On the dynamics of Bohmian measures

TL;DR

This paper investigates the dynamical behavior of Bohmian measures, establishing their solutions to nonlinear equations, exploring classical limits, and connecting them with Wigner measures, especially for semi-classical wave packets.

Contribution

It provides rigorous proofs of Bohmian measures solving nonlinear Vlasov-type equations and links them to Wigner measures in one dimension, advancing understanding of quantum-classical transition.

Findings

Bohmian measures solve a nonlinear Vlasov-type equation for smooth wave functions.

In one dimension, Bohmian and Wigner measures are connected through mono-kinetic measures.

Convergence of Bohmian trajectories to classical flow is established for semi-classical wave packets.

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.