Kinetic Limits of Many-Body Classical Systems

Projektbeschreibung

Die Lücken in der kinetischen Theorie schließen

Wie physikalische Gesetze sich über Skalen hinweg verändern, ist eine grundlegende Herausforderung in der mathematischen Physik. Daher werden im ERC-finanzierten Projekt KiLiM große Systeme interagierender Teilchen, insbesondere in Gasen und Plasmen, erforscht. Diese Systeme werden mit den Boltzmann- und Landau-Gleichungen beschrieben, die Schlüsselkomponenten der kinetischen Theorie sind. Bisher konnte jedoch noch keine solide Grundlage für diese Gleichungen geschaffen werden, insbesondere für realistische interatomare Interaktionen. Das wird das KiLiM-Team ändern, indem die Gültigkeit der kinetischen Theorie für gewöhnliche Interaktionsmodelle wie das Lennard-Jones-Potential und das abgeschirmte Coulomb-Potential bewiesen wird. Damit geht das Team über idealisierte Modelle harter Kugeln hinaus. Mit der Arbeit könnte das Wissen zu statistischer Mechanik im Nichtgleichgewicht und vom Übergang zum Gleichgewicht in komplexen Systemen vertieft werden.

Ziel

This project studies the transition between dynamical laws governing the physical world at different scales. Our focus will be on large systems of interacting particles with random initial data, underlying the kinetic theory of gases and dilute plasmas. Central to this theory are the Boltzmann equation, and its appropriate modification for charged particles given by Landau. Their description of approach to equilibrium and irreversible behaviour is a legendary success in the physics of time-dependent phenomena. Nevertheless, the rigorous foundation of such equations remains a largely immature aspect of the theory. This is a major problem in mathematical physics and non-equilibrium statistical mechanics. The effective equations of kinetic theory are an approximation of particle systems ruled by the time-reversible laws of classical mechanics. But their validity should become exact in a suitable limit of large system size.

In the last decade, there has been substantial progress in the derivation of kinetic equations from first principles. Such work is restricted to rarefied regimes. Results are available for models of interacting monatomic gases of identical particles. Besides the macroscopic equations leading the average behaviour, results have been obtained for fluctuations, large deviations, and for the random evolution of tracer particles. Equilibrium fluctuations are in itself of great interest, including results on long time scales which justify physically relevant applications.

Most of the results hold only for an overidealized model of hard-sphere interactions. None of them is, with the present techniques, extendable to realistic interatomic potentials. The goal is to bridge this gap by proving the validity of kinetic theory for some of the most common interaction models in physics: such as the Boltzmann equation for Lennard-Jones type forces, the Vlasov-Boltzmann equation for mixtures, and the Landau equation for screened Coulomb potentials.