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Physics of Particles and Nuclei

Kinetics of dense matter: Correlations and memory

V. G. Morozov

Published Nov 22, 2008• 0 citations
Publisher

Abstract

A link between correlation contributions to kinetic equations for dense quantum systems and the energy conservation is considered. In order that the energy be conserved by an approximate collision integral, the one-particle density matrix and the mean interaction energy are treated as independent state parameters. It is shown how the density operator method can be used to derive non-Markovian kinetic equations including the correlation effect associated with the energy conservation. A quantum generalization of the Enskog theory is discussed.

DOI
10.1134/s1063779608070058

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A link between correlation contributions to kinetic equations for dense quantum systems and the energy conservation is considered. In order that the energy be conserved by an approximate collision integral, the one-particle density matrix and the mean interaction energy are treated as independent state parameters. It is shown how the density operator method can be used to derive non-Markovian kinetic equations including the correlation effect associated with the energy conservation. A quantum generalization of the Enskog theory is discussed.
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.
We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.
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 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).
In this thesis, we discussed the derivation of quantum kinetic equations appropriate for applications in nuclear physics, both in a non-relativistic and in a relativistic context. In each case we obtained a kinetic equation together with an equation for the effective interaction. The latter serves as the dynamical input of the former. In the kinetic equation both mean-field and collision effects are described in a self-consistent way because both are expressed in terms of the effective interaction. This self-consistency enforces us to solve a statistical problem (kinetic equation) and a dynamical problem (effective interaction) simultaneously. The interaction, that we obtained as a result of our considerations, is a generalized version of the usual Brueckner G-matrix in the sense that single-particle energies are corrected for collision effects by the so called rearrangement-term. 0nly in this way kinetic theory and interaction are consistent with one another and therefore constitute a conserving approximation.... Zie epilogue
We derive a non-Markovian generalization to the quantum kinetic theory described by Walser et al. [Phys. Rev. A 59, 3878 (1999)] in the absence of a condensed fraction for temperatures above the Bose-Einstein condensation temperature i.e., T>Tc.T>{T}_{c}.T>Tc​. Within this framework, quasiparticle damping arises naturally due the finite duration of a binary collision and it leads to a systematic Markov approximation from the non-Markovian Born theory. Such a self-consistent theory conserves the total energy to second order in the interaction strength. By introducing an improved damping function, we demonstrate global energy conservation at the order of the perturbation theory. Finally, we apply this kinetic theory to a simple model of an inhomogeneous Bose gas that is confined in a spherical box. By studying numerically the real-time quantum evolution towards equilibrium, we obtain damping rates and frequencies of the collective modes and illustrate the emergence of differing time scales for correlation and relaxation.