Kinetic Theory for the Low-Density Lorentz Gas

Jens Marklof, Andreas Strombergsson (Autoren)

Buch | Softcover

136 Seiten

2024
American Mathematical Society (Verlag)
978-1-4704-6869-9 (ISBN)

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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.

Jens Marklof, University of Bristol, United Kingdom. Andreas Strömbergsson, Uppsala University, Sweden.

1. Introduction
2. Point sets, point processes and key assumptions
3. First collisions
4. Convergence to a random flight process
5. Examples, extensions, and open questions
Index of notation

Erscheinungsdatum	05.04.2024
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	111 g
Themenwelt	Mathematik / Informatik ► Mathematik
Themenwelt	Naturwissenschaften ► Physik / Astronomie
ISBN-10	1-4704-6869-7 / 1470468697
ISBN-13	978-1-4704-6869-9 / 9781470468699
Zustand	Neuware