Kinetic theory for the low-density Lorentz gas / Jens Marklof, Andreas Strömbergsson.

Format:

Book

Author/Creator:

Marklof, Jens, author.

Strömbergsson, Andreas, author.

Series:

Memoirs of the American Mathematical Society ; v. 294, no. 1464.

Memoirs of the American Mathematical Society , 0065-9266 ; v. 294, no.1464

Language:

English

Subjects (All):

Dynamics.

Mathematical physics.

Probabilities.

probability.

Physical Description:

v, 136 pages ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2024]

Summary:

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including semiconductor physics and radiative transfer. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of the present study is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strömbergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers' centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

Contents:

Chapter 1. Introduction ; 1.1. Outline of assumptions on the scatterer configuration ; 1.2. The Lorentz process for hard sphere scatterers ; 1.3. The Lorentz process for potentials ; 1.4. The linear Boltzmann equation and generalisations ; 1.5. Outline of the paper ; Acknowledgement

Chapter 2. Point sets, point processes and key assumptions ; 2.1. uniform convergence of families of probability measures ; 2.2. Point processes and marked point processes ; 2.3. The list of assumptions ; 2.4. First consequences of the assumptions ; 2.5. A limiting process for macroscopic initial conditions ; 2.6. Properties of the point process

Chapter 3. First collisions ; 3.1. The transition kernel ; 3.2. Limit theorem for the first collision ; 3.3. Relations for the transition kernels ; 3.4. Scattering maps ; 3.5. Collision kernels ; 3.6. Relations for the collision kernels ; 3.7. Post-collision velocity ; 3.8. Bounding the probability of grazing a scatterer or hitting E

Chapter 4. Convergence to a random flight process ; 4.1. Joint distributions of path segments ; 4.2. Auxiliary results ; 4.3. Proof of Theorem 4.1. ; 4.4. Macroscopic initial conditions ; 4.5. Random flight processes ; 4.6. Semigroups and kinetic transport equations

Chapter 5. Examples, extensions, and open questions ; 5.1. The Poisson case ; 5.2. Periodic point sets ; 5.3. Quasicrystals of cut-and-project type ; 5.4. Scattering potentials satisfying the conditions in Section 3.4 ; 5.5. More general scattering potentials ; 5.6. Open questions

Bibliography

Index of notation.

Notes:

"February 2024, volume 294, number 1464 (first of 5 numbers)."

Includes bibliographical references (pages 129-131) and index.

ISBN:

1470468697

9781470468699

OCLC:

1426995435