On the Schrodinger-Poisson--Vlasov-Poisson correspondence

TL;DR

This paper investigates the relationship between Schr"odinger-Poisson and Vlasov-Poisson equations, demonstrating convergence of potential fields and exploring quantum effects in superfluids as approximations to classical collisionless matter.

Contribution

It provides a detailed numerical and analytical study of the Schr"odinger-Poisson to Vlasov-Poisson correspondence, highlighting convergence properties and quantum phenomena.

Findings

Potential converges to classical as (/m)^2

Density oscillations persist due to quantum effects

Quantum pressure regularizes classical singularities

Abstract

The Schr\"odinger-Poisson equations describe the behavior of a superfluid Bose-Einstein condensate under self-gravity with a 3D wave function. As $\hbar/m\to 0$, $m$ being the boson mass, the equations have been postulated to approximate the collisionless Vlasov-Poisson equations also known as the collisionless Boltzmann-Poisson equations. The latter describe collisionless matter with a 6D classical distribution function. We investigate the nature of this correspondence with a suite of numerical test problems in 1D, 2D, and 3D along with analytic treatments when possible. We demonstrate that, while the density field of the superfluid always shows order unity oscillations as $\hbar/m\to 0$ due to interference and the uncertainty principle, the potential field converges to the classical answer as $(\hbar/m)^{2}$. Thus, any dynamics coupled to the superfluid potential is expected to recover the classical collisionless limit as $\hbar/m\to 0$. The quantum superfluid is able to capture rich phenomena such as multiple phase-sheets, shell-crossings, and warm distributions. Additionally, the quantum pressure tensor acts as a regularizer of caustics and singularities in classical solutions. This suggests the exciting prospect of using the Schr\"odinger-Poisson equations as a low-memory method for approximating the high-dimensional evolution of the Vlasov-Poisson equations. As a particular example we consider dark matter composed of ultra-light axions, which in the classical limit ($\hbar/m\to 0$) is expected to manifest itself as collisionless cold dark matter.