Derivation of Kinetic Equations from Particle Models

George Russell Stone

Research output: ThesisDoctoral Thesis

Abstract

The derivation of the Boltzmann equation from a particle model of a gas is currently a major area of research in mathematical physics. The standard approach to this problem is to study the BBGKY hierarchy, a system of equations that describe the distribution of the particles. A new method has recently been developed to tackle this problem by studying the probability of observing a specific history of events.
We further develop this method to derive the linear Boltzmann equation in the Boltzmann-Grad scaling from two similar Rayleigh gas hard-sphere particle models. In both models the initial distribution of the particles is random and their evolution is deterministic. Validity is shown up to arbitrarily large times and with only moderate moment assumptions on the non-equilibrium initial data.
The first model considers a Rayleigh gas whereby one tagged particle collides with a large number of background particles, which have no self interaction. The initial distribution of the background particles is assumed to be spatially homogeneous and at a collision between a background particle and the tagged particle only the tagged particle changes velocity.
In the second model we make two changes: we allow the background particles to have a spatially non-homogeneous initial data and we assume that at collision both the tagged particle and background particle change velocity.
The proof for each model follows the same general method, where we consider two evolution equations, the idealised and the empirical, on all possible collision histories. It is shown by a semigroup approach that there exists a solution to the idealised equation and that this solution is related to the solution of the linear Boltzmann equation. It is then shown that under the particle dynamics the distribution of collision histories solves the empirical equation. Convergence is shown by comparing the idealised and empirical equations.
Original languageEnglish
QualificationPh.D.
Awarding Institution
  • University of Bath
Supervisors/Advisors
  • Matthies, Karsten, Supervisor
Date of Award23 Oct 2017
StatePublished - Sep 2017

Fingerprint

Tagged particle
Collision
Linear Boltzmann equation
Rayleigh
Model
BBGKY hierarchy
Hard spheres
Kinetic equation
Boltzmann equation
Ludwig Boltzmann
Non-equilibrium
System of equations
Evolution equation
Semigroup
Physics
Scaling
Moment
Interaction

Cite this

Derivation of Kinetic Equations from Particle Models. / Stone, George Russell.

2017. 160 p.

Research output: ThesisDoctoral Thesis

Stone, GR 2017, 'Derivation of Kinetic Equations from Particle Models', Ph.D., University of Bath.
Stone, George Russell. / Derivation of Kinetic Equations from Particle Models. 2017. 160 p.
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AB - The derivation of the Boltzmann equation from a particle model of a gas is currently a major area of research in mathematical physics. The standard approach to this problem is to study the BBGKY hierarchy, a system of equations that describe the distribution of the particles. A new method has recently been developed to tackle this problem by studying the probability of observing a specific history of events. We further develop this method to derive the linear Boltzmann equation in the Boltzmann-Grad scaling from two similar Rayleigh gas hard-sphere particle models. In both models the initial distribution of the particles is random and their evolution is deterministic. Validity is shown up to arbitrarily large times and with only moderate moment assumptions on the non-equilibrium initial data. The first model considers a Rayleigh gas whereby one tagged particle collides with a large number of background particles, which have no self interaction. The initial distribution of the background particles is assumed to be spatially homogeneous and at a collision between a background particle and the tagged particle only the tagged particle changes velocity.In the second model we make two changes: we allow the background particles to have a spatially non-homogeneous initial data and we assume that at collision both the tagged particle and background particle change velocity. The proof for each model follows the same general method, where we consider two evolution equations, the idealised and the empirical, on all possible collision histories. It is shown by a semigroup approach that there exists a solution to the idealised equation and that this solution is related to the solution of the linear Boltzmann equation. It is then shown that under the particle dynamics the distribution of collision histories solves the empirical equation. Convergence is shown by comparing the idealised and empirical equations.

M3 - Doctoral Thesis

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