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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Kinetic equations for a nonideal quantum system

Th. Bornath, D. Kremp, W. D. Kraeft, M. Schlanges

Published Oct 1, 1996• 57 citations
Publisher

Abstract

In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.54.3274

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In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.
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.
The following issues are discussed inspired by the recent paper of Kadanoff (arXiv: 1403:6162): (a) Construction of a generalized one-particle Wigner distribution (GWD) function (analog of the classical distribution function) from which the quantum kinetic equation due to Kadanoff and Baym (KB) is derived, often called the Quantum Boltzmann Equation (QBE); (b) The equation obeyed by this has a collision contribution in the form of a two-particle Green function. This term is manipulated to have Kinetic Entropy in parallel to its counterpart in the classical Boltzmann kinetic equation for the classical distribution function. This proved to be problematic in that unlike in the classical Boltzmann kinetic equation, the contribution from the kinetic entropy term was non-positive; (3) Kadanoff surmised that this situation could perhaps be related to quantum entanglement that may not have been included in his theory. It is shown that GWD is not positive everywhere (indicating dynamical quantumness) just like the commonly recognized property of the Wigner function (negative property indicating quantumness of the state). The issue of non-positive feature appearing in approximate evaluation of patently positive entities in many particle systems is here pointed to an early discussion of this issue (Phys. Rev. A10, 1852 (1974)) in terms of a theorem on truncation of cumulant expansion of a probability distribution function due to Marcinkeiwicz. The last issue of presence or absence of entanglement in an approximate evaluation of a many particle correlation poses a new problem; it is considered here in terms of fermionic entanglement theory in the light of density matrix and Green function theory of many-fermion systems. The clue comes from the fact that the Hartree-Fock approximation exhbits no entantanglement in two-particle fermion density matrix and hence also in two-particle Green function.
A single-time quantum transport equation, which includes effects beyond the quasiparticle approximation, is derived for Fermi systems in the framework of nonequilibrium real-time Green's functions theory. Ternary correlations are incorporated in the kinetic description via a cluster expansion for the self-energies (e.g., the transport vertex and the width) truncated at the level of three-body scattering amplitudes. A finite temperature/density formulation of the three-body problem is given. Corresponding three-body equations reduce to the well-known Faddeev equations in the vacuum limit. In equilibrium the equation of state contains virial corrections proportional to the third quantum virial coefficient.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.
A recently developed method [Semkat et al., Phys. Rev. E 59, 1557 (1999); Kremp et al., in Progress in Nonequilibrium Green’s Functions (World Scientific, Singapore, 2000), p. 34] for incorporating initial binary correlations into the Kadanoff–Baym equations (KBE) is used to derive a generalized T-matrix approximation for the self-energies. It is shown that the T-matrix obtains additional contributions arising from initial correlations. Using these results and taking the time-diagonal limit of the KBE, a generalized quantum kinetic equation in binary collision approximation is derived. This equation is a far-reaching generalization of Boltzmann-type kinetic equations: It self-consistently includes memory effects (retardation, off-shell T-matrices) as well as many-particle effects (damping, in-medium T-matrices) and spin-statistics effects (Pauli-blocking).
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.