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Classical Fluids via Quantum Mechanics

Posted in The Universe and Stuff with tags , , , , , , on June 17, 2012 by telescoper

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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Power versus Pattern

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , on June 15, 2012 by telescoper

One of the challenges we cosmologists face is how to quantify the patterns we see in galaxy redshift surveys. In the relatively recent past the small size of the available data sets meant that only relatively crude descriptors could be used; anything sophisticated would be rendered useless by noise. For that reason, statistical analysis of galaxy clustering tended to be limited to the measurement of autocorrelation functions, usually constructed in Fourier space in the form of power spectra; you can find a nice review here.

Because it is so robust and contains a great deal of important information, the power spectrum has become ubiquitous in cosmology. But I think it’s important to realise its limitations.

Take a look at these two N-body computer simulations of large-scale structure:

The one on the left is a proper simulation of the “cosmic web” which is at least qualitatively realistic, in that in contains filaments, clusters and voids pretty much like what is observed in galaxy surveys.

To make the picture on the right I first  took the Fourier transform of the original  simulation. This approach follows the best advice I ever got from my thesis supervisor: “if you can’t think of anything else to do, try Fourier-transforming everything.”

Anyway each Fourier mode is complex and can therefore be characterized by an amplitude and a phase (the modulus and argument of the complex quantity). What I did next was to randomly reshuffle all the phases while leaving the amplitudes alone. I then performed the inverse Fourier transform to construct the image shown on the right.

What this procedure does is to produce a new image which has exactly the same power spectrum as the first. You might be surprised by how little the pattern on the right resembles that on the left, given that they share this property; the distribution on the right is much fuzzier. In fact, the sharply delineated features  are produced by mode-mode correlations and are therefore not well described by the power spectrum, which involves only the amplitude of each separate mode.

If you’re confused by this, consider the Fourier transforms of (a) white noise and (b) a Dirac delta-function. Both produce flat power-spectra, but they look very different in real space because in (b) all the Fourier modes are correlated in such away that they are in phase at the one location where the pattern is not zero; everywhere else they interfere destructively. In (a) the phases are distributed randomly.

The moral of this is that there is much more to the pattern of galaxy clustering than meets the power spectrum…

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My Even Newer Theory of the Universe

Posted in The Universe and Stuff with tags , , on June 13, 2012 by telescoper

I have decided this evening to unveil my new cosmological theory.

My previous work  was based on the idea that the Universe was obtained from the Swedish furniture and home accessory emporium IKEA. This “Easy Self Assembly” hypothesis dispenses with the need for creation from nothing, and also accounts naturally for the observed geometry of space (it came in a flat pack).

My subsequent study of this scenario has focussed on properties of the Universe that can’t be explained in the earlier version of the theory, specifically  the cosmic microwave background. However, making my supper just now I suddenly hit upon the answer to that particular puzzle. Clearly, wanting to achieve the best results possible, on his/her way back from IKEA the Divine Creator stopped off at Marks and Spencer …

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Astronomy’s Next Big Thing

Posted in Science Politics, The Universe and Stuff with tags , , , , on June 12, 2012 by telescoper

I woke up this morning to hear an item about astronomy on the 7 o’clock news on BBC Radio 3. That doesn’t happen very often so I thought I’d follow it up with a short post before I head off to work.

The news item I heard followed up an announcement yesterday that the governing Council of the European Southern Observatory (ESO) had  approved the European Extremely Large Telescope (E-ELT) programme – which is to produce what will be the world’s largest ground-based optical telescope. Extremely Large is putting in mildly, of course. Its main mirror will be a colossal 39 metres in diameter (with a collecting area of almost a thousand square metres) and will have to made in bits with a sophisticated adaptive optics system to ensure that it can counter the effects of the Earth’s atmosphere and the limitations  of its own structure to  reach a phenomenal angular resolution of 0.001 arc seconds.

For more details on the telescope, see the official website here or the wikipedia article here, where you can also read more about the science to be done with E-ELT.

This telescope has been in planning for many years, of course. In fact, it began as an even more ambitious concept, a 100-metre diameter monster which I used to call the FLT. Over the years, however, for a mixture of technical and financial reasons, this was progressively de-scoped.

Yesterday’s announcement doesn’t mean that work will start immediately on building the E-ELT. That won’t happen until sufficient funding is secured and in the case of some countries, governmental approval obtained. Recent decisions by the UK Science and Technology Facilities Council to close down telescopes in Hawaii clearly anticipated the need to make some headroom in future budgets to enable this to happen. The best-case scenario is probably for E-ELT to take a decade or so to complete.

Of course the concentration of funding in ever and ever larger international facilities – such as E-ELT and the Square Kilometre Array – does create tensions within the UK astronomical community. Many scientists do excellent work with relatively small facilities, including those about to be closed down to make room for E-ELT. In the near future, the only ground-based optical facilities to which UK astronomers will have access will be operated by the European Southern Observatory. With fewer but larger (and more expensive) facilities operated by international agencies carrying out projects run by vast consortia, observational astronomy is definitely going the way of particle physics…

The problem  comes when the Next Big Thing  is too big to be built.  We might have already seen X-ray astronomy bubble burst in this way. To quote my learned friend Andy Lawrence:

Fundamentally, the problem is that X-ray astronomy has hit the funding wall. Everything gets inexorably bigger and more ambitious. Eventually its all or nothing… so when the answer is nothing … ah.

What will come after the Large Hadron Collider, or the E-ELT?  Is Big Science about to get too big?

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Sic Transit Gloria Monday

Posted in The Universe and Stuff with tags , , , on June 11, 2012 by telescoper

I can never resist a terrible pun, so thought this would be an especially  good day to post this video from NASA’s Solar Dynamics Observatory,  showing views of last week’s Transit of Venus taken at several different wavelengths..

 

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Big Bang: Who’s the Daddy?

Posted in History, The Universe and Stuff with tags , , , , on June 8, 2012 by telescoper

Time, I think, for a frivolous Friday poll.

I stumbled across a post on the Physics World Blog concerning a radio broadcast about Georges Lemaître.

Here’s a description of said programme:

Few theories could claim to have a more fundamental status than Big Bang Theory. This is now humanity’s best attempt at explaining how we got here: A Theory of Everything. This much is widely known and Big Bang Theory is now one of the most recognisable scientific brands in the world. What’s less well known is that the man who first proposed the theory was not only an accomplished physicist, he was also a Catholic priest. Father Georges Lemaître wore his clerical collar while teaching physics, and not at Oxford, Cambridge or MIT but at the Catholic University of Leuven in Belgium. It was this unassuming Catholic priest in an academic backwater who has changed the way we look at the origins of the universe. His story also challenges the assumption that science and religion are always in conflict. William Crawley introduces us to the “Father” of the Big Bang.

The question is whether the word “Father” in the last sentence should be taken as anything more than a play on the title he’d be given as a Catholic priest?

Lemaître’s work was highly original and it undoubtedly played an important role in the development of the Big Bang theory, especially in Western Europe and in the United States. However, a far stronger claim to the title of progenitor of this theory belongs to Alexander Alexandrovich Friedman, who obtained the cosmological solutions of Einstein’s general theory of relativity, on which the Big Bang model is based, independently of and shortly before Lemaître did. Unfortunately the Russian Friedman died in 1925 and it was many years before his work became widely known in the West. At least in my book, he’s the real “father” of the Big Bang, but I’m well aware that this is the source of a great deal of argument at cosmology conferences, which makes it an apt topic for a quick poll:

P.S. I prefer to spell Friedman with one “n” rather than two. His name in his own language is Алекса́ндр Алекса́ндрович Фри́дман and the spelling “Friedmann” only arose because of later translations into German.

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Most Exciting Venus Transit Pictures Ever!

Posted in The Universe and Stuff with tags , , on June 6, 2012 by telescoper

Before

After

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Tinker Tailor Soldier…Astronomer?

Posted in Science Politics, The Universe and Stuff with tags , on June 4, 2012 by telescoper

Tinker Tailor Soldier…Astronomer?

andyxl's avatarThe e-Astronomer

Not often I write two posts in one day, but here is an unexpected piece of news. It seems that the US National Reconnaisance Office have given two free telescopes to NASA. Its all explained at this NY Times article. They are as big as HST but have a wider field of view. They were designed for looking down of course.  Apparently there has been a secret study team and their conclusion is that one of these beasts would be perfect WFIRST, which had seemed to be kicked into the long grass.

They don’t exactly have the rest of the money yet or an actual approval … but the WFIRST fans are talking about shooting for 2020 … a year behind Euclid.

Ooooo what fun. Spot of healthy competition.

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A Ghost of a Jet?

Posted in Astrohype, The Universe and Stuff with tags , , on June 4, 2012 by telescoper

Last week an article in Nature News caught my eye. Ghostly jets seen streaming from Milky Way’s core was the headline. It’s based on a paper by Su & Finkbeiner recently submitted to the arXiv. There’s even a picture showing the jets in glorious technicolour:

Wow! Impressive stuff. If the jets look like that it’s amazing nobody ever saw them before!

Oh, hang on. The picture is an “artist’s conception”. In other words, it’s what the jets might look like if they actually existed, as imagined by a bloke with a box of crayons.

And how strong is the evidence that they do exist? Here’s the last paragraph of the Nature article (my emphasis):

Although the emissions are dim and the observations don’t have the statistical significance that astronomers require for proof, Baganoff says that several properties make them compelling evidence of jets. They are brighter at higher γ-ray energies and also brighter than the surrounding interstellar medium. They also seem to be long and thin, as would be expected of jets. “Taking all of the evidence together, it appears highly plausible that the features are jets emanating from the Galactic Centre,“ he says.

If they “don’t have the statistical significance that astronomers require for proof” then one wonders why they’re being given so much publicity. In any case the “ghostly jets seen streaming from the Milky Way’s core” can’t be said to have really been “seen” for certain. But they are “highly plausible”. In other words, the authors would like them to be there.

All I can say is that it must have been a slow news day at Nature.

Still, nice drawing.

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Astronomy Jobs at Cardiff!

Posted in The Universe and Stuff with tags , , , , , on June 1, 2012 by telescoper

Just a quick post to advertise a couple of job opportunities in the School of Physics & Astronomy at Cardiff University. For further details you can look at the official website, but here is an outline:

Two Faculty Positions in Astrophysics

Observational and theoretical studies of star-formation and/or extrasolar planetary systems.

The School of Physics and Astronomy at Cardiff University has immediate vacancies for two permanent faculty appointments in Astrophysics.  We are seeking experts in observational and theoretical studies of star-formation and/or extra-solar planetary systems to conduct world-class research and research-led teaching at undergraduate and postgraduate level.  The appointments will be at any level from Lecturer to Professor depending on the experience of the candidate; we expect at least one of the appointments to be at a junior level.

Physics and Astronomy at Cardiff University has undergone substantial expansion in the past few years and has very strong research groups in gravitational-wave physics, astronomical instrumentation, extragalactic astronomy and cosmology, star-formation and condensed matter physics.  There are presently 18 academic staff involved in astrophysics and relativity, with 15 post-doctoral researchers and 22 PhD students.

The appointment will be made at a level commensurate with experience.

The advertisement is also available on the AAS Jobs Register, or will be when they get their act together and put it online. The AAS website is just one of a number that have been recently improved, with the result that they’re much less efficient than they were before.

Anyway, I’m just passing on the advertisement so please don’t send me your CVs! If you’d like to apply please do so using the official Cardiff University jobs page, which also has a lot of general information about the City and the University.

P.S. There have been quite a few job vacancies in astronomy around the UK recently – Edinburgh, Surrey, Liverpool, Exeter etc. I wonder why that is, and where the money is coming from?

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