So next, we're lucky enough to have stopped Bruno Bettini. Bruno did his defence in Oxford and then post stocks sister interest in Italy and the University of Libya honour. We're lucky enough to welcome him back as a university research fellow. These fellowships are given to the brightest and best of young scientists by the Royal Society. Unfortunately, Bruno is so good that he's ended up getting a permanent position in Nottingham, which is great for him in Nottingham.
But we're very sorry too, to lose him at the end of this year. Bruno's published 30 papers already on quantum, many body dynamics out of equilibrium and related, strongly correlated electron work. So, Bruno, you've got those slides. Chad, to you. Thanks very much. OK. So thank you very much, Julia, for the introduction. Can you confirm that everything is fine? Can you see my slides? Yes, that's fine. OK, thanks. So welcome back to the second talk of this morning.
So in this talk, I would like to discuss the emergence of hydrodynamics in a very special kind of systems that can be thought of systems for retaining an extensive memory about the initial condition. So let us start by reviewing what Steve just said. So he said that hydrodynamics is a very general theory. Can describe a large spectrum of different systems and basically to apply hydrodynamics, we only did two main conditions.
So we need to be in local equilibrium and we need to have few conservation laws. So what I want to start my talk with is the following question. So do we actually need these conditions? So during the question section, we already question the first condition. And these will be also discussed in the third talk of this money by Andre. But in this talk, instead, I will assume local equilibrium and I will investigate the second condition.
So few conservation laws. So what I want to ask you is, are there some relevant, interesting systems that have more than few conservation laws? And can we handle them? So to understand this, let us go back to the microscopic interpretation that Steve already presented. And let's try to understand what microscopically having few conservation laws mean. So let us consider precisely the same system that Steve already showed. So a system of classical hard spheres in two dimensions.
So this system, as we learnt, only conserves no or mass energy and momentum. And so the local equilibrium state is fully specified by the density of number or mass energy and momentum. And we can describe the hydrant mix of the system just by means of four simple equations in this case. OK, but let's see what happens microscopically. So let's just see again what Steve already showed us.
So if we start just by moving one particle and we wait for long enough, what happens is that the all the other particles are set in motion. And after the way and after awhile, their velocities are basically looking like random. So velocity is randomise. More precisely what this means is that if we consider the distribution of velocity, which is reported in this, Instagram's here.
So here on the X axis, I'm reporting the velocity anyon on the Y axis, I'm reporting the number of particles with that velocity. So what happens is that I have some initial distribution of velocity here. One particle with some velocity. Sixteen is units and all the other are steel. And if I wait for long enough, what happens is that the distribution of velocities changes.
And actually, here it is. These are only a few particles. But as Steve mentioned, this is just a if you look at larger numbers of particles, what you will see is that these distribution here will follow the Maxo distribution. OK, so basically few conservation laws means forgetting about the initial conditions space. OK, but let's ask now the following question. What happens if we squash the system?
So let's do it. Let's squash the system. And the bus, instead of looking at to the to the system, let's look at a one dimensional version of it. So what happens if we now look at not me? So, again, we will set into motion one particle and see what happens. OK. So you see, the dynamics here is completely different. At each point in time, we have only one particle movie and the absolute value of the velocity is concerned.
So if we block again the distribution of velocity's, we see that the distribution of losses here does not change. So then they lost the distribution of velocities in the initial state is the same as the one I will find at infinite times. OK, but this has an immediate implication. So if we define as NJ the number of particles with velocity, veejay and veejay is the velocity of the initial the initial velocity of the JF sphere.
Then we have that all of these engines are concerned with, which basically means if you go back to these histograms, that the number of particles in each one of these beans is a separately conserved. OK, but this means that if I want to describe local equilibrium, I have to look at a very large number of densities, a number that is equal in this case to the number of particles in the system.
And this means that if I want to study hydrodynamics, I would need to write a very large number of equations that becomes infinite in determining lead. So I think there is scope to be worried here. OK, so we just saw the example of a system that has these infinite memory property that has a very large number that becomes infinite in the time clearing. So a number that is extensive in the in the size of the system of conservation laws. But is this case completely special or even though special?
Because, of course, with all these conservation laws, it's not something very common. But even those special is something that characterises an entire class. And in particular, since at the fundamental level, reality is one, two, not classical. It is interesting to ask whether systems of these kinds are existing also at the quantum real. So before doing that, let me start with a very brief crash course of quantum mechanics.
So when we studied quantum mechanics of a system of particles to describe its state, we use the wave function. So these wave function depends on the coordinates of all the particles are one hour in and on time. To describe the evolution of these wave function, we use the glorious Schrodinger equation, which basically tells us that the time derivative of the way function is defined by the application of a certain operator,
the Hamiltonian on the way function. So the Hamiltonian defines the dynamics of the system. And in this framework, conserved charges are just operators that commute with the Hamiltonian. So let us start by asking the following simple question how many conserve charges does the generic quantum mechanical system have? Well, perhaps surprisingly, very many. To see this point, let us consider a very simple example. So let's.
Studied a case of particles that are confined to one dimension and can only occupy discrete positions on the lattice which we take to have Alcides. Okay, so they can only occupy these discrete positions here. So in this case, finding all the conservation knows of the of the system becomes a very simple problem in linear algebra. So we reasoned as follows. So if you take a single particle, then the weight function becomes just the Nele dimensional vector here.
So basically it tells me what is the probability amplitude of finding the particle in the first position in the second and so on. And in the same way, the Hamiltonian becomes just a simple L by L matrix.
So if I take two particles, then the wave function becomes a nela square dimensional vector and Hamiltonian in that square dimensional matrix at square Times Square matrix, I can continue in this way and I get that when I can see there and particles then you way function is an L to the N dimensional vector and the Hamiltonian is an L to the end times out to the matrix. OK, but now we basically have to find all the matrices that compute with a given one which is our Hamiltonian.
It is the very simple problem in October. So we we find a number of independent matrices that do that. That is equal to the size of the Matrix, two dimensional matrix. And this is just done by taking all the matrices that are diagonal in the same basis as having done so. In this case, we end up having a very large number of conservation laws. OK. So the next question is, we found all these conservation goals and are they all important to they all matter for our description.
The answer is no, they don't. Most of them, the very vast majority of them don't. And the reason for that is that the vast majority of these conserve charges will not have a local density, so will not be relevant for lack of physics. So these can be pictured in this very simple diagram here. So I can portray the quantum anybody's system as this blue blob. And then when we are interested in local physics, for example, we want to study the emergence of local equilibrium.
Then we look at only at a small portion of these large blob, which I hear cherry red. Is these subsystem. And what happens is that the density of most of these conserved charges will not only leave in the local subsystem, but it will spread over and over on the system. So if I just look from the perspective of the system, these are not even looking like conserve conserve densities and are not constraining the local physics.
So the relevant question to ask is whether there are some conservation laws that have a local debt. So then we should ask, are there a quantum mechanical systems with extensive romanic conserve charges with local density? And perhaps surprisingly, there are so there is Terek ceased a class of systems called Quantou Integral Systems that enjoy a special mathematical structure that is allowing them to have an extensive number of conservation deals with local debt.
So these systems are very interesting on the mathematical level because they allow us to perform exact recreations and find the exact results in many different instances. But they are also interesting from the physical point of view because they describe many interesting physical systems. For example, they can describe spin shades. So what our speed chase, speed chase are a collection of speeds that are aligned in one dimension and can interact with each other.
So if one chooses the interactions appropriately, then one finds that indeed these kind of systems can be integral. Other examples of integral models are found by looking at interacting particles. Again in one deep on a one dimensional lattice. So something like the drawing here. So what has a one dimensional lattice? Which is this black line? And there are charged particles. These aren't these blue and red balls that can interact to each other with each other and jump on the largest.
Again, if one chooses the interaction appropriately and the whole thing appropriately, then there are examples of these systems that are indeed integral. Then there are also interesting examples of inevitable quantum field theories. But I am not able to picture them. So here I just leave it alone. OK, but perhaps I didn't convince you yet because these integrity seems to be a very adhoc thing.
So it's a very mathematical, but it's still unclear whether there are some real quantum systems in the real world that actually look like. Having all these many conservation laws. So let's ask a different question. Are there any real quantum systems with this property? Well, yes, there are. And some of them are also here in Oxford, for example, in this picture here. He's portrayed the ultra called Quantum Matak Group of the Oxford University.
And these people, together with many colleagues around the world, are realising in the laboratory systems that can have a very large number of conservation goals. So these kind of systems have already been considered by Steve. Are these called atomic gases that are confined by optical lattices?
So what what they do is they consider clouds of atoms. They cool them down at very low temperatures and then they confine them using last laser beams and using basic in the electromagnetic fields coming from the laser beams. And by choosing the configuration of the lasers, they can basically construct simulate quantum anybody systems in many different dimensions. For example, they can simulate the quantum anybody's system into these here.
The balls are the atoms. And these grey partier describes the lattice generated by the lasers. They can also construct construct solids in three dimensions. And in one dimension in particular, the one dimensional case is the one relevant for us. So let me consider a specific experiment that has been realised. Considering gases in wonderments called atomic gaseous in one day. So this is it is probably the most famous experiment in my in my field.
And it's called a quantum Newton's Cradle. So before discussing what a quantum neutron scrabble is. Let me just remind you of what a standard normal neutron credibly is. So the Newton credit is this simple desktop toy that is designed to portray the conservation of momentum and energy. So one sets one bolt motion and then they start moving in this interesting way.
So for the quantum version of the problem, what the experimentalists did was to prepare a cloud of atoms inside a one dimensional harmonic sharp. And then they managed to give to half of these of the atoms in this cloud some velocity. That's a V and two the other half the negative velocity minus B. And then they let the system evolve for some time inside the top so that the cloud here split and started to oscillate inside the truck colliding and then going back and forth.
So here, here, here instead, I'm really reporting some real pictures of the experiment, so this is really a picture of the item dancing. But the remarkable fact that has been found in these experiments is that these clouds can oscillate inside this trap for very, very, very long times, up to 2000 periods of oscillations here.
Without showing any dumping. Furthermore, the experimentalists also measure the momentum distribution of the atoms in the truck, which is very similar to the velocity distribution it was considering before in my simple classical problem. And they saw that indeed, the momentum distribution is remembering is keeping information about the initial configuration. You see, this is the initial curve and this is the curve that they they measured after 15 periods of oscillation.
So this is very strongly reminiscent of our nice, simple example with which we started to talk as a comparison. I also should note that in their three dimensional case. So when one does the same thing but doesn't constrain diatoms to leave only one D, then what happens is that the system rapidly normalises. So if one measure, the momentum distribution here finds that after a couple of appearance of oscillations, it immediately looks like a Gaussian.
OK, so now we just found that there are some interesting systems in the real world that show these these interesting property. That have a macroscopic and extensive number of conservation goals. So now we move to the main question. So can we describe these systems using hydrodynamics? So before moving into that, let me just note an important point. So having it in a hydrodynamic description for a quantum system is crucial from the practical point of view.
Indeed, if you want to describe a system of quantum particles, typically one needs a wave function that depends on three and plus one variables. And this becomes extremely expensive for from the point of view of the resources needed for Lajon. For example, let's look at the simple case I was considering before. So particles living on the on their one dimensional lattice of lengths. And so with Alcides. So in this case, if I want to describe the wave function, I need L to the end numbers.
So if I take, for example, Iraqis or 10 sides and I take as end the number of electrons in a side which is of the order of the apple got the number we see immediately that are these these number here becomes incredibly large. But this also becomes very large. If I want to simulate the number of atoms in a in a cold atom experiment, which is approximately 10 to the five. So these can work. But instead, they had a description only requires a few functions.
If you in the in the in the simple case, a few functions of one plus one variable. So this is it in the normal simplification is a gigantic simplification that could really help us. So, OK, let's let's start to understand whether we can. So, course, the problem that we are having here is that since we have extensive many conservation laws, it seems that we need to write an extensive number of questions, which is not pleasant to work with.
So these Crusoe's Survation Irakere moment comes when we understand that these can be done by a smart change environment. So to describe what is this change of variables? Let's go back to the simple example of one dimensional spheres. So let's go back to the scales here. Just showing the scattering of two spheres. Time runs downwards. Well, from left to right, there is space. So we have a blue sphere scattering with the red one. And they just scatter exchanging the velocity.
So the first thing to note is that if once if one wants to trace the trajectory of a single sphere, that is not so easy. Already, after one scattering, I find that the trajectory of the sphere looks like that. So basically, if I want to write it down, I need to know exactly what is the time at which the the two spheres scatter. But there is some other thing that I can look at in this diagram that has a much simpler propagation.
And this is the so-called tracer. So instead of looking at a given sphere, I look at this sphere with a given velocity. So if I follow this sphere with Velocity V1, for example, in this plot, I see that it moves on on a unique follows a uniform motion then jumps by an amount which is the size of the sphere and then continues the the nice uniform. So I can write that trajectory very easily. And I don't need to know exactly the time p zero.
I just need to know that the scattering happens. But to see how this simplification is a much greater than one might expect. Let us look at more spheres. So now here I am picturing the dynamics of many spheres that are these white patches here while the black is the background. And I coloured in red the tracer of one of the velocities, for example, the one before. So we see that. The trajectory of a given sphere is very complicated, you see it.
It performs many scouting's and it's really hard to trace the position at at some large time. But what instead we see is that the tracer is basically moving along something that is a uniform, linear motion. The only real difference that we see is that because of the interaction, the velocity of these linear motion is different than expected. So if there were no interactions, then the tracer would end up here following you, just continuing with its free velocity.
But because of the scattering, the effective velocity of this motion is different. OK, but so this suggests as a way to treat the problem. So if we find what is this effective velocity, then we can just treat the problem, considering the straight sets as free particles that are not interacting with each other and are moving at this effective velocity. So the idea is to describe the system using tracer's instead of spheres. OK, but this is actually a very general fact, intrathecal physics.
So it happens in many instances that complex, interactive many body systems can be described by quasi particles. So because the particles are emergent degrees of freedom that behave as the free particles on the vacuum, but instead describe the dynamics of a very complex one. So Trace, it's these three sets of fixed velocities. In our case are just an example of these classic particles.
So to make the statement of these Eureka guy more precise here, we can say that yes, we can describe the system by using quasar particles. So let us now make this discussion a little bit more quantitative and write down some equations. So if we turn into equations, then what the Eureka guy is saying is that we should switch from a description based on densities of conservation, those dieser, when I call them here, to a description based on the density of these Gwisai particles.
So these are always telling me the density and position at the time of the particle traces that are tracing velocity v. And then we will use these identities to specify the state of the system. So now the question is how? We describe the evolution of these dances. Well, how does it differ? Let's steal the picture that Steve had in his stock and look at a fluid passel full of classic particles. And let us look as at how the number of particles in the parts of changes we type.
But here, the problem is extremely simple because these particles move as if they were free. So basically to change a number of particles in the particle is just due to the flux of particles going in and out of the box without interacting just because of their normal motion. So I can just immediately turn this condition here into a quantitative equation as follows. So this is just the equation. I get. Very simple. So now we have the evolution equation.
And the only thing that we have to find now is this effective velocity. How do I find these effective velocity? Well, also, that is not very hard to do in this case, because basically, by definition, these effective velocity times, speed is equal to the free velocity timestep plus the contribution coming from the scouting's. And but this contribution is actually very easy to compute. It's just a times the number of jumps of the particle.
So we can compute that explicitly and find the following formula. So the most important feature of this equation for the effective philosophy is that it depends here on the road. So it depends on the density of cosmic particles. So in other words, it depends on the state of the system. So in other words, again, these are the nature of these cosmic particles. Their velocity will depend on what is the state of the system.
OK. So these two equations. Give me fully the entire hydrodynamic description of this simple system. Okay. But now brace yourself, because I'm going to say what is probably the most surprising part of this talk. So actually, the same exact description applies to all Quantou integral role models. OK, so what do I mean by that?
Is that the state of the system in all quantum integral models be described by emergent classic particles that move like free particles, but with some effective velocities depending on the state and the equations. The actual quantitative equation that I'm that I use to describe it are really the very same that I wrote before. The only difference is that in general, the quantity that causes particle jumps when interacts with another depends on the velocity of the two particle.
So here A becomes the function of a V and W and enters. That's the that's the main difference. OK, great. So now we have these hydrodynamic description. Let us see whether it it agrees with the experiment. Right. Because we can make a statement. So this has been done recently, actually two years ago by an experiment that it was carried out in Paris. And what they did was to do something very similar. They created something that was very similar to the Newton cradle that I described before.
The idea is the very same, the only difference that they created an initial condition, which is easier to study with the hydrodynamics. So basically, instead of preparing the atoms in the middle of the truck and giving to them these opposite velocities, they prepared two clouds of atoms separated, and then they let them evolve into trap.
As we showed before. OK, but here in this picture, I'm portraying the density profile measured in the experiment, which is this violates line compared to the predictions of these generalised hydrodynamics. So these hydrodynamic period that describe systems with an extensive number of conservation goals. And you see that the prediction here works very well for the expected. OK, good. So let me know, I think this is a good point. To summarise the main the main ideas that I covered in stock.
So the first point that I would like to convey is that some interesting physical systems have an extensive number of conservation. The second point is that in these systems, we can still defined hydrodynamics by describing the local equilibrium state in terms of these emergent quasi particles. And the third important point is that the nature of these particles depends on the very state of the system.
OK, so before concluding, let me just very briefly mention some of the future directions that can be embraced based on these on these ideas. So one direction is concerning, higher order corrections or next door there corrections, as Steve called them in his talk. So the level of hydrodynamics here that I describe is that the one on the largest possible scale, which is in the Steve terminology, on the Euler's scale.
But we can consider whether in these kinds of systems, there are some corrections, these kind of either dynamics, all of the above your stall time. And it turns out, actually, that there are. And the idea behind them is actually very simple and nice. So previously I said that these tracers are performing emotion.
That is almost a uniform, linear motion that is almost here is crucial because actually, if one looks more closely, one sees that the tracer is not actually moving along these these lines described by the effective velocity. But is it moving randomly around it? And these random motion against around the DeMain trajectory is the one originating in other stock like terms. So diffusion like them in this kind of system, which is very surprising.
So other questions along these lines are, can we continue? Can we find a third order correction for the correction? And up to what order can we expect hydrodynamics toward the. And the second, probably even more interesting direction for future research here is based on the following pressure. So where did quantum mechanics go? So here I said that I have some quantum anybody systems that are described by some simple classical hydrodynamics.
So how how is that possible? Where did the quantum correction, the corrections go? So in other words, how and why does hydrodynamic emerge from the quantum anybody dynamics? So this is a very interesting question. And, of course, a very hard one. So there are many of us here at the department trying to understand actually how this happens by looking at some simple models where we can actually solve the full quantum anybody dynamics and see how the hydrodynamics emerges.
So this also is connected to some of the points that were asked in the questions. OK, so I think that at this point, I can thank you for your attention and I'm very happy to take any questions that you might have. Thank you, Bruno. Thank you. That was a great talk. Thank you very much indeed. I'm not sure what's happening about the questions and answers at the moment. Could people put questions into the questions and answers, please?
Yes. Yes, yes, yes. They're coming through all as well. Question from Chris again. Chris said. You said we could ignore the L to the end, conserve quantities in a generic el cite and particle quantum system because most of them don't have associated local densities. Is that specific? The real space basis is that the case is. Why is your space special or not? That's that's a very good question.
So, yes, it is indeed some special property of the real space basis, because something that I didn't mention in the crash course on quantum mechanics is that what is a very important property of the Hamiltonian that I'm considering here is local interactions, local real space. So this makes the real space spaces special. So, yeah, that's. Ken, a question from Chen Kids Law. Do we usually study quantum integral systems with the Hubbard model?
You seem to just put in a new in the figure you used on the slide, which reminded me of the Hubbard model. Yeah, indeed. The one dimensional Hubbard model is a prominent example of intercropping model in one dimension. So I can recommend a book written by Euphorbia Nestler on the subject. He has a monograph on that. Yeah. So. So you'd say it's one of the most important examples of integral role model. Okay, so Fabien's. But once you've read stage book, you have until Fabien's, right?
Yeah. Yeah, yeah. Okay. I actually have it here somewhere. I can show you just. It was look like a case to the next question is from James Lee. He asks, Does the tracer approach work when time itself is quantised? So, for example, you have quantum loop gravity models. Yeah. Well, I mean, the simple way to answer this question is that I don't know. I don't expect it to work, but I don't have much to say about that, unfortunately.
But it sounds like a very interesting question. No, I. I had a question, which was when you talked about the initial conditions in the Newton Cradle experiment. Yeah. Could you say how you get that in the first case? How you get the two clouds of particles going in opposite directions? Yeah. Some kind of laser. Right. Like it. Some sort of a laser pulse. It's some advanced experimental technique. So, yeah, I don't I don't know exactly the detail.
So I just assume that they can do it. It's basically magic for me. Is it. I mean, when do you then have a different initial condition in the second experience? Why is why does that make it easier? Yeah. Because if you want to treat the system with hydrodynamics, you need to be in some sort of local paper and that the first metre condition that they give is very far from that,
that the particles are evolving not. At least at the beginning, when they start to separate, the clouds are starting to separate. So to have a quantitative comparison is much easier to prepare. The clouds already separated where you can basically described them as with some sort of local density approximation where as already basically in an equilibrium state. So this makes it much easier for the for the theoretical description.
In the other case, basically, what you would need to do for the first experiment is to start applying hydrodynamics at a certain time when the local equilibrium kicked in. But you don't precisely know what is the distribution. What is that? For example, density profile at that time, because you don't know what that time is.
So if you want to compare with the experiment with some in some quantitative way, it's much easier to start already by you with a configuration that works that that's the idea. OK. And then there's a question from Mandy Watson and Sarah Gould. Does the emergence of hydrodynamics from quantum mechanics tell us anything about the emergence of standard classical mechanics from quantum mechanics? What did she say? An interesting question, isn't it? Yeah. Yeah.
So. Well, probably does in some sense, but I think that. These this fact here is probably more general. In the sense that here what we are saying is basically that at the microscopic level, quantum systems and classical systems do basically the same thing. So. Well, yes, in some sense it does. If you want, just because you are saying that if you have a microscopic quantum object object, that one actually should be described by classical physics. Good.
OK. Thank you very much. Thank you for a great talk.