A One Dimensional Stochastic Model of Coarsening View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1992

AUTHORS

W. W. Mullins

ABSTRACT

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. More... »

PAGES

101-105

Book

TITLE

On the Evolution of Phase Boundaries

ISBN

978-1-4613-9213-2
978-1-4613-9211-8

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-1-4613-9211-8_7

DOI

http://dx.doi.org/10.1007/978-1-4613-9211-8_7

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1016403566


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0104", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Statistics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, 8309 Wean Hall, 15213, Pittsburgh, PA, USA", 
          "id": "http://www.grid.ac/institutes/grid.147455.6", 
          "name": [
            "Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, 8309 Wean Hall, 15213, Pittsburgh, PA, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Mullins", 
        "givenName": "W. W.", 
        "id": "sg:person.013517176323.92", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013517176323.92"
        ], 
        "type": "Person"
      }
    ], 
    "datePublished": "1992", 
    "datePublishedReg": "1992-01-01", 
    "description": "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}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\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}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\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}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\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.", 
    "editor": [
      {
        "familyName": "Gurtin", 
        "givenName": "Morton E.", 
        "type": "Person"
      }, 
      {
        "familyName": "McFadden", 
        "givenName": "Geoffrey B.", 
        "type": "Person"
      }
    ], 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/978-1-4613-9211-8_7", 
    "inLanguage": "en", 
    "isAccessibleForFree": false, 
    "isPartOf": {
      "isbn": [
        "978-1-4613-9213-2", 
        "978-1-4613-9211-8"
      ], 
      "name": "On the Evolution of Phase Boundaries", 
      "type": "Book"
    }, 
    "keywords": [
      "dL/dt", 
      "intervals", 
      "interval length", 
      "long time", 
      "cases", 
      "length", 
      "dt", 
      "dimensional model", 
      "lines", 
      "theoretical interest", 
      "model", 
      "point", 
      "random walk", 
      "time", 
      "representation", 
      "results", 
      "adjacent intervals", 
      "values", 
      "growth", 
      "walk", 
      "better representation", 
      "interest", 
      "stochastic model", 
      "single point", 
      "quantity", 
      "density", 
      "detail", 
      "process", 
      "diffusion coefficient", 
      "coefficient", 
      "limit", 
      "analytic results", 
      "independent random walks", 
      "probability density", 
      "physical processes", 
      "dimensional stochastic model", 
      "boundary points", 
      "common diffusion coefficient", 
      "asymptotic limit", 
      "average interval length", 
      "most physical processes", 
      "coarsening", 
      "Louat", 
      "grain growth", 
      "coarsening of intervals", 
      "time-independent probability density"
    ], 
    "name": "A One Dimensional Stochastic Model of Coarsening", 
    "pagination": "101-105", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1016403566"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/978-1-4613-9211-8_7"
        ]
      }
    ], 
    "publisher": {
      "name": "Springer Nature", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/978-1-4613-9211-8_7", 
      "https://app.dimensions.ai/details/publication/pub.1016403566"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2022-01-01T19:25", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20220101/entities/gbq_results/chapter/chapter_449.jsonl", 
    "type": "Chapter", 
    "url": "https://doi.org/10.1007/978-1-4613-9211-8_7"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/978-1-4613-9211-8_7'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/978-1-4613-9211-8_7'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/978-1-4613-9211-8_7'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/978-1-4613-9211-8_7'


 

This table displays all metadata directly associated to this object as RDF triples.

111 TRIPLES      23 PREDICATES      72 URIs      65 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/978-1-4613-9211-8_7 schema:about anzsrc-for:01
2 anzsrc-for:0104
3 schema:author N0133695ca4a241f1872324cd6476d083
4 schema:datePublished 1992
5 schema:datePublishedReg 1992-01-01
6 schema:description 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.
7 schema:editor Nf134e387decb4973a14f9d502289c93e
8 schema:genre chapter
9 schema:inLanguage en
10 schema:isAccessibleForFree false
11 schema:isPartOf Ne3cdfa4f60b647ad9c0539b18b3aec97
12 schema:keywords Louat
13 adjacent intervals
14 analytic results
15 asymptotic limit
16 average interval length
17 better representation
18 boundary points
19 cases
20 coarsening
21 coarsening of intervals
22 coefficient
23 common diffusion coefficient
24 dL/dt
25 density
26 detail
27 diffusion coefficient
28 dimensional model
29 dimensional stochastic model
30 dt
31 grain growth
32 growth
33 independent random walks
34 interest
35 interval length
36 intervals
37 length
38 limit
39 lines
40 long time
41 model
42 most physical processes
43 physical processes
44 point
45 probability density
46 process
47 quantity
48 random walk
49 representation
50 results
51 single point
52 stochastic model
53 theoretical interest
54 time
55 time-independent probability density
56 values
57 walk
58 schema:name A One Dimensional Stochastic Model of Coarsening
59 schema:pagination 101-105
60 schema:productId N8f7bc1c7a91d46d98f50d46fe22b3503
61 Nc47d158efeea421c9ef66bc1f69b7ef4
62 schema:publisher N0529052d8ab747f3bd9134b48c5a2101
63 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016403566
64 https://doi.org/10.1007/978-1-4613-9211-8_7
65 schema:sdDatePublished 2022-01-01T19:25
66 schema:sdLicense https://scigraph.springernature.com/explorer/license/
67 schema:sdPublisher N95463f5194184a84b4273cbb5f4a1be1
68 schema:url https://doi.org/10.1007/978-1-4613-9211-8_7
69 sgo:license sg:explorer/license/
70 sgo:sdDataset chapters
71 rdf:type schema:Chapter
72 N0133695ca4a241f1872324cd6476d083 rdf:first sg:person.013517176323.92
73 rdf:rest rdf:nil
74 N0529052d8ab747f3bd9134b48c5a2101 schema:name Springer Nature
75 rdf:type schema:Organisation
76 N27884c6a7d4d499f99c218519f023c55 schema:familyName Gurtin
77 schema:givenName Morton E.
78 rdf:type schema:Person
79 N8cbfae3e218144c1b7c08f92174009fd rdf:first Nea2c4a226d31430e98cd8e3e8e2511aa
80 rdf:rest rdf:nil
81 N8f7bc1c7a91d46d98f50d46fe22b3503 schema:name doi
82 schema:value 10.1007/978-1-4613-9211-8_7
83 rdf:type schema:PropertyValue
84 N95463f5194184a84b4273cbb5f4a1be1 schema:name Springer Nature - SN SciGraph project
85 rdf:type schema:Organization
86 Nc47d158efeea421c9ef66bc1f69b7ef4 schema:name dimensions_id
87 schema:value pub.1016403566
88 rdf:type schema:PropertyValue
89 Ne3cdfa4f60b647ad9c0539b18b3aec97 schema:isbn 978-1-4613-9211-8
90 978-1-4613-9213-2
91 schema:name On the Evolution of Phase Boundaries
92 rdf:type schema:Book
93 Nea2c4a226d31430e98cd8e3e8e2511aa schema:familyName McFadden
94 schema:givenName Geoffrey B.
95 rdf:type schema:Person
96 Nf134e387decb4973a14f9d502289c93e rdf:first N27884c6a7d4d499f99c218519f023c55
97 rdf:rest N8cbfae3e218144c1b7c08f92174009fd
98 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
99 schema:name Mathematical Sciences
100 rdf:type schema:DefinedTerm
101 anzsrc-for:0104 schema:inDefinedTermSet anzsrc-for:
102 schema:name Statistics
103 rdf:type schema:DefinedTerm
104 sg:person.013517176323.92 schema:affiliation grid-institutes:grid.147455.6
105 schema:familyName Mullins
106 schema:givenName W. W.
107 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013517176323.92
108 rdf:type schema:Person
109 grid-institutes:grid.147455.6 schema:alternateName Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, 8309 Wean Hall, 15213, Pittsburgh, PA, USA
110 schema:name Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, 8309 Wean Hall, 15213, Pittsburgh, PA, USA
111 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...