Perturbative and numerical methods for stochastic nonlinear oscillators
Identifieur interne : 000749 ( Main/Corpus ); précédent : 000748; suivant : 000750Perturbative and numerical methods for stochastic nonlinear oscillators
Auteurs : Giuseppe Curci ; Erika D'AmbrosioSource :
- Physica A: Statistical Mechanics and its Applications [ 0378-4371 ] ; 1999.
Abstract
Interferometric gravitational wave detectors are devoted to pick up the effect induced on masses by gravitational waves. The variations of the length dividing two mirrors is measured through a laser interferometric technique. The Brownian motion of the masses related to the interferometer room temperature is a limit to the observation of astrophysical signals. It is referred to as thermal noise and it affects the sensitivity of both the projected and the future generation interferometers. In this paper, we investigate the relevance of small non-linear effects and point out their impact on the sensitivity curve of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, etc.) through perturbative methods and numerical simulations. We find that in the first-order approximation the constants characterizing the power spectrum density (PSD) are renormalized but it retains its typical shape. This is due to the fact that the involved Feynman diagrams are of tadpole type. Higher-order approximations are required to give rise to up-conversion effects. This result is predicted by the perturbative approach and is in agreement with the numerical results obtained by studying the system's non-linear response by numerically simulating its dynamics.
Url:
DOI: 10.1016/S0378-4371(99)00238-1
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CALTECH, Ligo Project, California Institute of Technology, MC 18-34, 1200 E, California Blvd., Pasadena, CA 91125, USA. 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The variations of the length dividing two mirrors is measured through a laser interferometric technique. The Brownian motion of the masses related to the interferometer room temperature is a limit to the observation of astrophysical signals. It is referred to as thermal noise and it affects the sensitivity of both the projected and the future generation interferometers. In this paper, we investigate the relevance of small non-linear effects and point out their impact on the sensitivity curve of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, etc.) through perturbative methods and numerical simulations. We find that in the first-order approximation the constants characterizing the power spectrum density (PSD) are renormalized but it retains its typical shape. This is due to the fact that the involved Feynman diagrams are of tadpole type. Higher-order approximations are required to give rise to up-conversion effects. This result is predicted by the perturbative approach and is in agreement with the numerical results obtained by studying the system's non-linear response by numerically simulating its dynamics.</div></front></TEI><istex><corpusName>elsevier</corpusName><author><json:item><name>Giuseppe Curci</name><affiliations><json:string>I.N.F.N. Sezione di Pisa, Via Livornese 1291, San Piero a Grado I-56010 Pisa, Italy</json:string><json:string>E-mail: curci@mailbox.difi.unipi.it</json:string></affiliations></json:item><json:item><name>Erika D'Ambrosio</name><affiliations><json:string>Dipartimento di Fisica dell'Università di Pisa, P.zza Torricelli 2, I-56126 Pisa, Italy</json:string><json:string>Corresponding author. CALTECH, Ligo Project, California Institute of Technology, MC 18-34, 1200 E, California Blvd., Pasadena, CA 91125, USA. Tel.: +1-625-395-8742; fax: +1-626-395-3814</json:string><json:string>E-mail: ambrosio@ligo.caltech.edu</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>02.50.-r</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>05.40.+j</value></json:item></subject><language><json:string>eng</json:string></language><originalGenre><json:string>Full-length article</json:string></originalGenre><abstract>Interferometric gravitational wave detectors are devoted to pick up the effect induced on masses by gravitational waves. The variations of the length dividing two mirrors is measured through a laser interferometric technique. The Brownian motion of the masses related to the interferometer room temperature is a limit to the observation of astrophysical signals. It is referred to as thermal noise and it affects the sensitivity of both the projected and the future generation interferometers. In this paper, we investigate the relevance of small non-linear effects and point out their impact on the sensitivity curve of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, etc.) through perturbative methods and numerical simulations. We find that in the first-order approximation the constants characterizing the power spectrum density (PSD) are renormalized but it retains its typical shape. This is due to the fact that the involved Feynman diagrams are of tadpole type. Higher-order approximations are required to give rise to up-conversion effects. This result is predicted by the perturbative approach and is in agreement with the numerical results obtained by studying the system's non-linear response by numerically simulating its dynamics.</abstract><qualityIndicators><score>7.232</score><pdfVersion>1.2</pdfVersion><pdfPageSize>408 x 660 pts</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>2</keywordCount><abstractCharCount>1265</abstractCharCount><pdfWordCount>6826</pdfWordCount><pdfCharCount>33146</pdfCharCount><pdfPageCount>23</pdfPageCount><abstractWordCount>186</abstractWordCount></qualityIndicators><title>Perturbative and numerical methods for stochastic nonlinear oscillators</title><pii><json:string>S0378-4371(99)00238-1</json:string></pii><genre><json:string>research-article</json:string></genre><host><volume>273</volume><pii><json:string>S0378-4371(00)X0115-X</json:string></pii><pages><last>351</last><first>329</first></pages><issn><json:string>0378-4371</json:string></issn><issue>3–4</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><title>Physica A: Statistical Mechanics and its Applications</title><publicationDate>1999</publicationDate></host><categories><wos><json:string>PHYSICS</json:string><json:string>PHYSICS, MULTIDISCIPLINARY</json:string></wos></categories><publicationDate>1999</publicationDate><copyrightDate>1999</copyrightDate><doi><json:string>10.1016/S0378-4371(99)00238-1</json:string></doi><id>350699F60102618FA2C7157F3F8176BB437F709C</id><score>1.2084253</score><fulltext><json:item><original>true</original><mimetype>application/pdf</mimetype><extension>pdf</extension><uri>https://api.istex.fr/document/350699F60102618FA2C7157F3F8176BB437F709C/fulltext/pdf</uri></json:item><json:item><original>false</original><mimetype>application/zip</mimetype><extension>zip</extension><uri>https://api.istex.fr/document/350699F60102618FA2C7157F3F8176BB437F709C/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/350699F60102618FA2C7157F3F8176BB437F709C/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Perturbative and numerical methods for stochastic nonlinear oscillators</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>ELSEVIER</publisher><availability><p>©1999 Elsevier Science B.V.</p></availability><date>1999</date></publicationStmt><notesStmt><note type="content">Fig. 1: Power density spectrum for a pendulum with a quality factor ∼100.</note><note type="content">Fig. 2: Bins correlation 〈| x ̃(f)|2| x ̃(f′)|2〉conn. The width of each bin is 2η=0.1 mHz and the adimensional coefficient is α=0.1. Such coefficient represents the level of non-linearity of the concerned system.</note><note type="content">Fig. 3: The expansion in α leads to good approximations of 〈x2〉. The approximated expression is no longer valid when α→1. There is an asymptotic limit 〈x2〉∼α−1/2KBT(mω02)−1 if α grows.</note><note type="content">Fig. 4: Graphical representation of x(t) and 〈x(t)x(t′)〉. When the average over the random force is taken, all crosses are joined in all possible ways. For example, one can get 〈x(t)x(t′)〉 by combining two crosses at a point in the corresponding product.</note><note type="content">Fig. 5: Second-order corrections for the two-point correlation function. There are contributions including tadpoles. A tadpole is nothing but a contraction of x with itself at the same time. Since 〈x(t)x(t)〉 is a constant tadpoles are insertions of constant terms.</note><note type="content">Fig. 6: In this figure the unperturbed theoretical curve related to the linear case and the power spectrum density corresponding to a numerical simulation of the non-linear one are both shown. Data are considered over a period T chosen as ∼12002π/ω0. In the frequency domain the variations due to non-linearities are concentrated in a band that is as narrow as Q is high. It seems that for Q→∞ even small non-linearities may be revealed.</note><note type="content">Fig. 7: In this figure a sketch of how the sensitivity curve for a pendulum is modified by a non-linear term is shown. As an example an effective reduction is achieved at low frequencies. The non-linear force causes the pendulum to oscillate with two important modes. In order to keep the amplitude of the up-converted fundamental frequency small, α (which is proportonal to T) is expected to be less than 0.1. These results are in accordance with the trend shown by our perturbative calculation.</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Perturbative and numerical methods for stochastic nonlinear oscillators</title><author xml:id="author-1"><persName><forename type="first">Giuseppe</forename><surname>Curci</surname></persName><email>curci@mailbox.difi.unipi.it</email><affiliation>I.N.F.N. Sezione di Pisa, Via Livornese 1291, San Piero a Grado I-56010 Pisa, Italy</affiliation></author><author xml:id="author-2"><persName><forename type="first">Erika</forename><surname>D'Ambrosio</surname></persName><email>ambrosio@ligo.caltech.edu</email><affiliation>Dipartimento di Fisica dell'Università di Pisa, P.zza Torricelli 2, I-56126 Pisa, Italy</affiliation><affiliation>Corresponding author. CALTECH, Ligo Project, California Institute of Technology, MC 18-34, 1200 E, California Blvd., Pasadena, CA 91125, USA. Tel.: +1-625-395-8742; fax: +1-626-395-3814</affiliation></author></analytic><monogr><title level="j">Physica A: Statistical Mechanics and its Applications</title><title level="j" type="abbrev">PHYSA</title><idno type="pISSN">0378-4371</idno><idno type="PII">S0378-4371(00)X0115-X</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1999"></date><biblScope unit="volume">273</biblScope><biblScope unit="issue">3–4</biblScope><biblScope unit="page" from="329">329</biblScope><biblScope unit="page" to="351">351</biblScope></imprint></monogr><idno type="istex">350699F60102618FA2C7157F3F8176BB437F709C</idno><idno type="DOI">10.1016/S0378-4371(99)00238-1</idno><idno type="PII">S0378-4371(99)00238-1</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1999</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Interferometric gravitational wave detectors are devoted to pick up the effect induced on masses by gravitational waves. The variations of the length dividing two mirrors is measured through a laser interferometric technique. The Brownian motion of the masses related to the interferometer room temperature is a limit to the observation of astrophysical signals. It is referred to as thermal noise and it affects the sensitivity of both the projected and the future generation interferometers. In this paper, we investigate the relevance of small non-linear effects and point out their impact on the sensitivity curve of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, etc.) through perturbative methods and numerical simulations. We find that in the first-order approximation the constants characterizing the power spectrum density (PSD) are renormalized but it retains its typical shape. This is due to the fact that the involved Feynman diagrams are of tadpole type. Higher-order approximations are required to give rise to up-conversion effects. This result is predicted by the perturbative approach and is in agreement with the numerical results obtained by studying the system's non-linear response by numerically simulating its dynamics.</p></abstract><textClass><keywords scheme="keyword"><list><head>PACS</head><item><term>02.50.-r</term></item><item><term>05.40.+j</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1999">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><original>false</original><mimetype>text/plain</mimetype><extension>txt</extension><uri>https://api.istex.fr/document/350699F60102618FA2C7157F3F8176BB437F709C/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Elsevier, elements deleted: ce:floats; body; 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:entity SYSTEM="gr1" NDATA="IMAGE" name="gr1"></istex:entity><istex:entity SYSTEM="gr2" NDATA="IMAGE" name="gr2"></istex:entity><istex:entity SYSTEM="gr3" NDATA="IMAGE" name="gr3"></istex:entity><istex:entity SYSTEM="gr4" NDATA="IMAGE" name="gr4"></istex:entity><istex:entity SYSTEM="gr5" NDATA="IMAGE" name="gr5"></istex:entity><istex:entity SYSTEM="gr6" NDATA="IMAGE" name="gr6"></istex:entity><istex:entity SYSTEM="gr7" NDATA="IMAGE" name="gr7"></istex:entity></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla"><item-info><jid>PHYSA</jid><aid>4896</aid><ce:pii>S0378-4371(99)00238-1</ce:pii><ce:doi>10.1016/S0378-4371(99)00238-1</ce:doi><ce:copyright type="full-transfer" year="1999">Elsevier Science B.V.</ce:copyright></item-info><head><ce:title>Perturbative and numerical methods for stochastic nonlinear oscillators</ce:title><ce:author-group><ce:author><ce:given-name>Giuseppe</ce:given-name><ce:surname>Curci</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>a</ce:sup></ce:cross-ref><ce:e-address>curci@mailbox.difi.unipi.it</ce:e-address></ce:author><ce:author><ce:given-name>Erika</ce:given-name><ce:surname>D'Ambrosio</ce:surname><ce:cross-ref refid="AFF2"><ce:sup>b</ce:sup></ce:cross-ref><ce:cross-ref refid="CORR1">*</ce:cross-ref><ce:e-address>ambrosio@ligo.caltech.edu</ce:e-address></ce:author><ce:affiliation id="AFF1"><ce:label>a</ce:label><ce:textfn>I.N.F.N. Sezione di Pisa, Via Livornese 1291, San Piero a Grado I-56010 Pisa, Italy</ce:textfn></ce:affiliation><ce:affiliation id="AFF2"><ce:label>b</ce:label><ce:textfn>Dipartimento di Fisica dell'Università di Pisa, P.zza Torricelli 2, I-56126 Pisa, Italy</ce:textfn></ce:affiliation><ce:correspondence id="CORR1"><ce:label>*</ce:label><ce:text>Corresponding author. CALTECH, Ligo Project, California Institute of Technology, MC 18-34, 1200 E, California Blvd., Pasadena, CA 91125, USA. Tel.: +1-625-395-8742; fax: +1-626-395-3814</ce:text></ce:correspondence></ce:author-group><ce:date-received day="22" month="2" year="1999"></ce:date-received><ce:abstract><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para>Interferometric gravitational wave detectors are devoted to pick up the effect induced on masses by gravitational waves. The variations of the length dividing two mirrors is measured through a laser interferometric technique. The Brownian motion of the masses related to the interferometer room temperature is a limit to the observation of astrophysical signals. It is referred to as thermal noise and it affects the sensitivity of both the projected and the future generation interferometers. In this paper, we investigate the relevance of small non-linear effects and point out their impact on the sensitivity curve of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, etc.) through perturbative methods and numerical simulations. We find that in the first-order approximation the constants characterizing the power spectrum density (PSD) are renormalized but it retains its typical shape. This is due to the fact that the involved Feynman diagrams are of tadpole type. Higher-order approximations are required to give rise to up-conversion effects. This result is predicted by the perturbative approach and is in agreement with the numerical results obtained by studying the system's non-linear response by numerically simulating its dynamics.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:keywords class="pacs"><ce:section-title>PACS</ce:section-title><ce:keyword><ce:text>02.50.-r</ce:text></ce:keyword><ce:keyword><ce:text>05.40.+j</ce:text></ce:keyword></ce:keywords></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Perturbative and numerical methods for stochastic nonlinear oscillators</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Perturbative and numerical methods for stochastic nonlinear oscillators</title></titleInfo><name type="personal"><namePart type="given">Giuseppe</namePart><namePart type="family">Curci</namePart><affiliation>I.N.F.N. Sezione di Pisa, Via Livornese 1291, San Piero a Grado I-56010 Pisa, Italy</affiliation><affiliation>E-mail: curci@mailbox.difi.unipi.it</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Erika</namePart><namePart type="family">D'Ambrosio</namePart><affiliation>Dipartimento di Fisica dell'Università di Pisa, P.zza Torricelli 2, I-56126 Pisa, Italy</affiliation><affiliation>Corresponding author. CALTECH, Ligo Project, California Institute of Technology, MC 18-34, 1200 E, California Blvd., Pasadena, CA 91125, USA. Tel.: +1-625-395-8742; fax: +1-626-395-3814</affiliation><affiliation>E-mail: ambrosio@ligo.caltech.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="Full-length article"></genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">1999</dateIssued><copyrightDate encoding="w3cdtf">1999</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Interferometric gravitational wave detectors are devoted to pick up the effect induced on masses by gravitational waves. The variations of the length dividing two mirrors is measured through a laser interferometric technique. The Brownian motion of the masses related to the interferometer room temperature is a limit to the observation of astrophysical signals. It is referred to as thermal noise and it affects the sensitivity of both the projected and the future generation interferometers. In this paper, we investigate the relevance of small non-linear effects and point out their impact on the sensitivity curve of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, etc.) through perturbative methods and numerical simulations. We find that in the first-order approximation the constants characterizing the power spectrum density (PSD) are renormalized but it retains its typical shape. This is due to the fact that the involved Feynman diagrams are of tadpole type. Higher-order approximations are required to give rise to up-conversion effects. This result is predicted by the perturbative approach and is in agreement with the numerical results obtained by studying the system's non-linear response by numerically simulating its dynamics.</abstract><note type="content">Fig. 1: Power density spectrum for a pendulum with a quality factor ∼100.</note><note type="content">Fig. 2: Bins correlation 〈| x ̃(f)|2| x ̃(f′)|2〉conn. The width of each bin is 2η=0.1 mHz and the adimensional coefficient is α=0.1. Such coefficient represents the level of non-linearity of the concerned system.</note><note type="content">Fig. 3: The expansion in α leads to good approximations of 〈x2〉. The approximated expression is no longer valid when α→1. There is an asymptotic limit 〈x2〉∼α−1/2KBT(mω02)−1 if α grows.</note><note type="content">Fig. 4: Graphical representation of x(t) and 〈x(t)x(t′)〉. When the average over the random force is taken, all crosses are joined in all possible ways. For example, one can get 〈x(t)x(t′)〉 by combining two crosses at a point in the corresponding product.</note><note type="content">Fig. 5: Second-order corrections for the two-point correlation function. There are contributions including tadpoles. A tadpole is nothing but a contraction of x with itself at the same time. Since 〈x(t)x(t)〉 is a constant tadpoles are insertions of constant terms.</note><note type="content">Fig. 6: In this figure the unperturbed theoretical curve related to the linear case and the power spectrum density corresponding to a numerical simulation of the non-linear one are both shown. Data are considered over a period T chosen as ∼12002π/ω0. In the frequency domain the variations due to non-linearities are concentrated in a band that is as narrow as Q is high. It seems that for Q→∞ even small non-linearities may be revealed.</note><note type="content">Fig. 7: In this figure a sketch of how the sensitivity curve for a pendulum is modified by a non-linear term is shown. As an example an effective reduction is achieved at low frequencies. The non-linear force causes the pendulum to oscillate with two important modes. In order to keep the amplitude of the up-converted fundamental frequency small, α (which is proportonal to T) is expected to be less than 0.1. These results are in accordance with the trend shown by our perturbative calculation.</note><subject><genre>PACS</genre><topic>02.50.-r</topic><topic>05.40.+j</topic></subject><relatedItem type="host"><titleInfo><title>Physica A: Statistical Mechanics and its Applications</title></titleInfo><titleInfo type="abbreviated"><title>PHYSA</title></titleInfo><genre type="journal">journal</genre><originInfo><dateIssued encoding="w3cdtf">19991115</dateIssued></originInfo><identifier type="ISSN">0378-4371</identifier><identifier type="PII">S0378-4371(00)X0115-X</identifier><part><date>19991115</date><detail type="volume"><number>273</number><caption>vol.</caption></detail><detail type="issue"><number>3–4</number><caption>no.</caption></detail><extent unit="issue pages"><start>217</start><end>510</end></extent><extent unit="pages"><start>329</start><end>351</end></extent></part></relatedItem><identifier type="istex">350699F60102618FA2C7157F3F8176BB437F709C</identifier><identifier type="DOI">10.1016/S0378-4371(99)00238-1</identifier><identifier type="PII">S0378-4371(99)00238-1</identifier><accessCondition type="use and reproduction" contentType="copyright">©1999 Elsevier Science B.V.</accessCondition><recordInfo><recordContentSource>ELSEVIER</recordContentSource><recordOrigin>Elsevier Science B.V., ©1999</recordOrigin></recordInfo></mods></metadata><enrichments><istex:catWosTEI uri="https://api.istex.fr/document/350699F60102618FA2C7157F3F8176BB437F709C/enrichments/catWos"><teiHeader><profileDesc><textClass><classCode scheme="WOS">PHYSICS</classCode><classCode scheme="WOS">PHYSICS, MULTIDISCIPLINARY</classCode></textClass></profileDesc></teiHeader></istex:catWosTEI></enrichments><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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|étape= Corpus
|type= RBID
|clé= ISTEX:350699F60102618FA2C7157F3F8176BB437F709C
|texte= Perturbative and numerical methods for stochastic nonlinear oscillators
}}
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