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Leonard Kreutz

Publications and source records attributed to Leonard Kreutz.

At least 19 recordsLinked to original sources
arXiv2026

The charge dependent hard-sphere model: Polycrystals as low-energy configurations

We investigate the emergence of rigid polycrystalline structures in atomistic ionic particle systems as low-energy configurations. The interaction between particles of opposite charge is modeled by hard spheres that interact when they are tangential. The interaction between particles of same charge is modeled as a hard repulsion that forces a minimal distance between them. The atomistic energy is frame invariant, and no underlying reference lattice is assumed on the ionic configurations. The asymptotic behavior of configurations with finite surface energy scaling is identified by means of $Γ$-convergence. The related continuum theory is described by piecewise constant fields that encode the local orientation of the configuration. The limiting energy is local and concentrates at grain boundaries, which correspond to the boundaries of the regions where the underlying configuration has a constant orientation. The limiting energy density is anisotropic and depends on the relative misorientation of the two grains, their translation misfit, and the normal to their interface. Furthermore, we perform a fine analysis of surface energies for solid-solid and solid-vacuum phase transitions and determine energetically favorable orientation mismatches. This relies on a structure result for our grain boundaries, which shows that, due to the rigid setup, interpolating layers near the grain interface are energetically not favorable.

cond-mat.stat-mech↗
arXiv2026

Emergence of rigid Polycrystals from atomistic Systems with general Interactions

We investigate the formation of polycrystalline structures in a class of particle systems. The atomistic energy is modeled as a sum of particle energies that favor atoms being locally isometric to a reference lattice. The discrete frame invariant energy allows for particle configurations in which no underlying lattice is assumed a priori. We prove a discrete-to-continuum limit for configurations with finite surface-energy scaling by means of $Γ$-convergence. The resulting continuum theory is described by piecewise constant fields encoding the local orientation of the configuration. The limiting energy is concentrated on grain boundaries, corresponding to the interfaces between regions where the microscopic configuration has constant orientation. The associated energy density depends on the orientations of the two grains as well as on the normal to the interface. Due to our assumptions on the rigid interactions, solid-solid phase transitions with interpolating boundary layers are not energetically favorable; the energy density therefore decomposes into twice the energy density for solid-vacuum transitions.

cond-mat.mes-hall↗
arXiv2026

Discrete Quantitative Isocapacitary Inequality: Fluctuation Estimates

The classical isocapacitary inequality states that, among all sets of fixed volume, the ball uniquely minimizes the capacity. While this result holds in the continuum, it fails in the discrete setting, where the isocapacitary problem may admit multiple minimizers. In this paper we establish quantitative fluctuation estimates for the discrete isocapacitary problem on subsets of $\mathbb{Z}^d$ as their cardinality diverges. Our approach relies on a careful extension of the associated variational problem from the discrete to the continuum setting, combined with sharp (continuum) quantitative isocapacitary inequalities.

math.AP↗
arXiv2025

Crystallization in the Winterbottom shape and sharp fluctuation laws

We address finite crystallization in two dimensions in the presence of a flat crystalline substrate. Particles interact through short-range two- and three-body potentials favoring local square-lattice arrangements. An additional interaction term of relative strength $β>0$ couples the particles and the substrate. Our first main result proves crystallization for all $β>0$, corresponding to the onset of discrete Winterbottom configurations. The proof relies on a stratification technique from [31], characterizing the topology of the bond graph of minimizing configurations. Our second main result concerns fluctuations estimates for $β\in (0,1)$. We obtain bounds on the distance between distinct minimizers with the same number $N$ of particles, showing a sharp scaling law $N^{3/4}$ when $β$ is rational, and $N^{1/3}$ when $β$ is irrational and algebraic. This reveals a genuine substrate-driven effect on fluctuation laws. As a corollary, we derive a discrete-to-continuum convergence of minimizers towards the Winterbottom equilibrium shape in the large-particle limit.

cond-mat.mes-hall↗
arXiv2025

The Quantitative Faber-Krahn Inequality for the Combinatorial Laplacian in $\mathbb{Z}^{d}$

While the classical Faber-Krahn inequality shows that the ball uniquely minimizes the first Dirichlet eigenvalue of the Laplacian in the continuum, this rigidity may fail in the discrete setting. We establish quantitative fluctuation estimates for the first Dirichlet eigenvalue of the combinatorial Laplacian on subsets of $\mathbb{Z}^{d}$ when their cardinality diverges. Our approach is based on a controlled discrete-to-continuum extension of the associated variational problem and the quantitative Faber-Krahn inequality.

math.FA↗
arXiv2025

Derivation of Kirchhoff-type plate theories for elastic materials with voids

We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective two-dimensional model comprises a classical elastic bending energy and surface terms which reflect the possibility that voids can persist in the limit, that the limiting plate can be broken apart into several pieces, or that the plate can be folded. Building upon and extending the techniques used in the authors' recent work on the derivation of one-dimensional theories for thin brittle rods with voids, the present contribution generalizes the results of Santili and Schmidt (2022), by considering general geometries on the admissible set of voids and constructing recovery sequences for all admissible limiting configurations.

math.AP↗
arXiv2025

Energy concentration in a two-dimensional magnetic skyrmion model: variational analysis of lattice and continuum theories

We investigate the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, we establish a weak compactness of its topological charge density, which converges to an atomic measure with quantized weights. We characterize the $Γ$-limit of the energies as the total variation of this measure. In the case of lattice type energies, we first need to carefully define a notion of discrete topological charge for $\mathbb{S}^2$-valued maps. We then prove a corresponding compactness and $Γ$-convergence result, thereby bridging the discrete and continuum theories.

math.AP↗
arXiv2025

Second-Order $Γ$-Limit for the Cahn-Hilliard Functional with Dirichlet Boundary Conditions, II

This paper continues the study of the asymptotic development of order 2 by $Γ$ -convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions initiated in [8]. While in the first paper, the Dirichlet data are assumed to be well separated from one of the two wells, here this is no longer the case. In the case where there are no interfaces, it is shown that there is a transition layer near the boundary of the domain.

math.AP↗
arXiv2024

A note on the Winterbottom shape

In this short note we review results on equilibrium shapes of minimizers to the sessile drop problem. More precisely, we study the Winterbottom problem and prove that the Winterbottom shape is indeed optimal. The arguments presented here are based on relaxation and the (anisotropic) isoperimetric inequality.

math.AP↗
arXiv2023

Derivation of effective theories for thin 3D nonlinearly elastic rods with voids

We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of $Γ$-convergence. Hereby, we generalize the results of the purely elastic setting [57] to a framework of free discontinuity problems. The effective one-dimensional model features a classical elastic bending-torsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a {discontinuity} of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [32].

math.AP↗
arXiv2023

A proof of finite crystallization via stratification

We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality in $\ell^1$. As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the $n^{3/4}$-law.

cond-mat.mes-hall↗
arXiv2023

Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces

We present a quantitative geometric rigidity estimate in dimensions $d=2,3$ generalizing the celebrated result by Friesecke, James, and Müller to the setting of variable domains. Loosely speaking, we show that for each $y \in H^1(U;\mathbb{R}^d)$ and for each connected component of a smooth open, bounded set $U \subset \mathbb{R}^d$, the $L^2$-distance of $\nabla y$ from a single rotation can be controlled up to a constant by its $L^2$-distance from the group $SO(d)$, with the constant not depending on the precise shape of $U$, but only on an integral curvature functional related to $\partial U$. We further show that for linear strains the estimate can be refined, leading to a uniform control independent of the set $U$. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation ($GSBD^2$) for sequences of displacements related to deformations with uniformly bounded elastic energy. As an application, we rigorously derive linearized models for nonlinearly elastic materials with free surfaces by means of $Γ$-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solids.

math.AP↗
arXiv2022

Emergence of Wulff-Crystals from atomistic systems on the FCC and HCP lattices

We consider a system of $N$ hard spheres sitting on the nodes of either the $\mathrm{FCC}$ or $\mathrm{HCP}$ lattice and interacting via a sticky-disk potential. As $N$ tends to infinity (continuum limit), assuming the interaction energy does not exceed that of the ground-state by more than $N^{2/3}$ (surface scaling), we obtain the variational coarse grained model by $Γ$-convergence. More precisely, we prove that the continuum limit energies are of perimeter type and we compute explicitly their Wulff shapes. Our analysis shows that crystallization on $\mathrm{FCC}$ is preferred to that on $\mathrm{HCP}$ for $N$ large enough. The method is based on integral representation and concentration-compactness results that we prove for general periodic lattices in any dimension.

math.AP↗
arXiv2022

From atomistic systems to linearized continuum models for elastic materials with voids

We study an atomistic model that describes the microscopic formation of material voids inside elastically stressed solids under an additional curvature regularization at the discrete level. Using a discrete-to-continuum analysis, by means of a recent geometric rigidity result in variable domains [27] and Γ-convergence tools, we rigorously derive effective linearized continuum models for elastically stressed solids with material voids in three-dimensional elasticity.

math.AP↗
arXiv2022

Crystallinity of the homogenized energy density of periodic lattice systems

We study the homogenized energy densities of periodic ferromagnetic Ising systems. We prove that, for finite range interactions, the homogenized energy density, identifying the effective limit, is crystalline, i.e. its Wulff crystal is a polytope, for which we can (exponentially) bound the number of vertices. This is achieved by deriving a dual representation of the energy density through a finite cell formula. This formula also allows easy numerical computations: we show a few experiments where we compute periodic patterns which minimize the anisotropy of the surface tension.

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

Emergence of rigid Polycrystals from atomistic Systems with Heitmann-Radin sticky disk energy

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the `sticky disk' interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of $Γ$-convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, i.e., on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum-solid and solid-solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that due to the extremely brittle setup interpolating boundary layers near cracks are energetically not favorable.

cond-mat.stat-mech↗