DCDS
An entropy based theory of the grain boundary character distribution
Katayun Barmak Eva Eggeling Maria Emelianenko Yekaterina Epshteyn David Kinderlehrer Richard Sharp Shlomo Ta'asan
Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains—a texture—appropriate for a desired set of material properties. Here we discuss the role of energy in texture development, measured by a character distribution. We derive an entropy based theory based on mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, this distribution behaves like the solution to a Fokker-Planck Equation.
keywords: Texture Development Kantorovich-Rubinstein-Wasserstein Metric. Large Metastable Networks Large scale simulation Critical Event Model Fokker-Planck Equation Free Energy Coarsening Entropy Based Theory
DCDS
Distortion and entropy for automorphisms of free groups
Richard Sharp
Recently, several numerical invariants have been introduced to characterize the distortion induced by automorphisms of a free group. We unify these by interpreting them in terms of an entropy function of a kind familiar in thermodynamic ergodic theory. We draw an analogy between this approach and the Manhattan curve associated to a pair of hyperbolic surfaces.
keywords: entropy. subshifts of finite typ automorphisms distortion spectrum thermodynamic formalism Free groups
DCDS
Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples
Richard Sharp
For conformal graph directed Markov systems, we construct a spectral triple from which one can recover the associated conformal measure via a Dixmier trace. As a particular case, we can recover the Patterson-Sullivan measure for a class of Kleinian groups.
keywords: Dixmier trace Spectral triple conformal graph directed Markov system conformal measure Patterson-Sullivan measure.

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