21364
J. Phys. Chem. B 2005, 109, 21364-21368
Thermodynamics and Phase Transitions in a Fluid Confined by a Harmonic Trap† Vı´ctor Romero-Rochı´n* Instituto de Fı´sica, UniVersidad Nacional Auto´ noma de Me´ xico, Apartado Postal 20-364, 01000 Me´ xico, D.F. Mexico ReceiVed: December 14, 2004; In Final Form: March 16, 2005
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.
1. Introduction The achievement of Bose-Einstein condensation (BEC)1-3 and other quantum phenomena that emerge due to the very low temperatures and very low densities reached in gases of alkaline atoms, see refs 4-7 as selected examples, has generated a vigorous theoretical activity to gain a better understanding of these systems. A topic important for the full understanding of trapped atoms is the thermodynamics of the system. So far, both theoretical8-12 and experimental work13-18 have been concentrated in studying the temperature dependence and structure of the condensed fraction of atoms, the energy and heat capacity, and the release energy in time-of-flight measurements. However, not enough attention has been paid to the study and measurement of the equation of state (EOS) per se. In a recent article19 the author has approached this issue for gases at very low temperatures and densities. Here, we review in more detail the thermodynamic aspects at hand and study a classical fluid at high temperatures and densities in order to elucidate the existence of phase transitions. We believe that the equation of state of fluids in harmonic traps has not been a topic of attention because the volume and the usual pressure are no longer thermodynamic variables in this type of sytems. These variables are appropriate only in systems confined in rigid-walled vessels. The current cold gases are not of this type. They are confined by a potential that interacts with the gas eVerywhere, not only at the walls. Actually, there are no walls in the magnetic traps, and as a result of this confinement the fluid is no longer uniform. The appreciation that volume and pressure are not the relevant thermodynamic variables is not new. It is known in the field of nonuniform fluids.20-22 In these systems the density is no longer uniform everywhere and the thermodynamics is ruled by the external potential. Thus, one reaches the conclusion that the pressure is also not a constant everywhere and the mechanical †
Part of the special issue “Irwin Oppenheim Festschrift”. * E-mail address:
[email protected] equilibrium condition is, then, expressed in terms of properties of the local pressure tensor.23-26 The density functional theory developed after the recognition of the latter facts,20-22 however, is a local theory that needs the prescription of a free energy density functional. Nevertheless, even in such specialized fields as in density functional theory, it has not been explicitely recognized that thermodynamics is a theory that deals with a few global variables. As we review in this article, one can deduce the role of the frequency of a harmonic trap as the thermodynamic variable analogous to the volume. More than that, we shall show that the cube of the inverse of the geometric average of the oscillator frequencies of the (anisotropic) trap is the extensiVe variable that replaces the volume. Then, we shall argue, there exists an intensiVe variable, conjugate to the cube of the inverse of the frequency, that is responsible for the mechanical equilibrium of the fluid with itself and with the external force of the trap. We shall refer to the cube of the inverse of the frequency as the “harmonic volume” V and to its conjugate variable as the “harmonic pressure” P. The relevance of the knowledge of the equation of state cannot be overstated. On one hand, it yields the dependence of measurable thermodynamic variables in terms of the independent ones that define the thermodynamic state; this dependence is particular to a given substance, and essentially characterizes it. On the other hand, however, and particularly suited to the current interest in trapped cold gases, knowledge of the equation of state opens a new window into the elucidation of the interatomic interactions. That is, measurement of the equation of state through, say, the virial coefficients gives direct information on the details of the interatomic interaction potential. Thus, besides the thermodynamics per se, the equation of state becomes an alternative and complementary tool to enquire into the details at the microscopic level. 2. Harmonic Volume and Pressure The system we consider is a fluid of N identical particles of mass m, confined by a harmonic anisotropic trap of frequencies
10.1021/jp0443052 CCC: $30.25 © 2005 American Chemical Society Published on Web 04/21/2005
Thermodynamics of Fluid in Harmonic Trap
J. Phys. Chem. B, Vol. 109, No. 45, 2005 21365
ω1, ω2, and ω3, with geometric average ω ) (ω1ω2ω3)1/3, in thermodynamic equilibrium at temperature T. If there is an interaction among the particles, we shall consider it to be pairwise additive via a short-range potential V(|r bi - b rj|). The Hamiltonian of the system is N
H)
b p i2
∑ ∑ i)1 2m i
