 Are the third is yours? Hello. Yes. Can you hear me? Yes. Okay. Okay. First day, I want to thank the organizers for organizing this event. Nice event and to get to know each other, what we did in this, in the room. And it's a very nice initiative. And hopefully, we'll get to know each other's work better in through Zoom, and maybe it starts new conversations. So, and so, so I want to, so today I'm going to like speak on revisiting mazut and Suzuki bound for financial systems. So, and so what I'm going to focus mostly on is, but the applications of it in for transport and integral systems. So this work is, is done by with my supervisor, and, and, and it's roughly based on the, on the work, which is shown here. So this is going to be the wrap overview of my talk. So first, I'm going to talk some bits about transport and integral systems. Talk about setup and common setup and, and typical features of transport and integral system. So I'm going to talk about, like auto correlation functions, and, and try to motivate from this, why the auto, why auto course functions are important. And I'm going to discuss mazut and equality and to the equality. Then I'm going to talk about how to go from the inequality to the equality and then I'm going to discuss some few simple examples of in the harmony and then I'm going to move on to many things. So, I mean, I'll explain each of what each of these terms is in the letter. And, okay, so, okay, so one thing I should mention here that I'm going to quite contrary to the other things I'm going to focus my whole talk on classical systems. So, and so, so first I'm going to introduce like, like, say, suppose we have a system, one D system. And, and, and you have n particles and then you have this, each of the particles have this position q and momentum B. And typically, a one D system will have the same domain with the quadratic momentum and potential. And say suppose, and they're interacting with some potential V. And, and say suppose you have to consider the system to be isolated is considered them to be in the periodic boundary conditions. And typically, typically the form of this potential determine what kind of transport you have in the system. So, for example, a special case of this potential is a special case can happen when when this is for a special choice of this potential, the system can be indigable. The integrable one means that it has ended with a freedom and to introduce a freedom with the momentum of the position and then for a special choice of potential, it has any independent constant solution. But that is that you can construct some function of this B and q. So the time derivative of it is zero. And independent means that between the two, two constants of motion the positive bracket is zero. So, so a typical example of a very common example of an many body system is just the plain and simple harmonic oscillator. And, but there is a, for example, another non trivial example of an integrable classical system which is, if you have this potential to be an exponential, and also nearest neighbor interactive. So this is what is called the total potential and it's also a typical example of a non non integrable system is that suppose if you take the near again nearest neighbor interaction, but with a quadratic and a quadratic term, so it's what's called an SPO beta system. And, and, and for this, for the fun integrable system, the typical concept quantities are, if you consider them in the periodic lattice, you have this total, total inverse density as constant, you have the total momentum as the constant and, of course, you have the Hamiltonian, which is the constant of motion, but other than that, you do not have any consistent motion, but for integrable system you have for any degrees of freedom you can construct microscope the number of conserved quantity. So, so suppose somebody gave you a system and asked you, asked you, like, if the system integrable. There's no, I mean, other than doing any calculation, it's not a very, very easy to say the system is integrable. So, but what you can typically integral system has typical signatures in transport properties, which you can put in a computer and see and test and see that if the system is showing shows some So, one of it is like, if you can consider equilibrium fluctuation. So suppose you take the integral system and you part of the system somewhere, and you observe how this perturbation spreads in space and time. And typically what one finds in an integrable system is that the perturbation spreads in both space and time. And then if you rescale the spread of correlation in space and time by X, Y, T and T here, they all collapse on to each other. So, this is what is called the ballistic scaling. So, the system typically has this ballistic scaling, although there are exceptions, but, but, but typically this is what is one, but what he's saying. So this is, this is one aspect. And the other aspect what what one can do is like study the non-equipment transport. So suppose you take this integral system, and you connect the system to hit bar. You take the, then put the connect the left end of the system at a higher temperature than the right in the lower temperature. And, and you're trying to system out of the computer. There's an energy flow from the heart rate to the cooler and and the system will be some kind of steady state. And typically, typically what one finds is that if the system is integrable, you have, say, suppose, for example, here you put in the left end a temperature of two and the right end temperature of one. The integrable system typically you find that the temperature in the bulk is just flat. It does not have any structure, which is like a very confident like contrasting to some other system paper, not typical non integrable system you will have a if you measure the temperature in the bulk will have a temperature which is continuously interpolating between the hotter end to the cooler end. So typically flat temperature profile indicates that there is some kind of underlying integrable structure in this system. And, and also, also, even given the setup, you can also construct this study the thermal conductivity of the system, which is this kappa and thermal conductivity is typically given by the amount of current flowing across the system divided by the applied temperature gradient that is there. And typically this gives the kappa and and for the integrable system, this j by delta t is just a constant and so, so the thermal conductivity basically diverges linearly. I mean, this is in contrast to say for a typical non integrable system that that the thermal conductivity is not dependent on system size, which is constant for regular material. So, so, so what I'm going to try to say that we can also construct this thermal conductivity to another kind of experiment to, to the, to the equilibrium frustration, which is to the, like the green kubo fonder. The linear response theory tells that say suppose you again now take the system in equilibrium, and you study how how the flag, you take the fluctuations of the current operator, the auto correlation of the current operator, and, and you try to understand, and, and you compute this integral, the, the, the correlation of the, the, the auto correlation function, and green kubo says that the thermal conductivity is given by this form. And, and, and say suppose in an integral, and I'm saying in an integrable system, if you plot this quantity as a function of time, what you see is it, it, it decays, it starts from some value and decays to some constant here. And, and this, this constant is not zero. And so if this is not zero, then this integral basically diverge and it diverges as, as, as in the system size. So what I've been trying to motivate is that this kind of articulation function is important for understanding transport in integrable systems. And, and for this, and, and say for our talk, what we'll try to focus on is this auto kind of auto function of some observable say a, which are functions of the momentum and the position, and, and we'll focus on this quantity C a, which is the, the time average auto correlation So, so the long time value of this is what is called the derivative and the literature and, and, and we will try to see that try to estimate basically what is this long time saturation value of this thing of this auto correlation function. In terms of the constant submission of the system. So, okay, so, okay, so, so this auto collision function also have like, we have already seen, seen that, that this auto collision function is important to understand the transport coefficient of the bin cube of home through the bin cube of formula. So by studying this auto collision function, you can also for, for, they're trying to put up a study that you can, you can basically distinguish if the system is integrable or not, by, by putting a bound, like if the system, if the auto collision function do not be case to a constant then it possibly has a ballistic component of the transport, and hence integrable, and also it, it, it provides insight into that it got the property of this variable a that you are considering for the system. So these are some of the references that the discusses this quantity. So, so, so, so, so what, what does, so let me come to the quality. So, so what does, suppose you have this system, which is, which has our concept quantity, undergoing Hamiltonian detail. And let's say you denote this our conduct quantities as I want to IR. And, and for, for simply like, and if I want is the Hamiltonian, Hamiltonians always the concept quantity. So, for simple, for some purpose, let's say that we subtract the average out. And because this average is anywhere time independent quality so we can always do that. And, and, and let's say we define this, this concept quantity, so that the product, the average of this concept quantity is zero. The average is denoted is with respect to this initial and symbol, and this is the Gibson symbol, and where with the temperature, the inverse temperature beta. And, and, and let's say we, and we construct this correlation functions of this, of this concept quantities I want to IR, and call this matrix the idea. So what, what must do say is that muscle constructed this quantity, which is, which we denote as da, and which basically says that that which basically is this projection of this operate of this observable and we are interested in this then long time dynamics of this observable a. So it says that this, this, this quantity as defined here is always is bound basically this, this large long time coalition functions of the observable a. So what it says is that the long time coalition function decay to some constant and that constant is always greater or equal to this quantity da. And this was proved long, long time back in 1969. And, and what one typically expects is that, like, it's in an integral system if one considers the set of all concept quantities. And if I have, if I have, like, like, like an integral to think I have 10 kinds of quantities. And if I include all the 10 kinds of quantities in the system, this should be an equality like the long time saturation should be equal to the value provided by this model. So what it was seen in some previous studies in classical systems is that it's, it was not enough to saturate the bound by including the all the concept quantity but also one had to include the bilinear combinations of the concept quantity in order to saturate the So, so this is the paper that I am aware of the first work in this direction. And this is also a recent work, which we can address this problem in quantum systems. And, okay, so, so, so mazu did this calculation for classical systems. And what, but what should be published in 1971 was that for a, what he did was like he took a quantum system with, with, with this big spectra and and, and, and in that case in, in a content with this picture that a trivial set of Eigen constant some motion at this projection operator of the energy Eigen state. And what Suzuki proved was that the long term saturation value of the population function which is the CA is exactly equal to this projection of this observable on this Eigen state. This is an exact equality. So, and so this basically. So, so let me explain to you a bit more about like, so this quantity Ckl is again, again, this correlation of this, this concept quantity in the initial ensemble and, and, and that this quantities are basically the expectation that expectation value of the content quantity with the observable that we are trying to find the large time value of. And, and, and this is this is quite quite interesting formula because you see that in this. So basically getting, we basically get the large time saturation value of a dynamical problem without ever having to like run the dynamic so this is a competition like one can compute this from the static. And basically it gives a bound to this large time behavior of this auto position. And for, for example, and for here, the, the, the number of Eigen states that one needs to consider is proportional to the dimension of the inverse space that one is considering, and for say suppose an infinite dimensional problem, you have to consider the sum to for all that I can say 20. But a more general theorem of the corresponding to this is the key bound where you have both this heat and functional spectra was proved here. And, and, and, and so now the question is that, how do we reconcile this two different. So, so, so the people that this contribution functions exactly the case to this thing, and this quantity here, and muscle groups that it is lower bounded by, by this quantity. So, so the question that we try to ask is that is that equivalent of the physical equality in classical system. And, and, and the, and the answer that we try to like, we find is that, like, it is possible to attend this equality, like, but for the equality what, what has seen in the previous study. We have to include. When computing this, this projection of the observable to the concept quantity, we not only have to consider the, the first, the, the, the, the concept quantity is the linearly independent concept quantities of the system, but we also need to consider the, the, the, the product of this kind of quantity. So we have the concept quantity I one I two to IR, but we have to also consider I one square I one I two so on so forth, and then the funding for infinite that, and if we use that means that, but one can show that then, then this long time, long time, long time value of this integration of that position function is equal to this, this, this quantity, where this, where we have now all this averages are now constructed over a general keeps ensemble, which, with, with all these kinds of quantities as I k and this lambda k is the corresponding reference multiplier of the system. And the main point here being the difference is that it's not sufficient to take the first R concept quantity of the system, but also take by linear or higher order combinations of the concept quantity in order to actually this one. So, so, yeah, so, in order to explain this details, I think, I mean, it would be, I first let us, like, consider this example of this, like a simple harmonic oscillator, it's a one simple harmonic oscillator in order to see that how this is true. So, so, so it has this Hamilton, it has this momentum and the position be square and extra square. And now let us, and you can solve it explicitly you have this solution in terms of initial condition, and, and say suppose you do take the system in in an initial and symbol of e to the power beta H with the beta as your inverse temperature. And let's say you are interested in this. And in this observable as well. Now, now, with this solution you can compute specifically what is this, this auto correlation function here, this is integration of zero to tau dp, and this quantity access where you can log it in and do all the integration and then take an average. And what you find is that it is equal to two access to a whole square. Now you can, you needed the whole solution of this problem with the time and everything to compute. Now, but you can also compute this, this is bound from by considering the set of the only the only kind of quantity as Hamiltonian. And using this formula the position of excess square into the Hamiltonian so you take excess square into this whole Hamiltonian and take the expectation value, and then divided by the Hamiltonian square and the expectation value, and you find that these two are exactly equal. So for for this quadratic observable to see a is equal to be it that is, that is the long term saturation value is exactly given by the model value, and hence, the equality is satisfied. But for example, you will try you consider an observable which is extra the power four, these are really simple like you can just do it by hand. Like extra the power four, and then what you see is that you can again do this integral and, and you'll find that the long term saturation value is 54 excess square whole to the power four. And now you try to see that if you get this number from by considering this muscle formula, but by concern by with only the Hamilton because the Hamilton is only the concept quantity. And you find that if you plug everything in, like you plug x four and eight here, and you find this quantity is 81 by two x square whole four, and they do not last. So, so. So, and but now you what you do is that you take to introduce also while computing this value, you introduce each and also each square, and you compute exclusively and you see that it matches, matches with the long time that. Okay, here it will be a forum with some mistake here. Okay, and it matches it. So, so the point is that like, it's necessary to consider not the not only the concept quantity but also the concept quantity square. Initially, it's not like look bit weird, but but also a concept quantity squared is also a concept quantity of the system. And so, and it kind of makes sense. And, okay, so, so the second example that we can consider is this like single aquatic oscillator. Again, you have this just one particle you have momentum and position, and this is not linear. And it's again prepared in this ensemble minus beta page. But now you cannot solve it explicitly so you can solve this dynamic and put it in a computer and solve it. And let's say suppose you take this observable access square. And, and you compute the autocollution functions autocollution function with something like this. There's some initial observations in the beginning and then such that some non zero value. And this, this non zero value here is if you do the integral is something like point six three. And, and, and what you find is that like, then you, you compute the masses value with the only the only the Hamilton this one here is included to subtract off the averages. But it's just a discount. So suppose you take the Hamiltonian, then you get the value to be point six zero something, and you take Hamiltonian Hamiltonian squared you improve the bound a bit. Then you take higher combinations and then you keep on increasing. So, so you see that the, the bound keeps on increasing as we include more than more higher parts of the sense of quantity. And finally, if you like, as in Suzuki's one Suzuki's bound, if you take any finite number of missing, this will be equal to this long time saturation value. And one can kind of show this. Yeah. So, so, so the main takeaway message from this study this body problem is that it's when the potential is harmonic. Although the finite number of contact quantity is sufficient to saturate the bound. And, and, but the number of consequences required to actually the bound depends on the degree of observable that one is ready. And as we have seen that for the x4 we needed a square but for it to satisfy by the by only the taking the Hamiltonian. And for the nonlinear case, what typically one needs an infinite number of variables to saturate the bound. But however, as we have seen, as we can see here like the convergence is really far like, like the effect of taking three kinds of quantities and four kinds of quantities the difference is very less so it converges quite to the long time value. So, so this this was a part. I hope this is more or less clear what I'm trying to do. So you send questions to the house now. Otherwise, I try to move on to the many body question many body system. Okay, so sorry, maybe I have one very short question that in some system. Some system but maybe not in one particle so maybe I'm getting ahead of myself there but some system cannot be completely characterized by concept quantity only and there are these quite quasi concept quantities that has to be taken into account as well. Do what you are about to say also concern this type of systems or not. I'm not very familiar with the quasi concept quantity system and I have not. I do not know what exactly means, but if it is conserved in the dynamics in some sense I mean this discussion can also be taken over to quantum systems and here I'm discussing classical system but I guess it should hold this more general statement. Okay. Thank you. Okay. So now, so we'll try to discuss what happens in our many body system. So what one can show is that if the is the many body system is like a quadratic many body system. There's a chain of harmonic oscillators near the sliver interacting and one is interested in just observing quadratic variable. Then, then the long time saturation value CA is always equal to the mother's value computed just by taking the Hamiltonian. And it's always equal to the fact that the system is classical or quantum. And, but, but, but, but this, I mean, this, this kind of like follows from the, from the single harmonic oscillator problem. Also, this is consistent with the one that we have seen in the single harmonic oscillator problem. But, but what is more complex is more interesting is that when we consider this, this kind of system which is the total lattice. We have we have we consider this in particle total lattice, where you have this momentum be, and they're interacting with each other with, with nearest neighbor interaction with an exponential form. So what one can show is that the, the, the, when one tries on the equation of motion of this system, it can be written in terms of this. The equation of motion of the system can be written as something like this, like, in terms of this metric L and M, where L is this metric with the diagonals as the momentum and the off diagonals as this exponential part. And, and this, and the, and the aim is constructed as the upper part of this triangle, minus the lower part of this triangle. And this is what is typically called the latch metric that's there, and the equation of motion can be written as the time evolution of this metric as a commutator of them and help. And so given this form of equation of motion, what one can show is that the system has conserved quantities which are trace of trace of L to the power n. So, so you take this metric L and raise it to power n and then trace over and these are all constants of motion so they do not evolve in time with this Hamiltonian. So, so this is what is written here so you take L to the power n and you take the stress, which is the IEI. And then the one that can take like if you do this, then then trivially the first concept quantity of the system is the diagonal element, which is the total momentum, the second one is the Hamiltonian. And also, you have if you take the system in a periodic boundary condition you have the total density to the concept. And, and now one can know what we can study. So we study this observable. These are important in term in the study of heat transport. So we studied the total momentum current of the system and the total energy current of the system. And we suppose we define this quantity in which is, which is this quantity. We say the momentum current minus the average value of it this average now is all taken over this general gives ensemble with e to the power minus beta h plus p into the density is kind of the pressure and beta is the inverse temperature. And, and for example, and, and then one can compute this. This, this auto correlation function of this quantity and using molecular dynamic simulation and then try to compare this with the value of, of the muscle bound that is provided by considering the concept or the concept quantity. And, and now we can start considering this to set the first set here consists of considering only the first in concept quantities of the system. And the second set consists of considering the first in concept quantity of the system and also the bilinear combinations of them. So you have, you have in and then you have this. And you can compute the expected the mother value of this observable a n using this set I one and also this this that I do. And we'll try to see numerically what that how well these two matches with this long time saturation value of the CA up to my molecular dynamic simulation. So, okay, so here is the first plot so so this a one is the momentum current and a two is the energy current. And this one by in comes from some considerations of, like, because we are taking here the total sum. And so what we see is that if we consider. So, so this is, this is the long time saturation value as obtained from this molecular simulation, this, this car is valid. And if, and the market value as obtained by taking just the, okay, here we are first we are considering a total lattice with four sides, and there are four particles. And for this four particles, you can, you can have on the concept quantity that you see. And if you just consider the first four countries, the mother bound is given something here. While there, while if you consider the four countries and the bilingual combination the mother value improves and becomes almost exact to this long time saturation. And the similar thing happens for the energy energy and autocorrelation function. So now what. Now if you increase the system size to eight. You see again, if you consider the value combination, the mother value is exactly equal to do the long time saturation value of this autocorrelation function but if you take just the first eight console qualities of the system, it's still far from the saturation. And a similar thing happens with n is equal to 10. But, but what you can see is that the gap between the two keeps on decreasing as we increase the system size for both. So, and you can compute this. And this gap basically decreases as with system size with increasing system size as one over N. And, and, and, and, and it this makes it like possible that in the thermodynamic limit possibly this and the bound like the saturation value is can be possibly given by, by, by just considering the thermo in the sense of motion or that's the constant for motion of the system, but for any finite system be that last time saturation value should, if you have to get an equality we have to include all the higher order combinations of the quantity, the CA is equal to DA, which is, which includes all the combinations of concept quantity. And, and, and it is possible that in the thermodynamic limit, we don't need to consider the, it's enough to consider just the concept quantity of the system and not binding the terms. And, but this is, this is just a numerical evidence, but one needs to do this more rigorously. And yes, so this brings an end to my talk so summarize briefly what I have discussed that. Firstly, I, we address the question like when in the system the mother inequality becomes an equality. That is, when the long time saturation value of population function is exactly equal to the bound computed in equilibrium. That is the projection to concept quantity. And we found that it is for a finite system, we need to, we need to consider, not the concept points, but also higher combinations of concept quantity in order to saturate the bound. And for a many body system, we, the, the, the difference between the, the, for considering the higher other contribution to the, to the bound decreases as one over N. So, so, so what is what one is interested in is that a more detailed understanding of this one over N contribution in both classical and quantum systems. Like, how does it decrease as we increase the system size and what is the effective reasoning for this one over N dependence. And I want to mention here that I discussed everything in classical system but this, this was also seen in this is also seen in the quantum systems, and which we also also discussed in the paper. Yeah, so this brings my name to my talk. And thank you for listening. Thank you for this nice nice talk I club on behalf of the audience. And we now have plenty of time for questions. So please go ahead. Yeah. So maybe I could start start. So is your estimation of this long time average. Is it always lower bound or, or is it not the case. In that if you keep on adding more and more terms to your set of consent quantities. Is it always increasing or that that's not the case. This was my observation from your close but So what we could prove is that if you keep on in a certain way if you construct this. If you keep on adding this more kinds of quantities like, like you have this any dependent content quantity, and then you take products and cubes and cross products. Then you basically always increase the ground, and you get more and more accurate. So, so, so this. Yeah, you basically get more and more accurate as you increase the volume, as you increase the number of points, but yeah. Does that answer the question. Yeah, yeah, okay, but then so is it always the case that the value you get is increasing with increasing number of possibilities or this is just a coincidence from from those plots. So if you let's say, you know, I mean, it always increases. Okay, so there is no no way that it could overestimate the long time value and then decrease if you add more. No, no, no. If you add, add, add the consent quantity in certain way, you cannot add a certain way, it's always improved about. Okay, so there is another question by Vladan. It's a little bit, if I'm crossing correctly, whether can you give an introduction, one to introductory references to your subject. Yeah, I think it's, sorry, I think it's best to first. Oh no. What happened. Did my slide close. Yeah, I believe you have to share your screen again to to share your screen again. Okay, I think my keynote. The introductory reference is I mean to study the papers of mazure and Suzuki and they're the best. So, I mean, I mean, I would say that that's a minute, let me try to share. And it doesn't. Yes. Yeah, I think. I mean, these two are very nice reference their old papers, but they are very nice, the mazure's paper original and this paper by van Kempen. And also this, and also this is a piece of paper. And if you, I mean, there are a lot of interesting things going on here and for more. Yeah, I do not. I mean, there are these studies which are now being done in quantum systems and in regarding the island state globalization hypothesis, which has similar things that are being studied there. But I, yeah, for, for, for, I mean, here they have studied. Yeah, I think. I mean, here they have studied this. They have studied the. This muscle law for, for the exit the chain. Use the muscle law to prove that that she transport index of the chain. This is also nice paper. And yeah, that's what I can say. And you can also look at the references given this paper. Yeah, so let us wait a little bit more for maybe another question because we still have. So maybe I have a brief question while some people might be typing it. It's when you move on to the quantum case. Are there some significant difference in the demonstration in this case where specifically overall does everything adapt without notable changes. In the quantum case, it's a bit more subtle because like for example, Suzuki proved the system for in quantum case, which you have, you can prove it for a system with a discrete spectra, but for the car. But in general, you can have discrete and continuous spectra both and then things becomes more complicated and more mathematically challenging and this was kind of what I do not understand it completely well but this was addressed in this, this paper by present and others. And, and it's not so straightforward. It's not so straightforward. Yes. I mean, and I think people are working on this activity in quantum systems and then around this. Thank you. Okay, thank you. There is I have a question. So please go ahead. Maybe shred I can. I can give you the right to talk give me a second. Now you should be able to talk. Yeah, for giving me the right to talk. So I have a very trivial question so it is just like this that if you have a long range in tracking system do you need less number of higher orders of these concept quantities or. I mean, can you comment about this maybe. The long range system. If you keep on increasing the range of interaction in the system maybe you need lesser number of concept quantities or lesser order of so rather than going to four orders you just need to be two orders of two higher orders of these quantities. I mean, that's an interesting question. I mean you're talking I guess you're talking about many body. And I do not know the answer. But I mean, I mean I thank you do not know the answer but I guess I do not know one has to just see I mean it depends on what you are considering which observable you're considering and what type of long range. I mean, suppose if you have an infinitely. Then sometimes this model can be like very simple, simply mapped to a single body problem. And then, then you would possibly need higher order conservation. Thanks. Thanks. All right, so I don't see further questions appearing. So, thank you for this talk and I would like to thank all of the speakers of this session.