Classical Fluid Analogies for Schrödinger-Newton Systems

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!

#AoibhinnGallagher #NavierStokesEquations #SchrödingerEquation #SchrödingerPoissonSystem #viscosity

The butterfly effect — that the flapping of a butterfly’s wings in Brazil can cause a tornado in Texas — expresses the sensitivity of a chaotic system to initial conditions. In essence, because we can’t possibly track every butterfly in Brazil, we’ll never perfectly predict tornadoes in Texas, even if the equations behind our weather forecast are deterministic.

But this interpretation doesn’t fully capture the subtleties of the situation. With fluid dynamics, the small scales of a flow — like the turbulence in an individual cloud — are linked to the largest scales in the flow — for example, a hurricane. For short times, we’re actually quite good at predicting those large scales; our weather forecasts can distinguish sunny days and cloudy ones a week out. But at smaller scales, the forecast errors pile up quickly. No one can forecast that an individual cloud will form over your house three days from now. And because the small scales are linked to the larger scales, the uncertainties from the small scale cascade upward, limiting how far into the future we can reliably predict the weather.

And, unfortunately, drilling down to capture smaller and smaller scales in our models can’t fix the problem, unless our initial uncertainties are identically zero. To get around this problem, weather forecasters instead use ensemble forecasting, where they run many simulations of the weather with slightly different initial conditions. Those differences in initial conditions let the forecasters play with those initial uncertainties — how accurate is the temperature reading from that station? How reliable is the instrument reporting that humidity? How old is the satellite data coming in? Once all the forecasts are run, they can see how many predicted sunny days versus rainy ones, which ones resulted in severe weather, and so on. Often the probabilities we see in our weather app — like 30% chance of rain — depend on factors including how many of the forecasts resulted in rain.

Unfortunately, this butterfly effect permanently limits just how far into the future we can predict weather — at least until we fully understand the nature of the Navier-Stokes equations. For much more on this interesting aspect of chaos, check out this Physics Today article. (Image credit: NASA; see also T. Palmer at Physics Today)

https://fyfluiddynamics.com/2024/07/the-real-butterfly-effect/

#atmosphericScience #butterflyEffect #chaos #chaosTheory #clouds #fluidDynamics #mathematics #meteorology #NavierStokesEquations #physics #science #turbulence #weather