William J. Mullin

Bose-Einstein condensation in a harmonic potential

AbstractWe examine several features of Bose-Einstein condensation (BEC) in an external harmonic potential well. In the thermodynamic limit, there is a phase transition to a spatial Bose-Einstein condensed state for dimensionD≥2. The thermodynamic limit requires maintaining constant average density by weakening the potential while increasing the particle numberN to infinity, while of course in real experiments the potential is fixed andN stays finite. For such finite ideal harmonic systems we show that a BEC still occurs, although without a true phase transition, below a certain “pseudo-critical” temperature, even forD=1. We study the momentum-space condensate fraction and find that it vanishes as

THEORY OF SPIN DIFFUSION OF DILUTE, POLARIZED FERMIONS FOR ALL TEMPERATURES

{1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}

ON THE SNIDER EQUATION

in any number of dimensions in the thermodynamic limit. InD≤2 the lack of a momentum condensation is in accord with the Hohenberg theorem, but must be reconciled with the existence of a spatial BEC inD=2. For finite systems we derive theN-dependence of the spatial and momentum condensate fractions and the transition temperatures, features that may be experimentally testable. We show that theN-dependence of the 2D ideal-gas transition temperature for a finite system cannot persist in the interacting case because it violates a theorem due to Chester, Penrose, and Onsager.

The Two-Dimensional Bose–Einstein Condensate

Spin dynamics for arbitrarily polarized and very dilute solutions of 3He in liquid 4He are described. We began at a very fundamental level by deriving a kinetic equation for arbitrarily polarized dilute quantum systems based on a method due to Boercker and Dufty. This approach allows more controlled approximations than our previous derivation based on the Kadanoff-Baym technique. Our previous work is here generalized to include T-matrix interactions rather than the Born approximation. Spin hydrodynamic equations are derived. The general equations are valid for both Fermi and Bose systems. By use of a well-known phenomenological potential to describe the 3He-3He T-matrix we calculate longitudinal and transverse spin diffusion coefficients D⊥ and D¦ and the identical-particle spin-rotation parameter Μ. We confirm that these two diffusion constants differ at low T with D⊥ approaching a constant as T → 0, and D¦~1/T2. Estimates of errors made by our approximations are considered in detail. Good agreement is found in comparison with data from both Cornell University and the University of Massachusetts. We find that the s-wave approximation is inadequate and that mean-field corrections are important. Comparison is also made between theory and the recent UMass viscosity measurements.

Bose–Einstein condensation, fluctuations, and recurrence relations in statistical mechanics

We describe the solution of the transport equation over the entire range of temperature from the Boltzmann to fully degenerate regimes for dilute, polarized Fermi systems. Since spin-polarized systems can show unusual quantum effects involving spin rotation in both Boltzmann and degenerate regimes, a solution of the kinetic equation over the whole temperature range is expected to be useful. Our results for the longitudinal spin diffusion coefficient reduce to the known limits in the Boltzmann and degenerate regimes and also to the expected form in the peculiar high-polarization regime in which one spin species is degenerate and the other described by classical statistics (the degenerate-classical case). We derive numerical results for the spin-rotation quality parameter μ over the full temperature range. Unlike experimental results that show μ diminishing anomalously as the temperature decreases toward the degenerate regime, our value for μ is monotonically increasing. However, the transition to the degenerate-classical regime is found to occur with a rounded-step jump in μ as a function of polarization.

The origin of the phase in the interference of Bose-Einstein condensates

Physicists often claim that there is an effective repulsion between fermions, implied by the Pauli principle, and a corresponding effective attraction between bosons. We examine the origins and validity of such exchange force ideas and the areas where they are highly misleading. We propose that explanations of quantum statistics should avoid the idea of an effective force completely, and replace it with more appropriate physical insights, some of which are suggested here.

Longitudinal relaxation time for dilute quantum gases

We study the physical content of the Snider quantum transport equation and the origin of a puzzling feature of this equation, which implies contradictory values for the one-particle density operator. We discuss in detail why the two values are in fact not very different provided that the studied particles have sufficiently large wave packets and only a small interaction probability, a condition which puts a limit on the validity of the Snider equation. In order to improve its range of application, we propose a reinterpretation of the equation as a “mixed” equation relating the real one-particle distribution function (on the left-hand side of the equation) to the “free” distribution (on the right-hand side), which we have introduced in a recent contribution. In its original form, the Snider equation is valid only when used to generate Boltzmann-type equations where collisions are treated as point processes in space and time (no range, no duration); in this approximation, virial corrections are not included, so that the real and free distributions coincide. If the equation is used beyond this approximation to generate nonlocal and density corrections, we conclude that the results are not necessarily correct.

Dipolar anisotropy effects in the spin relaxation of adsorbed3He monolayers

We study the Hartree–Fock–Bogoliubov mean-field theory as applied to a two-dimensional finite trapped Bose gas at low temperatures and find that, in the Hartree–Fock approximation, the system can be described either with or without the presence of a condensate; this is true in the thermodynamic limit as well. Of the two solutions, the one that includes a condensate has a lower free energy at all temperatures. However, the Hartree–Fock scheme neglects the presence of phonons within the system, and when we allow for the possibility of phonons we are unable to find condensed solutions; the uncondensed solutions, on the other hand, are valid also in the latter, more general scheme. Our results confirm that low-energy phonons destabilize the two-dimensional condensate.