La méthode de l’entropie relative pour les limites hydrodynamiques de modèles cinétiques
La scelta delle interazioni inerziali nei continui con microstruttura
Dedicando speciale attenzione all’esempio significativo dei cristalli liquidi di Ericksen [6], viene presentato un apparato assiomatico che consente di dedurre rappresentazioni coerenti delle interazioni d’inerzia e dell’energia cinetica per continui con microstruttura.
Large deviation principles and generalized Sherrington-Kirkpatrick models
Large deviations for partition functions of directed polymers in an IID field
Consider the partition function of a directed polymer in ℤd, d≥1, in an IID field. We assume that both tails of the negative and the positive part of the field are at least as light as exponential. It is well known that the free energy of the polymer is equal to a deterministic constant for almost every realization of the field and that the upper tail of the large deviations is exponential. The lower tail of the large deviations is typically lighter than exponential. In this paper we obtain sharp...
Large scale behavior of semiflexible heteropolymers
We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the disorder) are modeled in terms of random rotations. We focus on the quenched regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative...
Last passage percolation in macroscopically inhomogeneous media.
Le equazioni di evoluzione dei continui ferromagnetici
This expository paper is meant to be a faithful account the invited lecture I gave in Naples on September 14, 1999, during the 16th Congress of U.M.I., the Italian Mathematical Union. In Section 2, I consider the Gilbert equation, the parabolic equation that rules the evolution of the magnetization vector in a rigid ferromagnet. Among the issues I here discuss are the relations of the Gilbert equation to the harmonic map equation and its heat flow, the existence of global-in-time weak solutions,...
Limit laws for the energy of a charged polymer
In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy Hn=∑1≤j<k≤nωjωk1{Sj=Sk} of the polymer {S1, …, Sn} equipped with random electrical charges {ω1, …, ωn}. Our approach is based on comparison of the moments between Hn and the self-intersection local time Qn=∑1≤j<k≤n1{Sj=Sk} run by the d-dimensional random walk {Sk}. As partially needed for our main objective and partially motivated by their independent interest,...
Limites hydrodynamiques
Linear convergence in the approximation of rank-one convex envelopes
A linearly convergent iterative algorithm that approximates the rank-1 convex envelope of a given function , i.e. the largest function below which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.
Linear convergence in the approximation of rank-one convex envelopes
A linearly convergent iterative algorithm that approximates the rank-1 convex envelope of a given function , i.e. the largest function below f which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.
Line-energy Ginzburg-Landau models : zero-energy states
We consider a class of two-dimensional Ginzburg-Landau problems which are characterized by energy density concentrations on a one-dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero-energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero-energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful...
Local minimizers with vortex filaments for a Gross-Pitaevsky functional
This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filaments in a rotationally forced Bose-Einstein condensate. This functional is derived as a Γ-limit of scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence...
Local order at arbitrary distances in finite-dimensional spin-glass models
For a finite dimensional spin-glass model we prove low temperature local order i.e. the property of concentration of the overlap distribution close to the value 1. The theorem hold for both local observables and for products of observables at arbitrary mutual distance: when the Hamiltonian includes the Edwards-Anderson interaction we prove bond local order, when it includes the random-field interaction we prove site local order.
Local Rules for Pentagonal Quasi-Crystals.
Long memory from Sauerbrey equation: a case in coated quartz crystal microbalance in terms of ammonia.