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Annalen der Physik

Renormalized Perturbation Expansion in Kinetic Theory

Ralf Der

Published Jan 1, 1977• 6 citations
Publisher

Abstract

Abstract Ladder summation techniques are applied to the recently developed c‐number diagram expansion for kinetic equations describing the relaxation of quantum fluids. By way of the resummation, the kinetic equations are reformulated in terms of a T‐matrix which describes the scattering of two particles influenced by the remaining particles of the system via the particle statistics (Bose or Fermi). At sufficiently high temperatures (Maxwell‐Boltzmann statistics), the T‐matrix introduced coincides with the usual T‐matrix of conventional two‐body scattering theory in free space. In low‐temperature Fermi systems, the T‐matrix differs from the reaction matrix of the Brueckner‐Goldstone theory because the “healing” property of the two‐body wave function does not obtain.

DOI
10.1002/andp.19774890409

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Physical Review•Apr 5, 1968•29 citations
Abstract Ladder summation techniques are applied to the recently developed c‐number diagram expansion for kinetic equations describing the relaxation of quantum fluids. By way of the resummation, the kinetic equations are reformulated in terms of a T‐matrix which describes the scattering of two particles influenced by the remaining particles of the system via the particle statistics (Bose or Fermi). At sufficiently high temperatures (Maxwell‐Boltzmann statistics), the T‐matrix introduced coincides with the usual T‐matrix of conventional two‐body scattering theory in free space. In low‐temperature Fermi systems, the T‐matrix differs from the reaction matrix of the Brueckner‐Goldstone theory because the “healing” property of the two‐body wave function does not obtain.
A new approach to the dynamics of relaxation and kinetics of thermalization in a scalar field theory is presented that incorporates the relevant time scales through the resummation of hard thermal loops. An alternative derivation of the kinetic equations for the "quasiparticle" distribution functions is obtained that allows a clear understanding of the different "coarse-graining" approximations usually involved in a kinetic description. This method leads to a systematic perturbative expansion to obtain the kinetic equations including hard thermal loop resummation and to an improvement including renormalization, off-shell effects, and contributions that change chemical equilibrium on short time scales. As a by-product of these methods we establish the equivalence between the relaxation time scale in the linearized equation of motion of the quasiparticles and the thermalization time scale of the quasiparticle distribution function in the "relaxation time approximation" including hard thermal loop effects. Hard thermal loop resummation dramatically modifies the scattering rate for long wavelength modes as compared to the usual (semi)classical estimate. Relaxation and kinetics are studied both in the unbroken and broken symmetry phases of the theory. The broken symmetry phase also provides the setting to obtain the contribution to the kinetic equations from processes that involve decay of a heavy scalar into light scalar particles in the medium.
A many-body theory is presented here for the Van Hove pair (or density) correlation function, in a quantum fluid obeying Boltzmann statistics, of which the Fourier transform $S(q,\ensuremath{\omega})$ determines the differential cross section for inelastic coherent scattering of slow neutrons. The theoretical formalism of two-particle thermodynamic Green's functions is based on Baym and Kadanoff's conserving TTT-matrix approximation, developed for strong short-range (intermolecular) potentials. The only other approximations made are: (i) the smallness of the activity or fugacity $z\ensuremath{\ll}1$, which implies Boltzmann statistics for the single-particle states; and (ii) the TTT matrices are replaced by elastic scattering amplitudes. Our results for $S(q,\ensuremath{\omega})$ developed to order z2{z}^{2}z2 are found to consist of Boltzmann averages of Lorentzian distributions in the energy transfer $\ensuremath{\hbar}\ensuremath{\omega}$ (except for a three-particle scattering term which is developed in the $\ensuremath{\delta}$-function limit of the Lorentzians). Our results are general enough to satisfy detailed balancing at all stages and are found to generalize Van Leeuwen and Yip's results for the classical case. The only unknown in our results is the intermolecular scattering amplitude or phase shifts. Our results seem, however, to break down in the hydrodynamic limit $\ensuremath{\omega}$ and $q\ensuremath{\rightarrow}0$ since each term in our expansion diverges in this limit.
Abstract The history of many‐body perturbation theory (MBPT) and its impact on Quantum Chemistry is reviewed, starting with Brueckner's conjecture of a linked‐cluster expansion and the time‐dependent derivation by Goldstone of such an expansion. A central part of this article is the time‐independent formulation of quantum chemistry in Fock space and its diagrammatic representation including the particle‐hole picture and the inversion of a commutator. The results of the time‐independent derivation of MBPT are compared with those of Goldstone. It is analyzed which ingredients of Goldstone's approach are decisive. The connected diagram theorem is derived both in a constructive way based on a Lie‐algebraic formulation and a nonconstructive way making use of the separation theorem. It is discussed why the Goldstone derivation starting from a unitary time‐evolution operator, ends up with a wave operator in intermediate normalization. The Møller–Plesset perturbation expansions of Bartlett and Pople are compared. Examples of complete summations of certain classes of diagrams are discussed, for example, that which leads to the Bethe‐Goldstone expansion. MBPT for energy differences is analyzed. The paper ends with recent developments and challenges, such as the generalization of normal ordering to arbitrary reference states, contracted Schrödinger k‐particle equations and Brillouin conditions, and finally the Nakatsuji theorem and the Nooijen conjecture. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
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.
A dense interacting Bose or Fermi gas is considered. Within the Kadanoff-Baym real-time Green-function technique a generalized kinetic equation is derived in the nonequilibrium ladder approximation. As a Markovian limit the Boltzmann-Uehling-Uhlenbeck equation is recovered. The relevance of retardation effects in the kinetic equation of dense systems is shown. It turns out that the stationary solution of the generalized kinetic equation reproduces the second virial coefficient of the equation of state, including the Beth-Uhlenbeck quantum corrections. We establish a generalized kinetic equation describing correlation effects in a consistent approach to nonequilibrium for Bose and Fermi systems. In this way we give a generalization of the Beth-Uhlenbeck formula that applies to nonequilibrium and to finite systems. The derivation makes use of the time dependent T-matrix equation, which establishes a generalization of the Bethe-Goldstone equation. The correlated parts of the density and energy, which can be interpreted as virial corrections, are thermodynamically consistent. In this way global energy conservation is fulfilled.
A quantum kinetic equation is obtained which includes scattering events beyond the binary collision process of the Boltzmann equation. The time rate of change of the Wigner function due to collisions is given as the difference of two scattering rates analogous to the collisional side of the Boltzmann equation. These total rates are given in terms of partial rates which are identified with probability amplitudes for scattering events in the medium. Thus, the method is inherently free of the divergence difficulties of theories which calculate the total rates in terms of scattering events in vacuum. Special attention is devoted to the question in what sense does some form of Fermi's golden rule apply to the calculation of the partial rates for a given event. It is shown that certain modifications to this picture are necessary, but the changes may be understood from a simple physical point of view.