# B08Curvature Effects in Molecular and Spin Systems

## Understanding Crystallization

Many basic phenomena in solid mechanics like dislocations or plastic and elastic deformation are in fact discrete operations: small breakdowns of perfect crystalline order. The goal of this project is thereofore to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization.

#### Scientific Details+

How do the well known phenomenological continuum theories of solid mechanics (linear and nonlinear elasticity theory, plasticity theory, fracture mechanics) emerge from discrete, atomistic models? This fundamental question lies at the heart of a great deal of current research in materials science and materials engineering, yet remains very poorly understood on a mathematical level. A key bottleneck is that we don't understand crystallization, that is to say the fact that under many conditions, atoms self-assemble into crystalline order and special geometric shapes. This is a main bottleneck because all the basic phenomena in solid mechanics (dislocations, grains, fracture, plastic and elastic deformation) are small or localized breakdowns of perfect crystalline order. The goal of the project is to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization. In particular, we aim to extend available results on crystalline order and shape from purely combinatorial energies to soft potentials which allow for elastic modes, and develop methods for the rigorous passage from these discrete models to continuum surface energy functionals and elastic energy functionals. Our mathematical approach will rely on combining methods from three areas: (i) atomistic mechanics and its recently developed generalized convexity notions, (ii) Gamma convergence techniques from the calculus of variations, and - crucially and as far as we know for the first time in our context - (iii) discrete differential geometry, which is a central theme in other projects of the SFB-Transregio.

#### Publications+

##### Papers
###### Maximal fluctuations on periodic lattices: an approach via quantitative Wulff inequalities

Authors: Cicalese, M. and Leonardi, G.P.
Journal: Commun. Math. Phys.
Note: preprint
Date: Sep 2019

###### From the N-clock model to the XY model: emergence of concentration effects in the variational analysis

Authors: M. Cicalese, G. Orlando and Ruf, M.
Note: preprint
Date: Aug 2019

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

Authors: M. Cicalese, A. Bach and Braides, A.
Note: preprint
Date: Jul 2019

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

Authors: Bach, A. and Cicalese, M. and Ruf, M.
Note: preprint
Date: Feb 2019

###### Barycenters for the Hellinger--Kantorovich distance over $\mathbb{R}^d$

Authors: Friesecke, Gero and Matthes, Daniel and Schmitzer, Bernhard
Note: preprint at arXiv
Date: 2019

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

Authors: M. Cicalese, M. Forster and Orlando, G.
Journal: SIAM J. Math. Anal.
Note: preprint
Date: 2019

###### From statistical polymer physics to nonlinear elasticity

Authors: Cicalese, Marco and Gloria, Antoine and Ruf, Matthias
Journal: preprint
Date: Sep 2018

###### $\Gamma$-Convergence Analysis of a Generalized $XY$ Model: Fractional Vortices and String Defects

Authors: Badal, Rufat and Cicalese, Marco and De Luca, Lucia and Ponsiglione, Marcello
Journal: Communications in Mathematical Physics, 358(2):705--739
Date: Mar 2018
DOI: 10.1007/s00220-017-3026-3

###### Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Authors: Braides, A. and Cicalese, M. and Ruf, M.
Journal: Analysis & PDE, (2018), vol. 11, no.2, 499-553.
Date: Mar 2018

###### Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional

Authors: Cotar, C. and Friesecke, G. and Klüppelberg, C.
Journal: Archive for Rational Mechanics and Analysis, 228 (3):891-922
Date: 2018
DOI: 10.1007/s00205-017-1208-y

###### Motion of discrete interfaces in low-contrast random environments

Author: Ruf, Matthias
Journal: ESAIM: COCV, Volume 24, Number 3, July–September 2018
Date: Oct 2017
DOI: 10.1051/cocv/2017067

###### Crystallization in two dimensions and a discrete Gauss-Bonnet theorem

Author: L. De Luca, G. Friesecke
Journal: J Nonlinear Sci 28, 69-90, 2017
Date: Jun 2017

###### Classification of Particle Numbers with Unique Heitmann-Radin Minimizer

Author: L. De Luca, G. Friesecke
Journal: J. Stat. Phys. 167, Issue 6, 1586–1592, 2017
Date: Apr 2017

###### Interfaces, modulated phases and textures in lattice systems

Authors: Braides, A. and Cicalese, M.
Journal: Arch. Rat. Mech. Anal., 223, (2017), 977-1017
Date: Feb 2017

###### The Zak transform on strongly proper G-spaces and its applications

Author: Jüstel, D.
Journal: Journal of the London Math. Society, 97:47-76
Date: 2017
DOI: 10.1112/jlms.12097

###### Crystalline Motion of Interfaces Between Patterns

Authors: A. Braides, M. Cicalese and Yip, N.K.
Journal: Journal of Statistical Physics October 2016, Volume 165, Issue 2, pp 274–319
Date: Oct 2016

###### Chirality transitions in frustrated S2-valued spin systems

Authors: Cicalese, Marco and Ruf, Matthias and Solombrino, Francesco
Journal: Math. Models Methods Appl. Sci., 26, (2016), no. 8, 1481-1529
Date: Jul 2016
DOI: 10.1142/S0218202516500366

###### Domain formation in magnetic polymer composites: an approach via stochastic homogenization

Authors: Alicandro, Roberto and Cicalese, Marco and Ruf, Matthias
Journal: Archive for Rational Mechanics and Analysis, 218(2):945--984
Date: 2015
DOI: 10.1007/s00205-015-0873-y

###### Frustrated ferromagnetic spin chains: a variational approach to chirality transitions

Authors: Cicalese, M. and Solombrino, F.
Journal: Journal of Nonlinear Science, 25(291-313)
Date: 2015

###### Twisted x-rays: incoming waveforms yielding discrete diffraction patterns for helical structures

Authors: G. Friesecke, R. D. James and Jüstel, D.
Date: 2015

###### Metastability and dynamics of discrete topological singularities in two dimensions: a Gamma-convergence approach

Authors: R. Alicandro, L. De Luca, A. Garroni and Ponsiglione, M.
Journal: Archive for Rational Mechanics and Analysis, 214(1):269--330
Date: 2014

##### PhD thesis
###### Crystalline Order, Surface Energy Densities and Wulff Shapes: Emergence from Atomistic Models

Author: Au Yeung, Yuen
Date: 2013

#### Prof. Dr. Marco Cicalese   +

Projects: B08
University: TU München
E-Mail: cicalese[at]ma.tum.de
Website: http://www-m7.ma.tum.de/bin/view/Analysis/WebHome

#### Prof. Dr. Gero Friesecke   +

Projects: B08
University: TU München
E-Mail: gf[at]ma.tum.de
Website: http://www-m7.ma.tum.de/bin/view/Analysis/WebHome

#### Annika Bach   +

Projects: B08
University: TU München
E-Mail: annika.bach[at]ma.tum.de