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arXiv:nucl-th/0210046 · Journal of Physics Condensed Matter

Non-Markovian collision integral in Fermi systems

V A Plujko, S N Ezhov, O M Gorbachenko, M. Kavatsyuk

Published Oct 3, 2002• 7 citations
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Abstract

The non-Markovian collision integral is obtained on the base of the Kadanoff–Baym equations for Green functions in a form with allowance for small retardation effects. The collisional relaxation times and damping width of the giant isovector dipole resonances in nuclear matter are calculated. For an infinite Fermi liquid the dependence of the relaxation times on the collective vibration frequency and the temperature corresponds to the Landau prescription.

DOI
10.1088/0953-8984/14/41/304

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The non-Markovian collision integral is obtained on the base of the Kadanoff–Baym equations for Green functions in a form with allowance for small retardation effects. The collisional relaxation times and damping width of the giant isovector dipole resonances in nuclear matter are calculated. For an infinite Fermi liquid the dependence of the relaxation times on the collective vibration frequency and the temperature corresponds to the Landau prescription.
Linearizing the appropriate kinetic equation we derive general response functions including self-consistent mean fields or density functionals and collisional dissipative contributions. The latter ones are considered in relaxation time approximation conserving successively different balance equations. The effect of collisions is represented by correlation functions which are possible to calculate with the help of the finite temperature Lindhard random-phase approximation expression. The presented results are applicable to the finite temperature response of interacting quantum systems if the quasiparticle or mean-field energy is parametrized within Skyrme-type functionals including density, current, and energy dependencies which can be considered alternatively as density functionals. In this way we allow to share correlations between density functional and collisional dissipative contributions appropriate for special treatment. We present results for collective modes such as the plasmon in plasma systems and the giant resonance in nuclei. The collisions lead in general to an enhanced damping of collective modes. If the collision frequency is close to the frequency of the collective mode, resonance occurs and the collective mode is enhanced showing a collisional narrowing.
In a previous publication [H. S. K\"ohler, Phys. Rev. C 51, 3232 (1995)] equilibration rates were studied for an extended system of nonequilibrium nuclear matter. The two-time Green's functions in the Kadanoff-Baym (KB) formalism were solved numerically. These quantum-mechanical calculations were compared with the Markovian (semi)classical Uehling-Uhlenbeck collision term often used in the analysis of collisions between heavy ions. The equilibration was found to be substantially slowed down by quantum effects. The theoretical as well as experimental study of nonequilibrium systems is of course of great interest in many areas of physics, e.g., within the realm of quantum transport in solid state devices. On the theoretical side various methods and approximations are the subject of intensive study. By the KB ansatz, for example, the equations can be simplified to a one-time kinetic equation while preserving the non-Markovian character. The generalized KB (GKB) ansatz of Lipavsky, Spicka, and Velicky [Phys. Rev. B 34, 6933 (1986)] is often preferred. In this paper approximations based upon the GKB ansatz are discussed in relation to the exact KB formalism. Numerical comparisons are made for both the strongly interacting nuclear medium as well as an electron plasma. A non-Markovian collision integral, including memory effects in a quasiparticle approximation, is often discussed in the literature. It is found that this approximation appreciably overestimates the collision rates even more than the classical (Boltzmann) approximation. The inclusion of a width in the spectral function, i.e., going beyond the quasiparticle approximation, gives considerable improvement. The competition between the memory and correlation effects is found to lead to an anomalous saturation of the relaxtion rate. \textcopyright{} 1996 The American Physical Society.
On the basis of the Kadanoff-Baym (KB) version of the time-dependent Green's function method, an Ansatz for the approximation of a spectral function is offered. The Ansatz possesses all the advantages of quasiparticle and extended quasiparticle approximations and satisfies the KB equation for a spectral function in the case of slightly nonequilibrium system when disturbances in space and time are taken into consideration in the gradient approximation. This feature opens opportunities for the microscopic derivation of the Landau kinetic equation for the quasiparticle distribution function of the normal Fermi liquid and provides the widening of these equations' temperature range of validity.
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).