Please use this identifier to cite or link to this item: https://oar.tib.eu/jspui/handle/123456789/2243
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dc.rights.licenseThis document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.eng
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dc.contributor.authorAdams, Stefan
dc.contributor.authorCollevecchio, Andrea
dc.contributor.authorKönig, Wolfgang
dc.date.accessioned2016-03-24T17:38:35Z
dc.date.available2019-06-28T08:04:53Z
dc.date.issued2010
dc.identifier.urihttps://oar.tib.eu/jspui/handle/123456789/2243
dc.identifier.urihttp://dx.doi.org/10.34657/1934-
dc.description.abstractWe consider $N$ bosons in a box in $R^d$ with volume $N/rho$ under the influence of a mutually repellent pair potential. The particle density $rhoin(0,infty)$ is kept fixed. Our main result is the identification of the limiting free energy, $f(beta,rho)$, at positive temperature $1/beta$, in terms of an explicit variational formula, for any fixed $rho$ if $beta$ is sufficiently small, and for any fixed $beta$ if $rho$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrised trace of $rm e^-beta Hcal_N$, where $Hcal_N$ denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of $N$ interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving lq infinitely longrq cycles, and their possible presence is signalled by a loss of mass of the lq finitely longrq cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the lq finitely longrq cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyse it further in future.
dc.formatapplication/pdf
dc.languageeng
dc.publisherBerlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
dc.relation.ispartofseriesPreprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 1490, ISSN 0946-8633-
dc.subjectFree energy
dc.subjectinteracting many-particle systems
dc.subjectBose-Einstein condensation
dc.subjectBrownian bridges
dc.subjectsymmtresed distribution
dc.subjectlarge deviations
dc.subjectempirical stationary measure
dc.subjectvariational formula
dc.subject.ddc510
dc.titleA variational formula for the free energy of an interacting many-particle system
dc.typereport-
dc.typeText-
dc.description.versionpublishedVersioneng
local.accessRightsopenAccess-
wgl.contributorWIASger
wgl.subjectMathematikger
wgl.typeReport / Forschungsbericht / Arbeitspapierger
dcterms.bibliographicCitation.journalTitlePreprint / Weierstraß-Institut für Angewandte Analysis und Stochastik-
local.identifier.doihttp://dx.doi.org/10.34657/1934-
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