Analysis and Data-Driven Computation for Nonequilibrium Thermodynamic Models

The role of statistical mechanics is to study large assemblies of microscopic entities (such as atoms and molecules) and to bridge the gap between microscopic entities and macroscopic properties like temperature profile and thermal conductivity. Mathematical justifications of numerous problems in nonequilibrium statistical mechanics, such as, for example, Fourier’s law statement that the heat flux is proportional to the temperature gradient, remain highly challenging.

This project aims to develop novel analytical tools and data-driven computational methods to study a series of nonequilibrium thermodynamic models arising from statistical physics and wave turbulence. These nonequilibrium models are relevant not only for the statistical physics, but also for countless applied scientific problems that are intrinsically irreversible and multiscale, such as chemical reactions and neural dynamics.

The project also provides research training opportunities for graduate students and advanced undergraduate students.

In this project, the principal investigator (PI) will use a combination of analytical and computational approaches to investigate how thermodynamic properties are derived from a class of microscopic energy transfer models, including classical billiards-like systems and nonlinear oscillator chain models coming from nonequilibrium statistical physics and wave turbulence. The general approach is to use minimum computational work to bypass some difficulties and to derive mathematically tractable stochastic models.

Developing thermodynamic laws from those stochastic models are usually much easier. The development and application of data-driven computational methods is a constitutive element of the proposed research. The PI has developed a series of novel computational methods that combines traditional Monte Carlo simulation with tools like numerical partial differential equation solver, coupling method, and artificial neural network.

They overcome several disadvantages of traditional, discretization-based algorithms, especially for high-dimensional problems. When studying many high-dimensional problems in this project, we need computational results of high-dimensional invariant probability measure and its ergodicity to bypass difficulties that are beyond the reach of rigorous methods.

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.