1 00:00:19,290 --> 00:00:24,060 So next, we're lucky enough to have stopped Bruno Bettini. 2 00:00:24,060 --> 00:00:32,520 Bruno did his defence in Oxford and then post stocks sister interest in Italy and the University of Libya honour. 3 00:00:32,520 --> 00:00:36,540 We're lucky enough to welcome him back as a university research fellow. 4 00:00:36,540 --> 00:00:42,060 These fellowships are given to the brightest and best of young scientists by the Royal Society. 5 00:00:42,060 --> 00:00:48,760 Unfortunately, Bruno is so good that he's ended up getting a permanent position in Nottingham, which is great for him in Nottingham. 6 00:00:48,760 --> 00:00:52,800 But we're very sorry too, to lose him at the end of this year. 7 00:00:52,800 --> 00:01:02,490 Bruno's published 30 papers already on quantum, many body dynamics out of equilibrium and related, strongly correlated electron work. 8 00:01:02,490 --> 00:01:07,580 So, Bruno, you've got those slides. Chad, to you. Thanks very much. 9 00:01:07,580 --> 00:01:13,540 OK. So thank you very much, Julia, for the introduction. Can you confirm that everything is fine? 10 00:01:13,540 --> 00:01:21,160 Can you see my slides? Yes, that's fine. OK, thanks. So welcome back to the second talk of this morning. 11 00:01:21,160 --> 00:01:30,400 So in this talk, I would like to discuss the emergence of hydrodynamics in a very special kind of systems that 12 00:01:30,400 --> 00:01:37,150 can be thought of systems for retaining an extensive memory about the initial condition. 13 00:01:37,150 --> 00:01:41,200 So let us start by reviewing what Steve just said. 14 00:01:41,200 --> 00:01:46,150 So he said that hydrodynamics is a very general theory. 15 00:01:46,150 --> 00:01:56,960 Can describe a large spectrum of different systems and basically to apply hydrodynamics, we only did two main conditions. 16 00:01:56,960 --> 00:02:02,840 So we need to be in local equilibrium and we need to have few conservation laws. 17 00:02:02,840 --> 00:02:07,700 So what I want to start my talk with is the following question. 18 00:02:07,700 --> 00:02:14,960 So do we actually need these conditions? So during the question section, we already question the first condition. 19 00:02:14,960 --> 00:02:21,620 And these will be also discussed in the third talk of this money by Andre. 20 00:02:21,620 --> 00:02:28,550 But in this talk, instead, I will assume local equilibrium and I will investigate the second condition. 21 00:02:28,550 --> 00:02:38,480 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? 22 00:02:38,480 --> 00:02:50,360 And can we handle them? So to understand this, let us go back to the microscopic interpretation that Steve already presented. 23 00:02:50,360 --> 00:02:56,540 And let's try to understand what microscopically having few conservation laws mean. 24 00:02:56,540 --> 00:03:00,590 So let us consider precisely the same system that Steve already showed. 25 00:03:00,590 --> 00:03:05,990 So a system of classical hard spheres in two dimensions. 26 00:03:05,990 --> 00:03:12,830 So this system, as we learnt, only conserves no or mass energy and momentum. 27 00:03:12,830 --> 00:03:20,510 And so the local equilibrium state is fully specified by the density of number or mass energy and momentum. 28 00:03:20,510 --> 00:03:27,930 And we can describe the hydrant mix of the system just by means of four simple equations in this case. 29 00:03:27,930 --> 00:03:32,840 OK, but let's see what happens microscopically. 30 00:03:32,840 --> 00:03:36,720 So let's just see again what Steve already showed us. 31 00:03:36,720 --> 00:03:46,840 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. 32 00:03:46,840 --> 00:03:52,860 And after the way and after awhile, their velocities are basically looking like random. 33 00:03:52,860 --> 00:03:55,560 So velocity is randomise. 34 00:03:55,560 --> 00:04:03,360 More precisely what this means is that if we consider the distribution of velocity, which is reported in this, Instagram's here. 35 00:04:03,360 --> 00:04:10,890 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. 36 00:04:10,890 --> 00:04:15,240 So what happens is that I have some initial distribution of velocity here. 37 00:04:15,240 --> 00:04:20,820 One particle with some velocity. Sixteen is units and all the other are steel. 38 00:04:20,820 --> 00:04:26,880 And if I wait for long enough, what happens is that the distribution of velocities changes. 39 00:04:26,880 --> 00:04:36,180 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, 40 00:04:36,180 --> 00:04:40,920 what you will see is that these distribution here will follow the Maxo distribution. 41 00:04:40,920 --> 00:04:49,410 OK, so basically few conservation laws means forgetting about the initial conditions space. 42 00:04:49,410 --> 00:04:55,170 OK, but let's ask now the following question. What happens if we squash the system? 43 00:04:55,170 --> 00:05:06,690 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. 44 00:05:06,690 --> 00:05:15,980 So what happens if we now look at not me? So, again, we will set into motion one particle and see what happens. 45 00:05:15,980 --> 00:05:19,880 OK. So you see, the dynamics here is completely different. 46 00:05:19,880 --> 00:05:27,900 At each point in time, we have only one particle movie and the absolute value of the velocity is concerned. 47 00:05:27,900 --> 00:05:35,580 So if we block again the distribution of velocity's, we see that the distribution of losses here does not change. 48 00:05:35,580 --> 00:05:42,930 So then they lost the distribution of velocities in the initial state is the same as the one I will find at infinite times. 49 00:05:42,930 --> 00:05:46,860 OK, but this has an immediate implication. 50 00:05:46,860 --> 00:05:57,210 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. 51 00:05:57,210 --> 00:06:03,840 Then we have that all of these engines are concerned with, which basically means if you go back to these histograms, 52 00:06:03,840 --> 00:06:09,730 that the number of particles in each one of these beans is a separately conserved. 53 00:06:09,730 --> 00:06:18,610 OK, but this means that if I want to describe local equilibrium, I have to look at a very large number of densities, 54 00:06:18,610 --> 00:06:24,060 a number that is equal in this case to the number of particles in the system. 55 00:06:24,060 --> 00:06:27,030 And this means that if I want to study hydrodynamics, 56 00:06:27,030 --> 00:06:34,290 I would need to write a very large number of equations that becomes infinite in determining lead. 57 00:06:34,290 --> 00:06:38,760 So I think there is scope to be worried here. OK, 58 00:06:38,760 --> 00:06:45,030 so we just saw the example of a system that has these infinite memory property 59 00:06:45,030 --> 00:06:50,760 that has a very large number that becomes infinite in the time clearing. 60 00:06:50,760 --> 00:06:56,700 So a number that is extensive in the in the size of the system of conservation laws. 61 00:06:56,700 --> 00:07:01,500 But is this case completely special or even though special? 62 00:07:01,500 --> 00:07:06,510 Because, of course, with all these conservation laws, it's not something very common. 63 00:07:06,510 --> 00:07:10,890 But even those special is something that characterises an entire class. 64 00:07:10,890 --> 00:07:17,100 And in particular, since at the fundamental level, reality is one, two, not classical. 65 00:07:17,100 --> 00:07:23,730 It is interesting to ask whether systems of these kinds are existing also at the quantum real. 66 00:07:23,730 --> 00:07:30,120 So before doing that, let me start with a very brief crash course of quantum mechanics. 67 00:07:30,120 --> 00:07:38,250 So when we studied quantum mechanics of a system of particles to describe its state, we use the wave function. 68 00:07:38,250 --> 00:07:44,960 So these wave function depends on the coordinates of all the particles are one hour in and on time. 69 00:07:44,960 --> 00:07:52,440 To describe the evolution of these wave function, we use the glorious Schrodinger equation, 70 00:07:52,440 --> 00:08:00,300 which basically tells us that the time derivative of the way function is defined by the application of a certain operator, 71 00:08:00,300 --> 00:08:05,940 the Hamiltonian on the way function. So the Hamiltonian defines the dynamics of the system. 72 00:08:05,940 --> 00:08:12,870 And in this framework, conserved charges are just operators that commute with the Hamiltonian. 73 00:08:12,870 --> 00:08:21,270 So let us start by asking the following simple question how many conserve charges does the generic quantum mechanical system have? 74 00:08:21,270 --> 00:08:28,480 Well, perhaps surprisingly, very many. To see this point, let us consider a very simple example. 75 00:08:28,480 --> 00:08:30,790 So let's. 76 00:08:30,790 --> 00:08:41,320 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. 77 00:08:41,320 --> 00:08:45,100 Okay, so they can only occupy these discrete positions here. 78 00:08:45,100 --> 00:08:52,210 So in this case, finding all the conservation knows of the of the system becomes a very simple problem in linear algebra. 79 00:08:52,210 --> 00:09:00,100 So we reasoned as follows. So if you take a single particle, then the weight function becomes just the Nele dimensional vector here. 80 00:09:00,100 --> 00:09:06,760 So basically it tells me what is the probability amplitude of finding the particle in the first position in the second and so on. 81 00:09:06,760 --> 00:09:12,440 And in the same way, the Hamiltonian becomes just a simple L by L matrix. 82 00:09:12,440 --> 00:09:14,030 So if I take two particles, 83 00:09:14,030 --> 00:09:22,560 then the wave function becomes a nela square dimensional vector and Hamiltonian in that square dimensional matrix at square Times Square matrix, 84 00:09:22,560 --> 00:09:27,350 I can continue in this way and I get that when I can see there and particles then you way function 85 00:09:27,350 --> 00:09:33,490 is an L to the N dimensional vector and the Hamiltonian is an L to the end times out to the matrix. 86 00:09:33,490 --> 00:09:40,510 OK, but now we basically have to find all the matrices that compute with a given one which is our Hamiltonian. 87 00:09:40,510 --> 00:09:48,200 It is the very simple problem in October. So we we find a number of independent matrices that do that. 88 00:09:48,200 --> 00:09:51,980 That is equal to the size of the Matrix, two dimensional matrix. 89 00:09:51,980 --> 00:09:58,370 And this is just done by taking all the matrices that are diagonal in the same basis as having done so. 90 00:09:58,370 --> 00:10:04,600 In this case, we end up having a very large number of conservation laws. 91 00:10:04,600 --> 00:10:13,850 OK. So the next question is, we found all these conservation goals and are they all important to they all matter for our description. 92 00:10:13,850 --> 00:10:20,300 The answer is no, they don't. Most of them, the very vast majority of them don't. 93 00:10:20,300 --> 00:10:26,690 And the reason for that is that the vast majority of these conserve charges will not have a local density, 94 00:10:26,690 --> 00:10:34,070 so will not be relevant for lack of physics. So these can be pictured in this very simple diagram here. 95 00:10:34,070 --> 00:10:39,050 So I can portray the quantum anybody's system as this blue blob. 96 00:10:39,050 --> 00:10:44,720 And then when we are interested in local physics, for example, we want to study the emergence of local equilibrium. 97 00:10:44,720 --> 00:10:51,290 Then we look at only at a small portion of these large blob, which I hear cherry red. 98 00:10:51,290 --> 00:10:59,360 Is these subsystem. And what happens is that the density of most of these conserved charges will not only leave in the local subsystem, 99 00:10:59,360 --> 00:11:04,730 but it will spread over and over on the system. So if I just look from the perspective of the system, 100 00:11:04,730 --> 00:11:11,810 these are not even looking like conserve conserve densities and are not constraining the local physics. 101 00:11:11,810 --> 00:11:19,160 So the relevant question to ask is whether there are some conservation laws that have a local debt. 102 00:11:19,160 --> 00:11:27,610 So then we should ask, are there a quantum mechanical systems with extensive romanic conserve charges with local density? 103 00:11:27,610 --> 00:11:37,240 And perhaps surprisingly, there are so there is Terek ceased a class of systems called Quantou Integral Systems that enjoy a special 104 00:11:37,240 --> 00:11:44,530 mathematical structure that is allowing them to have an extensive number of conservation deals with local debt. 105 00:11:44,530 --> 00:11:49,540 So these systems are very interesting on the mathematical level because they allow us 106 00:11:49,540 --> 00:11:55,030 to perform exact recreations and find the exact results in many different instances. 107 00:11:55,030 --> 00:12:00,190 But they are also interesting from the physical point of view because they describe many interesting physical systems. 108 00:12:00,190 --> 00:12:04,390 For example, they can describe spin shades. 109 00:12:04,390 --> 00:12:12,760 So what our speed chase, speed chase are a collection of speeds that are aligned in one dimension and can interact with each other. 110 00:12:12,760 --> 00:12:21,550 So if one chooses the interactions appropriately, then one finds that indeed these kind of systems can be integral. 111 00:12:21,550 --> 00:12:27,160 Other examples of integral models are found by looking at interacting particles. 112 00:12:27,160 --> 00:12:31,900 Again in one deep on a one dimensional lattice. So something like the drawing here. 113 00:12:31,900 --> 00:12:36,520 So what has a one dimensional lattice? Which is this black line? And there are charged particles. 114 00:12:36,520 --> 00:12:44,020 These aren't these blue and red balls that can interact to each other with each other and jump on the largest. 115 00:12:44,020 --> 00:12:48,460 Again, if one chooses the interaction appropriately and the whole thing appropriately, 116 00:12:48,460 --> 00:12:52,840 then there are examples of these systems that are indeed integral. 117 00:12:52,840 --> 00:12:57,850 Then there are also interesting examples of inevitable quantum field theories. 118 00:12:57,850 --> 00:13:03,140 But I am not able to picture them. So here I just leave it alone. 119 00:13:03,140 --> 00:13:10,230 OK, but perhaps I didn't convince you yet because these integrity seems to be a very adhoc thing. 120 00:13:10,230 --> 00:13:21,070 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. 121 00:13:21,070 --> 00:13:26,260 Having all these many conservation laws. So let's ask a different question. 122 00:13:26,260 --> 00:13:31,480 Are there any real quantum systems with this property? Well, yes, there are. 123 00:13:31,480 --> 00:13:35,890 And some of them are also here in Oxford, for example, in this picture here. 124 00:13:35,890 --> 00:13:41,170 He's portrayed the ultra called Quantum Matak Group of the Oxford University. 125 00:13:41,170 --> 00:13:44,830 And these people, together with many colleagues around the world, 126 00:13:44,830 --> 00:13:52,450 are realising in the laboratory systems that can have a very large number of conservation goals. 127 00:13:52,450 --> 00:13:56,080 So these kind of systems have already been considered by Steve. 128 00:13:56,080 --> 00:14:01,030 Are these called atomic gases that are confined by optical lattices? 129 00:14:01,030 --> 00:14:09,220 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 130 00:14:09,220 --> 00:14:15,140 laser beams and using basic in the electromagnetic fields coming from the laser beams. 131 00:14:15,140 --> 00:14:24,940 And by choosing the configuration of the lasers, they can basically construct simulate quantum anybody systems in many different dimensions. 132 00:14:24,940 --> 00:14:28,480 For example, they can simulate the quantum anybody's system into these here. 133 00:14:28,480 --> 00:14:37,360 The balls are the atoms. And these grey partier describes the lattice generated by the lasers. 134 00:14:37,360 --> 00:14:42,100 They can also construct construct solids in three dimensions. 135 00:14:42,100 --> 00:14:48,880 And in one dimension in particular, the one dimensional case is the one relevant for us. 136 00:14:48,880 --> 00:14:54,520 So let me consider a specific experiment that has been realised. 137 00:14:54,520 --> 00:14:59,560 Considering gases in wonderments called atomic gaseous in one day. 138 00:14:59,560 --> 00:15:06,030 So this is it is probably the most famous experiment in my in my field. 139 00:15:06,030 --> 00:15:11,800 And it's called a quantum Newton's Cradle. So before discussing what a quantum neutron scrabble is. 140 00:15:11,800 --> 00:15:15,890 Let me just remind you of what a standard normal neutron credibly is. 141 00:15:15,890 --> 00:15:23,110 So the Newton credit is this simple desktop toy that is designed to portray the conservation of momentum and energy. 142 00:15:23,110 --> 00:15:29,260 So one sets one bolt motion and then they start moving in this interesting way. 143 00:15:29,260 --> 00:15:39,820 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. 144 00:15:39,820 --> 00:15:45,930 And then they managed to give to half of these of the atoms in this cloud some velocity. 145 00:15:45,930 --> 00:15:50,400 That's a V and two the other half the negative velocity minus B. 146 00:15:50,400 --> 00:15:55,470 And then they let the system evolve for some time inside the top so that the cloud here 147 00:15:55,470 --> 00:16:00,300 split and started to oscillate inside the truck colliding and then going back and forth. 148 00:16:00,300 --> 00:16:09,580 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. 149 00:16:09,580 --> 00:16:17,440 But the remarkable fact that has been found in these experiments is that these clouds can oscillate inside this trap for very, 150 00:16:17,440 --> 00:16:22,540 very, very long times, up to 2000 periods of oscillations here. 151 00:16:22,540 --> 00:16:32,140 Without showing any dumping. Furthermore, the experimentalists also measure the momentum distribution of the atoms in the truck, 152 00:16:32,140 --> 00:16:38,350 which is very similar to the velocity distribution it was considering before in my simple classical problem. 153 00:16:38,350 --> 00:16:45,850 And they saw that indeed, the momentum distribution is remembering is keeping information about the initial configuration. 154 00:16:45,850 --> 00:16:53,180 You see, this is the initial curve and this is the curve that they they measured after 15 periods of oscillation. 155 00:16:53,180 --> 00:17:03,050 So this is very strongly reminiscent of our nice, simple example with which we started to talk as a comparison. 156 00:17:03,050 --> 00:17:05,690 I also should note that in their three dimensional case. 157 00:17:05,690 --> 00:17:13,610 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. 158 00:17:13,610 --> 00:17:25,040 So if one measure, the momentum distribution here finds that after a couple of appearance of oscillations, it immediately looks like a Gaussian. 159 00:17:25,040 --> 00:17:36,870 OK, so now we just found that there are some interesting systems in the real world that show these these interesting property. 160 00:17:36,870 --> 00:17:42,360 That have a macroscopic and extensive number of conservation goals. 161 00:17:42,360 --> 00:17:48,230 So now we move to the main question. So can we describe these systems using hydrodynamics? 162 00:17:48,230 --> 00:17:52,170 So before moving into that, let me just note an important point. 163 00:17:52,170 --> 00:17:59,820 So having it in a hydrodynamic description for a quantum system is crucial from the practical point of view. 164 00:17:59,820 --> 00:18:11,280 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. 165 00:18:11,280 --> 00:18:18,090 And this becomes extremely expensive for from the point of view of the resources needed for Lajon. 166 00:18:18,090 --> 00:18:21,660 For example, let's look at the simple case I was considering before. 167 00:18:21,660 --> 00:18:26,170 So particles living on the on their one dimensional lattice of lengths. 168 00:18:26,170 --> 00:18:33,270 And so with Alcides. So in this case, if I want to describe the wave function, I need L to the end numbers. 169 00:18:33,270 --> 00:18:41,550 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 170 00:18:41,550 --> 00:18:48,450 apple got the number we see immediately that are these these number here becomes incredibly large. 171 00:18:48,450 --> 00:18:57,240 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. 172 00:18:57,240 --> 00:19:05,270 So these can work. But instead, they had a description only requires a few functions. 173 00:19:05,270 --> 00:19:10,700 If you in the in the in the simple case, a few functions of one plus one variable. 174 00:19:10,700 --> 00:19:18,330 So this is it in the normal simplification is a gigantic simplification that could really help us. 175 00:19:18,330 --> 00:19:21,540 So, OK, let's let's start to understand whether we can. 176 00:19:21,540 --> 00:19:27,270 So, course, the problem that we are having here is that since we have extensive many conservation laws, 177 00:19:27,270 --> 00:19:32,670 it seems that we need to write an extensive number of questions, which is not pleasant to work with. 178 00:19:32,670 --> 00:19:41,190 So these Crusoe's Survation Irakere moment comes when we understand that these can be done by a smart change environment. 179 00:19:41,190 --> 00:19:47,760 So to describe what is this change of variables? Let's go back to the simple example of one dimensional spheres. 180 00:19:47,760 --> 00:19:52,410 So let's go back to the scales here. Just showing the scattering of two spheres. 181 00:19:52,410 --> 00:19:57,120 Time runs downwards. Well, from left to right, there is space. 182 00:19:57,120 --> 00:20:03,420 So we have a blue sphere scattering with the red one. And they just scatter exchanging the velocity. 183 00:20:03,420 --> 00:20:13,350 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. 184 00:20:13,350 --> 00:20:19,140 Already, after one scattering, I find that the trajectory of the sphere looks like that. 185 00:20:19,140 --> 00:20:28,140 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. 186 00:20:28,140 --> 00:20:35,010 But there is some other thing that I can look at in this diagram that has a much simpler propagation. 187 00:20:35,010 --> 00:20:43,410 And this is the so-called tracer. So instead of looking at a given sphere, I look at this sphere with a given velocity. 188 00:20:43,410 --> 00:20:47,610 So if I follow this sphere with Velocity V1, for example, in this plot, 189 00:20:47,610 --> 00:20:53,280 I see that it moves on on a unique follows a uniform motion then jumps by an 190 00:20:53,280 --> 00:20:58,230 amount which is the size of the sphere and then continues the the nice uniform. 191 00:20:58,230 --> 00:21:03,190 So I can write that trajectory very easily. And I don't need to know exactly the time p zero. 192 00:21:03,190 --> 00:21:11,250 I just need to know that the scattering happens. But to see how this simplification is a much greater than one might expect. 193 00:21:11,250 --> 00:21:13,260 Let us look at more spheres. 194 00:21:13,260 --> 00:21:23,420 So now here I am picturing the dynamics of many spheres that are these white patches here while the black is the background. 195 00:21:23,420 --> 00:21:29,460 And I coloured in red the tracer of one of the velocities, for example, the one before. 196 00:21:29,460 --> 00:21:35,200 So we see that. The trajectory of a given sphere is very complicated, you see it. 197 00:21:35,200 --> 00:21:41,160 It performs many scouting's and it's really hard to trace the position at at some large time. 198 00:21:41,160 --> 00:21:51,270 But what instead we see is that the tracer is basically moving along something that is a uniform, linear motion. 199 00:21:51,270 --> 00:22:01,290 The only real difference that we see is that because of the interaction, the velocity of these linear motion is different than expected. 200 00:22:01,290 --> 00:22:09,510 So if there were no interactions, then the tracer would end up here following you, just continuing with its free velocity. 201 00:22:09,510 --> 00:22:15,650 But because of the scattering, the effective velocity of this motion is different. 202 00:22:15,650 --> 00:22:20,080 OK, but so this suggests as a way to treat the problem. 203 00:22:20,080 --> 00:22:24,710 So if we find what is this effective velocity, then we can just treat the problem, 204 00:22:24,710 --> 00:22:31,780 considering the straight sets as free particles that are not interacting with each other and are moving at this effective velocity. 205 00:22:31,780 --> 00:22:38,050 So the idea is to describe the system using tracer's instead of spheres. 206 00:22:38,050 --> 00:22:42,220 OK, but this is actually a very general fact, intrathecal physics. 207 00:22:42,220 --> 00:22:49,840 So it happens in many instances that complex, interactive many body systems can be described by quasi particles. 208 00:22:49,840 --> 00:22:57,460 So because the particles are emergent degrees of freedom that behave as the free particles on the vacuum, 209 00:22:57,460 --> 00:23:04,180 but instead describe the dynamics of a very complex one. So Trace, it's these three sets of fixed velocities. 210 00:23:04,180 --> 00:23:07,960 In our case are just an example of these classic particles. 211 00:23:07,960 --> 00:23:17,680 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. 212 00:23:17,680 --> 00:23:25,690 So let us now make this discussion a little bit more quantitative and write down some equations. 213 00:23:25,690 --> 00:23:34,400 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, 214 00:23:34,400 --> 00:23:40,870 those dieser, when I call them here, to a description based on the density of these Gwisai particles. 215 00:23:40,870 --> 00:23:49,160 So these are always telling me the density and position at the time of the particle traces that are tracing velocity v. 216 00:23:49,160 --> 00:23:53,560 And then we will use these identities to specify the state of the system. 217 00:23:53,560 --> 00:24:00,630 So now the question is how? We describe the evolution of these dances. 218 00:24:00,630 --> 00:24:10,080 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. 219 00:24:10,080 --> 00:24:16,720 And let us look as at how the number of particles in the parts of changes we type. 220 00:24:16,720 --> 00:24:21,970 But here, the problem is extremely simple because these particles move as if they were free. 221 00:24:21,970 --> 00:24:27,430 So basically to change a number of particles in the particle is just due to the flux of 222 00:24:27,430 --> 00:24:34,270 particles going in and out of the box without interacting just because of their normal motion. 223 00:24:34,270 --> 00:24:40,540 So I can just immediately turn this condition here into a quantitative equation as follows. 224 00:24:40,540 --> 00:24:47,080 So this is just the equation. I get. Very simple. So now we have the evolution equation. 225 00:24:47,080 --> 00:24:52,510 And the only thing that we have to find now is this effective velocity. How do I find these effective velocity? 226 00:24:52,510 --> 00:25:00,610 Well, also, that is not very hard to do in this case, because basically, by definition, these effective velocity times, 227 00:25:00,610 --> 00:25:07,240 speed is equal to the free velocity timestep plus the contribution coming from the scouting's. 228 00:25:07,240 --> 00:25:15,630 And but this contribution is actually very easy to compute. It's just a times the number of jumps of the particle. 229 00:25:15,630 --> 00:25:19,890 So we can compute that explicitly and find the following formula. 230 00:25:19,890 --> 00:25:26,070 So the most important feature of this equation for the effective philosophy is that it depends here on the road. 231 00:25:26,070 --> 00:25:33,300 So it depends on the density of cosmic particles. So in other words, it depends on the state of the system. 232 00:25:33,300 --> 00:25:37,110 So in other words, again, these are the nature of these cosmic particles. 233 00:25:37,110 --> 00:25:41,450 Their velocity will depend on what is the state of the system. 234 00:25:41,450 --> 00:25:50,480 OK. So these two equations. Give me fully the entire hydrodynamic description of this simple system. 235 00:25:50,480 --> 00:25:59,660 Okay. But now brace yourself, because I'm going to say what is probably the most surprising part of this talk. 236 00:25:59,660 --> 00:26:07,240 So actually, the same exact description applies to all Quantou integral role models. 237 00:26:07,240 --> 00:26:09,350 OK, so what do I mean by that? 238 00:26:09,350 --> 00:26:18,020 Is that the state of the system in all quantum integral models be described by emergent classic particles that move like free particles, 239 00:26:18,020 --> 00:26:22,640 but with some effective velocities depending on the state and the equations. 240 00:26:22,640 --> 00:26:30,950 The actual quantitative equation that I'm that I use to describe it are really the very same that I wrote before. 241 00:26:30,950 --> 00:26:33,380 The only difference is that in general, 242 00:26:33,380 --> 00:26:38,960 the quantity that causes particle jumps when interacts with another depends on the velocity of the two particle. 243 00:26:38,960 --> 00:26:44,150 So here A becomes the function of a V and W and enters. 244 00:26:44,150 --> 00:26:48,240 That's the that's the main difference. OK, great. 245 00:26:48,240 --> 00:26:53,940 So now we have these hydrodynamic description. 246 00:26:53,940 --> 00:26:58,110 Let us see whether it it agrees with the experiment. 247 00:26:58,110 --> 00:27:09,690 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. 248 00:27:09,690 --> 00:27:19,500 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. 249 00:27:19,500 --> 00:27:27,000 The idea is the very same, the only difference that they created an initial condition, which is easier to study with the hydrodynamics. 250 00:27:27,000 --> 00:27:32,880 So basically, instead of preparing the atoms in the middle of the truck and giving to them these opposite velocities, 251 00:27:32,880 --> 00:27:38,210 they prepared two clouds of atoms separated, and then they let them evolve into trap. 252 00:27:38,210 --> 00:27:48,140 As we showed before. OK, but here in this picture, I'm portraying the density profile measured in the experiment, 253 00:27:48,140 --> 00:27:55,150 which is this violates line compared to the predictions of these generalised hydrodynamics. 254 00:27:55,150 --> 00:28:01,340 So these hydrodynamic period that describe systems with an extensive number of conservation goals. 255 00:28:01,340 --> 00:28:06,920 And you see that the prediction here works very well for the expected. 256 00:28:06,920 --> 00:28:12,170 OK, good. So let me know, I think this is a good point. 257 00:28:12,170 --> 00:28:16,910 To summarise the main the main ideas that I covered in stock. 258 00:28:16,910 --> 00:28:25,370 So the first point that I would like to convey is that some interesting physical systems have an extensive number of conservation. 259 00:28:25,370 --> 00:28:27,920 The second point is that in these systems, 260 00:28:27,920 --> 00:28:36,530 we can still defined hydrodynamics by describing the local equilibrium state in terms of these emergent quasi particles. 261 00:28:36,530 --> 00:28:44,320 And the third important point is that the nature of these particles depends on the very state of the system. 262 00:28:44,320 --> 00:28:55,630 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. 263 00:28:55,630 --> 00:29:04,300 So one direction is concerning, higher order corrections or next door there corrections, as Steve called them in his talk. 264 00:29:04,300 --> 00:29:11,140 So the level of hydrodynamics here that I describe is that the one on the largest possible scale, 265 00:29:11,140 --> 00:29:16,840 which is in the Steve terminology, on the Euler's scale. 266 00:29:16,840 --> 00:29:26,850 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. 267 00:29:26,850 --> 00:29:34,530 And it turns out, actually, that there are. And the idea behind them is actually very simple and nice. 268 00:29:34,530 --> 00:29:38,940 So previously I said that these tracers are performing emotion. 269 00:29:38,940 --> 00:29:47,340 That is almost a uniform, linear motion that is almost here is crucial because actually, if one looks more closely, 270 00:29:47,340 --> 00:29:54,270 one sees that the tracer is not actually moving along these these lines described by the effective velocity. 271 00:29:54,270 --> 00:30:04,340 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. 272 00:30:04,340 --> 00:30:09,300 So diffusion like them in this kind of system, which is very surprising. 273 00:30:09,300 --> 00:30:13,050 So other questions along these lines are, can we continue? 274 00:30:13,050 --> 00:30:21,770 Can we find a third order correction for the correction? And up to what order can we expect hydrodynamics toward the. 275 00:30:21,770 --> 00:30:28,670 And the second, probably even more interesting direction for future research here is based on the following pressure. 276 00:30:28,670 --> 00:30:41,060 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. 277 00:30:41,060 --> 00:30:48,680 So how how is that possible? Where did the quantum correction, the corrections go? 278 00:30:48,680 --> 00:30:55,520 So in other words, how and why does hydrodynamic emerge from the quantum anybody dynamics? 279 00:30:55,520 --> 00:30:59,510 So this is a very interesting question. And, of course, a very hard one. 280 00:30:59,510 --> 00:31:06,020 So there are many of us here at the department trying to understand actually how this happens by looking at some 281 00:31:06,020 --> 00:31:14,060 simple models where we can actually solve the full quantum anybody dynamics and see how the hydrodynamics emerges. 282 00:31:14,060 --> 00:31:19,170 So this also is connected to some of the points that were asked in the questions. 283 00:31:19,170 --> 00:31:30,040 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. 284 00:31:30,040 --> 00:31:36,490 Thank you, Bruno. Thank you. That was a great talk. Thank you very much indeed. 285 00:31:36,490 --> 00:31:40,450 I'm not sure what's happening about the questions and answers at the moment. 286 00:31:40,450 --> 00:31:49,360 Could people put questions into the questions and answers, please? 287 00:31:49,360 --> 00:31:59,530 Yes. Yes, yes, yes. They're coming through all as well. Question from Chris again. 288 00:31:59,530 --> 00:32:03,880 Chris said. You said we could ignore the L to the end, 289 00:32:03,880 --> 00:32:14,200 conserve quantities in a generic el cite and particle quantum system because most of them don't have associated local densities. 290 00:32:14,200 --> 00:32:20,730 Is that specific? The real space basis is that the case is. 291 00:32:20,730 --> 00:32:25,180 Why is your space special or not? That's that's a very good question. 292 00:32:25,180 --> 00:32:29,740 So, yes, it is indeed some special property of the real space basis, 293 00:32:29,740 --> 00:32:35,670 because something that I didn't mention in the crash course on quantum mechanics is that what is 294 00:32:35,670 --> 00:32:40,030 a very important property of the Hamiltonian that I'm considering here is local interactions, 295 00:32:40,030 --> 00:32:44,590 local real space. So this makes the real space spaces special. 296 00:32:44,590 --> 00:32:51,630 So, yeah, that's. Ken, a question from Chen Kids Law. 297 00:32:51,630 --> 00:32:56,640 Do we usually study quantum integral systems with the Hubbard model? 298 00:32:56,640 --> 00:33:01,860 You seem to just put in a new in the figure you used on the slide, which reminded me of the Hubbard model. 299 00:33:01,860 --> 00:33:09,480 Yeah, indeed. The one dimensional Hubbard model is a prominent example of intercropping model in one dimension. 300 00:33:09,480 --> 00:33:14,950 So I can recommend a book written by Euphorbia Nestler on the subject. 301 00:33:14,950 --> 00:33:18,480 He has a monograph on that. Yeah. 302 00:33:18,480 --> 00:33:23,730 So. So you'd say it's one of the most important examples of integral role model. 303 00:33:23,730 --> 00:33:28,250 Okay, so Fabien's. But once you've read stage book, you have until Fabien's, right? 304 00:33:28,250 --> 00:33:39,220 Yeah. Yeah, yeah. Okay. I actually have it here somewhere. I can show you just. 305 00:33:39,220 --> 00:33:44,650 It was look like a case to the next question is from James Lee. 306 00:33:44,650 --> 00:33:50,050 He asks, Does the tracer approach work when time itself is quantised? 307 00:33:50,050 --> 00:33:56,890 So, for example, you have quantum loop gravity models. Yeah. 308 00:33:56,890 --> 00:34:01,600 Well, I mean, the simple way to answer this question is that I don't know. 309 00:34:01,600 --> 00:34:08,490 I don't expect it to work, but I don't have much to say about that, unfortunately. 310 00:34:08,490 --> 00:34:13,150 But it sounds like a very interesting question. No, I. 311 00:34:13,150 --> 00:34:19,350 I had a question, which was when you talked about the initial conditions in the Newton Cradle experiment. 312 00:34:19,350 --> 00:34:23,350 Yeah. Could you say how you get that in the first case? 313 00:34:23,350 --> 00:34:28,110 How you get the two clouds of particles going in opposite directions? 314 00:34:28,110 --> 00:34:33,070 Yeah. Some kind of laser. Right. Like it. Some sort of a laser pulse. 315 00:34:33,070 --> 00:34:40,510 It's some advanced experimental technique. So, yeah, I don't I don't know exactly the detail. 316 00:34:40,510 --> 00:34:45,110 So I just assume that they can do it. It's basically magic for me. 317 00:34:45,110 --> 00:34:51,280 Is it. I mean, when do you then have a different initial condition in the second experience? 318 00:34:51,280 --> 00:34:57,820 Why is why does that make it easier? Yeah. Because if you want to treat the system with hydrodynamics, 319 00:34:57,820 --> 00:35:05,180 you need to be in some sort of local paper and that the first metre condition that they give is very far from that, 320 00:35:05,180 --> 00:35:12,400 that the particles are evolving not. At least at the beginning, when they start to separate, the clouds are starting to separate. 321 00:35:12,400 --> 00:35:20,470 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 322 00:35:20,470 --> 00:35:25,510 of local density approximation where as already basically in an equilibrium state. 323 00:35:25,510 --> 00:35:32,100 So this makes it much easier for the for the theoretical description. 324 00:35:32,100 --> 00:35:34,110 In the other case, basically, 325 00:35:34,110 --> 00:35:43,800 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. 326 00:35:43,800 --> 00:35:47,220 But you don't precisely know what is the distribution. What is that? 327 00:35:47,220 --> 00:35:52,440 For example, density profile at that time, because you don't know what that time is. 328 00:35:52,440 --> 00:35:55,710 So if you want to compare with the experiment with some in some quantitative way, 329 00:35:55,710 --> 00:36:02,210 it's much easier to start already by you with a configuration that works that that's the idea. 330 00:36:02,210 --> 00:36:10,790 OK. And then there's a question from Mandy Watson and Sarah Gould. Does the emergence of hydrodynamics from quantum mechanics tell us anything 331 00:36:10,790 --> 00:36:22,320 about the emergence of standard classical mechanics from quantum mechanics? 332 00:36:22,320 --> 00:36:26,420 What did she say? An interesting question, isn't it? Yeah. Yeah. 333 00:36:26,420 --> 00:36:37,440 So. Well, probably does in some sense, but I think that. 334 00:36:37,440 --> 00:36:42,980 These this fact here is probably more general. 335 00:36:42,980 --> 00:36:48,980 In the sense that here what we are saying is basically that at the microscopic level, 336 00:36:48,980 --> 00:36:53,330 quantum systems and classical systems do basically the same thing. 337 00:36:53,330 --> 00:37:00,040 So. Well, yes, in some sense it does. 338 00:37:00,040 --> 00:37:06,850 If you want, just because you are saying that if you have a microscopic quantum object object, 339 00:37:06,850 --> 00:37:12,410 that one actually should be described by classical physics. Good. 340 00:37:12,410 --> 00:37:19,380 OK. Thank you very much. Thank you for a great talk.