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