LHC linking park

The Large Hadron Collider comes online on the 10^th and there is little else that is the topic of conversation at any place where there are more than two physicists. Higgs or no Higgs? That is the first thing we will know. And supposedly a whole lot more.

As with all else in the world, LHC has been associated with theories that tell you that the world will end when we turn the machine on. You can find out about these concerns from the horse’s mouth here. And I also came across this refutation/explanation by Sunil Mukhi (via).

If you wanted to know more about the LHC, you could try this. Or at 8pm on the history channel tomorrow there is a show on the LHC that I must have received 10 emails telling me to watch. The host of the show talks about it over at Cosmic Variance.

Or, you could watch this video I found recently…LHC rap…pretty cool I thought it was…

From birds and fish to cells

A physicist’s view of collective transport in biological systems

Let me motivate what I want to say today with a couple of videos. First up, an amateur video of a flock of starlings in Scotland.

Or see this one, where the flock cohesively responds to a predator. A Starling is a small bird, shown in the picture alongside, about the size and shape of a “myna” if you are familiar with it. They fly in flocks that do amazing things as a collective entity as you just saw in the above example. Understanding how they do this is field of active research as indicated by this cover of the October issue of Physics Today. I will tell you a little bit about how they do that subsequently. But you might say to me, “They are birds, and they have brains, albeit “bird brains”, so they see, process that information somehow and do stuff. Why would a physicist concern herself with that?” So, to make my point even more clear, let me show you one more video.

This one is a microscopic movie of a bacterial swarm (obtained from here). Do you see the complex flow patterns they exhibit? These guys clearly do not have brains. It might be that this rich collective behavior originates in more chemistry than physics, but clearly not biology. And to make my point that it is indeed just physics, I ask you to look at this other video (obtained from here)

Do you see the similarity in the flow pattern to that seen in the bacteria? Can you guess what you are looking at? It is just a vibrated monolayer of some centimeter long metal rods! Whatever is going on here is clearly just physics. Moreover, one can mathematically represent the motion of the bacteria/birds and those of the rods by the same set of equations! What I want to do in the rest of this post is to give you a flavor of some of the physics behind these and other collective phenomena in biological systems.

When looking at fish schools or bird flocks, the first postulate that comes to mind is that the phenomenon is “follow the leader”, with one bird/fish doing its own thing, and the others following. But as stated above, the things that birds/fish do is “mathematically similar” to what the bacteria do. So “follow the leader” seems an unlikely scenario. The logical next question to ask would be, “What are the minimal rules that can give rise to this kind of behavior?” We have known the answer to this question for a while now [1]. The rules are the following – Each member of the flock does each of three things a) Alignment : Adjust my direction of motion so that I am going in the same direction as my neighbors, b) Velocity matching : I adjust my speed so that I am going with the same speed as my neighbors c) Cohesion : I try to keep the distance from my neighbors the same at all times. With these basic rules and simple boundary conditions for entities at the edge of the flock, like “If there is food, turn towards it” and “If there is danger, turn away from it”, most of the complex patterns exhibited by these groups of organisms can be reproduced!

But these are just rules. So, the next question to ask would be, “Can these rules come about from just physical interactions?” Let us ignore the boundary conditions associated with food/predator for the moment, they clearly are chemistry and other higher processes and focus on the bulk flocking rules. What is a unifying thing between the birds, the fish the bacteria and so on? What they are, are objects that have a non-spherical shape that actively move through a medium (air/water etc.). Now what does that mean? They exert a force on the medium [2]. The medium responds, i.e., the fact that my bird/fish/bacterium is pushing on the fluid induces a flow in the fluid itself. This response now propagates through the fluid. So, a bird/fish/bacterium that is elsewhere will feel this change in the fluid, in terms of the local flow field and pressure gradients. And it will adjust its own force on the fluid accordingly, and this whole things feeds back to the other entities in the flock. This phenomenon is called hydrodynamic interaction. And this is the dominant interaction that produces the three aspects of flocks that is listed in the previous paragraph!

Further, I want to make the case that this quest for minimal mechanisms for collective behavior is not just restricted to animal group behavior on the different scales encompassed from birds to bacteria. For this, see the famous video below.

This is a video of a neutrophil chasing a bacterium and then gobbling it, the immune system of your body at work. I know what you are thinking, “This is one cell chasing one bacterium and the primary thing at play here is chemotaxis, so what is collective about this?” The collective aspect lies in how the cell crawls, i.e., at the sub-cellular scale. The interplay between membrane fluctuations, the stresses in the actin-microtubule network that makes up the cytoskeleton of the cell, the interaction of this network stress with the medium that the cell is in and many other things go into understanding how the cell crawls. The mathematical paradigm and the physics aspects of this question are not so different from those one uses to address animal group behavior we considered earlier! But for now, this is just a teaser. A separate post on this to follow later.

[1] Actually the first instance in literature was in the context of an algorithm for computer graphics, available here.

[2] If we want to be careful, then clearly third law tells us that the swimmer must at least be a force dipole. Since there is no mathematics displayed here, I fudge this point.

On the Torus

What is it ?
In layman’s terms the inner tube of a tire or if you prefer, a doughnut or a vadai. Technically, it is the surface of revolution of a circle about a coplanar axis outside itself (to put that in context, a sphere is a surface of revolution of a circle about one of its diameters, i.e., a coplanar axis through its center).

Why am I talking about it?
I always thought that a torus is a strange object, almost like a mobius strip or something. Mathematically it is a weird compact manifold. And so I imagined that a torus is as likely to occur in nature as a cube. But then, I found out today that certain dumbbell shaped surfactants self assemble into nanoscale tori [1]! Further digging and googling told me something I should have known, that people who make Carbon nanotubes have actually seen Carbon nanorings that form at the same time, that are essentially tori.

What does this mean?
So, the above facts tell us that self assembly can lead to tori. If it is easy to form and stays stable, then more likely than not, we will find it in nature. And indeed we do. They call it (rather unimaginatively) the torovirus.

Goes to show that one has to modify and augment one’s intuition with each new fact learnt!

[1] A Primer on self assembly of surfactants available here.

Dark Energy again

A short while ago I said a little something about dark energy over at the SC blog. Today I discovered this at the hubblesite. I did not like it too much, but it has more info than my post did on some aspects, but less info on others. Further reading available here as well. Also, Cosmic Variance has something to say on the presentation here.

The second law

A while back, I had written this post on irreversibility. But I was just talking text book thermodynamics and stat mech. Here, Sean Carroll is putting the arrow of time in a cosmological perspective. Do check it out. Also, while you are at it, stop by SC and read my post on elastomers and tell me what you think.

Theory of Everything?

First I saw this. Then, I went to the physics archive, the democratic forum where anyone can post and found this. I printed it, skimmed it and formed a minimal impression (to the effect that it is a good idea but ways to go before it can be a theory of everything). But, I am not well versed in the relevant jargon, so I need to spend some time and talk to people who are well versed (and that is the advantage of being in a physics department, all kind of help is just a stroll down the corridor away!) before I can tell you about it. But, in the mean time do go and check out this post, and scroll through the comment thread, we have Garret Lisi in there trying to explain this stuff himself.

PS: The pretty picture is the polytope with E8 symmetry, the group that has that nice property that the fuss is all about. I know, I know, I just put it in because it is pretty :)

Journal club – Immunization and terrorists

I was whiling away some time yesterday when I read this paper [1]. What the paper is about is “effective immunization strategy for scale free networks”. First let me tell you what each of these words mean. If you already know or are not interested, skip to the last two paragraphs.

What is a network? It is a collection of points (nodes) that are connected to each other in some way (links), so that you can get from one point to another though some path along these links. What are the real world systems a network can represent ? The Internet (each computer is a node and each hard wired connection is a link), the world wide web (each website is a node and each link is a link…), social networks of people, biological networks of chemical reactions, epidemiological networks that map out how a disease spreads through a social network of people in physical contact and so on and so forth. In fact, I think there is a subset of mathematicians and physicists who would have you believe that any problem in the world that is worth solving is actually reducible to a question about some property of some network!

What is immunization? It is exactly the same as in English. I give a vaccine to a node (person) so that it cannot be infected by a disease and hence cannot spread it. What is an immunization strategy? Given a network that you do not know much about (only the fact that it is a network) what scheme should I use to immunize nodes so that with least possible immunization I can guarantee that I will not have an epidemic?

And finally, what is a scale free network? You can find some info here, but in my mind, when somebody says scale free network, I imagine airports as nodes and flights that emanate from them as links. Supposing I draw this, it looks like the image alongside [2]. The first thing you notice is that there are hubs. Suppose you close your eyes and pick one point on the network, like as not, that point will be connected to one other point and that point will be a hub, i.e., connected to a whole bunch of other points. You can do this exercise in your mind with airports if you like and you will see the same thing. Mathematically, such networks turn out to have some nice properties, but for the purpose at hand let us leave it at that.

Alright, so much for preliminaries. What is this paper about? Well, it turns out that if you used a “random immunization” strategy, i.e., if you randomly chose a fraction of people that are connected by such a network and immunized them and asked “Is my population now protected from an epidemic?”, you find that you basically have to immunize everybody before you can be sure there is no epidemic. So this is no good as a strategy. So, what is a better strategy? The authors of this paper say that instead of randomly picking people and immunizing them, you do the following. You randomly pick a person, then randomly pick one of his/her acquaintances (a node that is linked to the node you picked) and immunize them instead. In this way, your network becomes protected from epidemics way sooner. This is their primary result [3].

Now if you think about it, it is easy to see why this is true in scale free networks, again using the airports analogy from earlier. Suppose I wanted to shut down air traffic in the USA. I have a list of airports. The way I want to accomplish my mission is by randomly picking airports to shut down. Now suppose I randomly picked an airport and shut down the airport that I picked, it will take me forever to do this for I will choose the hubs with the same probability as I would choose the hundreds of other itty-bitty airports around. Instead if I said I will pick an airport at random and shut down one of the airports it is connected to, I am way more likely to shut down hubs than in my first route. And hence the strategy described in this paper is indeed better for scale free networks.

But what made this paper priceless in terms of amusement obtained for the time spent was the following sentence…”As a final remark, we note that our approach may be relevant to other networks, such as ecological networks of predator prey [32,33], metabolic networks [34], networks of cellular proteins [35], and terrorist networks. For terrorist networks, our findings suggest that an efficient way to disintegrate the network is to focus more on removing individuals whose name is obtained from another member of the network.” Homeland security…are you listening?

Caveats and Disclaimers

[1] I know squat about networks theory, so if you are an expert, correct me and if you are not, do not trust me entirely.

[2] It is actually an image of stock trades in the new york stock exchange stolen from here…thanks to google’s image search, but same difference.

[3] As far as the mathematics go, there are other assumptions in here…the network must a) have a tree structure, b) be undirected, c) uncorrelated, d) unweighted. But for the resolution I have of such things…same difference.

Biomechanics – Cool stuff!

The physics posts that have shown up until now on this blog have been on boring things (albeit, close to my heart boring things) such as the second law and equilibrium self assembly. So, I have been thinking for a while now that I should write something on a cooler topic, even though I might know a lot less about it. The topic kind of chose itself this last week when my idle reading on collective intelligence (more on that in a subsequent post) took an interesting tangent.

The tangent is Biomechanics. As a discipline, in the broadest terms, this is a quest to understand the locomotion of living things. Of course, this quest can be addressed at many levels. But for the purposes of this post, we are going to restrict ourselves to the following question. Clearly, living things have lots of stuff on their mind (read neural control circuit) apart from locomotion. So, they have evolved in such a way that they must have a minimal feedback-limited control circuit that governs locomotion. If we could figure out what this minimal model is, we could apply it to robotics and hence be able to design robots that walk easily and hence have room left in them to build in other functions. Why would we want to do that? A standing example could be what happened to the Mars Rover Opportunity. The video below is a time lapsed footage of the rover extricating itself from some loose sand. In real time, it was stuck for a month [1].

You see, the Mars rover belonged to the old robotics paradigm of control that was not feedback-limited and certainly not minimal. This idea of minimality is something we learnt from watching nature solve complicated problems with ease [2]. So, people started asking, how do living things walk/run? A successful minimal model that came out of this study is what I call the “foot-forward strategy”. This model essentially said that the organism put half its feet down and kept the other half in the air (3 feet if you are a hexapod and one if you are human). It measured the resistance that the surface gave each of its feet and decided how far ahead to land the feet that are in the air and then keep repeating the process [3,4]. This model, that has just two ingredients could successfully explain the locomotion of a wide range of living things in a wide range (not exhaustive as you will see below) of terrains.

This strategy was implemented with great success in a robot called the Rhex. See below a video of Rhex zipping through all kinds of terrain, in a direction given to it by a guy with a remote control [5].

This is all well and good and a success to the method of scientific enquiry. But this model works only when the terrain is solid. Consider for example the following video of a spider moving on some uneven substrate. This substrate has holes that are larger than the foot of the spider and deeper than the length of its leg, but it still manages to zip across it (the video is slowed down 20 times).

So the question now is how do we understand this kind of motion. In attempt to study this question systematically, researchers decided to take this motion into the lab. Find below a video of a cockroach running across a wide mesh. Again the video is slowed down 50 times. It is actually moving very fast [6].

Now, there are several possible explanations for how the cockroach manages this. One possible explanation could be that this is an emergent (euphemism for “pleasantly unforeseen”?) consequence of the foot-forward strategy itself. But this the researchers can test readily, for they had the Rhex and we know for a fact that Rhex does not know anything other than the foot-forward strategy. So they did that and let Rhex run on the mesh to see what happens.

Oops! Rhex does not like the mesh! This tells us that there are two possibilities. The messier of the two is the possibility that the Cockroach has more than the simple foot forward strategy built into its neural circuit and we need to figure out what that is. But there is a simpler possibility. May be there is a physiological feature of the leg of the insect that we are missing. And this latter turns out to be the answer. Look at the cartoon along side of a cockroach. Its legs have spines or hairs or whatever you want to call them. These hairs have the property that they give easily in one direction (when pushed towards the leg) and are very stiff in the other direction, requiring loads greater than the weight of the insect in question to make them give. What the insect does when its foot lands ina hole is to use one of these pikes for leverage. You can go back to the video of the cockroach to see that this is indeed the case. So the claim now is the foot-forward strategy together with spikes or hairs are sufficient to negotiate terrain with gaps. The researchers tested this as well. They took Rhex and put spikes on his legs with the same properties as those on the legs of insects and put him back on the mesh. See the outcome in the video below.

It works! Rhex manages to get across, even though less elegantly than his real insect counterparts. The researchers of course performed other tests to verify the hypohesis. They took a cockroach and removed the hair from its legs and watched it stumble on the mesh. They took one of the fastest running creature on earth, the ghost crab (Ocypode quadrata) and let it run on the mesh. It struggled of course, because it runs on sand and hence has no spikes on its legs. And then they put spikes on its legs and watched it make it across the mesh successfully [7]. And so the researchers have successfully demonstrated that the foot forward strategy is enough even with holes in the terrain! This whole thread is a cool illustration of scientific methodology in general and that is one of the reasons I decided to write about it. The other of course are the cool videos. Are you as impressed by the coolness of it all as I am?

Asides, References and Disclaimers:

[1] I had the unique opportunity of watching the rover stuck on Mars live! I was at a NASA meeting at the Kennedy Space Center at the time and they were supposed to show us the rover in action live. But as it turned out, the Rover was stuck for the duration of the meeting.

[2] I looked for a reference on the control strategy for the rover, but could not find one. So, what I am saying here is hearsay. It must be classified or something. And the hearsay comes from a friend of mine that is Robotics researcher at UPenn, subject to my understanding of what he said.

[3] Notice that individual sensing gives an advantage to the multilegged creature. If the resistance on the front most leg is smaller than the hind ones, you know you are probably going onto softer terrain and step accordingly and keep your eyes, if you have them, on your food or predator or ipod or whatever.

[4] This of course is my minimal interpretation of the model. Find more details on this and other models in this Science review paper and the references there in.

[5] All the references on the development and implementation of the Rhex can be found in the website liked above. As an aside note that this project is funded by DARPA. So, unless we have secretly discovered an alien inhabited planet that the U.S is planning to invade, Rhex is more likely to be used in Iraq or whatever the next location for the war on terror is!

[6] This and all of the following is work done by Daniel Goldman and his collaborators. You can find all relevant references at his website. The videos are all stolen from there as well.

[7] The crab videos are on youtube as well. If you want to watch them, they are here and here.

PS: Apologies if this post showed up multiple times in your feed reader. Blogger screwed me over.

On Micelles, Vesicles and Artificial Cells

-The magic of directed self assembly

This post is a collection of thoughts on the principles of entropy, energy and equilibrium expressed in the context of self assembly of surfactant molecules [1]. Let us begin by asking what a surfactant is. For the purposes at hand, a surfactant is a molecule with a small head that likes water and a long tail that hates water as shown in the cartoon alongside. What “loves” means in the following is that the entity can lower its energy by being in contact with water and what “hates” means is that it costs the system a lot of energy when it is in contact with water [2]. Now, we put a bunch of these surfactant molecules in water and allow them to come to “equilibrium”. What do they do?

To understand this question, we have to first clarify what a system will like to do. The equilibrium state of the system will be one where it can do the maximum number of things it likes. The first thing the system likes to do is lower its energy as much as it can. On the other hand, the system likes to have as much disorder as possible, technically speaking, “maximize its entropy”. If I call the energy of the system E and the entropy of the system S, then the system likes to have a minimum value for the quantity F = E – TS, and this is called the free energy of the system. Don’t let this little jargon scare you. What follows is simple enough even if you don’t remember this. Also, before we can guess what the system will like to do, we need to know one more thing about the surfactant molecules. If two surfactant molecules come close to each other, what would they do? The tails of these molecules are such that they are happiest when they are as close to each other as they can get, for they lower their energy by reducing their interaction with water and increase their entropy as well [3]. The heads of these molecules are such that they want to stay as far away from each other, because these heads are usually charged and like charges repel right? So they lower their energy by staying away from each other.

With that, we have all the ingredients we need to answer the question we asked. It is now all about a competition between love and hate. Suppose the heads love water way more than the tails hate it. Then the equilibrium state of the system will be a solution of the surfactant molecules in water, with all the molecules well separated from each other and doing their own thing [4]. Next, suppose the circumstances are that the hate of the tail wins. Also, suppose that the heads are wide objects so that the overall shape of the surfactant molecule is a cone (see figure). Then, the molecules are happiest when they form micelles. Micelles are objects that are spheres, with the polar heads outside near the water and the tails inside, talking only to each other and protected from the water by the polar heads. Note that, in order to form micelles, you need a given amount of surfactant in the water (If you have fewer surfactant molecules, entropy wins and they stay in the form of the solution). The everyday situation under which micelles are formed is when you wash your clothes with soap. The dirt on the clothes form nucleating centers for the micelles and the micelle itself being water soluble, dissolves in the water when you rinse your clothes.

The more interesting case is when the heads are not fat, i.e., the surfactant molecule is a cylinder rather than a cone (see figure) and still the hate of the tails wins. In this case, the system forms what are called “lipid bilayers”. This is just two layers of surfactant molecules assembled such that the tails of each layer face each other (effectively, it is like having a layer of oil trapped between two layers of polar heads). Now, in this structure, the tails in the middle are clearly happy for all their neighbors are fellow hydrocarbons. But, just as clearly, the tails at the edge of the structure are unhappy because they have to talk to the surrounding water. One way to eliminate this is for this bilayer to fold on itself to form a spherical shell (see figure, which displays a cross section of such a structure). This way, there is no surface of tails talking to the water. But the trade off comes at the cost of forcing the heads in the inner layer to be more close to each other than they like. But, if the hate of the tails for water is large enough, this happens and the resulting stable structure is now a vesicle!

The interesting things to note here are twofold. One, in spite of the language I am using, in the actual experiment, all I did was take a spoonful of surfactant molecules and put it in water. All the structures mentioned above self assembled! I did not have to do a thing. The second thing to note is that, the above vesicle is essentially a minimal cell membrane, the first step towards the process that converts an auto catalytic chemical reaction into what we call now as life!

So, if we can make this membrane functional, namely, make sure that the chemical machinery required for life is trapped inside the vesicle, make appropriate “holes” so the membrane is suitably permeable (i.e., it lets some stuff in (raw material for making food) and some other stuff out (waste products) and not vice versa), we would have made an artificial cell! Some first steps in this direction have already been taken. See for example, this PNAS article reporting the use of “directed self assembly” to make a bio reactor, which is to say it is not quite a cell yet, and this article entitled “Towards an artificial cell based on gene expression in vesicles” . We are not very far from making what can only be termed as artificial life, in a physics lab, in a test tube. And the reason I started thinking along these lines was to be able to ask the question – “Intelligent design anyone??” :) [5].

Caveats and disclaimers

[1] The aim is to try and keep things simple, focusing on the primary ideas and suppressing all but the bare essentials in terms of details and subtleties.

[2] The jargon is that the head of a surfactant molecule is a polar group like sodium sulfate and hence this ionizes in water and hence is hydrophilic. The tail is a covalently bonded hydrocarbon polymer and hence hates the high dielectric constant medium of water and is hydrophobic.

[3] The entropy of a polymer would be given by the number of configurations they have. They can fluctuate better and sample their accessible phase space better in the lypophilic environment of other tails, than in water, where any fluctuation will result in an energy cost.

[4] It is clearly an oversimplification. The question is truly one of entropy versus energy. So, this will be a strong function of the concentration of the surfactant and the temperature of the water. At low enough concentrations or high enough temperature this will always be the default state with no possibility of self assembled structures.

[5] This is actually a frivolous statement. The stumbling block that people have to overcome is the complexity of a real biological membrane, which has embedded proteins and is active and what not. But from a physicist’s point of view, it is but self assembly, but takes a lot of time. I say this in spite of the fact that from what knowledge we have of the primordial soup and pre-life conditions on earth, there appear to have been singular events that precipitated the emergence of life in nature and we do not know either way, the probability of such singular events occurring from random initial conditions.

Irreversibility

Irreversibility is a fact of life. A hot cup of coffee sitting on the counter of your kitchen gets cold. The opposite never happens spontaneously. A gas released in vacuum quickly expands to fill the whole volume. But this simple fact of irreversiblity is actually a conceptually messy thing in physics. The following is my attempt to explain this rather obtuse concept in a non-mathematical and jargon free form.

In order to motivate what I want to say in this post, I ask you to try and do the following Gedanken experiment (German for thought experiment) in your heads. Imagine that I have turned off gravity for the moment. Imagine a big box in which I put two balls with some kinetic energy. What would they do? They would move with constant velocity in the direction of their velocity. Further I assume that if they hit one of the walls of the box or each other, they will rebound elastically (i.e., will not loose any of their energy). So, if I watch these balls for a while, they are just going to rattle around for ever. Now, I make a movie of this experiment and then show it to somebody else. But, I play the movie backwards in the rewind mode. Will they be able to tell that I am playing the movie backwards? The answer is no.

The system we used in the above experiment can be thought as a minimal model for a gas in a container, albeit a gas with only two molecules. Now, note two things about the above model system. The total amount of energy in the system is the same at all times, this is called “conservation of energy”. Next, note the claim that I cannot distinguish between the movie of the above experiment played forward in time and backward in time. This is called “time reversal invariance”. These two properties are fundamental properties of real physical interactions between molecules. All of microscopic physics has time reversal invariance [1].

Now, let us do another experiment. Again, I have turned off gravity. I take a big room, I seal it off and then vacuum it, in that I remove all the air from the room. In the middle of this room I have a little nozzle that when I start this experiment, releases a small amount of a pink gas. Now, if I wait a while, what will happen? The pink gas will expand and fill the whole room. And hence if I make a movie of this experiment and show it to somebody in the rewind mode, they will be able to tell immediately that I am showing them the movie backwards for a gas that fills the whole room will appear to spontaneously go back to a small volume in the middle of the room. And everybody knows that that cannot happen. Yes?

Next, let us reformulate the second experiment in terms of the model for a gas we used in the first experiment. So what am I doing? I am releasing a large number (about 10000 billion say [2]) of pink balls, each with some kinetic energy into a big box and watching them for a while. The microscopic physics for this system is the same as in the first experiment, i.e., it has “time reversal invariance”. But, when I look at a system with a large number of balls, I “know” the direction of time. So, what is wrong? What did I miss?

The first thing you might ask is “Did you do the calculation for the 10000 billion balls to see that the theory predicts that the system still has time reversal invariance?” The answer to this question is that nobody and no computer can do this calculation. Then you say “Ah! Then you don’t know what your theory says so the whole point is moot!” Well, you would be right but for the fact that the mathematician Poincare proved that if you take a finite system (i.e., a system in a box say) with finite energy, then it will always come back to where it started from [3]. In the case of our experiment, Poincare’s theorem says that the gas that fills the room will eventually all come back and sit in the middle of the room again. But we know this does not happen. So, ask again, what is wrong? The catch here is that you have to ask how long does it take for the system to come back to where it started from. The answer to this question, for the case of the billions and billions of balls that is our gas is a time greater than the age of the universe! So, no matter how long you watch it, it is never going to come and sit in the middle of the room because you can never watch long enough! So, the message here is that the irreversibility that we ubiquitously observe in the world around us is an accident of the time scales over which we conduct the said observations.

Ok, we explained the paradox. But so what? Physics is supposed to be a mathematical model for observed phenomena. And the observed phenomenon here is that the spontaneous expansion of a gas is irreversible on the time scales that are relevant to life on earth. Poincare Recurrence might be a fact of life, but is irrelevant for anything we observe. So, what physics should give me is a rule of thumb that some things will happen (eg., a gas released in vacuum will expand to fill all available space, or a hot cup of coffee placed in a cold room will get cooler) and some things won’t (eg., gas in a room will not all spontaneously go and sit in one corner of the room, or coffee will not get hotter by taking energy from the cold room and hence making it even colder). This rule of thumb is called the Second Law of Thermodynamics that we all learnt in high school.

In the form we learnt in high school, this law is usually stated as “Heat always flows from a body at a higher temperature to a body at a lower temperature”. But a more general statement of this law would be that any system evolves so as to maximize its randomness. It is easy to see that this law explains why a gas will expand to fill the whole room. A given number of molecules occupying a small volume is less random than the same number of molecules occupying a larger volume right? It is rather more obscure to see how the two forms of the law (the first one more naturally explains the coffee scenario while the latter more naturally explains our Gedanken experiments) are equivalent, but it can be shown that they indeed are. But the point to note is that as I said earlier, this law, as is all of thermodynamics, is a rule of thumb to explain observed reality on length scales and time scales relevant to us. Deriving this from fundamental theory is a major mathematical headache addressed by all manners of scientists, mathematicians studying dynamical systems, physicists studying statistical mechanics and so on, with various degrees of success [4]. But clearly, it works and works very well! So we use it anyway.

(For my few regular readers, don’t worry, this post is an anomaly, not the new norm!)

Caveats for experts

[1] CP violating weak forces are not in my picture, I am living in a QED+classical gravity world of macroscopic physicists.

[2] Actually, an Avogadro number of molecules.

[3] I know that Poincare Recurrence theorem talks about approach within an arbitrarily small neighborhood of the initial condition and is stated in terms of bounded orbits, but I did not know how else to say this in plain English.

[4] The basis of thermodynamics in Statistical Mechanics is well established and there is a well developed theory of Hamiltonian dynamics that lets you ask such things as ergodicity and mixing in the trajectory space to substantiate domain of validity of the postulates of Stat Mech, but all this is rather esoteric stuff is n’t it? Or can we state it in plain English in an accessible way?

Statistical Mechanics and Elasticity

What is Elasticity theory? Given the strain on a solid, i.e., for a given macroscopic deformation, it tells you what the stresses in the solid are. There, as always, are two levels at which one studies the theory of elasticity. Suppose the constitutive relationship between the stress and strain is known, for example Hooke's law is a constitutive relation that is linear. Then, the problem is one of solving a set of differential equations to obtain the stress. When the theory is linear, as in the above case, this is simple exercise. The other level of study is associated with deriving this constitutive relation. This is the issue addressed in the rest of this rambling.

First, let us consider the context of an ordered crystalline solid. Further, let us assume that the solid is a defect free single crystal. The energy required to deform such a solid is set by the bonding energy that is primarily electrostatic. So, the thermal energy is much smaller than the typical energy scale in the problem. Therefore, a very good model for the solid would be a lattice of balls interconnected by stiff springs. In this case, the elasticity is linear up to very large applied stresses and the problem of getting a constitutive relation reduces to a Newtonian N particle problem that can be readily solved by going to generalized coordinates that are the normal modes of the system. Next, we ask, if we superpose thermal fluctuations on this answer, how is it changed? The result? It is not changed at all. This result is surprising at first sight, but is a rather obvious artifact of the harmonic nature of the interaction. One can readily verify this by writing down the partition function. The conclusion then is that for a perfect crystalline solid, statistical mechanics is irrelevant for understanding the elastic response of the system! You can hardly ever say this for a finite temperature N particle problem!

So, when does stat mech play a role in understanding elasticity of a crystalline solid? When there are defects in the solid. What happens is that as you increase the applied force on the solid, well before you probe the limits of the harmonic approximation made on the interaction, the defect gives. So, the response of this system is governed by what the defects do, rather than what the background perfect crystal is doing. The dynamics of defects in a solid is an elegantly formulated problem. A well developed theory exists a la E & M for understanding this. I don't know much about this so I will stop by saying that the problem of defect dynamics and defect interactions is a nicely formulated one that one can look up. The point of interest here is that the defects will now have a statistics associated with the temperature of the system. Therefore, in order to understand the elasticity of this system, one needs to take into account the fluctuations of the defects in the system. I don't know much about this either. But I mean to look it up and will tell you when I do.

So much for crystalline solids. There is of course a whole class of amorphous solids whose entire elasticity is statistical in origin, rubber being the standing prototype of this class. They are the reason I started thinking about elasticity and stat mech in the first place. But, I guess I must take another idle morning to sort this stuff out for myself.