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Continuous Time Random Walk Based Theory for a One-Dimensional Coarsening Model

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Abstract

In this work we propose a master equation describing evolution of the velocity statistics in a one-dimensional coarsening model motivated by the studies of polycrystalline materials. The model postulates the dynamics of a large number of intervals—referred to as domains—on the real line. The length of the intervals changes during evolution and the intervals are removed from the system once their length reaches zero. The coarsening process observed in this model exhibits a number of interesting features, such as nonhomogeneous inter-arrival times between reconfiguration events and development of spatiotemporally self-similar distributions.

We generalize the standard continuous time random walk (CTRW) theory to include time-dependent jumps and subject it to time-dependent temporal rescaling to obtain an accurate non-homogeneous Poisson description of the coarsening process in the one-dimensional model. The theory leads to the evolution equation having self-similar solutions observed in simulations.

The new framework allows to accurately estimate coarsening rates and characterize resulting steady-state distribution for the domain energies described by a power law of a uniformly distributed quantity. Although derived here in the context of a one-dimensional systems, this work naturally extends to higher dimensional CTRW coarsening models.

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Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Diego Torrejon & Maria Emelianenko

  2. Department of Mathematics, The University of Akron, Akron, OH, 44325, USA

    Dmitry Golovaty

Authors
  1. Diego Torrejon
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  2. Maria Emelianenko
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  3. Dmitry Golovaty
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Correspondence to Diego Torrejon.

Additional information

DT was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1356109. ME was partially supported by National Science Foundation grant DMS-1056821.

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Torrejon, D., Emelianenko, M. & Golovaty, D. Continuous Time Random Walk Based Theory for a One-Dimensional Coarsening Model. J Elliptic Parabol Equ 2, 189–206 (2016). https://doi.org/10.1007/BF03377401

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  • DOI: https://doi.org/10.1007/BF03377401

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