 talk about diffusion from convection. So first of all, I would really like to thank the organizers for the invitation and also for the ingenuity in organizing this conference, and I'll discuss basically how diffusion can emerge from convective or ballistically propagating modes, which is the work ID together with Iacopo and Takato. So okay, the motivation behind doing this study is basically centuries old, so trying to understand microscopic origins of transport coefficients of our normal transport, ideal transport arises and what are really the kind of the physical properties of the system that influence it. And the thing that we wanted to do is basically tell something about these quantities without referencing or without dwelling into the intricate integrability structures such as form factor expansion. So and the reason for trying to do that is that possibly we might hope that applicability of the method that we developed goes beyond integrable systems, maybe also beyond single dimension, but also to reflect somehow from more physical perspective about some results that were previously derived in integrable systems. So at this point, I would also like to mention the work from Ben, where he puts some of the ideas I'll present in a more formal and more rigorous framework. So let me start by first discussing a couple of basic ingredients which basically underlie our approach without going into too many technical details. So the basic idea, as was the case in many preceding talks, is to somehow take a hydrodynamical limit. So and the way we do it is basically take, let's say an infinite system divided into small but still large cells and then look at what's happening to local operators that extend throughout these so-called fluid or hydrodynamical cells. So basically these are the operators that are just sums of some local or quasi local densities over these hydrodynamical cells. And an assumption that is usually used, that was also used on Monday at many instances is so-called assumption of local equilibration, where one assumes that locally the system, after long enough time, is described by some generalized Gibbs ensemble, where the description should be correct if one includes all of the quasi local conserved quantities. So one of the main properties, so GGE is actually a plastering property, which basically tells us that if you're considering some local properties which are encoded in local observables, basically you don't need to care what the value of chemical potentials is far away from your hydrodynamical cell. But this intuition expands from just local expectation values also to local connected correlation functions. And one can actually prove if one considers endpoint connected correlation function, that basically we can forget about all of the connected correlation functions that do not include quantities that are all localized at a single hydrodynamical cell. There are of course corrections due to the boundaries. So of course at boundaries there will be one over L corrections if L is the size of your hydrodynamical cell. So the second notion that is very important and was well, was basically already introduced on Monday by Takata is a notion of normal modes. So normal modes are just some excitations on top of a homogeneous background. So it would just say thermal state. Then you excite some local excitation and then you look what's happening. As Takata nicely explained what starts happening is that you see many peaks propagating throughout your system and basically how you can identify normal modes. You just look at the packets that are traveling with the different velocities and usually they are also spreading in some way, either diffusively or super diffusively. And how one can obtain these normal modes is just by diagonalizing the Drude matrix which is the time average current-current auto correlation function. Also the eigenvalues of these Drude matrix not only gives us the combinations of conservations corresponding to the normal modes, but also the velocities with which they are propagating through our system. So the second important quantity which will be at the center of my discussion is the Onsager matrix which carries more information than just the Drude weight. It tells us how these peaks that are propagating through our system are spreading in time. So basically it tells us what the width of this spread is. And in particular if this quantity is finite, I should also mention that the Onsager matrix is connected to diffusion matrix through the Einstein relation. So just by susceptibility of charges. So and if diffusion matrix and Onsager matrix are finite, of course transport is well okay the spreading is diffusive. But these two quantities are also directly related to what I motivated my talk with. So the transport coefficients namely Drude weight tell us about ideal conductivity while Onsager matrix tell us about normal conductivity in our system. So the primary idea behind obtaining the results which I'll show to you is something that we call hydrodynamic expansion. And the idea is to try to identify the modes or the operators, in some sense the operators of the level of correlation function, but this is already technical. So let's say just identify some operators which survive the hydrodynamic limit. So where we send the size of our hydrodynamic cells to infinity. And of course the operators that survive such a limit are quasi-local conserved quantities because they are actually conserved in the system. So if you increase it, there where we will not decrease. So what we did is we basically take a current in source as a current operator. We say okay we will look at what this current operator does to some density matrix. And basically what we do is is just expand the current operator in this equilibrium state in terms of this quasi-local densities, which also depends on the position of our hydrodynamic cell. In particular what turns out to be important is the first order in this expansion and the second order in this expansion, at least for what we want to do. There are also higher order terms, but one can show that they will not be important. And there is only a remainder and I will kind of, for most of the talk, I'll forget about this remainder. I'll only comment about it in the conclusion. So one can look at this hydrodynamic expansion also as expanding the density matrix around some homogeneous stationary density matrix in terms of local or quasi-local densities of charges. So after one done this expansion, one can simply take the expression for say through the weight matrix and insert the expansion into the definition. And what we actually can see is that the only contribution that is non-zero is the first order in this expansion when we are considering through the weights. And we were actually able to reproduce a well-known result which is obtained from hydrodynamic projection or Mazur bound, which basically tells us, so you can see here, it tells us that through the matrix is equivalent to the products of overlaps of charges and currents and then there should be a new source of sensitivity matrix. But one can of course always choose a nicer basis and in particular probably the nicest basis is normal mode basis which has a diagonal and normalized susceptibility matrix. So in the normal mode basis, through the weight is simply equivalent to the projection of current J, so Jk here and here, onto the conservation law Nk. Okay, here there should be some other. So this k is not the same as this k, sorry for this. And yeah, we should also project the second current to normal modes and this gives us the through the weight. So what's important here is basically that we have obtained some information about the dynamics in our system from considering purely statical quantity such as susceptibility or overlaps of the currents with our normal modes. But we can play a similar game by considering Onsager matrix. So asking ourselves, okay, how will these peaks that are propagating through our system spread with time? And it proves to be useful to redefine a bit our current basically taking away the ballistic contribution of our current. And if we do this, the Onsager matrix makes relatively simple. So we can play a similar game once again and obtain a result which is a bit more complicated. But in some sense, what's important is that actually from this result one can see the relation to the theory of non-linear fluctuating hydrodynamics, which on which I'll comment more later on. But once again, if we go to this nice normal mode basis, the Onsager matrix is simply equivalent to the projection of the current onto the products of conservation laws divided by the difference of their velocities. Okay, here I should comment that we're not claiming that this is a full Onsager matrix in general because we only took into account the part of our hydrodynamic expansion coming from the expansion of the density matrix or the current operator, which will expect the densities of local conservation laws. In principle, what could come into play is less local conservation laws. But for now, we don't care about it. So let's see how far actually this result took us. Well, I should also mention that this result is quite nice as well because basically we get the information, some part of the information in general, about how this normal mode spread simply by diagonalizing the Drude weight matrix, which tells us just about the velocities and the structure of these normal modes. So we see that we can learn some finer detail about the dynamics already just from Drude weights. Okay, so let me now clarify how our result manifests itself in different setups. Namely, there were quite a few results that preceded our study. So the first bound on diffusion constant, as far as I know, was one by Tomasz in 2014 where he lower bounds the diffusion constant, in particular the spin diffusion constant by quadratically extensive conserved quantities. And indeed, one can make a natural connection with our study because as I showed you before, what we have here is basically the products of conservation which is a quadratically extensive quantity. So because n is quasi local, the product that took quasi local is actually quite extensive. But one can even make a quantitative connection with this result. And then there was also a result which connected diffusion constant or lower bound with the diffusion constant by the curvature of the Drude weight. And once again, in special cases, one can immediately see this result from our study. So the systems that are under the most scrutiny in this conference are integrable systems and in integrable systems, the exact expression for the diffusion constant or Onsager matrix has been obtained using the thermodynamic form factor expansion and also via kinetic approach. And what we can show is that actually the expression that we obtained saturates the Onsager matrix, which was exactly computed in integrable systems, which means that the only contribution to diffusion constant or Onsager matrix actually comes from the fluctuations of local conservation laws. So the second order expansion of our current or of our density matrix in terms of densities of conservation laws. Okay, there is a further insight that we got from this result, which is understanding a peculiar relation between diffusion constant, which we observed year and a half ago, between diffusion constant and Drude self-weight. So the diffusion constant was shown to be, for some cases, connected to the curvature of the Drude self-weight. And indeed, if one analyzes our result in a bit more detail, one sees that this will happen whenever one of the normal modes, so whenever, so when the requirements for this overlap between the current and the products of two normal modes for it to be non-zero is some, let's say, symmetry restriction, which fixes one of the velocities to zero. So if this is the case, then what we claim is that the diffusion constant is equivalent to the Drude self-weight, which is a particular quantity similar to Drude weight, where instead of taking integral time average of current-current-alto-correlation function, one just looks at the current-current-alto-correlation function locally. So finally, there is also a relation to the non-linear fluctuating hydrodynamics, which has actually been an inspiration for doing this study, and in particular, performing the expansion of the current. But before I tell you a couple of words about non-linear fluctuating hydrodynamics, let me just mention KPC equation, which was already presented for integrable and non-integrable models on Monday by Vira Nyakopo. And what's important, the important point, one of the important points about this equation is that it actually corresponds to the universality class, which is not diffusing. So it's super diffusive, which means that the spreading of your normal modes is faster than in the diffusive case. And what has been observed historically is that actually anomalous behavior emerges in numerous classical one-dimensional systems, in particular unharmonic oscillators and so on. And this has been explained by Herbert with non-linear fluctuating hydrodynamics, so which, well, one of the statements probably the simplest one one can make is if there are no symmetry restrictions, if we are away from any special point, and if we have a conservation law, one-dimensional systems will exceed its super diffusive behavior. And the way one obtains this is in the spirit very similar to what we have done. So it's basically expanding the expectation value of the current in terms of conservation laws in stationary states up to the second border, and then adding some phenomenological noise and diffusion, which was already discussed by Acopo last week. And one can see that basically what we get if this G, which is just the second order expansion of the current with respect to the conservation laws is non-zero is something that's essentially K-PZ equation. So if this is the case, non-linear fluctuating hydrodynamics predicts the divergence of the diffusion constants, but even more it gives us the dynamical exponent. But also if we look at our result, we can see that whenever we have, so let me go back to our equation, so whenever we have two normal modes, let's say, which have a non-zero overlap with the current, and they have provided that they have the same velocity, we see that this expression will actually be divergence. So everything works out consistently. And so let me stress again, and this result is not specialized to just integrable systems. It holds also for non-integrrable systems. Okay, so let me now go back to our expansion and basically make an overlook of what I hope can be done in this direction. So the first question that was kind of put under the rug is what are the other contributions to diffusion constant? And it's clear that there should be other contributions to diffusion constant. In particular, if we consider models with zero due to weight, then the convective contribution to the diffusion constant, so the expression, the main expression that I was showing you is zero. But we know that in general, when we have non-integrrable models, at least we should expect diffusive behavior. So there should also be other mechanisms giving rise to diffusion with non-integrrable models at least. So one option is that there are some dramatically extensive conserved quantities which are not somehow simply related to local conservation laws. Because even if you take some generic model, usually symmetry restrictions will actually tell you that the convective contribution to diffusion, if you're making products of your local conservation law, will give you exactly zero. So there might exist some quadratic extensive quantities which are not simply given by the products of two local conservation laws, which might give rise to diffusion in non-integrrable systems or even noisy systems. Another question which is I guess more straightforward is to address the question of higher cumulants of the current. So we're able to obtain, through the weight, obtain some contribution to the diffusion constant, are there contributions to higher cumulants of the current. And finally, it would be very nice to go beyond one dimension, in particular at least to the flat length, to the two-dimensional one. Because we know that there are, first of all, a couple of very interesting results coming from ADS-CFT, some universal bounds on diffusion. And secondly, transporting two-dimensional systems is also related to one of the greatest questions in condensed matter physics, which is high temperature supercomputing. Okay, so with this I would like to finish and thank you for your attention. Thank you. Are there any questions we have? Yeah, let's say three minutes for questions. Just unmute yourself and ask. So can I ask? Yeah, I'm Lorenzo here. Yeah, so very naively, is there maybe any way where you can use this expansion of the reduced density matrix in terms of quasi-local conservation laws to compute one-point functions in given an equilibrium steady state, say? In equilibrium? Well, I mean, so any, say, Heisenberg chain you want to compute local one-point functions, say, sigma xj, sigma xj plus one. Well, in principle, you would have to, I mean, what I was assuming here is that you can anyway compute the overlaps of currents with charges. And in particular, that would mean if you wanted to calculate expectation values that you have to calculate also the overlaps, which is even harder. I mean, so in this case, I guess that the most of overlaps will be simply zero, because, I mean, all local concerns, well, I mean, so I just was wondering, because this is a different, a difficult problem if you address it directly. So maybe your method allows us to give some idea. I mean, I don't think it makes calculating the expectation values. You mean not the dynamics, it's just the expectation value, either thermal states or GGE states? No, it doesn't simplify it. I mean, I mean, calculating expectation values is basically zero order in our expansion. So I think, I mean, I don't think you can profit anything from this analysis. Thanks. Other questions? Hi, Marco, could I ask you a question? Yeah, of course. So the divergence of the diffusion constant you mentioned, is that cutoff dependent in any simple sense or sorry or not? The divergence of the diffusion constant that you described went with the copropagating mode. Yeah. Is there any simple sense in which that cutoff dependent or cutoff? You mean by in the number of conservation laws? For example, for like the like the spin diffusion constant. Okay. Yeah, so now we haven't dealt with the divergence and emergence of KPC using this in integrable spin chains. So what I was discussing in the scope of non-linear hydrodynamics is the case when you have just a couple of conservation laws in that. And if you have some diagonal contribution, the diffusion constant will diverge. So I think, I mean, as you know, I mean, this business with KPC and spin chains is quite technically involved and yeah, one should in principle introduce some cutoffs. Yeah, can I just add, so when you have these divergence, at least in the with fine number of conservation laws, you can actually work out a little bit more and bound the super diffusion exponent instead of trying to bound the diffusion constant, which you find something divergent, you can find the super diffusion exponent, but you have to use some kind of Hilbert space construction with that. I don't know how to do this in the in the form that Marco was explaining here. Yeah, so there is that that's explained in the paper that by Ben, which I showed in the show. Other question.