I thought I’d mention here a paper now on arXiv that I co-wrote with my PhD student Aoibhinn Gallagher. Here is the abstract:
The Schrödinger-Poisson formalism has found a number of applications in cosmology, particularly in describing the growth by gravitational instability of large-scale structure in a universe dominated by ultra-light scalar particles. Here we investigate the extent to which the behaviour of this and the more general case of a Schrödinger-Newton system, can be described in terms of classical fluid concepts such as viscosity and pressure. We also explore whether such systems can be described by a pseudo-Reynolds number as for classical viscous fluids. The conclusion we reach is that this is indeed possible, but with important restrictions to ensure physical consistency.
arXiv:2507.08583
It is based on work that his in her now-completed PhD thesis, along with another paper mentioned here. I have been interested for many years in the Schrödinger-Newton system (or, more specifically, the Schrödinger-Poisson system in the case where self-gravitational forces are involved). In its simplest form this involves a wave-mechanical representation, in the form of an effective Schrödinger equation, of potential flow described classically by an Euler equation. More recently we got interested in the extent to which such an approach could be used to model viscous fluids represented by a Navier-Stokes equation rather than an Euler equation. That was largely because the effective Planck constant that arises in this representation has the same dimensions as kinematic viscosity (but there’s more to it than that).
In the paper we explored a limited aspect of this, by looking at situations where there is no vorticity (so still a potential flow) but there is viscosity. There aren’t many examples of fluid flow in which there is viscosity but no vorticity, and most of those that do exist are about one-dimensional flow along channels or pipes with boundary conditions that don’t really apply to astrophysics, but one example we did look at in detail was the dissipiation of longitudinal waves in such a fluid.
One upshot of this work is that one can indeed describe some aspects of quantum-mechnical fluids such as ultra-light scalar matter in terms of classical fluid properties, such as viscosity, but you have to be careful. For more information, read the paper!
In the Dark 