We all do economics

The very interesting blog, Mind Hacks, has a post on a theory of a bipolar economy.

A 1935 Psychological Review article proposed a ‘manic-depressive psychoses’ theory of economic highs and lows based on the idea that the market has a form of monetary bipolar disorder.

I find it quite interesting how people like to reframe the problem of economic crashes in their own subject. In psychology it seems perfectly natural to ascribe the behaviour to individual human behaviour. As a physicist I'm completely convinced that it's a collective effect that arises from many relatively simple individuals, trying to win a game, interacting in a highly complex system. Of course one could possibly say the same about the brain itself.

I wonder if biochemists have some hormone explanation and neuroscientists some neurotransmitter reason. Perhaps all these perspectives are equally right (or wrong) – I guess the only thing for sure is that we don't really know!

A phase diagram in a jar

One of the things I love about colloids is just how visual they are. Be it watching them jiggling around under a confocal microscope, or the beautiful TEM images of crystal structures, I always find them quite inspirational, or at least instructional, for better understanding statistical mechanics.

Sedimentation

Just to prove I'm on the cutting edge of science, I recently discovered another neat example from 1993. At the liquid matter conference in Vienna Roberto Piazza gave a talk titled "The unbearable heaviness of colloids". As a side note there was a distinct lack of playful titles, maybe people were too nervous at such a big meeting. Anyway, the talk was about sedimentation of colloids.

Sedimentation is something I don't usually like to think about because gravity, as any particle physicist will agree, is a massive pain in the arse. Never-the-less, my experimental colleagues are somewhat stuck with it (well, most of them). As is often the way it turns out you can turn this into a big advantage. What Piazza did, and then others later, was to use the sedimentation profile of a colloidal suspension to get the full equation of state, in fact the full phase diagram, from a single sample.

The nicest example is from Paul Chaikin's lab (now in NYU, then in Princeton), where they used a colloidal suspension that was really close to hard spheres. They mixed a bunch of these tiny snooker balls in suspension, and then let it settle for three months. What they got is this lovely sample, with crystal at the bottom (hence the strange scattering of the light), and then a dense liquid which eventually becomes a low density gas at the top. It's as though the whole phase diagram is laid out before you.

Equation of State

This is a very beautiful illustration, but it's not the best bit. In the same way that atmospheric pressure is due to the weight of the air above you, if you can weigh the colloids above a particular point in the sample then you can calculate the pressure at that point. This is exactly what they did. There are many different ways to measure the density of colloids at a particular height, if you can do it accurately enough (which was the big breakthrough in Piazza's 1993 paper) then you can calculate the density as a function of pressure. In a system where temperature plays no role such as this, this is exactly the equation of state (EoS).

When compared with theoretical calculations for hard spheres the experimental data lies perfectly on the theory curves, complete with first order phase transition where it crystallises. This is really a lovely thing. EoSs are very sensitive to exact details, so in the same way that in my group we compare our simulation of the EoS to check our code, this showed very accurately that their colloids really were hard spheres.

So I think this is all very nice. I nicked the above images from Paul Chaikin's website, I recommend having a poke around, there's loads of great stuff (you really need to see the m&ms).

Back from the dead

Can't remember the number of times I've said I've been away because I've been busy, but this time it'll be different. Well it probably won't be different, it looks like I'm destined to be an inconsistent blogger!

It's now been three months since I arrived in the Netherlands for my new job and I'm enjoying it a lot here. The pace is much faster in the group than I'm used to but I'm enjoying the buzz of lots of interesting things getting done. Now I'm more settled I'm hoping for a spectacular return to blogging - there's certainly enough to talk about here!

The Dutch are good at science

In general the Netherlands has a fantastic history in the sciences. I was watching Carl Sagan's Cosmos the other day (best telly ever made), he loved the Netherlands it would seem. There's a whole episode where people dress up in pointy hats and reenact bits from Dutch scientific history.


I'm no historian so there's no point making a huge list. Some notable greats though include Cristiaan Huygens, famous for the wave theory of light, he worked on telescopes and even the pendulum clock. The microscope was invented in the Netherlands, allowing the Antonie van Leeuwenhoek to discover "a universe in a drop of water".

What about statistical mechanics?

Closer to the focus of this blog, the name Johannes van der Waals is never far away. His theories allowed us to begin to understand why matter undergoes phase transitions.Two names that are important for us here in Utrecht are Peter Debye and Leonard Ornstein.

Peter Debye is another one of those names that just seems to pop up all the time. It's littered through my thesis because of his work on phonons. Debye was professor at the university of Utrecht for a very short time. I believe the university didn't deliver on his startup money so he left. The picture is from our coffee room in the Debye Institute.

As well as working in the Debye Institute I also work in the Ornstein Lab, after Leonard Ornstein. For me his name is most famous from the Ornstein-Zernike relation in liquid state theory, however, I think he did a lot of varied stuff. He followed on from Debye at Utrecht in 1914 where he remained until 1940. Ornstein was Jewish and at the beginning of the war was dismissed from his position at the university. Only six months later he died. Seems to me it should be the Ornstein Institute, anyway, we also have his picture up.

Enough history

So the Dutch weren't too bad at science. The living ones aren't too shabby either. So hopefully lots of interesting things to be posted in the coming weeks.

Universality at the critical point

Time for more critical phenomena.

Another critical intro

I've talked about this a lot before so I will only very quickly go back over it. The phase transitions you're probably used to are water boiling to steam or freezing to ice. Now water is, symmetrically, very different from ice. So to go from one to the other you need to start building an interface and then slowly grow your new phase (crystal growth). This is called a first order phase transition and it's the only way to make ice.

Now water and steam are, symmetrically, the same. At most pressures the transition still goes the same way – build an interface and grow. However, if you crank up the pressure enough there comes a special point where the distinction between the two phases becomes a bit fuzzy. The cost of building an interface goes to zero so there's no need to grow anything. You just smoothly change between the two. This is a second order, or continuous, phase transition and it's what I mean by a critical point.

As I've demonstrated before, one of the consequences of criticality is a loss of a sense of scale. This is why, for instance, a critical fluid looks cloudy. Light is being scattered by structure at every scale. This insight is embodied in the theory of the renormalisation group, and it got lots of people prizes.

Universality

A second feature of critical phenomena is universality. Close to the critical point it turns out that the physics of a system doesn't depend on the exact details of what the little pieces are doing, but only on broad characteristics such as dimension, symmetry or whether the interaction is long or short ranged. Two systems that share these properties are in the same universality class and will behave identically around the critical point.

At this stage you may not have a good picture in your head of what I mean, it does sound a bit funny. So I've made a movie to demonstrate the point. The movie shows two systems at criticality. On the left will be an Ising model for a magnet. Each site can be up or down (north or south) and neighbouring sites like to line up. The two phases at the critical point are the opposite magnetisations represented here by black and white squares.

On the right will be a Lennard-Jones fluid. This is a model for how simple atoms like Argon interact. Atoms are attracted to one another at close enough range but a strong repulsion prevents overlap. The two phases in this case are a dense liquid and a sparse gas.

One of these systems lives on a lattice, the other is particles in a continuous space that are free to move around. Very different as you can see from the pictures. However, what happens when we look on a slightly bigger length scale? Role the tape!


At the end of the movie (which you can view HD) the scale is about a thousand particle diameters across containing about 350,000 particles and similar for the magnet. At this distance you just can't tell which is which. This demands an important point: These pictures I've been making don't just show a critical Ising model, they pretty much show you what any two-dimensional critical system looks like (isotropic, short range...). Even something complicated from outside of theory land. And this is why the theory of critical phenomena is so powerful, something that works for the simplest model we can think of applies exactly - not approximately - to real life atoms and molecules, or whatever's around the kitchen.

Meeting is good

Once again I find myself making some excuse as to why it's been over a month since my last post. My first reason is I'm finishing up my current postdoc. My other reason is I've been doing lots of travelling. This is much more exciting as I've been finding out more about all the cool soft matter / stat-mech work that is going on in the UK. Some of which I will blog about in time. I've also learned that half the people in soft matter in the UK have worked at some point in the Netherlands, which is handy because I'm moving to the Netherlands!

Getting to the point

All this travelling is related to the topic I wanted to get to today - the value of meeting. I was started off thinking about this thanks to Alice Bell's article in the THE on the value of the seminar. Here Alice calls for seminars to be posted online, something I agree with very much, as a way to reach more people (and to improve the standard a bit). From my experience I've had to use hundreds of pounds of grant money touring the country giving the same seminar. While I value that experience - meeting the people in the groups, direct interaction and so on - it's a shame that people at other universities can't see the talk as well.

Of course if people knew it was online they may not turn up, but hopefully not. I might start sticking mine up here.

The more efficient way of way of reaching many like-minded academics is of course the conference. A good conference can do wonders for your creativity and enthusiasm, it can give you an instant snapshot of the state-of-art and you can meet future employers/collaborators.

But they can be a bit stuffy and long. And expensive. So I'd like to fly the flag for a third kind of academic interaction, the informal science "retreat". Not long ago we had our annual Cornish Soft Matter weekend. A small group of physicists and chemists from a couple of universities got together for a more relaxed meeting. Talks were projected onto a sheet, we were sitting on sofas or the floor, and the start of a talk would be delayed due to people making a last minute cup of tea (usually this was me). All this in a really nice setting.

The demographic was largely PhD students and postdocs, and everyone had to give a short talk. If it overran, fine, if people had questions they'd asked them right away. Students were encouraged to ask as many questions as possible and academics resisted the urge to tear anyone to bits with their sharpened critical skills.

Scientifically it's great. I got to hear from the people who make all these synthetic colloids that I always cite. Their concerns weren't always about phase diagrams or dynamic arrest, sometimes it was simply how much stabiliser or chemical X do I need to get the polydispersity down. These are problems I don't usually get to hear about and it's particularly nice to get it from the people at the coal face.

Because the atmosphere is more relaxed you can give a different kind of talk. In a conference you're so worried about being jumped on that you tend to take out all the personality from a talk, all the wild speculation and, well, then fun side of science. Here we could kind of let rip. If we wanted.

Socially it is also a good thing. It's easy to get a little isolated with your own little problem, especially when your doing a PhD, so it's nice to mix a bit. Science, like most jobs, requires a degree of networking. While I hate this word and all that it implies, these informal gatherings are a much better way to get to know people than conferences. People at conferences are always trying to look smart and generally suck up to the established professors. Makes me shiver just thinking about it.

A snappy conclusion

The main thing that made this meeting nice was the atmosphere. I highly recommend anyone to organise something similar if it's possible. Sure, it was no Copenhagen, but the science was good and it helped create that sense of being in a scientific community.

While it's not free it's a lot cheaper than a conference. I guess you don't need to go all the way to Cornwall but it is nice to get out the department for a couple of days - especially when you usually sit at a desk all the time.

An early look at simulation

While I was putting together the post on 2D disks I came across a lovely paper from 1962 on 2D melting by Alder and Wainwright. From there I found this paper from 1959: Studies in Molecular Dynamics. I. General Method by the same authors.

They describe the "event driven" molecular dynamics (MD) algorithm. Normally, with MD you calculate forces, and thus accelerations, and update this way. Hard disks or spheres behave more like snooker balls, the forces are more or less instantaneous impulses that conserve momentum so it's better to deal with collision events and leave out the acceleration part.

The paper gives a fascinating insight into the early days of computer simulation (they still refer to them as "automatic computers"), what their limitations were and what details were worth worrying about. To give you an idea, in 1959 they say:

With the best presently available computers, it has been possible to treat up to five hundred molecules. With five hundred molecules it requires about a half-hour to achieve an average of one collision per molecule.

So in their case it was CPU speed that was the problem, they get about thousand collisions per hour. To put that in perspective, a modern event-driven simulation of a similar system will maybe hit about a billion collisions per hour on a reasonable desktop [source]. I don't say this to mock their efforts, these are the giants on whom's shoulders we stand. I'll come back to why that number is so comparatively big these days, first I want to look at visualisation.

Visualistion

In 1959 there were no jpegs or postscripts and certainly no Povray or VMD, I'm not sure they even had printers. So how do you visualise your simulation? Well they had a rather elegant answer to that. They could output the current state of the system to a cathode-ray tube as a bunch of dots in the positions of the particles. Then they pointed a camera at the screen and left the shutter open while they ran a simulation. What you get is these beautiful images below showing the particle trajectories. Firstly in a crystal phase you can see the particles rattling around their lattice sites

This is a projection of the FCC lattice (the squares confused me at first). In the fluid phase they do a little bit of cage rattling and then start to wander off.

[Figures reprinted with permission Alder and Wainwright, J. Chem. Phys. 31, 459 (1959). Copyright 1959, American Institute of Physics].

I honestly couldn't show it better today. Some people dismiss visualisations as pretty pictures that only exist to attract attention. Perhaps this is sometimes true but it only takes one look at this to see how they can stir the imagination and shape the intuition – and that's what creates new ideas.

Algorithms

I'd quickly like to come back to the speed difference between 1959 and today. A lot of the difference can be put down to Moore's law. After an annoying amount of Googling I can't really say how much faster modern CPUs are. A lot probably. However, I'd like to focus on an often overlooked factor – the development of algorithms.

A general event driven algorithm calculates the collision time for each pair of particles and, if it is under a cutoff, stores it in an event queue. It then fast forwards to the shortest time whereupon it will need to update the queue with new events that appear after the collision. Initially this requires checking all pairs, Alder and Wainwright call this the "long cycle", and this has complexity of order N-squared, O(N^2). This means that if you double the number of particles, N, then you have four times as many calculations to perform.

After a collision you only need to update events involving the particles that collided so you can get away with doing N updates. This is the "short cycle" and is O(N). It's not mentioned in this paper but I think there's an issue with sorting the event queue so this is probably still O(N^2). Either way, for their early simulations the total number of collisions per hour tanked as N was increased.

And this is where algorithms come in. You can use all sorts of tricks. In dense systems you can use a cell structure to rule out collisions between pairs far away. Modern algorithms focus largely on keeping the event queue properly sorted. A binary tree will sort with O(log(N)) and here they claim to have it O(1). Of course the complexity is not the only important factor, there may be other more important overheads, but it gives an idea of the limitations.

In equilibrium statistical mechanics specialised computer algorithms have made a spectacular impact. Techniques such as the Wolff algorithm, Umbrella sampling, and many many more, have outstripped any speed up by Moore's law by many orders of magnitude. I could go on about algorithms for hours (maybe a post brewing), instead I'll just make the point that it doesn't always pay to just sit and wait for a faster computer.

We've come a long way

These early simulation studies weren't just important for developing methods, they were able to answer some serious questions that were hopelessly out of reach at the time. Since then simulation has firmly established itself in the dance between theory and experiment, testing ideas and generating new ones. And it shows no sign of giving up that position.

Lipid membranes on the arXiv

A while ago I discussed lipid membranes and how they could exhibit critical behaviour. There were some lovely pictures on criticality on giant unilamellar vesicles (GUVs) which are sort of model cell walls. That work was done by Sarah Keller and friends in Seattle.

This morning on the arXiv I saw this new paper, also by Sarah:

Dynamic critical exponent in a 2D lipid membrane with conserved order parameter

They look at the critical dynamics of the GUV's surface. Being embedded in a 3D fluid does have its consequences so they've attempted to account for the effect of hydrodynamic interactions. I haven't poured over their model but the paper looks really nice.