Kinetic limits of dynamical systems

TL;DR

This paper develops a renormalisation technique using ergodic theory to derive macroscopic transport equations from microscopic dynamical systems like the Lorentz gas and kicked Hamiltonians.

Contribution

It introduces a novel renormalisation method based on homogeneous flow ergodic theory for deriving kinetic limits of simple dynamical systems.

Findings

Derived transport equations for Lorentz gas and kicked Hamiltonians.

Established connections between microscopic dynamics and macroscopic transport.

Provided a rigorous mathematical framework for kinetic limits.

Abstract

Since the pioneering work of Maxwell and Boltzmann in the 1860s and 1870s, a major challenge in mathematical physics has been the derivation of macroscopic evolution equations from the fundamental microscopic laws of classical or quantum mechanics. Macroscopic transport equations lie at the heart of many important physical theories, including fluid dynamics, condensed matter theory and nuclear physics. The rigorous derivation of macroscopic transport equations is thus not only a conceptual exercise that establishes their consistency with the fundamental laws of physics: the possibility of finding deviations and corrections to classical evolution equations makes this subject both intellectually exciting and relevant in practical applications. The plan of these lectures is to develop a renormalisation technique that will allow us to derive transport equations for the kinetic limits of two classes of simple dynamical systems, the Lorentz gas and kicked Hamiltonians (or linked twist maps). The technique uses the ergodic theory of flows on homogeneous spaces (homogeneous flows for short), and is based on joint work with Andreas Str\"ombergsson.