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.