Bernstein–Greene–Kruskal and Case–Van Kampen Modes for the Landau–Vlasov Equation

Fernando Haas,Rodrigo Vidmar

doi:10.3390/atoms10010028

Abstract

The one-dimensional Landau–Vlasov equation describing ultracold dilute bosonic gases in the mean-field collisionless regime under strong transverse confinement is analyzed using traditional methods of plasma physics. Time-independent, stationary solutions are found using a similar approach as for the Bernstein–Greene–Kruskal nonlinear plasma modes. Linear stationary waves similar to the Case–Van Kampen plasma normal modes are also shown to be available. The new bosonic solutions have no decaying or growth properties, in the same sense as the analog plasma solutions. The results are applied for real ultracold bosonic gases accessible in contemporary laboratory experiments.

Highlights

When the average collision time in ultracold dilute gases made of bosonic atoms is much larger than the relevant dynamics characteristic time scale, it is possible to have a model based on the Landau–Vlasov equation [1]
The Landau–Vlasov equation is obtained from the Boltzmann–Vlasov equation [1–4] neglecting the collision operator
Traditional methods from plasma theory have been adapted to the Landau–Vlasov equation

Summary

Introduction

When the average collision time in ultracold dilute gases made of bosonic atoms is much larger than the relevant dynamics characteristic time scale, it is possible to have a model based on the Landau–Vlasov equation [1]. The similarity between the Landau–Vlasov equation and the Vlasov–Poisson system describing collisionless electrostatic plasmas provides a stimulating scenario for the application of plasma techniques in a seemingly uncorrelated area such as in the study of ultracold bosonic gases. In this context, the present work is dedicated to the discussion of time-independent solutions and stationary wave solutions for the 1D Landau–Vlasov equation. The Case–Van Kampen modes are available for an ultracold boson gas described by the Landau–Vlasov equation.

The One-Dimensional Landau–Vlasov Equation

Bernstein–Greene–Kruskal Modes

Maxwell–Boltzmann Distribution

Water Bag Distribution

Case–Van Kampen Modes

Conclusions