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We present a method to prove convergence of gradient flows of families of energies that -converge to a limiting energy. It provides lower-bound criteria to obtain the convergence that correspond to a sort of C 1 -order -convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field.

We introduce a "Coulombian renormalized energy" W which is a logarithmic type of interaction between points in the plane, computed by a "renormalization." We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique minimizer. Its minimization in general remains open.Our motivation is the study of minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields H c 1 and H c 2 . In that regime, minimizing configurations exhibit densely packed triangular vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, W as a -limit of the Ginzburg-Landau energy. More precisely we show that the vortices of minimizers of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W , thus providing a first rigorous hint at the Abrikosov lattice. This is a next order effect compared to the mean-field type results we previously established.The derivation of W uses energy methods: the framework of -convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem, that we introduce.

We study the large-N limit of a system of N bosons interacting with a potential of intensity 1=N . When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and nondegenerate, and that there is complete Bose-Einstein condensation on this state. Using our method we then treat two applications: atoms with "bosonic" electrons on one hand, and trapped two-dimensional and three-dimensional Coulomb gases on the other hand.

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case β = ∞ corresponds to "weighted Fekete sets" and also falls within our analysis.It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni [BZ].By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W , a Coulombian interaction for points in the plane introduced in [SS1], which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When β → ∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W . In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W , which are conjectured to be "Abrikosov" triangular lattices.

Abstract. 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.

We study systems of n points in the Euclidean space of dimension d ≥ 1 interacting via a Riesz kernel |x| −s and confined by an external potential, in the regime where d − 2 ≤ s < d. We also treat the case of logarithmic interactions in dimensions 1 and 2. Our study includes and retrieves all cases previously studied in [SS4, SS5, 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 R d+1 . As n → ∞, we exhibit a next to leading order term in n 1+s/d in the asymptotic expansion of the total energy of the system, where the constant term in factor of n 1+s/d 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.

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