Kinetic Limits of Many-Body Classical Systems

Descripción del proyecto

Colmar las lagunas de la teoría cinética

Comprender cómo las leyes físicas se desplazan a través de las escalas es un reto fundamental de la física matemática. El equipo del proyecto KiLiM, financiado por el CEI, aborda esta cuestión estudiando grandes sistemas de partículas en interacción, especialmente en gases y plasmas. Estos sistemas se describen mediante las ecuaciones de Boltzmann y Landau, que son componentes clave de la teoría cinética. Sin embargo, establecer una base rigurosa para estas ecuaciones sigue siendo un problema sin resolver, en particular para las interacciones interatómicas realistas. En KiLiM se intenta colmar esta laguna demostrando la validez de la teoría cinética para modelos de interacción comunes, como las fuerzas de Lennard-Jones y los potenciales de Coulomb apantallados, yendo más allá de los modelos idealizados de esfera dura. Con este trabajo se podría mejorar nuestra comprensión de la mecánica estadística de no equilibrio y la transición al equilibrio en sistemas complejos.

Objetivo

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.