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    On the Dynamics of Bohmian Measures

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    Type
    Article
    Authors
    Markowich, Peter A. cc
    Paul, Thierry A.
    Sparber, Christof
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    KAUST Grant Number
    KUK-I1-007-43
    Date
    2012-05-08
    Preprint Posting Date
    2010-11-24
    Online Publication Date
    2012-05-08
    Print Publication Date
    2012-09
    Permanent link to this record
    http://hdl.handle.net/10754/562182
    
    Metadata
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    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.
    Citation
    Markowich, P., Paul, T., & Sparber, C. (2012). On the Dynamics of Bohmian Measures. Archive for Rational Mechanics and Analysis, 205(3), 1031–1054. doi:10.1007/s00205-012-0528-1
    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.
    Publisher
    Springer Nature
    Journal
    Archive for Rational Mechanics and Analysis
    DOI
    10.1007/s00205-012-0528-1
    arXiv
    1011.5361
    Additional Links
    http://arxiv.org/abs/arXiv:1011.5361v2
    ae974a485f413a2113503eed53cd6c53
    10.1007/s00205-012-0528-1
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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