arXiv:1207.0590 · Kinetic and Related Models

A kinetic equation for spin polarized Fermi systems

Leif Arkeryd, ,Mathematical Sciences, S-41296 Gothenburg

Published Dec 10, 2013• 3 citations

Abstract

This paper considers a kinetic Boltzmann equation, having a general type of collision kernel and modelling spin-dependent Fermi gases at low temperatures. The distribution functions have values in the space of positive hermitean $2\times2$ complex matrices. Global existence of weak solutions is proved in $L^1\cap L^{\infty}$ for the initial value problem of this Boltzmann equation in a periodic box.

DOI: 10.3934/krm.2014.7.1

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arXiv:1207.0590 · Kinetic and Related Models

A kinetic equation for spin polarized Fermi systems

Leif Arkeryd, ,Mathematical Sciences, S-41296 Gothenburg

Published Dec 10, 2013• 3 citations

arXiv HTML PDF

Abstract

2\times2

complex matrices. Global existence of weak solutions is proved in

L^1\cap L^{\infty}

for the initial value problem of this Boltzmann equation in a periodic box.

DOI: 10.3934/krm.2014.7.1

Related papers

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T. L. Andreeva, P. L. Rubin

We derive a kinetic equation for a polarized paramagnetic gas that is exact in the Boltzmann (binary) approximation. The equation is written in a compact form and applies to both Fermi and Bose gases in a wide temperature range as long as the Boltzmann approximation remains applicable. The derived equation is used to analyze the conditions for the propagation of spin waves in polarized Fermi and Bose gases. We deduce a universal criterion for the propagation of weakly damped spin waves in a wide temperature range. The criterion is reduced to the condition that the real parts of the particle zero-angle scattering amplitudes (or T matrices) be much larger than their imaginary parts. We derive dispersion equations for spin waves at high and low gas temperatures and show that spin waves can propagate in both these limiting cases.We derive a kinetic equation for a polarized paramagnetic gas that is exact in the Boltzmann (binary) approximation. The equation is written in a compact form and applies to both Fermi and Bose gases in a wide temperature range as long as the Boltzmann approximation remains applicable. The derived equation is used to analyze the conditions for the propagation of spin waves in polarized Fermi and Bose gases. We deduce a universal criterion for the propagation of weakly damped spin waves in a wide temperature range. The criterion is reduced to the condition that the real parts of the particle zero-angle scattering amplitudes (or T matrices) be much larger than their imaginary parts. We derive dispersion equations for spin waves at high and low gas temperatures and show that spin waves can propagate in both these limiting cases.

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2\times2

complex matrices. Global existence of weak solutions is proved in

L^1\cap L^{\infty}

for the initial value problem of this Boltzmann equation in a periodic box.

{\mathit{T}}^{2}

. Spin waves would thus continue to be damped even to T=0 K.

T=0.

The analogy between temperature dependences of spin wave attenuation and ultrasound absorption in degenerate Fermi liquid at arbitrary temperature is presented. We also discuss spin-polarized Fermi-liquid in the general context of the Fermi-liquid theory and compare it with ``Fermi liquid'' with spontaneous magnetization.

This paper is devoted to the dynamics of a weakly interacting Fermi gas at the kinetic time regime

t\sim λ^{-2}

where

λ\ll 1

is the strength of the interaction potential. We prove that if the initial state is close to equilibrium, then the two-point time correlation function of the many-body quantum dynamics can be computed effectively. In fact, we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of the quantum Boltzmann equation from many-body quantum mechanics.