arXiv subjects

Marco Cicalese

Publications and source records attributed to Marco Cicalese.

At least 37 records · Page 2Linked to original sources

arXiv2021

Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size

We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing $δ$, the discretized functionals $Γ$-converge to the Mumford-Shah functional only if $δ\ll\varepsilon$, $\varepsilon$ being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic and isotropic random lattices we prove this $Γ$-convergence result also for $δ\sim\varepsilon$, a regime at which the discretization with respect to a periodic lattice converges instead to an anisotropic version of the Mumford-Shah functional.

math.AP↗

arXiv2020

Topological singularities in periodic media: Ginzburg-Landau and core-radius approaches

We describe the emergence of topological singularities in periodic media within the Ginzburg-Landau model and the core-radius approach. The energy functionals of both models are denoted by $E_{\varepsilon,δ}$, where $\varepsilon$ represent the coherence length (in the Ginzburg-Landau model) or the core-radius size (in the core-radius approach) and $δ$ denotes the periodicity scale. We carry out the $Γ$-convergence analysis of $E_{\varepsilon,δ}$ as $\varepsilon\to 0$ and $δ=δ_{\varepsilon}\to 0$ in the $|\log\varepsilon|$ scaling regime, showing that the $Γ$-limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter (upon extraction of subsequences) $$ λ=\min\Bigl\{1,\lim_{\varepsilon\to0} {|\log δ_{\varepsilon}|\over|\log{\varepsilon}|}\Bigr\}, $$ we show that in a sense we always have a separation-of-scale effect: at scales less than $\varepsilon^λ$ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than $\varepsilon^λ$ the concentration process takes place "after" homogenization.

math.AP↗

arXiv2020

The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$

We study a nearest neighbors ferromagnetic spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of $N$ equi-spaced vectors, also known as $N$-clock model. We find a fast rate of divergence of $N$ with respect to the lattice spacing for which the $N$-clock model has the same discrete-to-continuum variational limit of the $XY$ model, in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of $N$ at which the coarse-grain limit does not detect topological defects, but it is instead a $BV$-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for $N$, whose analysis requires the aid of Cartesian currents.

math-ph↗

arXiv2020

The antiferromagnetic XY model on the triangular lattice: topological singularities

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice in the vortex regime. Within this regime, the spin system cannot overcome the energetic barrier of chirality transitions, hence one of the two chirality phases is prevalent. We find the order parameter that describes the vortex structure of the spin field in the majority chirality phase and we compute explicitly the $Γ$-limit of the scaled energy, showing that it concentrates on finitely many vortex-like singularities of the spin field.

math-ph↗

arXiv2020

Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle~$\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $Γ$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $\varepsilon \to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $\mathcal{S}_N$; at a second stage, we let $N \to +\infty$. The final result of this two-step limit process is an anisotropic total variation of $\mathbb{S}^1$-valued vector fields of bounded variation.

math-ph↗

arXiv2020

The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice. The system is fully frustrated and displays two families of ground states distinguished by the chirality of the spin field. We compute the Γ-limit of the energy in a regime which detects chirality transitions on one-dimensional interfaces between the two admissible chirality phases.

math.AP↗

arXiv2019

Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via Gamma-convergence. We consider energy densities that may depend on interactions between all points of the lattice and we give conditions that ensure compactness and integral-representation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk- and surface contributions decouple in the limit. The results are applied to long-range and multi-body interactions in the setting of weak-membrane energies.

math.AP↗

arXiv2019

Variational analysis of a two-dimensional frustrated spin system: emergence and rigidity of chirality transitions

We study the discrete-to-continuum variational limit of the $J_{1}$-$J_{3}$ spin model on the square lattice in the vicinity of the helimagnet/ferromagnet transition point as the lattice spacing vanishes. Carrying out the $Γ$-convergence analysis of proper scalings of the energy, we prove the emergence and characterize the geometric rigidity of the chirality phase transitions.

math.AP↗

arXiv2018

From statistical polymer physics to nonlinear elasticity

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the $Γ$-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the $Γ$-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).

math-ph↗

arXiv2016

$Γ$-convergence analysis of a generalized $XY$ model: fractional vortices and string defects

We propose and analyze a generalized two dimensional $XY$ model, whose interaction potential has $n$ weighted wells, describing corresponding symmetries of the system. As the lattice spacing vanishes, we derive by $Γ$-convergence the discrete-to-continuum limit of this model. In the energy regime we deal with, the asymptotic ground states exhibit fractional vortices, connected by string defects. The $Γ$-limit takes into account both contributions, through a renormalized energy, depending on the configuration of fractional vortices, and a surface energy, proportional to the length of the strings. Our model describes in a simple way several topological singularities arising in Physics and Materials Science. Among them, disclinations and string defects in liquid crystals, fractional vortices and domain walls in micromagnetics, partial dislocations and stacking faults in crystal plasticity.

math.AP↗

arXiv2016

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

We study the discrete-to-continuum limit of ferromagnetic spin systems when the lattice spacing tends to zero. We assume that the atoms are part of a (maybe) non-periodic lattice close to a flat set in a lower dimensional space, typically a plate in three dimensions. Scaling the particle positions by a small parameter $\varepsilon>0$ we perform a $Γ$-convergence analysis of properly rescaled interfacial-type energies. We show that, up to subsequences, the energies converge to a surface integral defined on partitions of the flat space. In the second part of the paper we address the issue of stochastic homogenization in the case of random stationary lattices. A finer dependence of the homogenized energy on the average thickness of the random lattice is analyzed for an example of magnetic thin system obtained by a random deposition mechanism.

math.AP↗

arXiv2016

On global and local minimizers of prestrained thin elastic rods

We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By $Γ$-convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations. Through some simple simulations we finally compare our results with real experiments.

math.AP↗

arXiv2015

Ground states of a two phase model with cross and self attractive interactions

We consider a variational model for two interacting species (or phases), subject to cross and self attractive forces. We show existence and several qualitative properties of minimizers. Depending on the strengths of the forces, different behaviors are possible: phase mixing or phase separation with nested or disjoint phases. In the case of Coulomb interaction forces, we characterize the ground state configurations.

math.AP↗

arXiv2015

Improved convergence theorems for bubble clusters. I. The planar case

We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this construction to isoperimetric problems for planar bubble clusters. In this setting we develop an improved convergence theorem, showing that a sequence of almost-minimizing planar clusters converging in $L^1$ to a limit cluster has actually to converge in a strong $C^{1,α}$-sense. Applications of this improved convergence result to the classification of isoperimetric clusters and the qualitative description of perturbed isoperimetric clusters are also discussed. Analogous results for three-dimensional clusters are presented in part two, while further applications are discussed in some companion papers.

math.AP↗

arXiv2015

Sharp stability inequalities for planar double bubbles

In this paper we address the global stability problem for double-bubbles in the plane. This is accomplished by combining the "improved convergence theorem" for planar clusters developed in arXiv:1409.6652 with an ad hoc analysis of the problem, which addresses the delicate interaction between the (possible) dislocation of singularities and the multiple-volumes constraint.

math.AP↗

arXiv2015

Chirality transitions in frustrated $S^{2}$-valued spin systems

We study the discrete-to-continuum limit of the helical XY $S^{2}$-spin system on the lattice $\mathbb{Z}^{2}$. We scale the interaction parameters in order to reduce the model to a spin chain in the vicinity of the Landau-Lifschitz point and we prove that at the same energy scaling under which the $S^{1}$-model presents scalar chirality transitions, the cost of every vectorial chirality transition is now zero. In addition we show that if the energy of the system is modified penalizing the distance of the $S^{2}$ field from a finite number of copies of $S^{1}$, it is still possible to prove the emergence of nontrivial (possibly trace dependent) chirality transitions.

math-ph↗

arXiv2014

Frustrated ferromagnetic spin chains: a variational approach to chirality transitions

We study the energy per particle of a one-dimensional ferromagnetic/anti-ferromagnetic frustrated spin chain with nearest and next-to-nearest interactions close to the helimagnet/ferromagnet transition point as the number of particles diverges. We rigorously prove the emergence of chiral ground states and we compute, by performing the Gamma-limits of proper renormalizations and scalings, the energy for a chirality transition.

math-ph↗

arXiv2013

Q-tensor continuum energies as limits of head-to-tail-symmetric spin systems

We consider a class of spin-type discrete systems and analyze their continuum limit as the lattice spacing goes to zero. Under standard coerciveness and growth assumptions together with an additional head-to-tail symmetry condition, we observe that this limit can be conveniently written as a functional in the space of $Q$-tensors. We further characterize the limit energy density in several cases (both in 2 and 3 dimensions). In the planar case we also develop a second-order theory and we derive gradient or concentration-type models according to the chosen scaling.

math.AP↗