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Progress of Theoretical Physics

On the Landau Theory of Fermi Fluids. B: Macroscopic Kinetic Equation

T. Morita

Published Feb 1, 1962• 0 citations
Publisher

Abstract

It is shown that the kinetic equation, in which the particles drift under the action of the external field and by interactions with other particles, can be written such that the particles are considered to be accelerated by a macroscopic field rather than individual interactions. Quantum and classical fluids are considered; the quantum case is derived in connection with the Landau theory of Fermi fluids. (L.N.N.)

DOI
10.1143/ptp.27.425

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Progress of Theoretical Physics•Feb 1, 1962•1 citation
It is shown that the kinetic equation, in which the particles drift under the action of the external field and by interactions with other particles, can be written such that the particles are considered to be accelerated by a macroscopic field rather than individual interactions. Quantum and classical fluids are considered; the quantum case is derived in connection with the Landau theory of Fermi fluids. (L.N.N.)
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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.
We consider a quantum Langevin kinetic equation for a system of fermions. We first derive the Langevin force noise correlation functions in Landau's Fermi-liquid kinetic theory from general considerations. We then use the resulting equation to calculate the equilibrium dynamic structure factor in the collisionless regime at low temperatures. The result is in agreement with the conventional many-body result. We then use the theory to derive both the fluctuating Navier-Stokes equations for a quantum fluid and the fluctuating hydrodynamic equations for fermions in the presence of quenched disorder. We also discuss the modifications needed for the fluctuating hydrodynamic equations to describe an electron fluid with long-ranged interactions, and we prove an HHH theorem for the nonlinear Landau kinetic equation.
Since the systems described by quantum fields are by definition extended, it is natural to think that in some limit they may reasonably well be approximated as fluids. This means that the state of the system is parameterized by a few locally well-defined fields such as temperature or energy density, obeying a set of hyperbolic equations of motion. A concrete example is the extensive use of fluid models to describe high-energy collisions [BelLan56, CarDuo73, CoFrSc74, Bjo83, CarZac83]. Our earlier derivation of quantum kinetic theory suggests a way to put this insight on a formal basis. Within its range of validity, the Boltzmann equation will drive the one-particle distribution function towards local thermal equilibrium. On scales much larger than the local thermalization scale, we expect to see hydrodynamical behavior [BeCoPa02]. This is, after all, the usual way of deriving hydrodynamics from kinetic theory [Hua87]. Beware, notwithstanding, that even at the level of classical kinetic theory there are still open questions regarding the cross-over from the kinetic to the hydro regime [KarGor02, KarGor03]. If we understand hydrodynamics as stated in the first paragraph of this chapter, then a system defined in terms of a quantum field may not have a hydrodynamic limit. This has been shown in [Elz02] for the case of a free Fermi field.
Close to equilibrium, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. The general expression derived for the kernel is evaluated to second order in the interparticle potential. The result is a wavevector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born approximation cross section. The theory is formulated in terms of second-quantized phase space operators whose equilibrium averages are the n-particle Wigner distribution functions. Convenient expressions for the commutators and anticommutators of the phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations of the classical pair distribution function h(k) and direct correlation function c(k). The kinetic equation is presented as the equation of motion of a two -particle correlation function, the phase space density-density anticommutator, and is derived by a formal closure of the quantum BBGKY hierarchy. An alternative derivation using a projection operator is also given. It is shown that the method used for approximating the kernel by a second order expansion preserves all the sum rules to the same order, and that the second-order kernel satisfies the appropriate positivity and symmetry conditions.