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SUMMARY:Discussions - 
DTSTART:20220324T140000Z
DTEND:20220324T150000Z
UID:TALK171716@talks.cam.ac.uk
DESCRIPTION:Over the last few years\, anomalous behaviors have been observ
 ed for one-dimensional chains of oscillators. The rigorous derivation of s
 uch behaviors from deterministic systems of Newtonian particles is very ch
 allenging\, due to the existence of conservation laws\, which impose very 
 poor ergodic properties to the dynamical system. A possible way out of thi
 s lack of ergodicity is to introduce stochastic models\, in such a way tha
 t the qualitative behaviour of the system is not modified. One starts with
  a chain of oscillators with a Hamiltonian dynamics\, and then adds a stoc
 hastic which keeps the fundamental conservation laws (energy\, momentum an
 d stretch\, usually).\nFor the unpinned harmonic chain where the velocitie
 s of particles can randomly change sign (and therefore the only conserved 
 quantities of the dynamics are the energy and the stretch)\, it is known t
 hat\, under a diffusive space-time scaling\, the energy profile evolves fo
 llowing a non-linear diffusive equation involving the stretch. Recently it
  has been shown that in the case of one-dimensional harmonic oscillators w
 ith noise that preserves the momentum\, the scaling limit of the energy fl
 uctuations is ruled by the fractional heat equation.\nThis talk aims at un
 derstanding the transition regime for the energy fluctuations. Let us cons
 ider the same harmonic Hamiltonian dynamics\, but now perturbed by two sto
 chastic noises: both perturbations conserve the energy\, but only the firs
 t one preserves the momentum. If the second one is null\, the momentum is 
 conserved\, the energy transport is superdiffusive and described by a L&ea
 cute\;vy process governed by a fractional Laplacian. Otherwise\, the volum
 e conservation is destroyed\, and the energy normally diffuses. What happe
 ns when the intensity of the second noise vanishes with the size of the ch
 ain? In this case\, we can show that the limit of the energy fluctuation f
 ield depends on the evanescent speed of the random perturbation\, we recov
 er the two very different regimes for the energy transport\, and we prove 
 the existence of a crossover between the normal diffusion regime and the f
 ractional superdiffusion regime.
LOCATION:Seminar Room 1\, Newton Institute
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