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On a Boltzmann equation for Haldane statistics

Leif Arkeryd, Anne Nouri

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

The paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong L1 \nsolutions are obtained for the Cauchy problem. The main results concern existence, uniqueness, and stability. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.

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The paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong L1 \nsolutions are obtained for the Cauchy problem. The main results concern existence, uniqueness, and stability. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.
We study a quantum Boltzmann-Condensation system that describes the evolution of the interaction between a well formed Bose-Einstein condensate and the quasi-particles cloud. The kinetic model is valid for a dilute regime at which the temperature of the gas is very low compared to the Bose-Einstein condensation critical temperature. In particular, our system couples the density of the condensate from a Gross-Pitaevskii type equation to the kinetic equation through the dispersion relation in the kinetic model and the corresponding transition probability rate from pre to post collision momentum states. We rigorously show the following three properties (1) the well-posedness of the Cauchy problem for the system in the case of a radially symmetric initial configuration, (2) find qualitative properties of the solution such as instantaneous creation of exponential tails and, (3) prove the uniform condensate stability related to the initial mass ratio between condensed particles and quasi-particles. The stability result from (3) leads to global in time existence of the initial value problem for the quantum Boltzmann-Condensation system.
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 2×22\times22×2 complex matrices. Global existence of weak solutions is proved in L1∩L∞L^1\cap L^{\infty}L1∩L∞ for the initial value problem of this Boltzmann equation in a periodic box.
The mathematical contributions by X.G. Lu (J. Statist. Phys. 98 (5/6) (2000) 1335–1394) and by M. Escobedo et al. (Electronic J. Differential Equations, Monograph 4 (2003)) presented in this Note constitute the first stage in the understanding of the superfluid dynamics, especially of the Bose–Einstein condensation, by means of kinetic models. The Boltzmann–Nordheim equation, which is physically relevant to describe dilute quantum Bose gases, sets important mathematical problems. Nevertheless, under an unphysical truncation of the collision cross-section at low energies, it has been proved that the spatially homogeneous Cauchy problem is well-posed. Furthermore, relaxation towards equilibrium holds in a weak sense, with the appearance of a singularity in infinite time if the initial mass is supercritical, which corresponds to the formation of a Bose–Einstein condensate.
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).
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this thesis a detailed comparison of Boltzmann and Kadanoff-Baym equations in the framework of relativistic quantum field theories in 3+1 space-time dimensions. In a first step, for simplicity we consider a real scalar Phi^4 quantum field theory and in a second step we generalize our results to a chirally invariant Yukawa-type quantum field theory including fermions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for Boltzmann equations.
In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.