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Towards a gradient flow for microstructure

Banff International Research Station for Mathematical Innovation and Discovery

A central problem of microstructure is to develop technologies capable of producing an arrangement, or ordering, of the material, in terms of mesoscopic parameters like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? We describe very briefly the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution, and illustrate why it should be considered a material property. Its identification as a gradient flow by our method is tantamount to exhibiting the harvested statistic as the iterates in a mass transport JKO implicit scheme, which we found astonishing. Consequently the GBCD is the solution, in some sense, of a Fokker-Planck Equation. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process. (joint work with P. Bardsley, K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, X.-Y. Lu and S. Ta'asan).

Mathematics; Partial differential equations; Statistical mechanics, structure of matter

video/mp4

Author affiliation: Carnegie Mellon University

2018-10-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67495

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/