 So the first talk for this second session of today is by Alvisi Bassanello, who will tell us about how do dynamics of images locally integrate with models Alvisi, for you. Okay, so yeah, thank you very much for organizing the workshop, it's really nice and I bet it has been quite a nightmare to organize so many talks and speakers. So yeah, let's start and please, if there are questions, just ask me, okay. So something that I very like, I like very much about thematic workshop is like everybody has been already exposed to the introduction, so more or less I can skip some basics of integrity. Okay. What are the topics of today? I want to show you how is it possible to include some in the hydrodynamic framework, some in homogeneities in the dynamics. You have already seen in the previous, sorry, I'm in a guest office. So, I already, you have already seen as impossible to describe, for example, in homogeneities like traps and so on, but actually it can be much more general. You can play with more or less arbitrary couplings that locally conserve the integrated model, okay. And then the second part of the talk, I want to show you something about what to say, a problem that escaped the first smooth picture. Indeed, there can be some homogeneities that apparently look like to be smooth, but they are not. In this case, you should supplement GHD with further consideration in order to gain, again, a productive power, okay. But let's start with the situation where everything goes smoothly. So, what I will tell you is quite general, but for the sake of a concreteness, I want to focus on a very specific model, for example, labeling. Now, in this model, in the homogenous case, there are some parameters that we can change within the integrated system. Of course, we can change the chemical potential, but we can also change the interaction, okay. And for every valid interaction, the system is integrable. Now, physically changing the external potential, the chemical potential just amounts to change the longitudinal trap of the system. But you can also measure directing on the interaction just squeezing the system. Also, if you squeeze the transverse direction, you're increasing the effective interaction with the other one in the system, okay. So, I would like to approach the problem of describing the system when we start having some smooth homogeneities in you and see. And of course, I want to do it with generalized dynamics. Now, this, I promise, that is the only general slide, introduction slide on generalized dynamics, okay. So, the idea of the game is always the same. You have some smooth homogeneities, so you can invoke local relaxation, the local stationary state of the homogenous system. And since we're integrable, the stationary state is a GG. And you are left out with a problem of doing together, GG is a different point in space and time, okay. This is actually the technically hard part. And what you gain, what you obtain after this considerations are GHD equations, okay. Here, I'm neglecting the physical actions. I just want to stay at the earlier scale. In this equation, we have the filling, in my notation, the previous talk has been like N several times. That is a function of space and time, because you want to be homogeneous, and of another variable called rapidity, or quasi momentum, lambda, in my notation, okay. In general, this equation describes particles going around the system. You see that there is a velocity, effective velocity, coupled to a spatial derivative, and just describes you how particles are moving in space. But then there can be also other things affecting forces, coupled to the translation of the space. Indeed, the translation of the space are just changing the quasi momentum of particles, and of course, changes in body momentum associated with forces. This is the general form, and of course we're going to extensively discuss the first terms. So, let me start with, I want to give you a flavor as possible to derive this kind of GHD equation, how they can be obtained. Let me focus on a very simple situation, where I'm fully homogeneous in space, and I just look at homogeneous in time. Since I have a minor vinegar, let me just change slowly in time the interaction, but of course you can play with more general situations. So these are given cards for the evolution, for the change in the coupling, but in order to approach the problem is very instructive to split it in a sequence of small branches. Instead of smooth evolution, I just look at sequence in these small branches. So I'm changing the coupling delta C, waiting for the time delta T, so on and so forth. And of course the full problem just amounts to solve the infinitesimal branch and repeat the solution. And of course I want to eliminate those low changes, so I can more or less assume delta T to be very, very large compared with some microscopic spaces. So let me solve the small branch. I start from an initial state that they can assume to be a GG. And I want to stress that is a GG at a given value of the coupling C. Indeed, since we are changing the coupling, we are also changing the integral models, the model we are referring to, because it's coupled in the family. I just excise the system. I wait for delta T timer that supposed to be long enough to get the relaxation to the new GG. And of course a new G now is complete on the model of C plus delta C, so different integral model. And let's start with the problem of doing together these two Gs. How you can do that? If you look in the holy book of branches, you will do what they are being told to be to do, like looking at the expectation value of the charges. So what I'm doing here, simply, I'm imposing that the expectation value of the charges of the post quench theory. So compute delta C plus delta C is equal to the final state of the initial state. Now, the right, right hand side is simple because it's just an expectation value of a charge computed on a GG. Instead, the left hand side member is not simple at all. And in China for arbitrary delta C is not suitable. But we just need the small that is easy because the quench is infinitesimal. And what you can guess so that if you start telling us finding this guy, what you need to compute in the end is just the derivative of the charge, arbitrary charges actually, which aspect of the coupling on the GG. And the, and the turning point of the computation like this guy can be excited to compute that using a generalization that I'm not fine. The Hermann finite here and just tells you how to compute the day, but you would just get to a copy of them. And it just uses that they're going to use a concept charge, and they can play exactly the same trick with other concepts. So this guy here, this delta C of QJ is computable for arbitrary values of the charges. And if you do that, you impose the conservation of the charge expectation value for arbitrary charges. And then you put into the game also special meeting is how the task was doable. You will find this kind of journalize the hydrodynamics equation. Well, of course, you have a false term, but also some are the last time, sorry, but also some false terms that actually are organized in two different times, one couple to time and homogenities and change your time and time and this and this and also to space and there are like two separate contributions. And again, these two forces can be divided categorized into different times. Did we have some single particle effects. And these are just due to the fact that if such engine dispersion law that single particle because you're changing the captain, of course you accept some force on the situation. This single particle effects that you get, for example, if you want to study maybe a new gene steps. However, in addition, we have also other types of collective effects. You can see that collective because of the specific proportion to the film, so they depends on the price of the other particles. And these times are there only if you're changing the scattering data, the scattering phase. Indeed, for example, if you look at the bringer, you want to change the direction, for example, you want to increase their portion. Of course, the party could be accelerated because they refer more. And this, this must be contained. And this time exactly does the job. Now, setting aside the possible checks. I just want to flash some possible applications. For example, now you can start the slow induction crunch and they'll be a model. Okay, we can follow GHE, the behavior, the, or the, or the feeling, and of course also complete the observables for which we have a examples, for example, this case is simply the last. Now, in order to move to the second part, I want to make a very trigger observations. But whose consequences are not real. Let me consider now the GC equation, the homogenous case in space. Okay, but with some new genes in homogeneity in time. Okay, so I got the special day, but just the simple GC with alpha summer. Now, as it is, I can absorb the time dependence, parameterizing the equation. Okay. And if you do so, you can some equations that are completely time dependent. They say they do not have any explicit time dependence. What does it mean? It means that this equation is describing some reversible. Because there is no any explicit deformation time. For example, if I start from initial gene, I start changing the captain somehow to some value and I go back. Every time I go back to the same value of the interaction. I'm also back to the initial state. Now, for example, if I remind the system, the system, you're in the ground state of the ground state is vector. You can, for example, use the derivative theorem. This case, you're changing the parameter. You follow the ground state. Okay, so you can go back to certain value and go back. And of course, it will be again, the ground state. But in this case, this holds true also for getting the situation. And this just do because you are not only, not have only the Hamiltonian, but infinite many charges. And so if there is not a gap in the Hamiltonian, there are gaps in the other charges. This is the physical interpretation. However, you can also ask, is it always the case? And as you can guess, if I'm posting this question, the answer is no. And this moved me to the second part, which I want to discuss this specific example of a flux quenching the except that it's pinching. Let me comment a bit on the system. So we have our favorite entire screen chain. Okay. This object is in terrible for arbitrary values, the magnetic field, the end of the interaction delta. In addition, I put another parameter capital C in this notation that has the meaning of being a magnetic flux piercing the spin chain. Okay. So what we do is to make this flux a dynamical variable start changing in time in super steps. Okay. Now, probably I'm sure that you are familiar with the except that's been changed. Maybe you're not so familiar with this in changing the presence of the flux. And the reason is trivia. In the sense that the Hamiltonian that are given value of the flux is actually completely equivalent to the Hamiltonian in the absence of flux. The two are connected by interest formation. So as long as you want to study them dynamics, correlation functions, changing delta B, you can be at arbitrary values, the flux, the field is always the same. But of course, in this case, since we are changing the value of the flux in time, we have to keep the formula, the three parameter flux. Okay. Now, if you focus on the reversibility or lack of it or this process, you find the point that the steam phase diagram in the slide. So here is totally depends on the value of the interaction. Indeed, if that is larger than one now, you are accepting the situation that I was discussing the previous slide, fully reversibility of the protocol, a change of flux also some value go back and then back to the same interest. And if you're looking at a situation where that is smaller than one, right, in absolute value, of course, this matters in model, you'll find that that the system lacks reversibility, and in addition to get some end of it. But of course, you don't have in the rest markets. Right. And then in the main part of the talk, I want to show to try to explain why this difference and to explain how we can still be predictive. So, let me present you a crash course in the TVLX. Okay. Now, in this situation, in this protocol, you can still buy down some of each equation. Well, here, I'm just the monetization of each string, because in this case, different from the bringer, we do not have only a single single kind of pieces of excitation but several of them. And actually, their number is still independent on the value of that. For data larger than one case, you have a few many strings that can be actually interpreted as bound state of element excitation discovered at first finger. The data smaller than one case is that the number is fine and dependent on that. But this is not the critical difference between the two situations. I want to double your attention on the fact that for data larger than one case. You actually have that the rapidities live in a Boolean zone. Now, the rapidities are just a parameterization of the momentum. You should actually look at the momentum as a function that you will discover that if you move the rapidity within, I don't see the point. If you move the rapidity within the definition domain, the momentum associated will cover the whole Boolean zone. Data smaller than one, of course, your analytics, so you still have a Boolean zone in terms of momentum. But what we discover is like if you look at the momentum as a function of the rapidity and you move the rapidity within the definition domain, the momentum will not cover the whole Boolean zone. So while for data larger than one, you can actually interpret this as a Boolean zone for the rapidities, this is not a Boolean zone for the rapidities. So this would be the crucial difference to explain the two different variables. So question, where does the entropy production come from? Because in the end, the GHD equation we brought are exactly as those that we slice. However, if you want to solve a differential equation, you need the differential equation that also provides boundary conditions. And the trick here, what really matters is the different boundary conditions you have to put at the boundaries of the definition of the rapidity space. So let me start first with a situation where everything goes smoothly. So data larger than one. In this case, as I told you about Boolean zone, and let me try to explain what happens with cutting picture. Of course, I'm a thermodynamic limiter, so I have a few links. I'm describing an infinite thermodynamic large number of particles, but let me play with these particles that are set on the different strings. In principle, I have infinite many of them, but there will be infinite many strings on a slide. So just for now. Before I start changing the flux, okay, I increase the flux and this from the GC equation means that I'm accepting some force. The rapidity space means that I'm just moving the particles in the rapidity space. In this case, if I increase the flux and make larger or the particles go in this direction. But now a certain point will be in a situation where you keep on increasing the flux and a certain point one the particle reaches the boundaries. And of course, you would like to keep on translating it. How can you do that? Well, in this case, the solution is simple because you have a Boolean zone. What you have to do is just a continuous isolation from minus by half and keep on going. And of course, these kind of boundary equations, boundary conditions are irreversible. If you are back to minus by half, and then you go back, you know exactly where you have to go. So in this case, we have some time irreversible GC equation, time irreversible boundary conditions, and fully time irreversible dynamics. Let me now instead look at the case where something goes wrong. So let me focus on that as more than one and delta equal to one half person visited. In this case, we have three strings. For other values of Delta, the number of strings is different. The situation is slightly more complicated, but the same picture holds. Okay. So in this case, I want to play the same game. So I start changing the flux, let me increase it and the particles will accelerate. Now, in this case, the accelerator in this way, the first strings on the right, the third on the left. Just because this is the way in which the strings parameterize the momentum. Okay. The strings I need like just a simple organization. But now again, you are finding a certain situation where you are keeping on translating particles and one of them reaches the boundaries. But now it's not so clear how you can continue this translation because you do have a Boolean zone and not under. So what can you do? Well, you will learn a lot. If you look at the charge again values at the boundaries of the repeated. If you look at it, you discover this, this relation that if I take for a given charge on a given string and send the repeated state to infinite. You get this something that is proportional to the same charge on the first thing. The proportionality is just the monetization. The monetization just tells you the number of elementary components that are forming the excitation described by the string J. Okay. So what, what is it happening from the point of view the charges. The strings, just at the boundaries, the previous definition plus infinity minus infinity, I completely distinguishable. You can't really distinguish any longer with the charges. If you have a bound state of n particles or n free particles. Now, this could be a little bit fishy, maybe because we are told that the charges can equally fix the population of these things. And actually, this is true for the terminal case, because when you do time of dynamics, a single point that plus infinity minus infinity is just measure zero doesn't really matter for time of dynamics. Okay. So effectively, the charges are completely fixed. The, the time of state and possible correlation function. However, in this case, we are forcing all the particles to move through, to move through this zero measure point that in this case, because the dynamics starts met. Okay. So, from the point of view the charges, we cannot tell apart the strings. And so, the idea is that we can move the particle from one string to another one and keep on with the revolution. Okay. So, now we have several different interesting possibilities. For example, we can have a little process. Particle for the first string, let's say that is just a element of excitation that jumps to the first string that's still describing just element of excitation. However, you can also be in a situation where a molecular two particle bond state jumps to another string that doesn't support bond state and the numbers. So, you have a break in a bond state. But even more intensely within a situation where you are forming a bond state, for example, imagine that I have two particles here on the first string through the mental excitation sorry in the first thing. And they can proceed with the revolution, according to the source, either the second or the first thing. This situation are equally possible from the point of view the strings, but of course, if what physics industries would like to be deterministic, and we have to understand which is the recognition rate and how to fix it. And this can't be told by the charges on. So, let me approach this problem. You just charge conservation and end up maximization. And we actually have already look at the charges we already fixed that. So, the only ingredient we are left with is and maximization. To be honest, in this case, the right ingredient is not really the end of itself, but it's great under the first solution. Let me consider now the young, young, young. This is just a definition of the function. Usually this guy is conserved with the under the generalize the dynamics equation. If you compute the time derivative or flux derivative in this case of the entropy insert the g h equation and find it is. However, doing that, you are doing some integration by parts, there are some boundary terms that usually they do not matter, but in this case, they matter. And indeed, if you computed the, the change of the entropy under the flux change, you find the two different contributions that are not necessarily zero. And this is set only with quantities of the repeat minus. Another one with quantities with a bit plus. These are boundary times. And so now you can just ask for charge conservation and impose that you are maximizing the end of period production. And this gives you some computer deterministic way to fix how to recombine the bound states. So you can keep on evolving the system and having to get the GHD equation plus this Monday. Let me see how it works. Now, in this case, we are comparing the GTI equation, the four lines with the dashed lines are accepted organizations. And here we are changing the flux from zero to five. And on different times case. That is this capital Steve, of course, larger T is slower we are proceeding. We are starting from the ground state with a fine monetization. Okay, so the initial state is just a pharmacy on the first finger. And then we are up. And here first column delta equal to one alpha, sorry to cross or 1.5 that is larger than one. And here second column, third column, delta equal to one half with different values, the initial. For so the energy second rule, the current, the spin. Now, Delta larger than one, everything goes very, very simple. And just to remind you that the larger than one is a case where everything goes fine. You need, you see all the cars one of the other. Instead, in the Delta equal to one half case, still this GHD seems to predict reason we were was happening, but the convergence is much slower. And not to understand this, let me fix. Let me focus on this plot and the spring. You see that the GHD equation predicts some. Here they just have been done. Actually, there is this point also in the plot of the energy just less evident and also in the other close. Now, of course, the microscopic pollution itself is fully smooth. So in order to reproduce a sample is no it easy to get to zoom out a lot. Okay, so this is why we are experiencing this very, very large tea in order to start approaching the GZ production. Now, you can ask her what, what is, what, what this is one and it is exactly when the feeling that is moving the first finger on the deflux on the deflux is when this feeling hits for the first time a boundary of the repeated domain. And so is where this non-trivial boundary condition started to be important. And guess what? If you look now at the entanglement of the system, this half of the entanglement of half of a chain, you see this behavior provides the flux where the Fermi C is not touching the boundaries. So when the user GHD with non-trivial production, the entropy just flat. This is actually the entropy minus the entropy resistance. And then suddenly when you hit this boundary, the entanglement of this growth. And that's because as soon as the young, young entropy is not zero any longer, you can accommodate for actually 10 young large entanglement of half of a chain. So these just, sorry, I always miss the point. So this entropy just grew. Seeing only that you're actually producing. Okay. So, yes. So, so can I, can I ask? Yes. So, so in this protocol, I still, I don't see whether there is a explicit time dependence on the variation on the sorry on the GHD equations. So before you show that there was no explicit independence. Yeah, but then I find it a bit strange because I could do the same protocol saying the time of the universe and approaching an adiabatic. You can do what? So I find a bit strange that also here you don't have an explicit time dependence because then I could do the same protocol in the time of the universe. So approaching adiabatic process. And in that case, I shouldn't find an increase of the entropy. Whereas since your protocol does not depend explicitly on time, you would predict still in any case an increase of the entropy. So I understand why in the previous case there was no increase in the entropy and it was reversible. But now I find a bit incompatible. The fact that you, you, you have an increase of the entropy and also equation that do not depend explicitly on time. Wait, wait, wait. The fact is that the we do not have. Okay. The question itself are not explicit independent. Okay. That just you can epimatize them in terms of the class changes. Okay. And the practice that in this case that I don't want no end of the production the sense that a young young empathy should tell you the saturation the entanglement of a human. I want to say is always the same. You're not changing. But here. At this point here, yeah, sorry for that. One is more than one, where the time evolution up to say where the dashed line is placed. You're still reversible in the sense that the feeling is not touching the boundaries. So the part of the maximization entropy doesn't take place. Then a certain point you hit these boundaries. And what does it happen? Even though you just hit it a bit with an infinitesimal change of the flux. This is what I find it very strange that even infinitesimal and that's a trick. If I will look at a finite interval, finite interval, the entropy will saturate the young young. That is the same infinitesimal. But I mean physically, so you're changing with an infinitesimally small quench for infinitesimally window parameters. So I mean, this is why I think it's strange that you can still have entropy production. Even though I'm allowed to do this infinitesimally slow. Because I mean, I understand what you're saying, but then once you hit this point of, of, of. Then I mean, so once again, so it seems that as soon as I do delta. Phi. Here for example, this is the entanglement of your heart for a chain that is infinite in this case. So even a small quench infinitesimal will end up in a just in the girl for the entanglement. Because it's not separating. It's not a finite system. So just keep on growing. Right. So, so I mean, imagine, okay. So just, and then I will close, but imagine that I give you a finite window of the variation for the angle phi. So I say I want to vary a phi from zero to pi in this protocol. And then imagine you repeat the same variation scheme for times that it's first one microsecond and 10 seconds then the angel of the universe. And then you telling me that it doesn't matter how long that it takes it will always produce. Of course, if you look at it here, these are curves for different times. Okay, we scaled on the same value of 15. So of course, if I do it slowly, the entanglement is larger is going more. But just because because in this case, I look at the entanglement of the system that is infinite. So this question point is actually if I would look at entanglement to be our finite interval with the rest of the chain. Okay. You're going to see a situation to some point. And that point is say delta phi closer to the initial value. Okay. Maybe I can just make a quick comment. Okay. Yes. So I think there's another time scale, which has to do with the thermodynamic limit of the system. So that's the time scale below which you would have the idea of a transformation. You know, this is what this is. Wait, in this case for the entanglement just have to what I can control with a young young entropy is S over L, L size of the system that is thermodynamic larger is infinite in this case because L is up for the system. So, of course, it keeps going with time. So if I wait for more time is going more just because the saturation pointer that is richer is indeed because there is an M in front of it. Okay. The story will be different if I look at the finite system. Yeah. I will say that to this more. I didn't mind. We can just consider it but thank you. Yes, whenever you wish. So, other questions. Yeah, I have a question. Can I ask you a question? Yes, for sure. It's about xxz and xxz you have this flux five and there is a special situation when I consider zero temperature be your magnetic field be equal to zero and delta smaller than one. And then on this, there is a special process when five this twist with your capital five changes from zero to four pi like double period. And then there's some very delicate current phenomena happen. I mean, the ground state. It goes down to the ground state. And maybe I just send a sense of the chat, the reference to the paper where all of this described, and then I would expect that in this special situation, the entity production will be lower because it's looks like Coherent to me. Maybe you can have a look at the paper and later when you have time, maybe you can reply. I'm just trying to keep throughout. It's in the chat. My name is Vladimir Corpin and I send it through the chat. I guess I send it to everyone actually. Yeah. Thank you very much. I will definitely look at it. I would like to discuss more. Okay, great. Thank you. Thank you. And so let me conclude. Then maybe there could be other questions, but I want just to flesh the last slide. Okay. So some I told you if it's possible to include some emojis and the dynamics of the integral model as well as long as they locally expect respect and they'll be to the month. And then I pointed out a situation where something that is actually it's not a parameter is not smooth and dynamics and sensitive escape the user GHD situation. Okay. Now, what is in my shopping list for the future. Actually, I would really like to have a general understanding how it's possible to play with. I'm a genius. The model that are. Where the new movies cannot be smooth. For example, imagine that I'm changing some interaction. The model. Even though the permit that changes move. It can be better. The response of the system to this change is not smooth at all. For example, in a bling. See larger than zero. You just have one single particle species. See smaller than zero. You have instead the infinite many bound states. Okay. So no matter how slow you are moving from C positive to C negative, you always cross a point where the system is moving from C positive to C negative, you always cross a point where the system is moving from C positive to C negative. The system is not responding as much fashion. Okay. And well, the being is a situation, but if you. This, why this and really in the future, because it's difficult. But for example, the sun Gordon model that is implemented and experiment again. Because of the intrinsic and emojis. So the trap is a ingenious model as well. And the same Gordon, the particle content, the excitation content really depends on the system. And so I think it is a primary interest to be able to control this framework with the ingenious model with emojis interactions. And so, yeah, he's actually working progress and we should have some results soon for the. So, yeah, thank you for attending my talk and of course thank you for my collaborators in this process. And if there are other questions but I think that maybe. Thank you. Thank you for this very nice talk. And yeah, I mean there are other quick questions there were already some questions. I have one, one very quick question. I mean, that's about entropy protection. I mean, if you look at ordinary hydrodynamics entropy is produced. Whenever the system develops a shock. Yes. And that's on the oiler scale and that's why I just wonder whether there's some analogy, you know, with your entropy reduction and the entropy, you know, coming from the shock. Well, I would really like not to have this neither my own talks, but there is some sort of shocker. Imagine that I'm plotting the feeling. Okay, the feeling first thing is just this fantasy, and this approach in the boundary and then I continue somehow. And you'll find that, for example, if you started from the ground stated feelings as a banner, you'll keep the stringer then the feeling is continuing the other stringer. Because of the different forces experiencing is not actually one anymore but is smaller. So there's a sort of jumper. If you try to put together the two things I do not know if this is directly related to your question but is the most evident kind of shocker can imagine this. Maybe I make an additional comment on that. So in GHD usually one says that there are no shocks that develop. I mean, there's no sustained entropy production contrary to standard hydro that like Herbert says you usually you have shock developing where the sustained So while you're you're the phenomenon phenomenon that you're mentioning seems to be producing a large amount of entropy. So that's a bit different from, yeah, so that's nice to investigate a bit more. In this case, it is difficult to understand what a shock is because they're not in a module in space at the same time. Yes, I understand. Like the feeling with a shocker space that is like a straight line. But this one is a shocker rapidity space, which is what happens always in GHD. No, but you need like you say to have a shocker, you want to write down the repeat and the space. You have this and it looks like a straight line. If I plot the profile as a function of the rapidity I always see a shock. I discontinued. But these are not shocks. This is a bit different. Exactly. So I mean, but this looks exactly like that to me. But here there's entropy production so that's a bit.