Classical Fluids via Quantum Mechanics

The subject of this post is probably a bit too technical to interest many readers, but I’ve been meaning to post something about it for a while and seem to have an hour or so to spare this morning so here goes. This is going to be a battle with the clunky WordPress latex widget too so please bear with me if it’s a little difficult to read.

The topic something I came across a while ago when thinking about the way the evolution of the matter distribution in cosmology is described in terms of fluid mechanics, but what I’m going to say is not at all specific to cosmology, and perhaps isn’t all that well known, so it might be of some interest to readers with a general physics background.

Consider a fluid with density $\rho= \rho (\vec{x},t)$ . The velocity of the fluid at any point is $\vec{v}=\vec{v}(\vec{x},t)$ . The evolution of such a fluid can be described by the continuity equation:

$\frac{\partial \rho}{\partial t} + \vec{\nabla}\cdot (\rho\vec{v})= 0$

and the Euler equation

$\frac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot\vec{\nabla})\vec{v} +\frac{1}{\rho} \vec{\nabla} P + \vec{\nabla} V = 0,$

in which $P$ is the fluid pressure (pressure gradients appear in the above equation) and $V$ is a potential describing other forces on the fluid (in a cosmological context, this would include its self-gravity). To keep things as simple as possible, consider a pressureless fluid (as might describe cold dark matter) and restrict consideration to the case of a potential flow, i.e. one in which

$\vec{v} = \vec{\nabla}\phi$

where $\phi=\phi(\vec{x},t)$ is a velocity potential; such a flow is curl-free. It is convenient to take the first integral of the Euler equation with respect to the spatial coordinates, which yields an equation for the velocity potential (cf. the Bernoulli equation):

$\frac{\partial \phi}{\partial t} + \frac{1}{2} (\nabla \phi)^{2} + V=0.$

The continuity equation becomes

$\frac{\partial \rho}{\partial t} + \vec{\nabla}\cdot(\rho\vec{\nabla}\phi) = 0$

This is all standard basic classical fluid mechanics. Now here’s the interesting thing. Introduce a new quantity $\Psi$ defined by

$\Psi(\vec{x},t) \equiv R\exp(i\phi/\nu),$

in which $R=R(\vec{x},t)$ and $\nu$ is a constant. Using this construction, it turns out that

$\rho = \Psi\Psi^{\ast}= |\Psi|^2=R^2$ .

After a little bit of fiddling around putting this in the previous equation you can obtain the following:

$i\nu \frac{\partial \Psi}{\partial t} = -\frac{\nu^2}{2} \nabla^2{\Psi} + V\Psi + Q\Psi$

which, apart from the last term $Q$ and a slightly different notation, is identical to the Schrödinger equation of quantum mechanics; the term $\nu$ would be proportional to Planck’s constant $h$ in that context, but in this context is a free parameter.

The mysterious term $Q$ is pretty horrible:

$Q = \frac{\nu^2}{2} \frac{\nabla^2 R}{R},$

and it turns the Schrödinger equation into a non-linear equation, but its role can be understood by seeing what happens if you start with the normal single-particle Schrödinger equation and work backwards; this is the approach taken historically by David Bohm and others. In that case the term $Q$ appears as a strange extra potential term in the Bernoulli equation which is sometimes called the quantum potential. In the context of fluid flow, however, the term describes the the effect of pressure gradients that would arise if the fluid were barotropic. In the approach I’ve outlined, going in the opposite direction, this term is consequently sometimes called the “quantum pressure”. The parameter $\nu$ controls the size of this term, which has the effect of blurring out the streamlines of the purely classical solution.

This transformation from classical fluid mechanics to quantum mechanics is not a new idea; in fact it goes back to Madelung who, in the 1920s, was trying to find a way to express quantum theory in the language of classical fluids.

What interested me about this approach, however, is more practical. It might seem strange to want transform relatively simple classical fluid-mechanical setup into a quantum-mechanical framework, which isn’t the obvious way to make progress, but there are a number of advantages of doing so. Perhaps chief among them is that the construction of $\Psi$ means that the density $\rho$ is guranteed positive definite; this means that a perturbation expansion of $\Psi$ will not lead to unphysical negative densities in the same way that happens if perturbation theory is applied to $\rho$ directly. This approach also has interesting links to other methods of studying the growth of large-scale structure in the Universe, such as the Zel’dovich approximation; the “waviness” controlled by the parameter $\nu$ is useful in ensuring that the density does not become infinite at shell-crossing, for example.

Anyway, here are some links to references with more details:

http://adsabs.harvard.edu/abs/1993ApJ…416L..71W
http://adsabs.harvard.edu/abs/1997PhRvD..55.5997W
http://adsabs.harvard.edu/abs/2002MNRAS.330..421C
http://adsabs.harvard.edu/abs/2003MNRAS.342..176C
http://adsabs.harvard.edu/abs/2006JCAP…12..012S
http://adsabs.harvard.edu/abs/2006JCAP…12..016S
http://adsabs.harvard.edu/abs/2010MNRAS.402.2491J

I think there are many more ways this approach could be extended, so maybe this will encourage someone out there to have a look at it!

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This entry was posted on June 17, 2012 at 12:33 pm and is filed under The Universe and Stuff with tags Bernoulli equation, Continuity equation, David Bohm, Euler equation, Madelung Transformation, Quantum Mechanics, Schrodinger Equation. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

3 Responses to “Classical Fluids via Quantum Mechanics”

Anton Garrett Says:
June 17, 2012 at 4:15 pm

I knew this mathematics starting from the other way round, ie begin with the Schroedinger equation and derive (coupled, nonlinear) equations for the amplitude and phase of the wavefunction. David Bohm attempted to give these a physical interpretation.

In fluid mechanics the Navier-Stokes equations (or whatever approximation to them is used) need to be supplemented by the condition that the pressure cannot go negative and that, whenever their solution is on the point of doing so, you get a vacuum with P=0, as in the cavitation phenomenon around ship propellers. I don’t know if that condition has any analogy in cosmology, or even in nonrelativistic quantum theory where it might be tested.

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- telescoper Says:
  June 17, 2012 at 5:16 pm
  
  Yes, I should have mentioned the Bohm approach because that’s the more familiar way of doing this. That interprets the term Q as a quantum potential for a single-particle.
  
  Incidentally, googling about I see there’s been some work on this approach in the context of Clifford Algebra. Very interesting.
  
  Reply
Paul Stevenson Says:
June 18, 2012 at 11:03 am

I haven’t tried to integrate it with WordPress, but mathjax is a wonderful way of getting nicely typeset on the web. I stumbled across it on the PRL website when there was a news item about the APS supporting the project. Now the difficult battle is to get my University to integrate it with their VLE. Sorry for all the TLAs.

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