Johannes Zimmer

Crystal symmetry and the reversibility of martensitic transformations

Martensitic transformations are diffusionless, solid-to-solid phase transitions, and have been observed in metals, alloys, ceramics and proteins. They are characterized by a rapid change of crystal structure, accompanied by the development of a rich microstructure. Martensitic transformations can be irreversible, as seen in steels upon quenching, or they can be reversible, such as those observed in shape-memory alloys. In the latter case, the microstructures formed on cooling are easily manipulated by loads and disappear upon reheating. Here, using mathematical theory and numerical simulation, we explain these sharp differences in behaviour on the basis of the change in crystal symmetry during the transition. We find that a necessary condition for reversibility is that the symmetry groups of the parent and product phases be included in a common finite symmetry group. In these cases, the energy barrier to lattice-invariant shear is generically higher than that pertaining to the phase change and, consequently, transformations of this type can occur with virtually no plasticity. Irreversibility is inevitable in all other martensitic transformations, where the energy barrier to plastic deformation (via lattice-invariant shears, as in twinning or slip) is no higher than the barrier to the phase change itself. Various experimental observations confirm the importance of the symmetry of the stable states in determining the macroscopic reversibility of martensitic transformations.

Escaping Points of Exponential Maps

The points which converge to ∞ under iteration of the maps z↦λexp(z) for λ ∈ C/{0} are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ. It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpinska for specific choices of λ.

GENERIC formalism of a Vlasov–Fokker–Planck equation and connection to large-deviation principles

In this paper we discuss the connections between a Vlasov–Fokker–Planck equation and an underlying microscopic particle system, and we interpret those connections in the context of the GENERIC framework (Ottinger 2005 Beyond Equilibrium Thermodynamics (New York: Wiley-Interscience)). This interpretation provides (a) a variational formulation for GENERIC systems, (b) insight into the origin of this variational formulation, and (c) an explanation of the origins of the conditions that GENERIC places on its constitutive elements, notably the so-called degeneracy or non-interaction conditions. This work shows how the general connection between large-deviation principles on one hand and gradient-flow structures on the other hand extends to non-reversible particle systems.

Large deviations and gradient flows

In recent work we uncovered intriguing connections between Otto’s characterization of diffusion as an entropic gradient flow on the one hand and large-deviation principles describing the microscopic picture (Brownian motion) on the other. In this paper, we sketch this connection, show how it generalizes to a wider class of systems and comment on consequences and implications. Specifically, we connect macroscopic gradient flows with large-deviation principles, and point out the potential of a bigger picture emerging: we indicate that, in some non-equilibrium situations, entropies and thermodynamic free energies can be derived via large-deviation principles. The approach advocated here is different from the established hydrodynamic limit passage but extends a link that is well known in the equilibrium situation.

Existence of Dynamic Phase Transitions in a One-Dimensional Lattice Model with Piecewise Quadratic Interaction Potential

The existence of travelling waves in an atomistic model for martensitic phase transitions is the focus of this study. The elastic energy is assumed to be piecewise quadratic, with two wells representing two stable phases. We develop a framework such that the existence of subsonic heteroclinic waves in a bi-infinite chain of atoms can be proved rigorously. The key is to represent the solution as a sum of a (here explicitly given) profile and a corrector in

Three-dimensional model of martensitic transformations with elasto-plastic effects

L^2(\mathbb{R})

Upscaling from particle models to entropic gradient flows

. It is demonstrated that the kinetic relation can be easily inferred from this framework.

Calculation of long time classical trajectories: algorithmic treatment and applications for molecular systems

Martensitic transformations with elasto-plastic effects caused by the formation of dislocations in a parent austenite phase are studied by using a phase-field description. The method presented in this paper extends an existing microelastic model for the simulation of coherent martensitic transformations by taking into account the dislocation dynamics. Computational results show the difference between coherent and partially coherent martensitic transformation and illuminate elasto-plastic effects of transformation dislocations on the final martensitic microstructure.

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

We prove that, for the case of Gaussians on the real line, the functional derived by a time discretization of the diffusion equation as entropic gradient flow is asymptotically equivalent to the rate functional derived from the underlying microscopic process. This result strengthens a conjecture that the same statement is actually true for all measures with second finite moment.

Stability of travelling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects

We study the problem of finding a path that joins a given initial state with a final one, where the evolution is governed by classical (Hamiltonian) dynamics. A new algorithm for the computation of long time transition trajectories connecting two configurations is presented. In particular, a strategy for finding transition paths between two stable basins is established. The starting point is the formulation of the equation of motion of classical mechanics in the framework of Jacobis principle; a shortening procedure inspired by Birkhoffs method is then applied to find geodesic solutions. Numerical examples are given for Mullers potential and the collinear reaction H(2) + H --> H + H(2).