Mathematical theory of non-equilibrium statistical mechanics

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This proposal concerns mathematical theory of  non-equilibrium statistical mechanics (classical and quantum) in dynamical systems setting. It is a continuation of a research program whose origin can be traced back to early 1990's and the discovery of the so called Fluctuation Relations (FR) and Fluctuation Theorems (FT). The FR is a universal property of the statistics of entropy production linked to a time-reversal and the probability of violation of the Second Law of Thermodynamics, while FT refers to the corresponding mathematical result in theory of Large Deviations.  The discovery of FR goes back to numerical experiments of Evans-Cohen-Morris (1993) and physics-theoretical works of Evans and Searles (1994). The first mathematical formulation and a mathematical proof of FT was given by Gallavotti and Cohen in 1995 in context of Anosov diffeomorphisms of compact Riemannian manifolds. The extensions of FR to quantum mechanical setting goes back to  early 2000's and works of Kurchan, Tasaki, and Tasaki-Matsui, involving some fundamentally new ideas compared to the previously studied  classical setting. These discoveries generated an enormous body physics-theoretical, numerical, and experimental works which have fundamentally improved our understanding of non-equilibrium physics with applications extending to chemistry and biology. In spite of this spectacular progress, there are comparatively very few mathematically rigorous results on the derivations of FR and FT. The mathematical progress has been hampered by the lack of clear mathematical structures behind various formulations of the FR in the physics literature and a lack of clarity what constitutes the proof of FT in a given model. This proposal continues with my long-term research program dealing with uncovering mathematical structures behind the FR and the FT, refining our general mathematical understanding of non-equilibrium statistical mechanics, and then using these results as a starting point in mathematical analysis of concrete physically relevant models in non-equilibrium statistical mechanics. In the  last six years  this research program has led to twenty three scientific publications,  and has generated an interconnected web of research   sub-directions and projects that are impossible to describe all in the allotted space. In the research proposal I will focus on three that go beyond boundaries of non-equilibrium statistical mechanics and which I  expect will have the largest impact on mathematics and mathematical  physics. 1. Ergodicity, Large Deviation Principle, and Fluctuation Relation  for Lagrangian trajectories of randomly forced Navier-Stokes systems. 2. Non-equilibrium statistical mechanics   of repeated quantum measurement processes. 3. Vanishing noise limit for the entropy production for diffusion processes.