On the dynamics of Bohmian measures

Peter Markowich (1) , Thierry Paul (2) , Christof Sparber (3)

1 DAMTP - Department of Applied Mathematics and Theoretical Physics
2 CMLS - Centre de Mathématiques Laurent Schwartz
3 UIC - Department of Mathematics, Statistics and Computer Science [Chicago]

We revisit the concept of Bohmian measures recently introduced by the authors in \cite{MPS}. 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.

Mots clés: Quantum dynamics; Bohmian mechanics; Classical limit; Kinetic equations; Wigner measure; Quantum trajectories

Domaines: Equations aux dérivées partielles [math.AP]

Peter Markowich, Thierry Paul, Christof Sparber. On the dynamics of Bohmian measures. Archive for Rational Mechanics and Analysis, 2012, 205 (3), pp.1031-1054. ⟨10.1007/s00205-012-0528-1⟩. ⟨hal-00539227⟩