C-star algebras and non-equilibrium statistical mechanics

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This PhD research aims at developing C-star algebra and functional-analytic methods for the study of quantum and classical statistical mechanics out of equilibrium and hydrodynamics. The goal is to obtain mathematically rigorous results in the emergent fields of the non-equilibrium statistical mechanics of closed systems, especially in view of recent activity in the context of hydrodynamics. The research will look for understanding the Boltzmann-Gibbs projection principle, non-equilibrium steady states, and the hydrodynamic equations, at the various scales of hydrodynamics (Euler, diffusive), in many-body systems of various types including quantum and classical lattices and gases of particles. Deriving hydrodynamics for Hamiltonian, reversible systems is one of the most important open problems of mathematical physics, and this PhD research aims at making progress in this direction.