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Physical Review

Failure of the Weak-Coupling Model in the Transport Theory of Dense Real Gases

Rachel Williams, Jerome Weinstock

Published May 5, 1968• 6 citations
Publisher

Abstract

It is shown that the weak-coupling limit is not valid for the third-order term in the density expansion of quantum or classical transport coefficients. The weak-coupling limit is not valid in the sense that it does not preserve the qualitative features of real gases and, hence, is useless within the context of the transport divergence problem. In fact, it is shown that, although the weak-coupling limit of the recollision part of the quantum four-particle collision operator diverges, its contribution to transport coefficients actually converges (whereas this contribution is known to diverge logarithmically when no weak-coupling limit is taken). A rather curious behavior is uncovered for the quantum weak-coupling limit of the recollision part of the four-particle collision operator. It is found that the matrix elements of this operator grow as ${t}^{\frac{1}{2}}$ and oscillate rapidly. This growth, which is faster than logarithmic in $t$, is associated with a spatial overlapping of broadening wave packets. Because of the accompanying oscillation, however, it does not present any divergence problem to transport theory.

DOI
10.1103/physrev.169.196

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arXiv:2210.09712•Physical Review A•Dec 21, 2022•2 citations
It is shown that the weak-coupling limit is not valid for the third-order term in the density expansion of quantum or classical transport coefficients. The weak-coupling limit is not valid in the sense that it does not preserve the qualitative features of real gases and, hence, is useless within the context of the transport divergence problem. In fact, it is shown that, although the weak-coupling limit of the recollision part of the quantum four-particle collision operator diverges, its contribution to transport coefficients actually converges (whereas this contribution is known to diverge logarithmically when no weak-coupling limit is taken). A rather curious behavior is uncovered for the quantum weak-coupling limit of the recollision part of the four-particle collision operator. It is found that the matrix elements of this operator grow as t12{t}^{\frac{1}{2}}t21​ and oscillate rapidly. This growth, which is faster than logarithmic in ttt, is associated with a spatial overlapping of broadening wave packets. Because of the accompanying oscillation, however, it does not present any divergence problem to transport theory.
A formal density expansion of transport coefficients expressed in terms of momentum autocorrelation functions is derived for a degenerate quantum gas (Bose-Einstein or Fermi-Dirac statistics) at any frequency and for noncentral as well as central forces. The result to lowest order in density is reduced to the solution of a well-defined quantum two-body problem, and the first density correction is reduced to the solution of a well-defined quantum three-body problem. It is confirmed that at zero frequency one only needs the asymptotic forms of the collision operator to calculate quantum transport coefficients. It is further pointed out that the coefficients of the third- and higher order terms of the density expansion diverge at zero frequency, in analogy with the classical case. A "renormalization" is suggested which, it is believed, leads to a nonanalytic density dependence for quantum gases distinct from the nonanalytic behavior associated with degeneracy statistics.
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 ldquomixedrdquo equation relating the real one-particle distribution function (on the left-hand side of the equation) to the ldquofreerdquo 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.
Renormalized expressions for the transport coefficients of a normal quantum fluid are derived from a nonlocal kinetic equation. As was shown in the classical case by Forster and Martin and by R\'esibois, the expressions for the transport coefficients separate naturally into "kinetic" and "direct" parts. The kinetic terms are proportional to matrix elements of the inverse of the kinetic kernel, and have the same general structure as the "bare" transport coefficients obtained from a local Boltzmann equation. The direct terms are proportional to matrix elements of the kernel itself, and have no counterpart in calculations based on a local kinetic equation, as in the kinetic theory of gases. We evaluate the transport coefficients using the weak-coupling approximation to the kernel derived in a previous paper. Results are given, first, for both Bose and Fermi fluids at arbitrary temperature, and then for the Fermi fluid near T=0T=0T=0, where complete solutions are obtained. It is found that the direct parts of the shear viscosity and thermal conductivity are of higher order in TTT than the kinetic parts, and are therefore negligible at very low temperature. The kinetic parts have the same leading temperature dependence as the predictions of the Landau theory. For the bulk viscosity, however, the direct and kinetic contributions begin at the same order (T2{T}^{2}T2) in the temperature.
The white noise approach to the investigation of the dynamics of a quantum particle interacting with a dilute and in general non-equilibrium gaseous environment in the low density limit is outlined. The low density limit is the kinetic Markovian regime when only pair collisions (i.e., collisions of the test particle with one particle of the gas at one time moment) contribute to the dynamics. In the white noise approach one first proves that the appropriate operators describing the gas converge in the sense of appropriate matrix elements to certain operators of quantum white noise. Then these white noise operators are used to derive quantum white noise and quantum stochastic equations describing the approximate dynamics of the total system consisting of the particle and the gas. The derivation is given ab initio, starting from the exact microscopic quantum dynamics. The limiting dynamics is described by a quantum stochastic equation driven by a quantum Poisson process. This equation then applied to the derivation of quantum Langevin equation and linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper describes the approach which was developed by L. Accardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. The second part presents the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fock-antiFock representation. This approach simplifies the derivation and allows to express the strength of the quantum number process directly in terms of the one-particle SSS-matrix.
This is the first of a series of articles where we develop a theory for the description of the dynamics of dilute quantum gases, at or out of equilibrium, with an accurate treatment of the correlation effects introduced by binary collisions (these correlations are not taken into account when the Boltzmann equation is used). For this purpose, we propose the introduction of a variant of the Wigner transform, the « free Wigner transform », which wipes out the effects of the potential at short distances. If initially factorized, the entering part of the free transform remains exactly so during all collision, so that it is better adapted to a molecular chaos assumption than the ordinary transform. Once the free transform is known, the effects of the short distance correlations can be reconstructed from it. Therefore, the problem is to find an appropriate equation of evolution which gives its time variations. With this aim in mind, we study in the present article the evolution of the free Wigner transform in a binary collision. We show that its evolution equation includes coefficients which can be exactly obtained from the « on shell » terms of the T matrix (S matrix, phase shifts), even while the particles are interacting. In addition to the usual terms which involve the scattering cross sections, as usual with the Boltzmann equation, the calculation provides other terms which correspond to collision retardation effects and « refraction » effects (refraction of the wave function of the test atom by density gradients of the collision partners). These terms will play an important role in the following article, and introduce the second virial corrections to the equilibrium and transport properties of the gas.
We continue to study frequency-dependent complex bulk viscosities of one-dimensional Bose and Fermi gases with contact interactions, which exhibit the weak-strong duality according to our recent work. Here we show that they are contributed to by Drude peaks divergent at zero frequency as typical for transport coefficients of quantum integrable systems in one dimension. In particular, their Drude weights are evaluated based on the Kubo formula in the high-temperature limit at arbitrary coupling as well as in the weak-coupling and strong-coupling limits at arbitrary temperature, where systematic expansions in terms of small parameters are available. In all three limits, the Drude peaks are found at higher orders compared to the finite regular parts.