TY - JOUR
T1 - On the Dynamics of Bohmian Measures
AU - Markowich, Peter A.
AU - Paul, Thierry A.
AU - Sparber, Christof
N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: 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.
PY - 2012/5/8
Y1 - 2012/5/8
N2 - 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.
AB - 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.
UR - http://hdl.handle.net/10754/562182
UR - http://arxiv.org/abs/arXiv:1011.5361v2
UR - http://www.scopus.com/inward/record.url?scp=84865498658&partnerID=8YFLogxK
U2 - 10.1007/s00205-012-0528-1
DO - 10.1007/s00205-012-0528-1
M3 - Article
SN - 0003-9527
VL - 205
SP - 1031
EP - 1054
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
IS - 3
ER -