Journal
RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI
Volume 28, Issue 4, Pages 777-805Publisher
EUROPEAN MATHEMATICAL SOC
DOI: 10.4171/RLM/785
Keywords
Coarsening; texture development; large metastable networks; critical event model; entropy-based theory; free energy; Fokker-Planck equation; Kantorovich-Rubinstein-Wasserstein metric
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Funding
- NSF [DMR 0520425, DMS 0405343, DMS 0305794, DMS 0806703, DMS 0635983, DMS 0915013, DMS 1112984, OISE-0967140]
- Simons Foundation [415673]
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A central problem of microstructure is to develop technologies capable of producing an arrangement, or ordering, of a polycrystalline material, in terms of mesoscopic parameters, like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? Our goal is to describe the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution discovered recently, and to illustrate why it should be considered a material property. For the GBCD statistic, we have developed a theory that relies on mass transport and entropy. The focus of this paper is its identification as a gradient flow in the sense of De Giorgi, as illustrated by Ambrosio, Gigli, and Savare. In this way, the empirical texture statistic is revealed as a solution of a Fokker-Planck type equation whose evolution is determined by weak topology kinetics and whose limit behavior is a Boltzmann distribution. The identification as a gradient flow by our method is tantamount to exhibiting the harvested statistic as the iterates in a JKO implicit scheme. This requires several new ideas. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process.
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