3,036 research outputs found
Long-time behavior of a finite volume discretization for a fourth order diffusion equation
We consider a non-standard finite-volume discretization of a strongly
non-linear fourth order diffusion equation on the -dimensional cube, for
arbitrary . The scheme preserves two important structural properties
of the equation: the first is the interpretation as a gradient flow in a mass
transportation metric, and the second is an intimate relation to a linear
Fokker-Planck equation. Thanks to these structural properties, the scheme
possesses two discrete Lyapunov functionals. These functionals approximate the
entropy and the Fisher information, respectively, and their dissipation rates
converge to the optimal ones in the discrete-to-continuous limit. Using the
dissipation, we derive estimates on the long-time asymptotics of the discrete
solutions. Finally, we present results from numerical experiments which
indicate that our discretization is able to capture significant features of the
complex original dynamics, even with a rather coarse spatial resolution.Comment: 27 pages, minor change
The large deviation approach to statistical mechanics
The theory of large deviations is concerned with the exponential decay of
probabilities of large fluctuations in random systems. These probabilities are
important in many fields of study, including statistics, finance, and
engineering, as they often yield valuable information about the large
fluctuations of a random system around its most probable state or trajectory.
In the context of equilibrium statistical mechanics, the theory of large
deviations provides exponential-order estimates of probabilities that refine
and generalize Einstein's theory of fluctuations. This review explores this and
other connections between large deviation theory and statistical mechanics, in
an effort to show that the mathematical language of statistical mechanics is
the language of large deviation theory. The first part of the review presents
the basics of large deviation theory, and works out many of its classical
applications related to sums of random variables and Markov processes. The
second part goes through many problems and results of statistical mechanics,
and shows how these can be formulated and derived within the context of large
deviation theory. The problems and results treated cover a wide range of
physical systems, including equilibrium many-particle systems, noise-perturbed
dynamics, nonequilibrium systems, as well as multifractals, disordered systems,
and chaotic systems. This review also covers many fundamental aspects of
statistical mechanics, such as the derivation of variational principles
characterizing equilibrium and nonequilibrium states, the breaking of the
Legendre transform for nonconcave entropies, and the characterization of
nonequilibrium fluctuations through fluctuation relations.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text,
figures and appendices added, many references cut, close to published versio
Entropic Ricci curvature bounds for discrete interacting systems
We develop a new and systematic method for proving entropic Ricci curvature
lower bounds for Markov chains on discrete sets. Using different methods, such
bounds have recently been obtained in several examples (e.g., 1-dimensional
birth and death chains, product chains, Bernoulli-Laplace models, and random
transposition models). However, a general method to obtain discrete Ricci
bounds had been lacking. Our method covers all of the examples above. In
addition, we obtain new Ricci curvature bounds for zero-range processes on the
complete graph. The method is inspired by recent work of Caputo, Dai Pra and
Posta on discrete functional inequalities.Comment: Published at http://dx.doi.org/10.1214/15-AAP1133 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Modified log-Sobolev inequalities and two-level concentration
We consider a generic modified logarithmic Sobolev inequality (mLSI) of the
form for some difference operator , and show how it implies
two-level concentration inequalities akin to the Hanson--Wright or Bernstein
inequality. This can be applied to the continuous (e.\,g. the sphere or bounded
perturbations of product measures) as well as discrete setting (the symmetric
group, finite measures satisfying an approximate tensorization property,
\ldots).
Moreover, we use modified logarithmic Sobolev inequalities on the symmetric
group and for slices of the hypercube to prove Talagrand's convex
distance inequality, and provide concentration inequalities for locally
Lipschitz functions on . Some examples of known statistics are worked out,
for which we obtain the correct order of fluctuations, which is consistent with
central limit theorems
Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models
We calculate a Ricci curvature lower bound for some classical examples of
random walks, namely, a chain on a slice of the n-dimensional discrete cube
(the so-called Bernoulli-Laplace model) and the random transposition shuffle of
the symmetric group of permutations on n letters
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