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
Discrete & Continuous Dynamical Systems - A 2011, 30(2): 427-454 doi: 10.3934/dcds.2011.30.427
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
Distortion and entropy for automorphisms of free groups
Richard Sharp
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 347-363 doi: 10.3934/dcds.2010.26.347
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

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