From Deterministic Dynamics to Thermodynamic Laws

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Statistical mechanics attempts to explain the thermodynamic behavior of large systems and to bridge the gap between the microscopic and macroscopic worlds. Equilibrium statistical mechanics is relatively well-developed, but many problems in non-equilibrium (i.e., irreversible) statistical mechanics have not been well answered yet. In particular, the derivation of macroscopic transport laws from microscopic deterministic dynamics is a century-old challenge (for example, Fourier's law of heat conduction from microscopic heat conduction). In this project, the investigator studies the derivation of thermodynamic properties of a gas that is not at thermal equilibrium, starting from a microscopic particle description of the gas. The microscopic heat conduction problem under study is representative of numerous physical systems that are intrinsically high-dimensional, multiscale, noisy, and non-equilibrium. Mathematical advances on this problem will be of value for other areas of science where non-equilibrium dynamics is an important feature, including materials science, neuroscience, molecular dynamics, fluid dynamics, and bio-chemical networks. The project includes research experience opportunities for undergraduate students. In this project, the investigator studies how thermodynamic properties are derived from non-equilibrium microscopic heat conduction models. The aim is to develop a series of results that connect microscopic deterministic particle systems, mesoscopic stochastic differential equations, and macroscopic thermodynamic laws. Starting from a deterministic kinetic particle system that models the heat conduction among gas particles in a tube with different temperatures at two ends, the project looks numerically for a mathematically tractable stochastic energy exchange process that preserves the main dynamical features of the deterministic problem, examines its stochastic stability and the mesoscopic limit problem when the scale of local dynamics increases, and derives macroscopic thermodynamic properties related to thermal conductivity, long-range correlation, the local thermodynamic equilibrium, and the fluctuation theorem. 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.