coarsening coefficient long time length random walk diffusion coefficient dimensional model single point better representation lines process walk grain growth interval length false probability density https://scigraph.springernature.com/explorer/license/ intervals Louat theoretical interest 1992 stochastic model model detail en 2022-05-20T07:47 A One Dimensional Stochastic Model of Coarsening boundary points average interval length 1992-01-01 asymptotic limit A one dimensional model of the coarsening of intervals on a line is considered in which the boundary points between adjacent intervals execute independent random walk with a common diffusion coefficient D/2; when two boundary points meet, they coalesce into a single point that continues to execute random walk. We calculate the following quantities in the asymptotic limit of long times: 1) The average interval length (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\;\left\langle l \right\rangle = \sqrt {\pi Dt}$$\end{document} , 2) the time-independent probability density for the reduced length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = 1/\left\langle l \right\rangle$$\end{document} , and 3) the expected value of dl/dt for a given l, which is positive for and negative for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l > {l_c} = \sqrt {2/\pi } \left\langle l \right\rangle$$\end{document} and negative for l < lc. The model is similar to one proposed by Louat for grain growth. Although it is not a good representation of the details of most physical processes of coarsening, it is of theoretical interest since it is one of the few cases for which analytic results can be obtained. dL/dt common diffusion coefficient interest independent random walks time values physical processes growth 101-105 analytic results dt most physical processes representation quantity point density https://doi.org/10.1007/978-1-4613-9211-8_7 adjacent intervals cases limit dimensional stochastic model results chapter chapters W. W. Mullins doi 10.1007/978-1-4613-9211-8_7 On the Evolution of Phase Boundaries 978-1-4613-9213-2 978-1-4613-9211-8 Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, 8309 Wean Hall, 15213, Pittsburgh, PA, USA Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, 8309 Wean Hall, 15213, Pittsburgh, PA, USA Springer Nature - SN SciGraph project pub.1016403566 dimensions_id Statistics Springer Nature McFadden Geoffrey B. Gurtin Morton E. Mathematical Sciences