On the Dynamics of Bohmian Measures

Handle URI:
http://hdl.handle.net/10754/562182
Title:
On the Dynamics of Bohmian Measures
Authors:
Markowich, Peter A. ( 0000-0002-3704-1821 ) ; Paul, Thierry A.; Sparber, Christof
Abstract:
The present work is devoted to the study of dynamical features of Bohmian measures, recently introduced by the authors. We rigorously prove that for sufficiently smooth wave functions the corresponding Bohmian measure furnishes a distributional solution of a nonlinear Vlasov-type equation. Moreover, we study the associated defect measures appearing in the classical limit. In one space dimension, this yields a new connection between mono-kinetic Wigner and Bohmian measures. In addition, we shall study the dynamics of Bohmian measures associated to so-called semi-classical wave packets. For these type of wave functions, we prove local in-measure convergence of a rescaled sequence of Bohmian trajectories towards the classical Hamiltonian flow on phase space. Finally, we construct an example of wave functions whose limiting Bohmian measure is not mono-kinetic but nevertheless equals the associated Wigner measure. © 2012 Springer-Verlag.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Applied Mathematics and Computational Science Program
Publisher:
Springer Nature
Journal:
Archive for Rational Mechanics and Analysis
Issue Date:
8-May-2012
DOI:
10.1007/s00205-012-0528-1
ARXIV:
arXiv:1011.5361
Type:
Article
ISSN:
00039527
Sponsors:
The authors want to thank the referees for helpful remarks in improving the paper and in particular for the short, direct argument given in the proof of Lemma 3. This publication is based on work supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST). C. SPARBER has been supported by the Royal Society via his University research fellowship. P. MARKOWICH acknowledges support from the VPP Office of KSU in Riyadh, KSA, and from the Royal Society through his Wolfson Research Merit Award.
Additional Links:
http://arxiv.org/abs/arXiv:1011.5361v2
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorMarkowich, Peter A.en
dc.contributor.authorPaul, Thierry A.en
dc.contributor.authorSparber, Christofen
dc.date.accessioned2015-08-03T09:46:41Zen
dc.date.available2015-08-03T09:46:41Zen
dc.date.issued2012-05-08en
dc.identifier.issn00039527en
dc.identifier.doi10.1007/s00205-012-0528-1en
dc.identifier.urihttp://hdl.handle.net/10754/562182en
dc.description.abstractThe present work is devoted to the study of dynamical features of Bohmian measures, recently introduced by the authors. We rigorously prove that for sufficiently smooth wave functions the corresponding Bohmian measure furnishes a distributional solution of a nonlinear Vlasov-type equation. Moreover, we study the associated defect measures appearing in the classical limit. In one space dimension, this yields a new connection between mono-kinetic Wigner and Bohmian measures. In addition, we shall study the dynamics of Bohmian measures associated to so-called semi-classical wave packets. For these type of wave functions, we prove local in-measure convergence of a rescaled sequence of Bohmian trajectories towards the classical Hamiltonian flow on phase space. Finally, we construct an example of wave functions whose limiting Bohmian measure is not mono-kinetic but nevertheless equals the associated Wigner measure. © 2012 Springer-Verlag.en
dc.description.sponsorshipThe authors want to thank the referees for helpful remarks in improving the paper and in particular for the short, direct argument given in the proof of Lemma 3. This publication is based on work supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST). C. SPARBER has been supported by the Royal Society via his University research fellowship. P. MARKOWICH acknowledges support from the VPP Office of KSU in Riyadh, KSA, and from the Royal Society through his Wolfson Research Merit Award.en
dc.publisherSpringer Natureen
dc.relation.urlhttp://arxiv.org/abs/arXiv:1011.5361v2en
dc.titleOn the Dynamics of Bohmian Measuresen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.identifier.journalArchive for Rational Mechanics and Analysisen
dc.contributor.institutionCNRS and CMLS, Ecole Polytechnique 91, 128 Palaiseau cedex, Franceen
dc.contributor.institutionDepartment of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607, United Statesen
dc.identifier.arxividarXiv:1011.5361en
kaust.authorMarkowich, Peter A.en
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