§ All authors contributed equally to this work.Martensitic transformations are diffusionless solid-to-solid phase transitions characterized by a rapid change of crystal structure, observed in metals, alloys, ceramics, and proteins 1,2 . Phenomenologically, they come in two widely different classes. In steels, quenching generates a microstructure which remains essentially unchanged upon subsequent loading or heating; the transformation is not reversible 1 . In shape-memory alloys on the other hand the microstructures formed on cooling are easily manipulated by loads and disappear upon reheating, and the transformation is reversible 3,4 . Here we explain these sharp differences on the basis of the change in crystal symmetry during the transition. In particular, we show that martensitic transformations fall into two categories. In one case 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. These transformations are therefore irreversible, as observed in steels. In the other case, the energy barrier to lattice-invariant shears can be much higher than that pertaining to the phase change. Consequently, transformations of this type can occur with virtually no plasticity and can be reversible, as for shape-memory alloys.Martensitic transformations are at the basis of numerous technological applications. Most notable amongst these is in steel, where the transformation induced by quenching (fast cooling) is exploited for enhancing the alloy's strength 1 . Another is the fascinating shape-memory effect in alloys like Nitinol, used in medical and engineering devices 3 . Martensitic phase changes are also exploited to toughen structural ceramics 5 such as zirconia, and observed in biological systems such as the tail sheath of the T4 bacteriophage virus 6 . Ideas originating from the study of these transformations have led to improved materials for actuation (ferromagnetic shape-memory alloys 7,8 and ferroelectrics 9 ) and to candidates for artificial muscles 10 . Finally, the rich microstructure (distinctive patterns developed at scales ranging from a few nanometers to a few microns) that accompanies these transformations, has made this a valuable theoretical sand-box for the development of multi-scale modeling tools 11 .

The points which converge to ∞ under iteration of the maps z −→ λ exp(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 Karpińska for specific choices of λ.

We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional J h characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional K h . We establish a new connection between these systems by proving that J h and K h are equal up to second order in h as h → 0.This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.

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.