On the Dynamics of Bohmian Measures

Markowich, Peter A.; Paul, Thierry A.; Sparber, Christof

On the Dynamics of Bohmian Measures

Type

Article

Authors

Markowich, Peter A.

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

Preprint Posting Date

2010-11-24

Online Publication Date

2012-05-08

Print Publication Date

2012-09

Date

2012-05-08

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

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.