Effective long range order and phase transitions in finite, macroscopic one and two dimensional systems
Identifieur interne : 001793 ( Istex/Corpus ); précédent : 001792; suivant : 001794Effective long range order and phase transitions in finite, macroscopic one and two dimensional systems
Auteurs : Y. ImrySource :
- Annals of Physics [ 0003-4916 ] ; 1969.
English descriptors
- Teeft :
- Bogoljubov inequality, Bose, Bose condensation, Bose einstein condensation, Bose systems, Boundary condition, Boundary points, Condensation temperature, Const, Cornell university, Correlation range, Correlation ranges, Corresponding lattice, Critical region, Curie weiss, Differential equation, Dimensional bose, Dimensional systems, Enhancement, Film thickness, Finite, Finite system, Finite systems, Finite temperature, Free energy, Good approximation, Ground state, Harmonic, Harmonic crystal, Harmonic oscillators, Heisenberg, Heisenberg model, High temperature expansion, High temperature susceptibility, High temperatures, Imry, Infinite range correlations, Integral equation, Interaction, Interaction range, Interaction ranges, Interaction strength, Interesting effects, Long range, Long range enhancement, Long range interaction, Long range interactions, Long range order, Magnetic field, Magnetic systems, Magnetization, Momentum state, Nearest neighbours ising chain, Oscillator, Other hand, Other systems, Other terms, Other words, Partition function, Pergamon press, Periodic boundary conditions, Phase equilibrium, Phase transition, Phase transitions, Phys, Physical picture, Power series expansion, Quadratic form, Range interaction, Range interactions, Range order, Recent proof, Second order, Second order term, Short range, Short range interactions, Similar considerations, Temperature range, Temperature region, Thermodynamic, Thermodynamic limit, Thin bose, Thin films, Tunneling problem.
Abstract
Abstract: The concept of long range order is discussed for finite macroscopic systems. It is shown that if the interaction range or strength, in units of kT, is of O(log N), long range order is obtained in one dimensional systems. Short range and long range interactions are able to combine to cause long range order even when each of the two interactions by itself is much too weak to do so. A physical picture of the effects of the long range interactions is presented and supported by detailed calculations. We also show that the theorem stating that the two dimensional Heisenberg Model does not exhibit long range order is inapplicable when the interaction strength in units kT is of O(log N), or when the range of the interaction is of O(log N)14. The two dimensional Bose gas will condense at temperatures of O(T0log N), where T0 is the condensation temperature in three dimensions. The recent proof that superfluid long range order is impossible in two dimensions, is shown to be based, for finite N, on the analogous fact in the free gas, which is invalid according to the above. Consideration of thin Bose films shows that the transition from three dimensional (condensation at T0) to two dimensional (condensation below T0) behavior occurs at film thickness of order log N. This result can be of practical importance.
Url:
DOI: 10.1016/0003-4916(69)90345-5
Links to Exploration step
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Imry</name><affiliation><mods:affiliation>Israel Atomic Energy Commission, Soreq Nuclear Research Centre, Yavne, Israel</mods:affiliation></affiliation><affiliation><mods:affiliation>Department of Applied Physics, Cornell University, Ithaca, New York 14850 USA∗∗Work performed at this address was supported by the U.S.A.E.C. under contract AT(30-1)3326.</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:46E569F392F87602167CD8B8F00DE3470D97165B</idno><date when="1969" year="1969">1969</date><idno type="doi">10.1016/0003-4916(69)90345-5</idno><idno type="url">https://api.istex.fr/ark:/67375/6H6-T1038W0K-6/fulltext.pdf</idno><idno type="wicri:Area/Istex/Corpus">001793</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001793</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Effective long range order and phase transitions in finite, macroscopic one and two dimensional systems</title><author><name sortKey="Imry, Y" sort="Imry, Y" uniqKey="Imry Y" first="Y" last="Imry">Y. Imry</name><affiliation><mods:affiliation>Israel Atomic Energy Commission, Soreq Nuclear Research Centre, Yavne, Israel</mods:affiliation></affiliation><affiliation><mods:affiliation>Department of Applied Physics, Cornell University, Ithaca, New York 14850 USA∗∗Work performed at this address was supported by the U.S.A.E.C. under contract AT(30-1)3326.</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Annals of Physics</title><title level="j" type="abbrev">YAPHY</title><idno type="ISSN">0003-4916</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1969">1969</date><biblScope unit="volume">51</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="27">27</biblScope></imprint><idno type="ISSN">0003-4916</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0003-4916</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="Teeft" xml:lang="en"><term>Bogoljubov inequality</term><term>Bose</term><term>Bose condensation</term><term>Bose einstein condensation</term><term>Bose systems</term><term>Boundary condition</term><term>Boundary points</term><term>Condensation temperature</term><term>Const</term><term>Cornell university</term><term>Correlation range</term><term>Correlation ranges</term><term>Corresponding lattice</term><term>Critical region</term><term>Curie weiss</term><term>Differential equation</term><term>Dimensional bose</term><term>Dimensional systems</term><term>Enhancement</term><term>Film thickness</term><term>Finite</term><term>Finite system</term><term>Finite systems</term><term>Finite temperature</term><term>Free energy</term><term>Good approximation</term><term>Ground state</term><term>Harmonic</term><term>Harmonic crystal</term><term>Harmonic oscillators</term><term>Heisenberg</term><term>Heisenberg model</term><term>High temperature expansion</term><term>High temperature susceptibility</term><term>High temperatures</term><term>Imry</term><term>Infinite range correlations</term><term>Integral equation</term><term>Interaction</term><term>Interaction range</term><term>Interaction ranges</term><term>Interaction strength</term><term>Interesting effects</term><term>Long range</term><term>Long range enhancement</term><term>Long range interaction</term><term>Long range interactions</term><term>Long range order</term><term>Magnetic field</term><term>Magnetic systems</term><term>Magnetization</term><term>Momentum state</term><term>Nearest neighbours ising chain</term><term>Oscillator</term><term>Other hand</term><term>Other systems</term><term>Other terms</term><term>Other words</term><term>Partition function</term><term>Pergamon press</term><term>Periodic boundary conditions</term><term>Phase equilibrium</term><term>Phase transition</term><term>Phase transitions</term><term>Phys</term><term>Physical picture</term><term>Power series expansion</term><term>Quadratic form</term><term>Range interaction</term><term>Range interactions</term><term>Range order</term><term>Recent proof</term><term>Second order</term><term>Second order term</term><term>Short range</term><term>Short range interactions</term><term>Similar considerations</term><term>Temperature range</term><term>Temperature region</term><term>Thermodynamic</term><term>Thermodynamic limit</term><term>Thin bose</term><term>Thin films</term><term>Tunneling problem</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: The concept of long range order is discussed for finite macroscopic systems. It is shown that if the interaction range or strength, in units of kT, is of O(log N), long range order is obtained in one dimensional systems. Short range and long range interactions are able to combine to cause long range order even when each of the two interactions by itself is much too weak to do so. A physical picture of the effects of the long range interactions is presented and supported by detailed calculations. We also show that the theorem stating that the two dimensional Heisenberg Model does not exhibit long range order is inapplicable when the interaction strength in units kT is of O(log N), or when the range of the interaction is of O(log N)14. The two dimensional Bose gas will condense at temperatures of O(T0log N), where T0 is the condensation temperature in three dimensions. The recent proof that superfluid long range order is impossible in two dimensions, is shown to be based, for finite N, on the analogous fact in the free gas, which is invalid according to the above. Consideration of thin Bose films shows that the transition from three dimensional (condensation at T0) to two dimensional (condensation below T0) behavior occurs at film thickness of order log N. 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It is shown that if the interaction range or strength, in units of kT, is of O(log N), long range order is obtained in one dimensional systems. Short range and long range interactions are able to combine to cause long range order even when each of the two interactions by itself is much too weak to do so. A physical picture of the effects of the long range interactions is presented and supported by detailed calculations. We also show that the theorem stating that the two dimensional Heisenberg Model does not exhibit long range order is inapplicable when the interaction strength in units kT is of O(log N), or when the range of the interaction is of O(log N)14. The two dimensional Bose gas will condense at temperatures of O(T0log N), where T0 is the condensation temperature in three dimensions. The recent proof that superfluid long range order is impossible in two dimensions, is shown to be based, for finite N, on the analogous fact in the free gas, which is invalid according to the above. Consideration of thin Bose films shows that the transition from three dimensional (condensation at T0) to two dimensional (condensation below T0) behavior occurs at film thickness of order log N. 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It is shown that if the interaction range or strength, in units of kT, is of O(log N), long range order is obtained in one dimensional systems. Short range and long range interactions are able to combine to cause long range order even when each of the two interactions by itself is much too weak to do so. A physical picture of the effects of the long range interactions is presented and supported by detailed calculations. We also show that the theorem stating that the two dimensional Heisenberg Model does not exhibit long range order is inapplicable when the interaction strength in units kT is of O(log N), or when the range of the interaction is of O(log N)14. The two dimensional Bose gas will condense at temperatures of O(T0log N), where T0 is the condensation temperature in three dimensions. The recent proof that superfluid long range order is impossible in two dimensions, is shown to be based, for finite N, on the analogous fact in the free gas, which is invalid according to the above. Consideration of thin Bose films shows that the transition from three dimensional (condensation at T0) to two dimensional (condensation below T0) behavior occurs at film thickness of order log N. This result can be of practical importance.</p></abstract></profileDesc><revisionDesc><change when="1969">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/ark:/67375/6H6-T1038W0K-6/fulltext.txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Elsevier, elements deleted: tail"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//ES//DTD journal article DTD version 4.5.2//EN//XML" URI="art452.dtd" name="istex:docType"></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla" xml:lang="en"><item-info><jid>YAPHY</jid><aid>69903455</aid><ce:pii>0003-4916(69)90345-5</ce:pii><ce:doi>10.1016/0003-4916(69)90345-5</ce:doi><ce:copyright type="unknown" year="1969"></ce:copyright></item-info><head><ce:title>Effective long range order and phase transitions in finite, macroscopic one and two dimensional systems</ce:title><ce:author-group><ce:author><ce:given-name>Y</ce:given-name><ce:surname>Imry</ce:surname><ce:cross-ref refid="AFF1"><ce:sup loc="post">a</ce:sup></ce:cross-ref><ce:cross-ref refid="AFF2"><ce:sup loc="post">b</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="AFF1"><ce:label>a</ce:label><ce:textfn>Israel Atomic Energy Commission, Soreq Nuclear Research Centre, Yavne, Israel</ce:textfn></ce:affiliation><ce:affiliation id="AFF2"><ce:label>b</ce:label><ce:textfn>Department of Applied Physics, Cornell University, Ithaca, New York 14850 USA<ce:cross-ref refid="FN1"><ce:sup loc="post">∗</ce:sup></ce:cross-ref><ce:footnote id="FN1"><ce:label>∗</ce:label><ce:note-para>Work performed at this address was supported by the U.S.A.E.C. under contract AT(30-1)3326.</ce:note-para></ce:footnote></ce:textfn></ce:affiliation></ce:author-group><ce:date-received day="31" month="5" year="1968"></ce:date-received><ce:abstract class="author"><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para view="all" id="simple-para.0010">The concept of long range order is discussed for finite macroscopic systems. It is shown that if the interaction range or strength, in units of <ce:italic>kT</ce:italic>, is of <ce:italic>O</ce:italic>(log <ce:italic>N</ce:italic>), long range order is obtained in one dimensional systems. Short range and long range interactions are able to combine to cause long range order even when each of the two interactions by itself is much too weak to do so. A physical picture of the effects of the long range interactions is presented and supported by detailed calculations. We also show that the theorem stating that the two dimensional Heisenberg Model does not exhibit long range order is inapplicable when the interaction strength in units <ce:italic>kT</ce:italic> is of <ce:italic>O</ce:italic>(log <ce:italic>N</ce:italic>), or when the range of the interaction is of <math altimg="si1.gif">O(<rm>log</rm> N)<sup loc="post"><fr shape="sol" align="c" style="s"><nu>1</nu><de>4</de></fr></sup></math>. The two dimensional Bose gas will condense at temperatures of <math altimg="si2.gif">O(<fr shape="sol" align="c" style="s"><nu>T<inf loc="post">0</inf></nu><de><rm>log</rm> N</de></fr>)</math>, where <ce:italic>T</ce:italic><ce:inf loc="post">0</ce:inf> is the condensation temperature in three dimensions. The recent proof that superfluid long range order is impossible in two dimensions, is shown to be based, for finite <ce:italic>N</ce:italic>, on the analogous fact in the free gas, which is invalid according to the above. Consideration of thin Bose films shows that the transition from three dimensional (condensation at <ce:italic>T</ce:italic><ce:inf loc="post">0</ce:inf>) to two dimensional (condensation below <ce:italic>T</ce:italic><ce:inf loc="post">0</ce:inf>) behavior occurs at film thickness of order log <ce:italic>N</ce:italic>. This result can be of practical importance.</ce:simple-para></ce:abstract-sec></ce:abstract></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Effective long range order and phase transitions in finite, macroscopic one and two dimensional systems</title></titleInfo><titleInfo type="alternative" lang="en" contentType="CDATA"><title>Effective long range order and phase transitions in finite, macroscopic one and two dimensional systems</title></titleInfo><name type="personal"><namePart type="given">Y</namePart><namePart type="family">Imry</namePart><affiliation>Israel Atomic Energy Commission, Soreq Nuclear Research Centre, Yavne, Israel</affiliation><affiliation>Department of Applied Physics, Cornell University, Ithaca, New York 14850 USA∗∗Work performed at this address was supported by the U.S.A.E.C. under contract AT(30-1)3326.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="Full-length article" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">1969</dateIssued><copyrightDate encoding="w3cdtf">1969</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><abstract lang="en">Abstract: The concept of long range order is discussed for finite macroscopic systems. It is shown that if the interaction range or strength, in units of kT, is of O(log N), long range order is obtained in one dimensional systems. Short range and long range interactions are able to combine to cause long range order even when each of the two interactions by itself is much too weak to do so. A physical picture of the effects of the long range interactions is presented and supported by detailed calculations. We also show that the theorem stating that the two dimensional Heisenberg Model does not exhibit long range order is inapplicable when the interaction strength in units kT is of O(log N), or when the range of the interaction is of O(log N)14. The two dimensional Bose gas will condense at temperatures of O(T0log N), where T0 is the condensation temperature in three dimensions. The recent proof that superfluid long range order is impossible in two dimensions, is shown to be based, for finite N, on the analogous fact in the free gas, which is invalid according to the above. Consideration of thin Bose films shows that the transition from three dimensional (condensation at T0) to two dimensional (condensation below T0) behavior occurs at film thickness of order log N. This result can be of practical importance.</abstract><relatedItem type="host"><titleInfo><title>Annals of Physics</title></titleInfo><titleInfo type="abbreviated"><title>YAPHY</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">1969</dateIssued></originInfo><identifier type="ISSN">0003-4916</identifier><identifier type="PII">S0003-4916(00)X0369-7</identifier><part><date>1969</date><detail type="volume"><number>51</number><caption>vol.</caption></detail><detail type="issue"><number>1</number><caption>no.</caption></detail><extent unit="issue-pages"><start>1</start><end>202</end></extent><extent unit="pages"><start>1</start><end>27</end></extent></part></relatedItem><identifier type="istex">46E569F392F87602167CD8B8F00DE3470D97165B</identifier><identifier type="ark">ark:/67375/6H6-T1038W0K-6</identifier><identifier type="DOI">10.1016/0003-4916(69)90345-5</identifier><identifier type="PII">0003-4916(69)90345-5</identifier><identifier type="ArticleID">69903455</identifier><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-HKKZVM7B-M">elsevier</recordContentSource></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/ark:/67375/6H6-T1038W0K-6/record.json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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