# 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
• M. Cicalese and G.P. Leonardi.
Maximal fluctuations on periodic lattices: an approach via quantitative Wulff inequalities.
Commun. Math. Phys., September 2019. preprint.
URL: http://cvgmt.sns.it/paper/4195/.
• G. Orlando M. Cicalese and M. Ruf.
From the N-clock model to the XY model: emergence of concentration effects in the variational analysis.
preprint, August 2019.
URL: http://cvgmt.sns.it/paper/4432/.
• A. Bach M. Cicalese and A. Braides.
Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions.
preprint, July 2019.
URL: http://cvgmt.sns.it/paper/4406/.
• A. Bach, M. Cicalese, and M. Ruf.
Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size.
preprint, February 2019.
arXiv:1902.08437.
• Gero Friesecke, Daniel Matthes, and Bernhard Schmitzer.
Barycenters for the Hellinger–Kantorovich distance over $\mathbb R^d$.
preprint at arXiv, 2019.
arXiv:1910.14572.
• M. Forster M. Cicalese and G. Orlando.
Variational analysis of a two-dimensional frustrated spin system: emergence and rigidity of chirality transitions.
SIAM J. Math. Anal., 2019. preprint.
arXiv:1904.07792.
• Marco Cicalese, Antoine Gloria, and Matthias Ruf.
From statistical polymer physics to nonlinear elasticity.
preprint, September 2018.
arXiv:1809.00598.
• Rufat Badal, Marco Cicalese, Lucia De Luca, and Marcello Ponsiglione.
Γ-Convergence Analysis of a Generalized $XY$ Model: Fractional Vortices and String Defects.
Communications in Mathematical Physics, 358(2):705–739, March 2018.
doi:10.1007/s00220-017-3026-3.
• A. Braides, M. Cicalese, and M. Ruf.
Continuum limit and stochastic homogenization of discrete ferromagnetic thin films.
Analysis & PDE, (2018), vol. 11, no.2, 499-553., March 2018.
URL: https://msp.org/apde/2018/11-2/p06.xhtml.
• C. Cotar, G. Friesecke, and C. Klüppelberg.
Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional.
Archive for Rational Mechanics and Analysis, 228 (3):891–922, 2018.
doi:10.1007/s00205-017-1208-y.
• Matthias Ruf.
Motion of discrete interfaces in low-contrast random environments.
ESAIM: COCV, Volume 24, Number 3, July–September 2018, October 2017.
doi:10.1051/cocv/2017067.
• G. Friesecke L. De Luca.
Crystallization in two dimensions and a discrete Gauss-Bonnet theorem.
J Nonlinear Sci 28, 69-90, 2017, June 2017.
• G. Friesecke L. De Luca.
Classification of Particle Numbers with Unique Heitmann-Radin Minimizer.
J. Stat. Phys. 167, Issue 6, 1586–1592, 2017, April 2017.
• A. Braides and M. Cicalese.
Interfaces, modulated phases and textures in lattice systems.
Arch. Rat. Mech. Anal., 223, (2017), 977-1017, February 2017.
• D. Jüstel.
The Zak transform on strongly proper G-spaces and its applications.
Journal of the London Math. Society, 97:47–76, 2017.
doi:10.1112/jlms.12097.
• M. Cicalese A. Braides and N.K. Yip.
Crystalline Motion of Interfaces Between Patterns.
Journal of Statistical Physics October 2016, Volume 165, Issue 2, pp 274–319, October 2016.
• Marco Cicalese, Matthias Ruf, and Francesco Solombrino.
Chirality transitions in frustrated S2-valued spin systems.
Math. Models Methods Appl. Sci., 26, (2016), no. 8, 1481-1529, July 2016.
doi:10.1142/S0218202516500366.
• Roberto Alicandro, Marco Cicalese, and Matthias Ruf.
Domain formation in magnetic polymer composites: an approach via stochastic homogenization.
Archive for Rational Mechanics and Analysis, 218(2):945–984, 2015.
doi:10.1007/s00205-015-0873-y.
• M. Cicalese and F. Solombrino.
Frustrated ferromagnetic spin chains: a variational approach to chirality transitions.
Journal of Nonlinear Science, 25(291-313), 2015.
• G. Friesecke, R. D. James and D. Jüstel.
Twisted x-rays: incoming waveforms yielding discrete diffraction patterns for helical structures.
2015.
arXiv:1506.04240.
• A. Garroni R. Alicandro, L. De Luca and M. Ponsiglione.
Metastability and dynamics of discrete topological singularities in two dimensions: a Gamma-convergence approach.
Archive for Rational Mechanics and Analysis, 214(1):269–330, 2014.

##### PhD thesis
• Yuen Au Yeung.
Crystalline Order, Surface Energy Densities and Wulff Shapes: Emergence from Atomistic Models.
Dissertation, Technische Universität München, München, 2013.
URL: http://mediatum.ub.tum.de/node?id=1142127.

#### 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

#### Arseniy Tsipenyuk   +

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