Florian Theil

Relaxation of rate-independent evolution problems

The relaxation of certain time-evolution problems is investigated. As a conceptually simple example, we study elastically deformable bodies that undergo martensitic phase transformations. The movement of the phase boundaries is hindered by dry friction. The fundamental problem is that the phase distribution forms a highly oscillatory microstructure in space. Therefore, it is desirable to derive a coarse-grained system that describes the effective properties. We introduce a concept of relaxation of the evolution system and apply it to the case where only two phases occur and the elastic energy is quadratic. Finally, we present a candidate for the relaxation in the general case.

Packing twelve spherical caps to maximize tangencies

The maximum number of non-overlapping unit spheres in R3 that can simultaneously touch another unit sphere is given by the kissing number, k(3)=12. Here, we present a proof that the maximum number of tangencies in any kissing configuration is 24 and that, up to isomorphism, there are only two configurations for which this maximum is achieved. The result is motivated by a three-dimensional crystallization problem.

A Study of a Hamiltonian Model for Martensitic Phase fransformations Including Microkinetic Energy

How can a system in a macroscopically stable state explore energetically more favorable states, which are far away from the current equilibrium state? Based on continuum mechanical considerations, the authors derive a Boussinesq-type equation [ILLEGIBLE] which models the dynamics of martensitic phase transformations. Here p > 0 is the mass density, 3O2fu is a regularization term that models the inertial forces of oscillations within a representative volume of length /, and of is a nonmonotone stress-strain relation. The solutions of the system, which the authors refer to as the micrmkinetically regularized wave equation, exhibit strong oscillations after times of order / and relaxation of spatial averages can be confirmed. This means that macroscopic fluctuations of the solutions decay, to the benefit of microscopic fluctuations. From the macroscopic point of view, this can be interpreted as a transformation of macroscopic kinetic energy into heat, i.e., as energy dissipation (despite the fact the authors consider a conservative system). The mathematical analysis for the microkinetically regularized wave equation consists of two parts. First, the authors present some analytical and numerical results on the propagation of phase boundaries and relaxation effects. Despite the fact that the model is conservative, it exhibits hysteretic behavior. Such behavior is usually interpreted in macroscopic models in terms of a dissipative threshold, which the driving force must overcome to ensure that the phase transformation proceeds. The threshold value depends on the volume of the transformed phase as observed in known experiments. Second, the authors investigate the dynamics of oscillatory solutions. Their mathematical tools are Young measures, which describe the one-point statistics of the fluctuations. They present a formalism that allows them to describe the effective dynamics of rapidly fluctuating solutions. The extended system has nontrivial equilibria that are only visible when oscillatory solutions are considered. The new method enables them to derive a numerical scheme for oscillatory solutions based on particle methods.

OPTIMAL UNCERTAINTY QUANTIFICATION FOR LEGACY DATA OBSERVATIONS OF LIPSCHITZ FUNCTIONS

We consider the problem of providing optimal uncertainty quantification (UQ) – and hence rigorous certification – for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.

Γ-Limit for Transition Paths of Maximal Probability

Chemical reactions can be modeled via diffusion processes conditioned to make a transition between specified molecular configurations representing the state of the system before and after the chemical reaction. In particular the model of Brownian dynamics—gradient flow subject to additive noise—is frequently used. If the chemical reaction is specified to take place on a given time interval, then the most likely path taken by the system is a minimizer of the Onsager-Machlup functional. The Γ-limit of this functional is determined explicitly in the case where the temperature is small and the transition time scales as the inverse temperature.

A Semigroup Approach to the Justification of Kinetic Theory

This paper develops a method to rigorously show the validity of continuum description for the deterministic dynamics of many interacting particles with random initial data. We consider a hard sphere flow where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density

The derivation of the linear Boltzmann equation from a Rayleigh gas particle model

f_0(u,v)

Dissipative Systems in Contact with a Heat Bath: Application to Andrade Creep

depending on

Energy Transport in Harmonic Lattices

d

On rate-independent hysteresis models

-dimensional position