Kinetically Constrained A place to keep the longer thoughts of Doug Ashton.
Posts with the tag stat-mech:

The Renormalisation Group

A new video which more or less completes the critical phenomena series. Jump straight to it if you want to skip the background. One of my favourite topics is the critical point. I’ve posted many times on it, so to keep this short you can go back here for a summary. In brief, we’re looking at a small point on the phase diagram where two phases begin to look the same. The correlation length diverges and all hell breaks loose. Well, lots of things diverge. At the critical point all length scales are equivalent and, perhaps most remarkably, microscopic details become almost irrelevant.

The thermodynamic limit

This post has been at the back of mind for a while and written in small, most likely disjoint pieces. I wanted to think about connecting some of the more formal side of statistical mechanics to our everyday intuitions. It’s probably a bit half baked but this is a blog not a journal so I’ll just write a follow-up if I think of anything. I’m often accused of living in a rather idealised world called the thermodynamic limit. This is completely true. To see why this is a good thing or a bad thing I should probably say something about what I think it is.

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.

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.

Colloids are just right

All being good it looks like I’ve secured employment for a tiny while longer. Hooray! The place I’m moving to is a big place for synthetic colloids, so it seems like a good time to go through what I know about colloids. If nothing else it’ll be interesting to compare this to what I’ll know in a year’s time! So, here is a theorists perspective on colloid science. I’ll spare the usual introduction about how colloids are ubiquitous in nature, you can go to Wikipedia for that. The type of colloids I’m interested in here are synthetic colloids made in the lab.

Statistical mechanics of tetris

I’m finding that I’m becoming increasingly fascinated by shape. It seems such a simple thing yet scratch the surface only a little and the complexity comes pouring out. Take simple tiling problems; I can tile my floor with squares or regular hexagons, but not regular octagons - they’ll always leave annoying gaps. From a statistical mechanics point of view those gaps are very important, little sources of entropy that you can’t get rid of. In three dimensions understanding the packing of tetrahedra has proved no simple task. But that’s a story for another day. So it came as no surprise that I was very taken with Lev Gelb’s talk on polyominoes at the Brno conference.

An unintuitive probability problem

Probability can do strange things to your mind. This week I had a probability problem where every time I tried to use intuition to solve it I ended up going completely wrong. I thought I’d share it as I think it’s interesting. Consider a one dimensional random walk. At each time step my walker will go left with probability \(p_l\), and right with probability \(p_r\). It stays where it is with probability \(1 - p_l - p_r\). Furthermore these probabilities are dependent on the walker’s position in space, so it’s really \(p_l(x)\) and \(p_r(x)\). I’m imagining I’m on a finite line of length, L, although it doesn’t matter too much.

Biological Membranes

It’s been ages since my last post. This is because I’ve been busy doing lots of interesting physics, met a bunch of interesting physicists, maybe I’ll write something about it. For now, something I’ve been meaning to write about for a while, and for once it’s something that’s timely. The journal Soft Matter has an issue out with a membrane biophysics theme. You can read the editorial for yourself if you have access, otherwise make do with my ropey understanding of it. Soft Matter is a relatively new journal that I think is looking really good. Their website needs work but I’ll leave that for my science 2.

Critical Point

I’m finally getting around to sharing what, for me, is the most beautiful piece of physics we have yet stumbled upon. This is the physics of the critical point. It doesn’t involve enormous particle accelerators and it’s introduction can border on the mundane. Once the consequences of critical behaviour are understood it becomes truly awe inspiring. First, to get everyone on the same page, I must start with the mundane - please stick with it, there’s a really cool movie at the bottom… Most people are quite familiar with the standard types of phase transition. Water freezes to ice, boils to water vapour and so on.

Entropy

I’ve been meaning to post something interesting about stat-mech about once a fortnight and so far I’m not doing so well. For today I thought I’d share my perspective on entropy. If you ask the (educated) person in the street what entropy is they might say something like “it’s a measure of disorder”. This is not a bad description, although it’s not exactly how I think about it. As a statistical mechanition I tend to think of entropy in a slightly different way to say, my Dad. He’s an engineer and as such he thinks of entropy more in terms of the second law of thermodynamics.