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Hydrodynamic limits from classical and quantum microscopic dynamics Olla, Stefano


One of the main problems in statistical mechanics is the mathematical derivation, through space-time scaling limits, of macroscopic conservation laws, like compressible Euler equations or the heat equation, from the microscopic dynamics of molecules. Different space-time scalings, that we denote hydrodynamic limits, can lead to different macroscopic equations: the Euler system of equations governs the convergence to mechanical equilibrium (towards constant pressure) in a hyperbolic scaling, while at diffusive scaling (larger time) heat equation governs the convergence to thermal equilibrium (constant temperature), if thermal conductivity is finite. Deriving macroscopic limits from a deterministic (classical Hamiltonian or quantum) dynamics is a famous arduous problem. Some results can be obtained by adding to the dynamics some random terms such that momentum, energy and volume are still conserved but all other integrals of the motion are destroyed. The harmonic chain, despite the fact to be completely integrable, is an exception and, for the deterministic dynamics in thermal (global) equilibrium, the Euler equation are valid in the hyperbolic scaling. If the masses are random, thanks to Anderson localization, Euler equations are valid even out of thermal equilibrium. Furthermore, in the random mass case, the temperature profile remains constant in time at every later time scale, including the diffusive one, giving a further proof that thermal conductivity of this system is null. This results on the harmonic chain are extended to the corresponding quantum dynamics. (Works in collaboration with Cedric Bernardin, Francois Huveneers, Amirali Hannani)

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