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Analysis of gradient descents in random energies and heat baths

By Timothy John Sullivan


This thesis concerns the mathematical analysis of random gradient descent\ud evolutions as models for rate-independent dissipative systems under the influence\ud of thermal effects. The basic notions of the theory of gradient descents\ud (especially rate-independent evolutions) are reviewed in chapter 2.\ud Chapters 3 and 4 focus on the scaling regime in which the microstructure\ud dominates the thermal effects and comprise a rigorous justification of rateindependent\ud processes in smooth, convex energies as scaling limits of ratedependent\ud gradient descents in energies that have rapidly-oscillating random\ud microstructure: chapter 3 treats the one-dimensional case with quite a broad\ud class of random microstructures; chapter 4 treats a case in which the microstructure\ud is modeled by a sum of “dent functions” that are scattered in\ud Rn using a suitable point process. Chapters 5 and 6 focus on the opposite\ud scaling regime: a gradient descent system (typically a rate-independent process)\ud is placed in contact with a heat bath. The method used to “thermalize”\ud a gradient descent is an interior-point regularization of the Moreau–Yosida\ud incremental problem for the original gradient descent. Chapter 5 treats\ud the heuristics and generalities; chapter 6 treats the case of 1-homogeneous\ud dissipation (rate independence) and shows that the heat bath destroys the\ud rate independence in a controlled and deterministic way, and that the effective\ud dynamics are a gradient descent in the original energetic potential\ud but with respect to a different and non-trivial effective dissipation potential.\ud The appendices contain some auxiliary definitions and results, most of them\ud standard in the literature, that are used in the main text

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