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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Discussions -
DTSTART;TZID=Europe/London:20220324T153000
DTEND;TZID=Europe/London:20220324T163000
UID:TALK171719AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/171719
DESCRIPTION:Over the last few years\, anomalous behaviors have
been observed for one-dimensional chains of oscil
lators. The rigorous derivation of such behaviors
from deterministic systems of Newtonian particles
is very challenging\, due to the existence of cons
ervation laws\, which impose very poor ergodic pro
perties to the dynamical system. A possible way ou
t of this lack of ergodicity is to introduce stoch
astic models\, in such a way that the qualitative
behaviour of the system is not modified. One start
s with a chain of oscillators with a Hamiltonian d
ynamics\, and then adds a stochastic which keeps t
he fundamental conservation laws (energy\, momentu
m and stretch\, usually).\nFor the unpinned harmon
ic chain where the velocities of particles can ran
domly change sign (and therefore the only conserve
d quantities of the dynamics are the energy and th
e stretch)\, it is known that\, 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 th
e case of one-dimensional harmonic oscillators wit
h noise that preserves the momentum\, the scaling
limit of the energy fluctuations is ruled by the f
ractional heat equation.\nThis talk aims at unders
tanding the transition regime for the energy fluct
uations. Let us consider the same harmonic Hamilto
nian dynamics\, but now perturbed by two stochasti
c noises: both perturbations conserve the energy\,
but only the first one preserves the momentum. If
the second one is null\, the momentum is conserve
d\, the energy transport is superdiffusive and des
cribed by a Lé\;vy process governed by a fra
ctional Laplacian. Otherwise\, the volume conserva
tion is destroyed\, and the energy normally diffus
es. 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 f
luctuation field depends on the evanescent speed o
f the random perturbation\, we recover the two ver
y different regimes for the energy transport\, and
we prove the existence of a crossover between the
normal diffusion regime and the fractional superd
iffusion regime.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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