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Mathematical Physics Seminar

Order Metrics and Diffusion Spreadability to Characterize Two-Phase Microstructures Across Length Scales

Salvatore Torquato - Princeton University

Location:  Zoom
Date & time: Wednesday, 14 September 2022 at 10:45AM - 11:45AM

The capacity to devise order metrics to characterize and classify microstructures of multiphase heterogeneous media across length scales is an outstanding but highly challenging task, given the richness of the possible geometries and topologies of the phases that can arise. I describe an initial investigation to formulate order metrics to characterize the degree of order/disorder of hyperuniform and nonhyperuniform microstructures of two-phase media in d-dimensional Euclidean space across length scales via the local volume-fraction variance [1].
In the second part of my talk, I consider the so-called spreadability, S(t), which is a measure of the spreadability of diffusion information in two-phase media as a function of time t when a solute is contained in one of the phases at t=0 [2]. I derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of S(t ) enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, I show that the “excess” spreadability, S(?) ? S(t ), decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the “slowest” being antihyperuniform media. The time-dependent spreadability is a powerful, dynamic-based figure of merit to probe and classify the spectrum of possible microstructures of two-phase media across length scales. I also establish a connection between the spreadability and an outstanding problem in discrete geometry, namely, microstructures with “fast” spreadabilities are also those that can be derived from efficient “coverings” of space.
1. S. Torquato, M. Skolnick, and J. Kim, ``Local order metrics for two-phase media across length scales,"
J. Phys. A: Math. & Theoret., 55 274003 (2022).
2. S. Torquato, ``Diffusion spreadability as a probe of the microstructure of complex media across length scales,"
Phys. Rev. E, 104 054102 (2021).

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