Physics for the 21st Century
Emergent Behavior in Quantum Matter Interview with Featured Scientist Paul Chaikin
Interviewer: What is emergence?
PAUL: Well, emergence is when you have basic fundamental entities that have properties on their own, which you understand, and yet when you bring them together, you get new different sorts of properties—different sorts of behaviors that you might not expect just from knowing what these single entities would do. So, even if you know, for example, all of the attractions, all of the forces between particles, if you bring them together, a number of them can act cooperatively in new ways. There can be phase transitions, there can be cooperative phenomena that you wouldn’t expect just from knowing the elementary forces. And emergence is the properties that you get from the ensemble of particles, from them all acting together rather than the individual particles. And, of course, this can be some simple things like what we might imagine is simple like going from a gas to a liquid to a solid, or originating life. All of that is emergent.
Interviewer: What’s a complicated case of emergence?
Paul: You, life. That’s an emergent phenomenon that’s well beyond what we can understand. We do understand crystallization pretty well now, although not everything about it. But, we do understand a lot of it. We understand a lot about broken symmetry; it happens all over. It was essentially discovered, I think, first, in condensed matter—in something like the formation of crystals or in magnetism. The magnetization picks out a direction in space rather than being pointed everywhere with no magnetization. And that fundamental breaking of symmetry occurs in nature and is used in high energy physics and string theory, everywhere. And that’s an emergent phenomena that came from condensed matter, from simple observation of matter.
Interviewer: What are some of the big questions that you’re trying to answer right now?
Paul: The biggest question is: what are the organization principles that nature uses for things? So, once you know all the forces between particles, and there are four fundamental forces, some of which we understand, some of which we understand less. But nonetheless suppose we knew them all. It turns out that if you knew all the forces, then you could tell exactly what would happen if you had two particles interacting with those forces. If you have even three, you can’t really solve the problem completely. You can predict certain motions. Sometimes they’ll be periodic. Sometimes they’ll be chaotic. But, anything more than two particles is a different story. And what we typically deal with here is 1024 particles or something, an immense number like that, that we have to have new organizational principles for. We can’t think of them on just an atomic level, just from the fundamental forces.
And that’s the many-body problem and there are lots of aspects of the many-body problem. People study them as you’ve heard already by understanding what elementary electrons do when there are many of them interacting with one another, when it’s not just two electrons. But when it’s many electrons then the problem is that the forces on any particle depend on the positions and the history of all the other particles that it’s interacting with. And that’s a very, very complex problem, too complex, for instance, for the more sophisticated computers we have to handle. It probably will always be too difficult for computers to handle when you have 1024 particles. So, you need new organizational principles to understand the behavior of collections of particles not just two particles or three. This is the many-body problem, this is what condensed matter physics exists to explain, to try and understand.
Interviewer: Let’s just talk for a second about what reductionism is, and where the usefulness is for both of these approaches in the work that you do.
Paul: Well, you have to be reductionist to some extent. You have to understand the fundamentals, the fundamental interactions that you’re dealing with before you deal with the bigger problem. Emergence comes from the fact that you know the elementary properties of the particles of the materials you’re dealing with but you get emergence, things that you don’t expect from those properties. And in the end, you have to be able to take the thing apart, to take the idea apart or the matter apart, in order to see what fundamentally is going on. Then you start putting it back together and you see how the properties, which you found interesting to begin with, came about, how they emerged from these basics. But you have to reduce the problem at first to something where you really understand what’s going on, on a fundamental, on an elementary level and then you can build on that.
Interviewer: Let’s talk about your work with M & Ms and ellipsoids. First tell me how this began.
Paul: Well, this basic question of how you break symmetry. For example, how you form a crystal is an interesting problem on its own. One of the most interesting things about it is to really get down to understand it at a very elementary level. You just want the essence of the problem. The simplest particle, the particle that people, physicists especially, but mathematicians as well, love to deal with is a sphere, right? That’s nice and isotropic and symmetric and everything else. And what’s been known forever, that is for millennia, is that if you dump spheres together, you can do it two ways. You can dump them together in an ordered array, like you’d see at a grocery store with oranges stacked up. Grocers and farmers and people thousands of years ago in Mesopotamia, when they first had grain and they first cultivated vegetables, knew that this is the densest packing. There was order. It was periodic. It was actually a crystal. It’s actually the crystal structure that you find for most elements. It’s what’s called a facecentered cubic, or a close pack structure.
On the other hand, what was also known thousands of years ago is that if you take spheres or oranges and you pour them into a container, they don’t pack that way. They pack randomly. The essence of the fact that things crystallize rather than always stay in a liquid an amorphous state has to do simply with the packing differences between this random arrangement and the periodic arrangement. And the essence of it, which is mathematics now and was only proven about six years ago, is that spheres pack more densely in an ordered array—in this crystal packing, in this FCC lattice—than they do randomly. It’s seventy-four percent for the crystal, and sixty-four percent, that’s how much they fill space, for the random arrangement. And this was very interesting.
And since in those days, I had this habit of having lunch every day of coffee and M & Ms, we decided; well, if we know how spheres pack already, then why not try ellipsoids? And so M & Ms, it turns out, are great. They taste good. They’re cheap. They’re all the same in shape. So, we tried packing them and it was really cool because we had an apparatus already set up to do packing of spheres. So, we just replaced the couscous with M & Ms and did the same packing. M & Ms don’t swell, of course, they just dissolve when you put them in water and heat them. But in any event, we measured them. And at first, it turns out, the reason M & Ms…the reason ellipsoids are a really good shape is because you know something about them. You always like to go from something that you know to something that’s a little bit different, but you know something about. And the interesting thing about ellipsoids is that an ellipsoid is just a sphere where you squashed one of the axes. It’s just a coordinate transformation. If you just take x, y, z, your coordinate system, and you put a sphere in it, and now you collapse the z axis by a factor of two, you essentially have an M & M. That’s what it is. Therefore you’d think that it would behave exactly the same. As a matter of fact, you could mathematically prove that the packing of M & Ms and the packing of spheres should be the same. What you do is you take a box of spheres, completely packed, and you squash one of the axes. It doesn’t change the packing fraction at all because the volume of each sphere changes by a factor of two when you change the coordinate. The volume of the whole thing changes by a factor of two because you’ve changed the coordinate. So, the packing fraction remains the same.
So, when my student came to me and said “no, the M & Ms pack better than spheres did,” I said, “that’s just wrong, I can prove it to you.” I was about to show him the proof, and I said, “well, actually, I only know that that works when they’re ordered, when they’re in this crystal.” If I do it when they’re random, it’s true that if I squash it, okay, if I change the axis by a factor of two, I’m not going to change the volume fraction. But, when I looked at the picture when I actually did this on my computer, it was clear that once you did that with the random arrangement, it’s unstable. Then I immediately knew that he was right and I knew why he was right. The reason he was right was that when the ellipsoids are not in a periodic array held in by their neighbors, there are actually twelve neighbors in that array. When they’re in a random array, they can rotate because they have an additional degree of freedom that spheres don’t. If you rotate a sphere, it doesn’t do anything. If you rotate an M & Ms, it changes its orientation, it changes its contact with its neighbors, it’s got an additional degree of freedom, this rotation. And it turns out because of that, it can rotate into a configuration where it packs denser. Once I saw that, it was clear.
Interviewer: So, what are some of the implications of this work?
Paul: Well, one thing it means is that a sphere is not necessarily the optimal shape for everything. And for a physicist who likes to think of everything in terms of spheres because it’s easiest, it tells you that that’s not the whole story. They know that. But it makes you think of doing other shapes for getting different structures now which might self-assemble to make things that are interesting and useful. Directly what does it mean? Well, directly, for example, what it means is, if you wanted to make a ceramic, lots of things are ceramic, what you really want to do is to start out with a powder that is as dense as possible and has as many contacts to neighbors as possible. So, if you did this with spherical particles, you could get twelve neighbors and a packing density of seventy-four percent. If you took every single microscopic, spherical particle and arranged it in a lattice—of course you would never do that, it would take forever to do it. Now you might try and self-assemble that. You could do it, but it would take some time. On the other hand, if you take ellipsoids that are not M & Ms, but ellipsoids with three axis of rotation, with three different dimensions. Then they actually pack to seventy-seven percent because they have six degrees of freedom, they have twelve neighbors, which is just the same as the spheres do in a lattice, in a crystal lattice. And therefore you could take these guys, pour them into a mold, and you would have a powder, which is denser and with the same number of neighbors as if you’d put spheres in an ordered array. So, you could make much better ceramics, much better materials. That’s sort of a gross practical application of it but nonetheless you could do it. More generally, it means that shape matters a lot in making different structures, random or ordered. And that it opens up just a whole new world for investigation, which a lot of people are taking up as it turns out.
Interviewer: How is the packing of M&Ms an example of emergence?
Paul: Because it’s the packing properties, the number of neighbors, the organization of the structures, the packing density, the stability are all properties of the group, of the ensemble of particles, rather than just the properties that you’d imagine just from the interaction of two of them, just from the forces between them. All you know from the forces between them, in the case of hard spheres, which is what we were dealing with spheres, is that they don’t interpenetrate.
Now, knowing just from the fact they don’t interpenetrate, that they will pack better in an ordered array than a disordered array, what the packing fraction is, or the number of contacts is something you couldn’t get just from this elementary force. So the emergent phenomena is the order, the density, the coordination, the correlations in both the disordered and the ordered phases. Those are all emergent phenomena. Even the number of contacts is an emergent phenomenon. It comes from the properties of the ensemble, not from the properties just of the two particles.
Interviewer: Is the idea that we can use these simple systems to come up with principles that can be applied to more complex systems?
Paul: Absolutely. I mean, what you want to do is before you try and tackle life, you’d like to know how particles come together and organize in a simpler way, right? And then you want to build upon that. You want to get the organizational principles. And one of the things that emergence sort of tells us, or teaches us, or we discover about it is that there are different scales, different numbers of particles that interact. And as you go from one scale to the next, from one, from ten, to a thousand, to a million, to Avogadro’s number, you have to think about things differently. There are emergent phenomena on each of those different scales.
For the particles, for the packing of these particles, that’s something that emerges on a scale you can see the basis of it when you have hundreds or thousands of particles. When you’ve got larger arrangements, larger clusters of particles, larger assemblies of particles, there are other phenomena that you’re going to find that build upon those phenomena, build on the packing properties at the scale of a thousand, or so. Which already you couldn’t have known from two.
Interviewer: So let’s talk a little bit about the work you did with colloids that started with David Pine.
Paul: There’s this problem, which is well known in colloidal physics, that at a low Reynolds number, that is, for slow motion in a viscous fluid, the motion is reversible. What it means is that if you do something in one direction, and then you go in the opposite direction, you get the same result as if you ran a movie in one direction, and then you just ran the movie backwards. Okay? That’s what the reversibility is. Going, reversing the direction is the same as going backwards in time from the original motion. And that’s sort of unusual. Not many things do that.
So this is well known. There’s a beautiful demonstration of it by G.I. Taylor in a famous movie that he made to demonstrate the effect, where he takes two concentric cylinders with a viscous fluid between them. He puts a drop of ink in between. He winds the inner cylinder four times one way, and it looks like the ink has just disappeared, as if it’s dissolved. And then he winds it back four times, and the drop of ink reappears. It’s an amazing movie. Anybody that hasn’t seen it has to see it.
Interviewer: So what did David Pine do?
Paul: So Dave Pine, and a colleague of his, Jerry Gollub, from Haverford, knew about this experiment.
So they set up G.I. Taylor’s experiment: two concentric cylinders, with a viscous fluid between them. But instead of putting ink in it, they put in particles, which are like ball bearings only much smaller. They’re colloidal size. But they’re big enough so they don’t really diffuse around, like conventional colloidal particles do. They did G.I. Taylor’s experiment. They sheared. They rotated the inner cylinder and then they rotated it back and then they did this in an oscillation. They just oscillated it back and forth. And the question was: would the particles come back to their original position each time they rotated back to the initial position or would they just diffuse away? And what they found is below a certain amplitude of motion. The particles would come back to precisely the position they had before every time. You could do this forever and the particles would just stay there. Whereas, if you were above a threshold, if you were above a certain amplitude of rotation of the inner cylinder, the particles would just diffuse away. They would look like they were alive, just diffusing away, just swimming away.
So the fact that they saw a threshold, rather than simply that the more they rotated the more particles were active—the fact that they saw a threshold really, really bothered me. I just didn’t understand it at all. And so I worried about it. I worried about it for a while. I said, “I just don’t understand this problem.” So I went and I wrote a little program on my computer. The program was trivial. It didn’t have any of the complexities that really occur in their system. In their system there’s hydrodynamic flows, there’s liquid flowing, everything. I just put in a bunch of particles that were randomly placed and I wrote down a simple rule. I said, “okay, these particles, when I shear them, I distort the structure. When these particles get within a certain distance of one another, I’m going to imagine if they’ve touched. And if they touch, I’m going to say there’s something irreversible that happens. So when I bring them back to their initial position, if they touched I’ll give them each a little displacement.”
And I set up this very simple program. It’s sort of the simplest possible dynamical problem you can imagine. There are no forces, there’s nothing. I just move them and if they come close to one another when I bring them back, I give them each a random displacement. So I set this up. And sure enough, when I put in a big displacement, everything was just diffusing around. Fine. That’s just what I expected. So now what I did is I reduced the amplitude of the displacement in the simulation.
And, sure enough, when I reduced the amplitude, I started out with a random distribution and there were particles jiggling around. They jiggled around for a little while and then they stopped. The first thing that occurred to me is, “Oh, I must have screwed up the program, or my computer’s gone dead.” So since I’m an experimentalist, the first thing I did is I smacked the computer on the side. And that didn’t do anything. So I just ran it again and the same thing happened. A different set, you know, a different configuration, a random configuration of particles, but they just moved around, and then they stopped. And as soon as that happened, I got it.
What had happened is the particles had moved around until they found a configuration where the next time that I made this distortion they no longer hit one another. And as long as they no longer hit one another, they wouldn’t move because I didn’t give them any displacement. So this is something I’d never seen before. This idea that just the irreversibility, just the fact that if things collided, they moved to new positions could be an organizing principle on its own. That could lead to organization. That’s really emergence. That’s without knowing anything, just the random motion of these particles, and the fact that it’s irreversible, that they come back to a different position is enough to organize it. That is really an emergent phenomena, and it’s something I’d never seen before.
And one of the things that’s interesting and is one of the forefronts of understanding the many-body problem, of understanding emergence, of understanding what happens for collections of particles, is not what their ground state is, is not what their equilibrium state is. But we really lack knowledge of an organizing principle of what happens when you drive something, when you drive a system which dissipates energy, which this system does.
Interviewer: So are you looking for the origin of life? Are you trying to find the principles of how things work, or both?
Paul: Yes. I’m trying to understand how nature organizes things. I’m trying to figure out how things work dynamically, in dynamic-driven systems. One of the reasons I’m doing the self-replication is because I want to understand driven dynamical systems. The only way you understand them is if you isolate something and you work on that. You say, “Okay, I’m going to look at this problem, this one problem. And if I can understand this problem, maybe I can generalize it, or maybe I can find some general principles,” like we did with this random organization. I think that’s a general principle, a general organizing principle, even though we did it just with driven colloids at first.
I’m hoping that the self-replication will be similar, that we’ll find something else about it. If I happen to stumble upon the origin of life, I won’t mind! Right? If that’s the way it works, I won’t mind. And, in fact, matter of fact, after coming up with that idea, I’m working with somebody who is taking a prebiotic soup, and we’re cycling temperature and light to see whether it will give something that self-replicates. It probably won’t, but I don’t think the experiment’s been tried before, so it’s worth doing.
Interviewer: So it seems like there are a lot of parallel paths that science is going down to answer some of the same questions.
Paul: What you can learn about nature—the more general physics—is, essentially, about discovering things about nature that are broadly applicable. I mean, most people would say universal, but I’m happy with broadly applicable. So principles that you have which aren’t just, “well, this does this because of one thing.” I mean, physics is not so much about… It’s a different way of looking at things, but it’s supposed to be general. So if you ask a biologist why something happens, he’ll often say, “Well, it’s because there’s a specific protein that does this,” which is actually often the case. A physicist wouldn’t like that so much. He’d rather have, “Well, the way proteins work in general is this.” And it’s broadly applicable. But if you find some fundamental organizing principles, they often are useful in many, many, many different contexts. Things are applicable all over the place. Physics is great. It’s all over.
8.2 Emergent Behavior in Quantum Matter – Video
Reductionism—breaking things into their component parts to study how they work—is an effective tool in physics. But many real-world challenges resist this approach. All too often, large scale behavior emerges in ways that are difficult to predict from the behavior of individual components. When the computational requirements are too massive or the theories that govern the component parts are inadequate, many complex systems have yielded to the physics of emergence, which seeks organizing principles at the system level. Find out how many different phenomena—superconductors, hydrodynamics, and even the formation of structure in the universe—are all fruitful areas where the physics of emergence is leading to new understanding.
Supplementary: Unit 8: Emergent Behavior in Quantum Matter — Printable Online Text
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