From the Kac Model to Non-Equilibrium Statistical Mechanics

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The discipline of thermodynamics arose from the engineering need to design better mechanisms of the transformation of heat into mechanical work in the newly invented steam engine. It soon became clear that it was necessary to understand statistical mechanics, i.e., how the mechanical motion and interactions of the extremely large number of atoms and molecules determine macroscopic thermodynamic parameters of the gas, such as the pressure or temperature. Since then, the ideas and methods initially set forth to understand the heat to work transformation have been extended to fluid motion, electrical conduction, chemical reactions and many more settings. In the last 30 years, they have been applied to a diverse variety of system formed by a large number of interacting elementary constituents like neural network, animal populations, large human networks and economic systems. These successes notwithstanding, the deep conceptual and mathematical foundations of statistical mechanics are incomplete, especially so for the non-equilibrium case. This project is aimed to improve understanding of these foundations.Mark Kac introduced a very simplified model to study convergence to equilibrium and emergence of effective macroscopic equations (the Boltzmann equation) in a dilute gas of colliding hard spheres. Thanks to the simplicity of this model one can compute exactly the spectral gap of the evolution and prove uniform convergence of the one-particle distribution to the solution of the Boltzmann-Kac equation. This comes at the price of losing all spatial structure and thus the ability to investigate crucial phenomena such as local equilibrium and approach to non-equilibrium steady states. This project will extend the Kac model by reintroducing a space structure and connecting it with heat and particle reservoirs in such a way as to obtain a model that is closer to a real gas but still is rigorously treatable. By improving the mathematical techniques developed to study complex dynamics of large systems, like scaling limits, hyperbolic dynamics, and functional inequalities, the Principal Investigator will extend the results known for the Kac system to this richer and more realistic model. Although most of the work will be of analytic nature, numerical simulation will be used as a guide toward more rigorous understanding. The final goal is to have a model where one can rigorously describe the transition from microscopic collision dynamics to a mesoscopic Boltzmann type equation and finally to a macroscopic thermodynamic/hydrodynamic fluid evolution.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.