Mathematical Theory of Non-Equilibrium Statistical Mechanics

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The research proposal concerns continuation of the research program on which I have worked for over a decade. The specific goals are the following.(I) Completion of the Entropic Fluctuation Program. This massive research program has been my main focus over the last five years and has already led to nearly 500 journal pages in print. The completion of the so-called ”quantum Evans-Searles” part of the program (fluctuation theory with respect to the reference state) requires an additional year of work and completion of two major papers (”Non-equilibrium statistical mechanics of Pauli-Fierz systems” (estimated around 100 pages) and ”Entropic fluctuations in statistical mechanics II. Quantum dynamical systems” (estimated around 400 pages)) and a completion of a research monograph ”Non-equilibrium statistical mechanics of locally interacting fermionic systems” (estimated around 400 pages). (II) Thermodynamics of non-equilibrium steady states. This research project is a natural continuation of the Entropic Fluctuation program. It concerns the problematic concept of ”entropy” for physical systems far from equilibrium. I believe that in various special situations (like open quantum systems) a satisfactory result with possibly far reaching physical and mathematical implications can be obtained by combining the geometric ideas of Ruelle concerning ”entropic connection and curvature” with the ideas of geometric parameter estimation theory (Efron).(III) Rare events and fluctuation symmetries in the theory of stochastic PDE’s. This project is devoted to study of large-time asymptotics (and in particular large deviation theory) for some stochastic PDE’s arising in mathematical physics. The principal motivation is non-equilibrium statistical mechanics and the ultimate goal is mathematically rigorous understanding of the Gallavotti-Cohen Fluctuation Relation for physical systems described by stochastic PDE’s. The motivating example are Navier–Stokes equations describing the motion of an incompressible viscous fluid. I also plan to study the complex Ginzburg–Landau equation and damped–driven dispersive PDE’s.(IV) Localization for interacting Fermi gases on a lattice. The Anderson localization for random Schrodinger operators describing the motion of an electron moving under the influence of a random external potential is very well understood in the large disorder regime. In contrast, virtually nothing is known about the Anderson localization in the physically important case where the interaction between electrons is not neglected. The traditional approach based on the spectral theory appears unsuitable and new ideas are needed. I plan to study this problem using the ideas and techniques that has recently emerged in mathematically rigorous literature on non-equilibrium quantum statistical mechanics. The main idea is to link the localization theory of a disordered sample of interacting fermions to the absence of the Landauer-Buttiker non-equilibrium steady state transport when thermal reservoirs are attached to the sample.(V) Open XY spin chains and spectral theory of Jacobi matrices. This project concerns a surprising link between the non-equilibrium statistical mechanics of XY chains and the spectral/scattering theory of Jacobi matrices. I have several papers on this subject and I plan to continue with the exploration of this link. The immediate specific goals are the new proof of Kotani theory and study of the regularity properties of Landauer-Buttiker formula for XY chain associated to Harper's equation. (VI) Shannon-McMillan-Breiman theorem and non-equilibrium statistical mechanics. The project concerns exploration of the link between recent developments in quantum information theory and quantum statistical mechanics.