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Reviews in Mathematical Physics

ON THE WEAK COUPLING LIMIT FOR A FERMI GAS IN A RANDOM POTENTIAL

Tin-Lun Ho, Lawrence J. Landau, A. J. WILKINS

Published Jun 1, 1993• 33 citations

Abstract

General conditions are derived for the essential self-adjointness of –∆ + V, where V is a translation-invariant random potential, and for the existence of the perturbation expansion. A sequence of graphs is exhibited violating Dell'Antonio's bound for skeleton graphs. For a translation-invariant and clustering Gaussian random potential V, and a translation-invariant and clustering initial state S of the Fermi gas, uncorrelated with the random potential, the weak coupling limit (Van Hove limit) yields increase of entropy, propagation of chaos, convergence of the state for sufficiently small values of the parameter τ to a gauge-invariant and quasi-free asymptotic state, and the semigroup describing the evolution of the two-point function. The asymptotic system is Bernoulli. Results are obtained not only for the average over the random potential but also with probability one. If the random potential V′ is absolutely continuous with respect to V, and if the state S′ is given by a density matrix in the GNS representation for S, then the weak coupling limit is the same as for V and S.

DOI: 10.1142/s0129055x93000061

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Abstract. In this note, we discuss some basic aspects of the dynamics of a homogenous Fermi gas in a weak random potential, under simplifying assumptions on the particle pair interactions. We are particularly interested in studying the delocalizing effects due to the Pauli principle. We derive the kinetic hydrodynamic limit, determined by a linear Boltzmann equation, for the momentum distribution function. Moreover, we prove that if the initial state is quasifree, then the time evolved state averaged over the randomness, which is by itself not quasifree, has a quasifree hydrodynamic limit. We show that the momentum distributions determined by the Gibbs states of a free fermion field persist into the diffusive time scale; this includes the limit of zero temperature. 1.

The asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].

We study the dynamics of many-body Fermi systems, for a class of initial data which are close to quasi-free states exhibiting a nonvanishing pairing matrix. We focus on the mean-field scaling, which for fermionic systems is naturally coupled with a semiclassical scaling. Under the assumption that the initial datum enjoys a suitable semiclassical structure, we give a rigorous derivation of the time-dependent Hartree-Fock-Bogoliubov equation, a nonlinear effective evolution equation for the generalized one-particle density matrix of the system, as the number of particles goes to infinity. Our result holds for all macroscopic times, and provides bounds for the rate of convergence.

We give the first rigorous derivation of a kinetic equation in a nontrivial translation invariant situation. We consider the heavy particle (mass scaled as me −δ , δ≥0) in the ideal fermi gas with translation invariant interaction eV between them. In the Heisenberg representation we prove convergence to the strongly continuous completely-positive semigroup T s for any value of scaled time s, t=s/e 2 and for δ>2. When δ=2 T s is not a semigroup. For 0≤δ<2 we announce some more weak results

The thermodynamic properties for a gas of one-dimensional fermi particles interacting via a repulsive delta-function potential have been studied under the limit of infinite repulsive coupling using the theory of grand canonical ensemble. All the thermodynamic functions have been found explicitly and the velocity of sound is derived as a function of the fugacity. In particular, the entropy is not zero at the absolute zero temperature for this model. Finally, some results are given for the arbitrary spin gas.

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}^{*}

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

a

. 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

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.