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Physica Scripta

Transport coefficients for a slow Fermi-particle motion

J. P. Błocki, A. G. Magner

Published May 1, 2013• 4 citations
Publisher

Abstract

The collective dynamics of a quantum gas of independent particles was studied for a slow collective motion within the nuclear model based on the time-dependent mean-field approach. Leading smooth semiclassical corrections to the friction and inertia coefficients for small multipole distortions of the infinitely deep square-well potential were obtained within the periodic orbit theory. The smooth inertia is mostly larger than its irrotational flow value.

DOI
10.1088/0031-8949/2013/t154/014006

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The collective dynamics of a quantum gas of independent particles was studied for a slow collective motion within the nuclear model based on the time-dependent mean-field approach. Leading smooth semiclassical corrections to the friction and inertia coefficients for small multipole distortions of the infinitely deep square-well potential were obtained within the periodic orbit theory. The smooth inertia is mostly larger than its irrotational flow value.
We use Feynman's influence-functional theory to study the quantum dynamics of a heavy particle moving in a free Fermi gas with arbitrary average velocity. A semiclassical expansion yields a nonlinear Langevin equation with the exact friction coefficient as derived in an earlier publication. The fluctuations around a steady state far from equilibrium are due to a nonclassical state-dependent noise term and can be described by a diffusion constant. In the limit of zero average velocity, the Einstein relation is fulfilled for arbitrary temperatures. For finite velocities the diffusion around the steady state is different in longitudinal and transverse directions and can be expressed in terms of the transport cross section and a ``diffusion'' cross section. In the case where the frictional force exhibits a maximum as a function of velocity and thus an unstable branch for v>vc{\mathit{v}}_{\mathit{c}}vc​, the longitudinal diffusion constant diverges on approaching vc{\mathit{v}}_{\mathit{c}}vc​ from below. Numerical results for the noise spectrum and the temperature and velocity dependence of the diffusion constants are presented for simple repulsive interaction potentials in one and three dimensions.
In this article, I provide a simple review on conventional methods and models on the transport phenomenon of nuclear many-body systems. By exploiting the basic idea of time-dependent projection operator, I recommend a novel method to derive the transport equation for collective motion which is embedded on the microscopic dynamics of timedependent single-particle motion. It is emphasized that the microscopic dynamics of single-particle motion should play an important role for understanding the dynamics of nuclear reaction and the synthesis mechanisms of new superheavy elements. The numerical results of energy dissipation and entropy production indicate that the collective motion passes through three stages, such as dephasing/decoherence, statistical relaxation and stationary state. The statistical relaxation is a complex anomalous diffusion process in general. With the aid of above analysis and results, a possible way to define the collective and single-particle variables for the realistic nuclear many-body systems is proposed.
At finite temperature, the hydrodynamic collective modes of superfluid trapped Fermi gases are coupled to the motion of the normal component, which in the BCS limit behaves like a collisionless normal Fermi gas. The coupling between the superfluid and the normal components is treated in the framework of a semiclassical transport theory for the quasiparticle-distribution function, combined with a hydrodynamic equation for the collective motion of the superfluid component. We develop a numerical test-particle method for solving these equations in the linear-response regime. As a first application we study the temperature dependence of the collective quadrupole mode of a Fermi gas in a spherical trap. The coupling between the superfluid collective motion and the quasiparticles leads to a rather strong damping of the hydrodynamic mode already at very low temperatures. At higher temperatures the spectrum has a two-peak structure, the second peak corresponding to the quadrupole mode in the normal phase.
Abstract This chapter develops a quantal transport theory for non-linear nuclear collective motion, the coordinates of which are functionals of the nucleonic degrees of freedom, the latter being interpreted as ‘heat bath’. Introduced on the mean field level the collective variables must be quantized. This is accomplished through the Bohm-Pines method, which leads to an extended Hamiltonian for the entire system. On this basis, transport equations are deduced through a non-perturbative Nakajima-Zwanzig procedure. This scheme is carried out within a locally harmonic approximation, in which all collective forces are linearized locally and in which Gaussian forms are used for propagators of Wigner functions. The diffusion coefficients are determined by the quantal fluctuation dissipation theorem, which for unstable modes requires appropriate analytic continuations. This scheme breaks down below a critical value of temperature identical to that found in the Caldeira-Leggett model for real time propagation. Fundamental properties of these propagators are studied.
Abstract A general method of discussing quantum-mechanical problems involving collective motion is proposed, in which the emphasis is placed on consideration of sets of states rather than single states, and in which the additional collective co-ordinates are not redundant but used to describe the sets. The method is applied to a number of relatively simple examples: plasma oscillations of an electron gas; some problems of nuclear structure including the α-particle model and the collective model; and to two problems in meson field theory, concerning nucleon isobars in strong-coupling theory and concerning nucleon recoil. One of the main aims is to determine as well as possible the parameters of the collective motion; in particular, a formula is given for the moment of inertia of a rotating system.
Quantum tunneling in many-body systems is the subject of many experimental and theoretical studies in fields ranging from cold atoms to nuclear physics. However, theoretical description of quantum tunneling with strongly interacting particles, such as nucleons in atomic nuclei, remains a major challenge in quantum physics. An initial-value approach to tunneling accounting for the degrees of freedom of each interacting particle is highly desirable. Inspired by existing methods to describe instantons with periodic solutions in imaginary time, we investigate the possibility to use an initial value approach to describe tunneling at the mean-field level. Real-time and imaginary-time Hartree dynamics are compared to the exact solution in the case of two particles in a two-well potential. Whereas real-time evolutions exhibit a spurious self-trapping effect preventing tunneling in strongly interacting systems, the imaginary-time-dependent mean-field method predicts tunneling rates in excellent agreement with the exact solution. Being an initial-value method, it could be more suitable than approaches requiring periodic solutions to describe realistic systems such as heavy-ion fusion.