Stationary non-equilibrium states of infinite harmonic systems View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1977-06

AUTHORS

Herbert Spohn, Joel L. Lebowitz

ABSTRACT

We investigate the existence, properties and approach to stationary non-equilibrium states of infinite harmonic crystals. For classical systems these stationary states are, like the Gibbs states, Gaussian measures on the phase space of the infinite system (analogues results are true for quantum systems). Their ergodic properties are the same as those of the equilibrium states: e.g. for ordered periodic crystals they are Bernoulli. Unlike the equilibrium states however they are not “stable” towards perturbations in the potential. We are particularly concerned here with states in which there is a non-vanishing steady heat flux passing through “every point” of the infinite system. Such “superheat-conducting” states are of course only possible in systems in which Fourier's law does not hold: the perfect harmonic crystal being an example of such a system. For a one dimensional system, we find such states (explicitely) as limits, whent→∞, of time evolved initial states μi in which the “left” and “right” parts of the infinite crystal are in “equilibrium” at different temperatures, βL−L≠βR−1, and the “middle” part is in an arbitrary state. We also investigate the limit of these stationary (t→∞) states as the coupling strength λ between the “system” and the “reservoirs” goes to zero. In this limit we obtain a product state, where the reservoirs are in equilibrium at temperatures βL−1 and βR−1 and the system is in the unique stationary state of the reduced dynamics in the weak coupling limit. More... »

PAGES

97-120

References to SciGraph publications

  • 1974-06. Markovian master equations in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1974-03. Ergodic properties of an infinite system of particles moving independently in a periodic field in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1976-02. Stability and equilibrium states of infinite classical systems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1976-06. Markovian master equations. II in MATHEMATISCHE ANNALEN
  • 1973-09. The harmonic oscillator in a heat bath in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1972-09. Approach to equilibrium of free quantum systems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1975. Partial Differential Equations and Related Topics, Ford Foundation Sponsored Program at Tulane University, January to May, 1974 in NONE
  • 1974-09. The Bloch equations in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1974-09. Stability and equilibrium states in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1975. Time evolution and ergodic properties of harmonic systems in DYNAMICAL SYSTEMS, THEORY AND APPLICATIONS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf01614132

    DOI

    http://dx.doi.org/10.1007/bf01614132

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1015305894


    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/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }, 
          {
            "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"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Yeshiva University", 
              "id": "https://www.grid.ac/institutes/grid.268433.8", 
              "name": [
                "Belfer Graduate School of Science, Yeshiva University, 10033, New York, NY, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Spohn", 
            "givenName": "Herbert", 
            "id": "sg:person.0762212765.01", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0762212765.01"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Yeshiva University", 
              "id": "https://www.grid.ac/institutes/grid.268433.8", 
              "name": [
                "Belfer Graduate School of Science, Yeshiva University, 10033, New York, NY, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Lebowitz", 
            "givenName": "Joel L.", 
            "id": "sg:person.015317013331.48", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015317013331.48"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "https://doi.org/10.1016/0031-8914(73)90249-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1003426129"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/0031-8914(73)90249-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1003426129"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/s0031-8914(53)80008-x", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1007482832"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01667915", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012013256", 
              "https://doi.org/10.1007/bf01667915"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01667915", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012013256", 
              "https://doi.org/10.1007/bf01667915"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01609407", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1014667636", 
              "https://doi.org/10.1007/bf01609407"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01609407", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1014667636", 
              "https://doi.org/10.1007/bf01609407"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01651541", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1015777029", 
              "https://doi.org/10.1007/bf01651541"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01651541", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1015777029", 
              "https://doi.org/10.1007/bf01651541"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01877712", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032950474", 
              "https://doi.org/10.1007/bf01877712"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01877712", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032950474", 
              "https://doi.org/10.1007/bf01877712"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/3-540-07171-7_3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1033377347", 
              "https://doi.org/10.1007/3-540-07171-7_3"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bfb0070592", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1035922206", 
              "https://doi.org/10.1007/bfb0070592"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bfb0070592", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1035922206", 
              "https://doi.org/10.1007/bfb0070592"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01351898", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1037136830", 
              "https://doi.org/10.1007/bf01351898"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01351898", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1037136830", 
              "https://doi.org/10.1007/bf01351898"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01651544", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039887132", 
              "https://doi.org/10.1007/bf01651544"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01651544", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039887132", 
              "https://doi.org/10.1007/bf01651544"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01608389", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1044026527", 
              "https://doi.org/10.1007/bf01608389"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/s0031-8914(56)90009-x", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1047614754"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01646030", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1051838867", 
              "https://doi.org/10.1007/bf01646030"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01646030", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1051838867", 
              "https://doi.org/10.1007/bf01646030"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.1665793", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1057743872"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.1665794", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1057743873"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.1666713", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1057744781"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.1705319", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1057774969"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.523188", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1058100208"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physreva.1.1086", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060464382"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physreva.1.1086", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060464382"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1143/ptp.39.236", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1063132803"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1143/ptps.45.56", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1063144934"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "1977-06", 
        "datePublishedReg": "1977-06-01", 
        "description": "We investigate the existence, properties and approach to stationary non-equilibrium states of infinite harmonic crystals. For classical systems these stationary states are, like the Gibbs states, Gaussian measures on the phase space of the infinite system (analogues results are true for quantum systems). Their ergodic properties are the same as those of the equilibrium states: e.g. for ordered periodic crystals they are Bernoulli. Unlike the equilibrium states however they are not \u201cstable\u201d towards perturbations in the potential. We are particularly concerned here with states in which there is a non-vanishing steady heat flux passing through \u201cevery point\u201d of the infinite system. Such \u201csuperheat-conducting\u201d states are of course only possible in systems in which Fourier's law does not hold: the perfect harmonic crystal being an example of such a system. For a one dimensional system, we find such states (explicitely) as limits, whent\u2192\u221e, of time evolved initial states \u03bci in which the \u201cleft\u201d and \u201cright\u201d parts of the infinite crystal are in \u201cequilibrium\u201d at different temperatures, \u03b2L\u2212L\u2260\u03b2R\u22121, and the \u201cmiddle\u201d part is in an arbitrary state. We also investigate the limit of these stationary (t\u2192\u221e) states as the coupling strength \u03bb between the \u201csystem\u201d and the \u201creservoirs\u201d goes to zero. In this limit we obtain a product state, where the reservoirs are in equilibrium at temperatures \u03b2L\u22121 and \u03b2R\u22121 and the system is in the unique stationary state of the reduced dynamics in the weak coupling limit.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1007/bf01614132", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1136216", 
            "issn": [
              "0010-3616", 
              "1432-0916"
            ], 
            "name": "Communications in Mathematical Physics", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "2", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "54"
          }
        ], 
        "name": "Stationary non-equilibrium states of infinite harmonic systems", 
        "pagination": "97-120", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "48ece84b67e9df156c0ce55cd1590b84806f60b1b7b0c1baa43bb1c6aee53c17"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/bf01614132"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1015305894"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/bf01614132", 
          "https://app.dimensions.ai/details/publication/pub.1015305894"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-11T13:31", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000370_0000000370/records_46757_00000000.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "http://link.springer.com/10.1007/BF01614132"
      }
    ]
     

    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/bf01614132'

    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/bf01614132'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf01614132'

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

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf01614132'


     

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

    141 TRIPLES      21 PREDICATES      48 URIs      19 LITERALS      7 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/bf01614132 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N6328baf652d64604a88910e9013535f9
    4 schema:citation sg:pub.10.1007/3-540-07171-7_3
    5 sg:pub.10.1007/bf01351898
    6 sg:pub.10.1007/bf01608389
    7 sg:pub.10.1007/bf01609407
    8 sg:pub.10.1007/bf01646030
    9 sg:pub.10.1007/bf01651541
    10 sg:pub.10.1007/bf01651544
    11 sg:pub.10.1007/bf01667915
    12 sg:pub.10.1007/bf01877712
    13 sg:pub.10.1007/bfb0070592
    14 https://doi.org/10.1016/0031-8914(73)90249-8
    15 https://doi.org/10.1016/s0031-8914(53)80008-x
    16 https://doi.org/10.1016/s0031-8914(56)90009-x
    17 https://doi.org/10.1063/1.1665793
    18 https://doi.org/10.1063/1.1665794
    19 https://doi.org/10.1063/1.1666713
    20 https://doi.org/10.1063/1.1705319
    21 https://doi.org/10.1063/1.523188
    22 https://doi.org/10.1103/physreva.1.1086
    23 https://doi.org/10.1143/ptp.39.236
    24 https://doi.org/10.1143/ptps.45.56
    25 schema:datePublished 1977-06
    26 schema:datePublishedReg 1977-06-01
    27 schema:description We investigate the existence, properties and approach to stationary non-equilibrium states of infinite harmonic crystals. For classical systems these stationary states are, like the Gibbs states, Gaussian measures on the phase space of the infinite system (analogues results are true for quantum systems). Their ergodic properties are the same as those of the equilibrium states: e.g. for ordered periodic crystals they are Bernoulli. Unlike the equilibrium states however they are not “stable” towards perturbations in the potential. We are particularly concerned here with states in which there is a non-vanishing steady heat flux passing through “every point” of the infinite system. Such “superheat-conducting” states are of course only possible in systems in which Fourier's law does not hold: the perfect harmonic crystal being an example of such a system. For a one dimensional system, we find such states (explicitely) as limits, whent→∞, of time evolved initial states μi in which the “left” and “right” parts of the infinite crystal are in “equilibrium” at different temperatures, βL−L≠βR−1, and the “middle” part is in an arbitrary state. We also investigate the limit of these stationary (t→∞) states as the coupling strength λ between the “system” and the “reservoirs” goes to zero. In this limit we obtain a product state, where the reservoirs are in equilibrium at temperatures βL−1 and βR−1 and the system is in the unique stationary state of the reduced dynamics in the weak coupling limit.
    28 schema:genre research_article
    29 schema:inLanguage en
    30 schema:isAccessibleForFree false
    31 schema:isPartOf N6a58ab09425d4754812ae62d8b33a273
    32 Ndcd46f0a6ce0479aac149bfa25728d9c
    33 sg:journal.1136216
    34 schema:name Stationary non-equilibrium states of infinite harmonic systems
    35 schema:pagination 97-120
    36 schema:productId N1a4003e322994aee9e56ff092e243903
    37 Nb1f346e918aa4beb9ab3b5f602ae8ba9
    38 Neae7b786ece64873a54dad5c772967a9
    39 schema:sameAs https://app.dimensions.ai/details/publication/pub.1015305894
    40 https://doi.org/10.1007/bf01614132
    41 schema:sdDatePublished 2019-04-11T13:31
    42 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    43 schema:sdPublisher N1619e7e1ff5943d4b5a352b66f814bed
    44 schema:url http://link.springer.com/10.1007/BF01614132
    45 sgo:license sg:explorer/license/
    46 sgo:sdDataset articles
    47 rdf:type schema:ScholarlyArticle
    48 N1619e7e1ff5943d4b5a352b66f814bed schema:name Springer Nature - SN SciGraph project
    49 rdf:type schema:Organization
    50 N1a4003e322994aee9e56ff092e243903 schema:name readcube_id
    51 schema:value 48ece84b67e9df156c0ce55cd1590b84806f60b1b7b0c1baa43bb1c6aee53c17
    52 rdf:type schema:PropertyValue
    53 N6328baf652d64604a88910e9013535f9 rdf:first sg:person.0762212765.01
    54 rdf:rest N9f43cc2f50c244699b01a58b9fbe222d
    55 N6a58ab09425d4754812ae62d8b33a273 schema:volumeNumber 54
    56 rdf:type schema:PublicationVolume
    57 N9f43cc2f50c244699b01a58b9fbe222d rdf:first sg:person.015317013331.48
    58 rdf:rest rdf:nil
    59 Nb1f346e918aa4beb9ab3b5f602ae8ba9 schema:name doi
    60 schema:value 10.1007/bf01614132
    61 rdf:type schema:PropertyValue
    62 Ndcd46f0a6ce0479aac149bfa25728d9c schema:issueNumber 2
    63 rdf:type schema:PublicationIssue
    64 Neae7b786ece64873a54dad5c772967a9 schema:name dimensions_id
    65 schema:value pub.1015305894
    66 rdf:type schema:PropertyValue
    67 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    68 schema:name Mathematical Sciences
    69 rdf:type schema:DefinedTerm
    70 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    71 schema:name Pure Mathematics
    72 rdf:type schema:DefinedTerm
    73 sg:journal.1136216 schema:issn 0010-3616
    74 1432-0916
    75 schema:name Communications in Mathematical Physics
    76 rdf:type schema:Periodical
    77 sg:person.015317013331.48 schema:affiliation https://www.grid.ac/institutes/grid.268433.8
    78 schema:familyName Lebowitz
    79 schema:givenName Joel L.
    80 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015317013331.48
    81 rdf:type schema:Person
    82 sg:person.0762212765.01 schema:affiliation https://www.grid.ac/institutes/grid.268433.8
    83 schema:familyName Spohn
    84 schema:givenName Herbert
    85 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0762212765.01
    86 rdf:type schema:Person
    87 sg:pub.10.1007/3-540-07171-7_3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1033377347
    88 https://doi.org/10.1007/3-540-07171-7_3
    89 rdf:type schema:CreativeWork
    90 sg:pub.10.1007/bf01351898 schema:sameAs https://app.dimensions.ai/details/publication/pub.1037136830
    91 https://doi.org/10.1007/bf01351898
    92 rdf:type schema:CreativeWork
    93 sg:pub.10.1007/bf01608389 schema:sameAs https://app.dimensions.ai/details/publication/pub.1044026527
    94 https://doi.org/10.1007/bf01608389
    95 rdf:type schema:CreativeWork
    96 sg:pub.10.1007/bf01609407 schema:sameAs https://app.dimensions.ai/details/publication/pub.1014667636
    97 https://doi.org/10.1007/bf01609407
    98 rdf:type schema:CreativeWork
    99 sg:pub.10.1007/bf01646030 schema:sameAs https://app.dimensions.ai/details/publication/pub.1051838867
    100 https://doi.org/10.1007/bf01646030
    101 rdf:type schema:CreativeWork
    102 sg:pub.10.1007/bf01651541 schema:sameAs https://app.dimensions.ai/details/publication/pub.1015777029
    103 https://doi.org/10.1007/bf01651541
    104 rdf:type schema:CreativeWork
    105 sg:pub.10.1007/bf01651544 schema:sameAs https://app.dimensions.ai/details/publication/pub.1039887132
    106 https://doi.org/10.1007/bf01651544
    107 rdf:type schema:CreativeWork
    108 sg:pub.10.1007/bf01667915 schema:sameAs https://app.dimensions.ai/details/publication/pub.1012013256
    109 https://doi.org/10.1007/bf01667915
    110 rdf:type schema:CreativeWork
    111 sg:pub.10.1007/bf01877712 schema:sameAs https://app.dimensions.ai/details/publication/pub.1032950474
    112 https://doi.org/10.1007/bf01877712
    113 rdf:type schema:CreativeWork
    114 sg:pub.10.1007/bfb0070592 schema:sameAs https://app.dimensions.ai/details/publication/pub.1035922206
    115 https://doi.org/10.1007/bfb0070592
    116 rdf:type schema:CreativeWork
    117 https://doi.org/10.1016/0031-8914(73)90249-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1003426129
    118 rdf:type schema:CreativeWork
    119 https://doi.org/10.1016/s0031-8914(53)80008-x schema:sameAs https://app.dimensions.ai/details/publication/pub.1007482832
    120 rdf:type schema:CreativeWork
    121 https://doi.org/10.1016/s0031-8914(56)90009-x schema:sameAs https://app.dimensions.ai/details/publication/pub.1047614754
    122 rdf:type schema:CreativeWork
    123 https://doi.org/10.1063/1.1665793 schema:sameAs https://app.dimensions.ai/details/publication/pub.1057743872
    124 rdf:type schema:CreativeWork
    125 https://doi.org/10.1063/1.1665794 schema:sameAs https://app.dimensions.ai/details/publication/pub.1057743873
    126 rdf:type schema:CreativeWork
    127 https://doi.org/10.1063/1.1666713 schema:sameAs https://app.dimensions.ai/details/publication/pub.1057744781
    128 rdf:type schema:CreativeWork
    129 https://doi.org/10.1063/1.1705319 schema:sameAs https://app.dimensions.ai/details/publication/pub.1057774969
    130 rdf:type schema:CreativeWork
    131 https://doi.org/10.1063/1.523188 schema:sameAs https://app.dimensions.ai/details/publication/pub.1058100208
    132 rdf:type schema:CreativeWork
    133 https://doi.org/10.1103/physreva.1.1086 schema:sameAs https://app.dimensions.ai/details/publication/pub.1060464382
    134 rdf:type schema:CreativeWork
    135 https://doi.org/10.1143/ptp.39.236 schema:sameAs https://app.dimensions.ai/details/publication/pub.1063132803
    136 rdf:type schema:CreativeWork
    137 https://doi.org/10.1143/ptps.45.56 schema:sameAs https://app.dimensions.ai/details/publication/pub.1063144934
    138 rdf:type schema:CreativeWork
    139 https://www.grid.ac/institutes/grid.268433.8 schema:alternateName Yeshiva University
    140 schema:name Belfer Graduate School of Science, Yeshiva University, 10033, New York, NY, USA
    141 rdf:type schema:Organization
     




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


    ...