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Diffusion Limit for a Slow-Fast Standard Map
Consider the map ( x , z ) ↦ ( x + ϵ - α sin ( 2 π x ) + ϵ - ( 1 + α ) z , z + ϵ sin ( 2 π x ) ) , which is conjugate to the Chirikov standard map with a large parameter. The parameter value α = 1 is related to “scattering by resonance” phenomena. For suitable α , we obtain a central limit theorem f...
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| Published in: | Communications in mathematical physics 2020-02, Vol.374 (1), p.187-210 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| container_end_page | 210 |
| container_issue | 1 |
| container_start_page | 187 |
| container_title | Communications in mathematical physics |
| container_volume | 374 |
| creator | Blumenthal, Alex De Simoi, Jacopo Zhang, Ke |
| description | Consider the map
(
x
,
z
)
↦
(
x
+
ϵ
-
α
sin
(
2
π
x
)
+
ϵ
-
(
1
+
α
)
z
,
z
+
ϵ
sin
(
2
π
x
)
)
, which is conjugate to the Chirikov standard map with a large parameter. The parameter value
α
=
1
is related to “scattering by resonance” phenomena. For suitable
α
, we obtain a central limit theorem for the slow variable
z
for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a “finite-time” decay of correlations result. |
| doi_str_mv | 10.1007/s00220-019-03600-7 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2358711367</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2358711367</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-a798be8f81b32184dbf529f8370b48ef6c5f1f1ed6b6f854d767638bc22e808e3</originalsourceid><addsrcrecordid>eNp9kEFLxDAQRoMoWFf_gKeAV6MzSZukR1ldFSoeVs8hbRPpstvWpEX899u1gjdPc3nvG3iEXCLcIIC6jQCcAwPMGQgJwNQRSTAVnEGO8pgkAAhMSJSn5CzGDQDkXMqEXN833o-x6VpaNLtmoL4L1NL1tvtiKxsHuh5sW9tQ0xfbn5MTb7fRXfzeBXlfPbwtn1jx-vi8vCtYxRUMzKpcl057jaXgqNO69BnPvRYKylQ7L6vMo0dXy1J6naW1kkoKXVacOw3aiQW5mnf70H2OLg5m042hnV4aLjKtEIVUE8VnqgpdjMF504dmZ8O3QTCHKmauYqYq5qeKOUhiluIEtx8u_E3_Y-0B6u9icQ</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2358711367</pqid></control><display><type>article</type><title>Diffusion Limit for a Slow-Fast Standard Map</title><source>Springer Nature</source><creator>Blumenthal, Alex ; De Simoi, Jacopo ; Zhang, Ke</creator><creatorcontrib>Blumenthal, Alex ; De Simoi, Jacopo ; Zhang, Ke</creatorcontrib><description>Consider the map
(
x
,
z
)
↦
(
x
+
ϵ
-
α
sin
(
2
π
x
)
+
ϵ
-
(
1
+
α
)
z
,
z
+
ϵ
sin
(
2
π
x
)
)
, which is conjugate to the Chirikov standard map with a large parameter. The parameter value
α
=
1
is related to “scattering by resonance” phenomena. For suitable
α
, we obtain a central limit theorem for the slow variable
z
for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a “finite-time” decay of correlations result.</description><identifier>ISSN: 0010-3616</identifier><identifier>EISSN: 1432-0916</identifier><identifier>DOI: 10.1007/s00220-019-03600-7</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Classical and Quantum Gravitation ; Complex Systems ; Diffusion rate ; Mathematical and Computational Physics ; Mathematical Physics ; Parameters ; Physics ; Physics and Astronomy ; Quantum Physics ; Relativity Theory ; Resonance scattering ; Theorems ; Theoretical</subject><ispartof>Communications in mathematical physics, 2020-02, Vol.374 (1), p.187-210</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2019</rights><rights>2019© Springer-Verlag GmbH Germany, part of Springer Nature 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-a798be8f81b32184dbf529f8370b48ef6c5f1f1ed6b6f854d767638bc22e808e3</cites><orcidid>0000-0002-6777-7848</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Blumenthal, Alex</creatorcontrib><creatorcontrib>De Simoi, Jacopo</creatorcontrib><creatorcontrib>Zhang, Ke</creatorcontrib><title>Diffusion Limit for a Slow-Fast Standard Map</title><title>Communications in mathematical physics</title><addtitle>Commun. Math. Phys</addtitle><description>Consider the map
(
x
,
z
)
↦
(
x
+
ϵ
-
α
sin
(
2
π
x
)
+
ϵ
-
(
1
+
α
)
z
,
z
+
ϵ
sin
(
2
π
x
)
)
, which is conjugate to the Chirikov standard map with a large parameter. The parameter value
α
=
1
is related to “scattering by resonance” phenomena. For suitable
α
, we obtain a central limit theorem for the slow variable
z
for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a “finite-time” decay of correlations result.</description><subject>Classical and Quantum Gravitation</subject><subject>Complex Systems</subject><subject>Diffusion rate</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Parameters</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Resonance scattering</subject><subject>Theorems</subject><subject>Theoretical</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLxDAQRoMoWFf_gKeAV6MzSZukR1ldFSoeVs8hbRPpstvWpEX899u1gjdPc3nvG3iEXCLcIIC6jQCcAwPMGQgJwNQRSTAVnEGO8pgkAAhMSJSn5CzGDQDkXMqEXN833o-x6VpaNLtmoL4L1NL1tvtiKxsHuh5sW9tQ0xfbn5MTb7fRXfzeBXlfPbwtn1jx-vi8vCtYxRUMzKpcl057jaXgqNO69BnPvRYKylQ7L6vMo0dXy1J6naW1kkoKXVacOw3aiQW5mnf70H2OLg5m042hnV4aLjKtEIVUE8VnqgpdjMF504dmZ8O3QTCHKmauYqYq5qeKOUhiluIEtx8u_E3_Y-0B6u9icQ</recordid><startdate>20200201</startdate><enddate>20200201</enddate><creator>Blumenthal, Alex</creator><creator>De Simoi, Jacopo</creator><creator>Zhang, Ke</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-6777-7848</orcidid></search><sort><creationdate>20200201</creationdate><title>Diffusion Limit for a Slow-Fast Standard Map</title><author>Blumenthal, Alex ; De Simoi, Jacopo ; Zhang, Ke</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-a798be8f81b32184dbf529f8370b48ef6c5f1f1ed6b6f854d767638bc22e808e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Classical and Quantum Gravitation</topic><topic>Complex Systems</topic><topic>Diffusion rate</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Parameters</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Resonance scattering</topic><topic>Theorems</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Blumenthal, Alex</creatorcontrib><creatorcontrib>De Simoi, Jacopo</creatorcontrib><creatorcontrib>Zhang, Ke</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Blumenthal, Alex</au><au>De Simoi, Jacopo</au><au>Zhang, Ke</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Diffusion Limit for a Slow-Fast Standard Map</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2020-02-01</date><risdate>2020</risdate><volume>374</volume><issue>1</issue><spage>187</spage><epage>210</epage><pages>187-210</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><abstract>Consider the map
(
x
,
z
)
↦
(
x
+
ϵ
-
α
sin
(
2
π
x
)
+
ϵ
-
(
1
+
α
)
z
,
z
+
ϵ
sin
(
2
π
x
)
)
, which is conjugate to the Chirikov standard map with a large parameter. The parameter value
α
=
1
is related to “scattering by resonance” phenomena. For suitable
α
, we obtain a central limit theorem for the slow variable
z
for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a “finite-time” decay of correlations result.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00220-019-03600-7</doi><tpages>24</tpages><orcidid>https://orcid.org/0000-0002-6777-7848</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0010-3616 |
| ispartof | Communications in mathematical physics, 2020-02, Vol.374 (1), p.187-210 |
| issn | 0010-3616 1432-0916 |
| language | eng |
| recordid | cdi_proquest_journals_2358711367 |
| source | Springer Nature |
| subjects | Classical and Quantum Gravitation Complex Systems Diffusion rate Mathematical and Computational Physics Mathematical Physics Parameters Physics Physics and Astronomy Quantum Physics Relativity Theory Resonance scattering Theorems Theoretical |
| title | Diffusion Limit for a Slow-Fast Standard Map |
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