312 research outputs found
Donsker-Varadhan asymptotics for degenerate jump Markov processes
We consider a class of continuous time Markov chains on a compact metric
space that admit an invariant measure strictly positive on open sets together
with absorbing states. We prove the joint large deviation principle for the
empirical measure and flow. Due to the lack of uniform ergodicity, the zero
level set of the rate function is not a singleton. As corollaries, we obtain
the Donsker-Varadhan rate function for the empirical measure and a variational
expression of the rate function for the empirical flow
Transport in the sine-Gordon field theory: from generalized hydrodynamics to semiclassics
The semiclassical approach introduced by Sachdev and collaborators proved to
be extremely successful in the study of quantum quenches in massive field
theories, both in homogeneous and inhomogeneous settings. While conceptually
very simple, this method allows one to obtain analytic predictions for several
observables when the density of excitations produced by the quench is small. At
the same time, a novel generalized hydrodynamic (GHD) approach, which captures
exactly many asymptotic features of the integrable dynamics, has recently been
introduced. Interestingly, also this theory has a natural interpretation in
terms of semiclassical particles and it is then natural to compare the two
approaches. This is the objective of this work: we carry out a systematic
comparison between the two methods in the prototypical example of the
sine-Gordon field theory. In particular, we study the "bipartitioning protocol"
where the two halves of a system initially prepared at different temperatures
are joined together and then left to evolve unitarily with the same
Hamiltonian. We identify two different limits in which the semiclassical
predictions are analytically recovered from GHD: a particular non-relativistic
limit and the low temperature regime. Interestingly, the transport of
topological charge becomes sub-ballistic in these cases. Away from these limits
we find that the semiclassical predictions are only approximate and, in
contrast to the latter, the transport is always ballistic. This statement seems
to hold true even for the so-called "hybrid" semiclassical approach, where
finite time DMRG simulations are used to describe the evolution in the internal
space.Comment: 30 pages, 6 figure
Large deviations of the empirical flow for continuous time Markov chains
We consider a continuous time Markov chain on a countable state space and
prove a joint large deviation principle for the empirical measure and the
empirical flow, which accounts for the total number of jumps between pairs of
states. We give a direct proof using tilting and an indirect one by contraction
from the empirical process.Comment: Minor revision, to appear on Annales de l'Institut Henri Poincare (B)
Probability and Statistic
A gradient flow approach to linear Boltzmann equations
We introduce a gradient flow formulation of linear Boltzmann equations. Under
a diffusive scaling we derive a diffusion equation by using the machinery of
gradient flows
Large deviation principles for nongradient weakly asymmetric stochastic lattice gases
We consider a lattice gas on the discrete d-dimensional torus
with a generic translation invariant, finite range
interaction satisfying a uniform strong mixing condition. The lattice gas
performs a Kawasaki dynamics in the presence of a weak external field E/N. We
show that, under diffusive rescaling, the hydrodynamic behavior of the lattice
gas is described by a nonlinear driven diffusion equation. We then prove the
associated dynamical large deviation principle. Under suitable assumptions on
the external field (e.g., E constant), we finally analyze the variational
problem defining the quasi-potential and characterize the optimal exit
trajectory. From these results we deduce the asymptotic behavior of the
stationary measures of the stochastic lattice gas, which are not explicitly
known. In particular, when the external field E is constant, we prove a
stationary large deviation principle for the empirical density and show that
the rate function does not depend on E.Comment: Published in at http://dx.doi.org/10.1214/11-AAP805 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Large deviations for a stochastic model of heat flow
We investigate a one dimensional chain of harmonic oscillators in which
neighboring sites have their energies redistributed randomly. The sites
and are in contact with thermal reservoirs at different temperature
and . Kipnis, Marchioro, and Presutti \cite{KMP} proved that
this model satisfies {}Fourier's law and that in the hydrodynamical scaling
limit, when , the stationary state has a linear energy density
profile , . We derive the large deviation
function for the probability of finding, in the stationary
state, a profile different from . The function
has striking similarities to, but also large differences from, the
corresponding one of the symmetric exclusion process. Like the latter it is
nonlocal and satisfies a variational equation. Unlike the latter it is not
convex and the Gaussian normal fluctuations are enhanced rather than suppressed
compared to the local equilibrium state. We also briefly discuss more general
model and find the features common in these two and other models whose
is known.Comment: 28 pages, 0 figure
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