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FD2W02 - Fractional kinetics, hydrodynamic limits and fractals

Over the last few years, anomalous behaviors have been observed for one-dimensional chains of oscillators. The rigorous derivation of such behaviors from deterministic systems of Newtonian particles is very challenging, due to the existence of conservation laws, which impose very poor ergodic properties to the dynamical system. A possible way out of this lack of ergodicity is to introduce stochastic models, in such a way that the qualitative behaviour of the system is not modified. One starts with a chain of oscillators with a Hamiltonian dynamics, and then adds a stochastic which keeps the fundamental conservation laws (energy, momentum and stretch, usually). For the unpinned harmonic chain where the velocities of particles can randomly change sign (and therefore the only conserved quantities of the dynamics are the energy and the stretch), it is known that, under a diffusive space-time scaling, the energy profile evolves following a non-linear diffusive equation involving the stretch. Recently it has been shown that in the case of one-dimensional harmonic oscillators with noise that preserves the momentum, the scaling limit of the energy fluctuations is ruled by the fractional heat equation. This talk aims at understanding the transition regime for the energy fluctuations. Let us consider the same harmonic Hamiltonian dynamics, but now perturbed by two stochastic noises: both perturbations conserve the energy, but only the first one preserves the momentum. If the second one is null, the momentum is conserved, the energy transport is superdiffusive and described by a Lévy process governed by a fractional Laplacian. Otherwise, the volume conservation is destroyed, and the energy normally diffuses. What happens when the intensity of the second noise vanishes with the size of the chain? In this case, we can show that the limit of the energy fluctuation field depends on the evanescent speed of the random perturbation, we recover the two very different regimes for the energy transport, and we prove the existence of a crossover between the normal diffusion regime and the fractional superdiffusion regime.

This talk is part of the Isaac Newton Institute Seminar Series series.

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