Dynamics and Kinetics

A deterministic system, such as a double pendulum, can exhibit chaotic or unpredictable behavior. The principal investigator will study the finite time evolution of chaotic deterministic systems. Traditionally it was assumed that the subtle dynamics of such systems can be analyzed only in the limit when time tends to infinity (that is to say when chaotic fluctuations settle down).

This project aims to extend previous studies of finite time evolution of such systems to a much larger class of systems, which will include some models with practical applications. Another part of this research deals with systems where the motion of a point particle is substituted by a particle of nonzero size. The study of such systems has led to some new findings in quantum chaos and new interpretations of experimental data.

The research will extend this theory to dimensions greater than two, which would allow application to interpretations of a much larger class of numerical and experimental data on physical systems. A third topic of the research deals with understanding whether and how probabilistic time-irreversible laws of statistical mechanics could be rigorously derived from deterministic time-reversible laws of classical mechanics. The project includes the training of graduate students.

A rigorous theory of finite time dynamics makes possible the analysis of transport in the phase space of chaotic and random dynamical systems. Namely, it allows the prediction of which events will likely occur first, e. g., which subset of the phase space the orbits will more likely visit. Thus far, this theory has been developed for Bernoulli shifts with equal probabilities of states.

Similar results will be proved for Bernoulli shifts with non-equal probabilities and for Markov shifts. The principles for constructing chaotic (hyperbolic) particle-like flows will be reconsidered, and new ones will be created. Physical particles where a finite size hard sphere moves instead of a point particle will be analyzed in dimensions higher than two.

A simple model with two discs localized in "cells" but allowed to collide with each other and with two pistons will be studied. In the model, each cell is connected to (bounded by) a thermostat, the thermostats have different temperatures, and the pistons do not allow the discs to stop for a long time or move very slowly for a long time. A goal is to prove that on average (in time) the disc in a cell with a higher temperature thermostat will have a higher energy. This will be a main step to a proof of the Fourier law.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.