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Journal of Physics A Mathematical and General

A convergent virial expansion for the quantum Lorentz gas with point interactions

C Carvalho

Published Mar 21, 1990• 1 citation
Publisher

Abstract

The author studies the quantum Lorentz gas with point interactions. The author uses the explicit form of the resolvent of this system to set up an expansion for the pressure and the central limit theorem to sum the many random contributions to the scattering of the light particle due to the heavy particles. The author also discusses the convergence of the expansion.

DOI
10.1088/0305-4470/23/6/017

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The author studies the quantum Lorentz gas with point interactions. The author uses the explicit form of the resolvent of this system to set up an expansion for the pressure and the central limit theorem to sum the many random contributions to the scattering of the light particle due to the heavy particles. The author also discusses the convergence of the expansion.
The authors calculate the second virial coefficient of the quantum Lorentz gas with point interaction. It can be evaluated in closed form for arbitrary strength of the interaction. Particular attention is given to the critical coupling which gives rise to a zero-energy resonance. Point interactions satisfy a modified Levinson's theorem; as a consequence they show that there is no cancellation between the bound state and the scattering contributions in the high-temperature expansion of the second virial coefficient, contrary to the regular potentials case. They also derive high- and low-energy sum rules for the point interaction. As an application of the sum rules they rederive a high-temperature expansion for the second virial coefficient.
A multiple scattering model of a quantum particle interacting with a random Lorentz gas of fixed point scatterers is established in an Euclidean space of arbitrary dimension. At the core of the model, the scattering amplitude for the point scatterers is derived in detail and expressed in terms of the scattering length. The fundamental properties of the model, such as the cross section and the scattering matrix, are calculated. In addition, the model is shown to verify the optical theorem and thus probability conservation. Finally, the differential and total cross sections are numerically computed in two situations whether the Lorentz gas is smaller or larger than the mean free path. A distinct Airy diffraction peak is obtained for a large enough number of scatterers. This observation is related to the extinction paradox.
In the framework of the Gross-Pitaevskii equation, we consider the problem of the expansion of quantum gases into a vacuum. For them, the chemical potential μ has a power-law dependence on the density n with the exponent ν = 2/D, whereD is the space dimension. For gas condensates of Bose atoms as the temperature T → 0, s scattering gives the main contribution to the interaction of atoms in the leading order in the gas parameter. Therefore, the exponent ν = 1 for an arbitrary D. In the three-dimensional case, ν = 2/3 is realized for condensates of Fermi atoms in the so-called unitary limit. For ν = 2/D, the Gross-Pitaevskii equation has an additional symmetry under Talanov transformations of the conformal type, which were first found for the stationary self-focusing of light. A consequence of this symmetry is the virial theorem relating the average size R of an expanding gas cloud to its Hamiltonian. The quantity R asymptotically increases linearly with time as t →∞. In the semiclassical limit, the equations of motion coincide with those of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1+2/D. In this approximations, self-similar solutions describe angular deformations of the gas cloud against the background of the expanding gas in the framework of equations of the Ermakov-Ray-Reid type.
The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality ddd. In particular, the convergence radius ρc(d)ρ_c(d)ρc​(d) of the expansion is particulary large at {\em exactly\/} d=3d=3d=3 dimensions, ρc(3)=7.1068…×lim⁡d→3ρc(d)ρ_c(3) = 7.1068\ldots \times \lim_{d\to3} ρ_c(d)ρc​(3)=7.1068…×limd→3​ρc​(d). The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.
We use the virial expansion to investigate the behavior of the two-component, attractive Fermi gas in the high-temperature limit, where the system smoothly evolves from weakly attractive fermions to weakly repulsive bosonic dimers as the short-range attraction is increased. We present a formalism for computing the virial coefficients that employs a diagrammatic approach to the grand potential and allows one to easily include an effective range R∗{R}^{*}R∗ in the interaction. In the limit where the thermal wavelength $\ensuremath{\lambda}\ensuremath{\ll}{R}^{*}$, the calculation of the virial coefficients is perturbative even at unitarity and the system resembles a weakly interacting Bose-Fermi mixture for all scattering lengths aaa. By interpolating from the perturbative limits $\ensuremath{\lambda}/|a|\ensuremath{\gg}1$ and ${R}^{*}/\ensuremath{\lambda}\ensuremath{\gg}1$, we estimate the value of the fourth virial coefficient at unitarity for R∗=0{R}^{*}=0R∗=0 and we find that it is close to the value obtained in recent experiments. We also derive the equations of state for the pressure, density, and entropy, as well as the spectral function at high temperatures.
An expansion for quantum statistical mechanics is derived that gives classical statistical mechanics as the leading term. Each quantum correction comes from successively larger permutation loops, which arise from the factorization of the symmetrization of the wave function with respect to localized particle interchange. Explicit application of the theory yields the full fugacity expansion for the quantum ideal gas, and the second fugacity coefficient for interacting quantum particles, which agree with known results. Compared to the Lee-Yang virial cluster expansion, the present expansion is expected to be more rapidly converging and the individual terms appear to be simpler to evaluate. The results obtained in this paper are intended for practical computer simulation algorithms for terrestrial condensed matter quantum systems.