimensional anisotropic harmonic potential, similar to those produced by the magnetic traps used to confine cold atoms. We show that instead of the usual thermodynamic variables pressure and volume, no longer existing in this case, there appear "new" variables: the volume is replaced by (the inverse cube of) the geometric average of the oscillator frequencies of the trap, and the hydrostatic pressure is replaced by an intensive variable, conjugate to the previous one, and responsible for the mechanical equilibrium of the fluid in the trap. We discuss the origin and physical meaning of these new variables. With the aid of molecular dynamics simulations we show the emergence of novel liquid, vapor and solid-like phases in a classical fluid. In particular, we calculate the liquid-vapor-like coexistence curve and show evidence for the appearance of a critical point. These phase transitions should be observable in fluids of not-so-cold alkaline atoms.
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 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 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.
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.
Abstract The critical properties displayed by an ideal 2D Bose gas trapped in a harmonic potential are determined and characterized in an exact numerical fashion. Beyond thermodynamics, addressed in terms of the global pressure and volume which are the appropriate variables of a fluid confined in a non-uniform harmonic potential, the density-density correlation function is also calculated and the corresponding correlation length is found. Evaluation of all these quantities as Bose–Einstein condensation (BEC) is approached manifest its critical continuous phase transition character. The divergence of the correlation length as the critical temperature is reached, unveils the expected spatial scale invariance proper of a critical transition. The logarithmic singularities of this transition are traced back to the non-analytic behavior of the thermodynamic variables at vanishing chemical potential, which is the onset of BEC. The critical exponents associated with the ideal BEC transition in the 2D inhomogeneous fluid reveals its own universality class.
We consider a Bose gas confined by a harmonic potential. We define an appropriate thermodynamic limit and analyze the properties of the phases and phase transition in this limit. Critical properties in the presence of the potential are found to be different from, though simply related to, those in the usual translationally invariant case. We argue that the properties of magnetically trapped rubidium and sodium gases (in which Bose-Einstein condensation has recently been observed) are well approximated by our thermodynamic limit except in a narrow window of temperature around the critical temperature. We also consider the effect of the confining potential on the non-equilibrium dynamics following a rapid quench to the ordered side and give a scaling description of the late-time universal dynamics.