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Bulletin of the American Physical Society

Finite Temperature Scaling of the Non-interacting Free Energy Density Functional

James W. Dufty, Valentin V. Karasiev, Samuel B. Trickey

Published Mar 21, 2011• 0 citations
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The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
We consider variational (density functional) models of fluids confined in parallel-plate geometries (with walls situated in the planes z = 0 and z = L respectively) and focus on the structure of the pair correlation function . We show that for local variational models there exist two non-trivial identities relating both the transverse Fourier transform and the zeroth moment at different positions and . These relations form an algebra which severely restricts the possible form of the function . For the common situations in which the equilibrium one-body (magnetization/number density) profile exhibits an odd or even reflection symmetry in the plane, the algebra simplifies considerably, and is used to relate the correlation function to the finite-size excess free energy . We rederive non-trivial scaling expressions for the finite-size contribution to the free energy at bulk criticality and for systems where large-scale interfacial fluctuations are present. Extensions to non-planar geometries are also considered.