Kinetic Limits of Many-Body Classical Systems

Introduction

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.

Theoretical Framework

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.

Kinetic Theory and Classical Mechanics

The effective equations of kinetic theory are an approximation of particle systems ruled by the time-reversible laws of classical mechanics. However, their validity should become exact in a suitable limit of large system size.

Recent Progress

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.

Macroscopic and Fluctuation Results

Besides the macroscopic equations leading the average behaviour, results have been obtained for:

1. Fluctuations
2. Large deviations
3. The random evolution of tracer particles

Equilibrium fluctuations are of great interest, including results on long time scales which justify physically relevant applications.

Limitations of Current Models

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.

Project Goal

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
• The Landau equation for screened Coulomb potentials.