Quantum corrections and multioccupancy in a semi-classical gas

A. A. Bakasov, G. Denardo

Loris Ferrari

The quantum corrections to the behavior of a semi-classical gas can be expressed as a power series of the ratio $eta$ between the cube of De Broglie's thermal wavelength and the specific volume. The connection between $eta$ and the multioccupancy of quantum states is the aim of the present work. By means of a chemical/physical approach it is possible to associate $eta$ to the concrete realization of multioccupancy in momentum space, through the formation of what we call "pseudo-molecules", i.e. multiplets of particles sharing momenta whose differences are negligeable in the continuous spctrum approximation. The pseudo-molecules result as casual consequences of multiple scattering processes among non interacting (scattering apart) particles.

Tommaso Coletta, Nicolas Laflorencie, Frédéric Mila

Motivated by some inconsistencies in the way quantum fluctuations are included beyond the classical treatment of hard-core bosons on a lattice in the recent literature, we revisit the large-S semi-classical approach to hard-core bosons on the square lattice at T=0. First of all, we show that, if one stays at the purely harmonic level, the only correct way to get the 1/S correction to the density is to extract it from the derivative of the ground state energy with respect to the chemical potential, and that to extract it from a calculation of the ground state expectation value of the particle number operator, it is necessary to include 1/\sqrt{S} corrections to the harmonic ground state. Building on this alternative approach to get 1/S corrections, we provide the first semiclassical derivation of the momentum distribution, and we revisit the calculation of the condensate density. The results of these as well as other physically relevant quantities such as the superfluid density are systematically compared to quantum Monte Carlo simulations. This comparison shows that the logarithmic corrections in the dilute Bose gas limit are only captured by the semi-classical approach if the 1/S corrections are properly calculated, and that the semi-classical approach is able to reproduce the 1/k divergence of the momentum distribution at k=0. Finally, the effect of 1/S^2 corrections is briefly discussed.

Erik Tillema

C. J. Pethick, H. Smith

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 thermo dynamic 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, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).