# Research Area C1

# Mathematical modeling of matter and materials

## PIs: Conti, Disertori, Müller, Neitzel, Niethammer

## Contributions by Nota, Ortiz, Schweitzer, Velázquez

## Topic and goals

Mathematical methods to analyze and connect descriptions of matter at different scales are crucial to understanding and designing advanced materials. Through strategic hiring, Bonn has built a powerful team to address key new challenges of the next decade in a close interaction of modeling, analysis, simulation, and optimization: from toy models in statistical mechanics with discrete symmetries to more realistic models with continuous symmetries and phase transitions; from equilibrium, asymptotic self-similarity and local interactions to non-equilibrium, complex space-time behavior and long-range effects in kinetic equations; from energy based methods in statics to the dynamics of defects and microstructure; from the simulation and analysis of single physics models to more realistic coupled real-world models, involving the full modeling–analysis–simulation–optimization cycle.

## State of the art, our expertise

**Statistical mechanics of elasticity and of systems with continuous symmetries**. Compared to the well known theory for fluids and gases, a rigorous statistical mechanics description of solids remains a major challenge. Continuum theory suggests that new notions of convexity, such as quasiconvexity, which have been hardly explored in statistical mechanics, will play a crucial role. If the Gibbs measure has a discrete symmetry, the problem can sometimes be reformulated as a system with a finite single spin set, e.g., as an Ising or Potts model, for which powerful tools such as Pirogov–Sinai theory, cluster expansions and renormalization group methods are available. A major long-standing problem is to extend these tools to the case of continuous symmetry, when massless Goldstone modes may arise. The Gibbs measure is expected to concentrate in the vicinity of a nonlinear sigma model, but this phenomenon is still far from being rigorously understood. Multiscale analysis and renormalization methods can be extended to such systems, but they currently require strong additional regularity conditions, such as analytic perturbations or reflection positivity. In the last years, powerful new methods such as the new renormalization approach by Brydges et al. have led to breakthroughs, among them an understanding of the self-avoiding random walk in the critical dimension. In a different direction, Adams–Buchholz–Kotecký–Müller are using this approach to study convexity and polyconvexity of the free energy of mass and spring models with realistic elastic interactions at low temperature. Regarding quantum diffusion for disordered materials, Disertori et al. [DSZ10] constructed a new multiscale approach, based on a family of Ward identities, that allowed them to study a spin model where no renormalization arguments or convexity estimates seem to apply.

**Kinetic models**. Niethammer, Nota, and Velázquez investigate fundamental models such as the Boltzmann equation, the Landau equation, or the coagulation equation. Key phenomena include concentration of mass in singular points [EV15] and self-similar long-time behavior. A fully rigorous picture for the coagulation equation exists only for three specific rate kernels, while for general kernels only partial results are available (see, e.g., [NV13]). For the Boltzmann equation, situations far from equilibrium pose many challenges. There are special homoenergetic solutions by Truesdell and Galkin, which can be used to describe Boltzmann gases subject to external shear, dilatation or compression, but their long time asymptotics is not well understood, due to a lack of detailed balance. Existence, uniqueness and stability of the distribution of velocities for these solutions is a completely open problem. Erbar’s gradient flow interpretation of the spatially homogeneous Boltzmann equation provides a link to RA B3. A rigorous derivation of kinetic equations from particle systems, such as Boltzmann or Lorentz gases, where a single, tagged particle moves in force fields produced by a random distribution of scatterers, is so far only available for potentials that decrease sufficiently fast.

**Microstructure in solids**. Calculus of variations methods led to a new understanding of material microstructure, with profound applications such as the discovery of ultra-low hysteresis transformations. In this context, Müller et al. [ZJM09] systematically analyzed energy barriers in incompatibility induced hysteresis. Conti and Ortiz studied the rigorous derivation of continuum models for plasticity from discrete variational models for dislocations and the relation between microstruc- ture and material failure [CGO15, CGM16, CO16]. It will be crucial for applications to develop a time-dependent relaxation theory, especially in situations where microstructures occur. A major trend in modeling, simulation and optimization is to move closer to real-world applications. In optimal control of partial differential equations (PDEs), for example, there is a growing interest in the analysis of nonconvex problems governed by nonlinear equations. Results for optimal control of fracture propagation and of quasilinear heat equations were obtained by Neitzel et al. [NWW17, BN18]. Regarding modeling, new data-driven descriptions are being developed (see IRU D1). In numerics, there is a growing interest in physics-based approximation approaches such as reduced-order to cope with the challenges induced by the required discrete model sizes. These challenges are even more pronounced in optimization, since the optimality systems consist of coupled forward and backward Problems.

## Research program

**Statistical mechanics of elasticity and of systems with continuous symmetries**. We will develop new tools to tackle long-standing open problems, such as solid-solid phase transitions, ’quasicrystalline’ phases as in liquid crystals, or the metal-insulator transition. A standard way to describe the presence of disorder is to add randomness in the Gibbs measure, e.g., in the Gaussian part. Here we will build on tools for random Schrödinger operators or random conductance models and stochastic homogenization, the latter providing a link to our expertise in microstructure and PDEs. We will also consider a random reference lattice to model crystals with defects such as grain boundaries and dislocations, and make connections to the continuum description of microstructures. To analyze quasi-crystalline phases one must replace ’atoms’ by non-isotropic molecules. Rigorous results are very limited even for discrete symmetries [DG13]. To consider continuous symmetries we will study nonlinear sigma models with target space Sn-1; as in the XY and Heisenberg model.

**Kinetic models**. While all explicitly known examples show self-similar long-time behavior for the coagulation equation, we conjecture that it is in general not universal, but that for certain rate 47kernels, solutions exhibit oscillatory behavior in self-similar variables [HNV17]. Our goal is to rigorously establish such oscillatory behavior in coagulation equations. Furthermore, we will investigate the gelation transition for kernels different from the solvable product kernel. For Boltzmann gases subject to strong shear, we will study the long-time asymptotics of homoenergetic solutions. Recent work of Nota–James–Velázquez suggests that the long-time behavior depends significantly on the competition between the collision kernel and applied forces. We will also address the major challenge of deriving kinetic equations in systems with longrange interactions in a few key examples, starting with Lorentz gases. In particular, we will explore the rigorous derivation of the linear Landau equation for Coulomb potentials, as well as problems analogous to Lorentz gases for quantum particles, specifically fermions. Here the combination of expertise in kinetic equations and in renormalization will be crucial. Furthermore, we plan to derive the nonlinear Landau equation from interacting particle systems, starting from a truncated Boguliubov hierarchy, and the coagulation equation for a system of particles moving and coalescing in a shear flow, thus extending work in [NV17].

**Evolution and control of defects and microstructure in solids**. While many problems in the static theory of microstructure are still open, the largest challenge is the study of the evolution of defects and microstructure. This involves formulating kinetic equations with appropriate mobility laws, deriving the driving forces from knowledge of the energetics, and understanding the predictions of the models, including the derivation of effective equations for the macroscopic material behavior. For example, understanding the motion of dislocations in solids is crucial for the study of plasticity. They can be described as measures supported on curves; the kinetic equation has to take into account this specific kinematics as well as the singularity of the Peach–Köhler force. This will involve a close interaction of methods in kinetic equations and variational methods for elastic microstructures. A very different kinematics arises for the motion of point defects, such as hydrogen atoms absorbed in metals, which directly influence brittleness. To carry out the full modeling–analysis–simulation–optimization cycle, Neitzel and Schweitzer will develop fast problem-dependent approximation techniques and linear and nonlinear solver technology for strongly coupled nonlinear systems of PDEs. This requires advanced numerical analysis of PDE-constrained optimization problems and optimization algorithms.

## Summary

We will address key challenges in the modeling, analysis, simulation and design of advanced materials, such as phase transitions for models with continuous symmetries, non-equilibrium and long-range effects in kinetic equations, dynamics of defects and microstructure, and the efficient simulation of multi-physics models. The multiscale character of these problems provides a natural link to RA C4. The expertise in microstructure of solids in this research area will also be crucial for IRU D1.

## Bibliography

[BN18] L. Bonifacius and I. Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic PDEs. *Math. Control Relat. Fields, 8(1):1–34, 2018*.

[CGM16] S. Conti, A. Garroni, and S. Müller. Dislocation microstructures and strain-gradient plasticity with one active slip plane. *J. Mech. Phys. Solids, 93:240–251, 2016*.

[CGO15] S. Conti, A. Garroni, and M. Ortiz. The line-tension approximation as the dilute limit of linearelastic dislocations.* Arch. Ration. Mech. Anal., 218(2):699–755, 2015*.

[CO16] S. Conti and M. Ortiz. Optimal scaling in solids undergoing ductile fracture by crazing.* Arch. Ration. Mech. Anal., 219(2):607–636, 2016*.

[DG13] M. Disertori and A. Giuliani. The nematic phase of a system of long hard rods. *Comm. Math. Phys., 323(1):143–175, 2013*.

[DSZ10] M. Disertori, T. Spencer, and M. R. Zirnbauer. Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. *Comm. Math. Phys., 300(2):435–486, 2010*.

[EV15] M. Escobedo and J. J. L. Velázquez. Finite time blow-up and condensation for the bosonic Nordheim equation. *Invent. Math., 200(3):761–847, 2015*.

[HNV17] M. Herrmann, B. Niethammer, and J. J. L. Velázquez. Instabilities and oscillations in coagulation equations with kernels of homogeneity one. *Quart. Appl. Math., 75(1):105–130, 2017*.

[NV13] B. Niethammer and J. J. L. Velázquez. Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels. *Comm. Math. Phys., 318(2):505–532, 2013*.

[NV17] A. Nota and J. J. L. Velázquez. On the growth of a particle coalescing in a Poisson distribution of obstacles.* Comm. Math. Phys., 354(3):957–1013, 2017*.

[NWW17] I. Neitzel, T. Wick, and W. Wollner. An optimal control problem governed by a regularized phase-field fracture propagation model. *SIAM J. Control Optim., 55(4):2271–2288, 2017*.

[ZJM09] Z. Zhang, R. D. James, and S. Müller. Energy barriers and hysteresis in martensitic phase transformations.* Acta Mater., 57(15):4332–4352, 2009*.