103 research outputs found
Systems of Points with Coulomb Interactions
Large ensembles of points with Coulomb interactions arise in various settings
of condensed matter physics, classical and quantum mechanics, statistical
mechanics, random matrices and even approximation theory, and give rise to a
variety of questions pertaining to calculus of variations, Partial Differential
Equations and probability. We will review these as well as "the mean-field
limit" results that allow to derive effective models and equations describing
the system at the macroscopic scale. We then explain how to analyze the next
order beyond the mean-field limit, giving information on the system at the
microscopic level. In the setting of statistical mechanics, this allows for
instance to observe the effect of the temperature and to connect with
crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201
Ginzburg-Landau vortices, Coulomb Gases, and Renormalized Energies
This is a review about a series of results on vortices in the Ginzburg-Landau
model of superconductivity on the one hand, and point patterns in Coulomb gases
on the other hand, as well as the connections between the two topics.Comment: review paper, submitted to J. Stat. Phy
Large Deviation Principle for Empirical Fields of Log and Riesz Gases
We study a system of N particles with logarithmic, Coulomb or Riesz pairwise
interactions, confined by an external potential. We examine a microscopic
quantity, the tagged empirical field, for which we prove a large deviation
principle at speed N. The rate function is the sum of an entropy term, the
specific relative entropy, and an energy term, the renormalized energy
introduced in previous works, coupled by the temperature. We deduce a
variational property of the sine-beta processes which arise in random matrix
theory. We also give a next-to-leading order expansion of the free energy of
the system, proving the existence of the thermodynamic limit.Comment: 80 pages, final version, to appear in Inventiones Mat
Next Order Asymptotics and Renormalized Energy for Riesz Interactions
We study systems of points in the Euclidean space of dimension
interacting via a Riesz kernel and confined by an external
potential, in the regime where . We also treat the case of
logarithmic interactions in dimensions and . Our study includes and
retrieves all cases previously studied in \cite{ss2d,ss1d,rs}. Our approach is
based on the Caffarelli-Silvestre extension formula which allows to view the
Riesz kernel as the kernel of a (inhomogeneous) local operator in the extended
space .
As , we exhibit a next to leading order term in in
the asymptotic expansion of the total energy of the system, where the constant
term in factor of depends on the microscopic arrangement of the
points and is expressed in terms of a "renormalized energy." This new object is
expected to penalize the disorder of an infinite set of points in whole space,
and to be minimized by Bravais lattice (or crystalline) configurations. We give
applications to the statistical mechanics in the case where temperature is
added to the system, and identify an expected "crystallization regime." We also
obtain a result of separation of the points for minimizers of the energy
From the Ginzburg-Landau model to vortex lattice problems
We study minimizers of the two-dimensional Ginzburg-Landau energy with
applied magnetic field, between the first and second critical fields. In this
regime, minimizing configurations exhibit densely packed hexagonal vortex
lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a
limiting interaction energy between points in the plane, , which we prove
has to be minimized by limits of energy-minimizing configurations, once
blown-up at a suitable scale. This is a next order effect compared to the
mean-field type results we previously established. The limiting "Coulombian
renormalized energy" is a logarithmic type of interaction, computed by a
"renormalization," and we believe it should be rather ubiquitous. We study
various of its properties, and show in particular, using results from number
theory, that among lattice configurations the hexagonal lattice is the unique
minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its
minimization in general remains open. The derivation of uses energy
methods: the framework of -convergence, and an abstract scheme for
obtaining lower bounds for "2-scale energies" via the ergodic theorem.Comment: 107 page
Lorentz Space Estimates for the Ginzburg-Landau Energy
In this paper we prove novel lower bounds for the Ginzburg-Landau energy with
or without magnetic field. These bounds rely on an improvement of the "vortex
balls construction" estimates by extracting a new positive term in the energy
lower bounds. This extra term can be conveniently estimated through a Lorentz
space norm, on which it thus provides an upper bound. The Lorentz space
we use is critical with respect to the expected vortex profiles
and can serve to estimate the total number of vortices and get improved
convergence results.Comment: 52 pages, 1 figur
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