 I am Zala Lenarchic from Joza Stefan Institute in Slovenia and I welcome you, everyone, on behalf of the other organizers, to this school on the out-of-equilibrium phenomenon. So I hope you'll enjoy it, learn a lot, meet new colleagues and have fun. So I mean this program is online, as you've for sure noticed. So let me just remind you that today we start with the first poster session. It will be held like here inside, essentially in this corridors around this lecture room so you can hang your posters anytime. In principle it's today, it's from names from A to L. And after that, at six, we'll have get-together cafeteria, so at the Adriatico guest house which is down at the seaside, essentially. There's another hotel that's more down so if you don't know where it is, ask colleagues or information or me, so you essentially need to walk down. So we'll just have some finger food drinks down there. And let me just introduce our first speaker, Anatoly Perconico, from Boston University that will probably give some more introductory lecture on chaos and ergodicity in quantum and classical systems. And I guess you're encouraged to ask many questions during the lecture, so don't hesitate. Let's be interactive, so let's learn more. All right, it's in order to be the first speaker, welcome everyone. So when I asked Alessandro Silva what they should talk about, he said that I'm supposed to talk about chaos close to equilibrium. I was not exactly sure what it means, so I thought I will use it as sort of excuse to talk about chaos and ergodicity and sort of our recent works, understanding them using adiabatic transformations, and I'll come to that in later lectures. But I first will start from some basic overview, I know some of you probably knew to these topics and I will try to give some overview of developments in the last, I don't know, 30 years. So before I start, I do encourage you to really ask questions, just interrupt. If I don't see your hand, just say question loudly. If I don't do it, I think many of you will be lost very quickly. I hope that it doesn't happen, and questions help not just you, they help other students to catch up because if I speak I might not notice that I'm said too fast and so on. So another disclaimer I want to make is that of course chaos itself has very, very long history and I'm not an expert in many of the topics and there are actually experts in the audience who know answers to some of your potential questions much better than I do. So but I'll try to do my best. So I will try to start from some really basic concepts of chaos and I will only talk about like chaos in physics and Hamiltonian systems. There is of course like huge chaos theory in other dynamical systems, I'm not even going to mention those. And I also will not mention some of the other topics which other speakers will discuss like circuits and so on. I'm not an expert there but also there will be other talks about this. So I will talk a little bit about chaos, ergodicity and determinism and then I will spend more time on I guess quantum ergodicity against state civilization hypothesis. I will try to distinguish chaos and ergodicity in small classical systems. We know about this distinction for a long time but it's actually this distinction is always there and then in the last lectures I think I will mostly talk about our own work in the last I don't know five plus years about how we can sort of merge concept of quantum and classical chaos in a single framework so we'll see how it works. So yeah even if you look into popular literature definition of chaos is still not unique, there is more precise definition of mathematical definition of chaos but when we talk about chaos in general then usually we talk about chaos as a lack of determinism. So and of course there is a very long history as I said like of what is chaos where it comes from. So we know like for now I will try to use chaos and ergodicity interchangeably but later I will sort of try to separate these two notions. So we know that statistical mechanics and thermodynamics is founded on ideas of chaos. In a way you can say that chaos gives us a measure for probability right when we talk about Gibbs weight or micro canonical weight and so on. So and there are many laws in statistical mechanics but one of this I guess main laws is the second law of thermodynamics which basically tell us that any sufficiently complex system an isolated system if left alone means that it's Hamiltonian is time independent it reaches an equilibrium state and this state is basically maximally random so maximizing entropy within the constraints that say we have fixed volume, fixed energy, fixed number of particles and so on. So and then this is example I stole from the internet of a chaotic motion in a closed but very complex system so it's a glass of water where we put an ink of drop and then what happens is that as time evolves the motion of the blue molecules expands through the glass and the system becomes more and more chaotic right because our probabilities to find blue molecules in the glass become more and more delocalized so we can less and less predict where blue molecules are but if you really stare in this picture you will see no no this is not the most chaotic state it's a very simple state right because we can describe it with a uniform concentration of blue ink so most chaotic state is somewhere here so and this is sort of in my opinion at least this simple picture kind of underlies why chaos and ergodicity in some sense lead to simplicity and I will try to talk a bit more about this so maximally random state is actually simple it's more predictable so the second law has of course many implications and many of you know them let me just show a couple of examples so when the ship sails from Boston to Trieste or Venice it has to use fuel and why there is so much energy in the ocean right it's warm there are lots of molecules they move fast and so on why don't we take this energy for free and the reason is precisely that the motion is chaotic so if we take energy from molecules of the ocean and put it into motion of the ship we actually reduce entropy of the system right because motion of molecules is more random than motion of the ship so this is the reason we have to use the fuel and another of course very famous example is that heat always flows from hot to cold gain an isolated system without adding extra work and this is also as you all know leads this also leads to entropy increase so now let's go to physics and start from maybe the simplest example I can imagine just particle in one dimension this is predictable stable that's all what we cover in beginning even high school lectures maybe I'll talk about Hamiltonian so maybe not high school but first year of undergraduate mechanics so in one dimension we know motion is stable solve various systems where well harmonic oscillators like a nonlinear oscillator and we all know that the key reason for stability is the fact that you have energy conservation so if you have some potential v of x and energy which is p squared over time plus v of x is conserved which means that we uniquely know what momentum is without even solving equations of motion up to a sign so like let's take an example in nonlinear oscillating a quartic potential so then it's very convenient to express the motion as a line in phase space right so we have vertical axis momentum horizontal x and the particle has some kind of circular motions not exactly a circle it's it's a constant energy curve but anyway the particle moves on on on this well-defined orbit so now we can ask how stable this motion is so we can compare we can take for example small nonlinearity so suppose the potential is almost harmonic but where it's small nonlinearity I took all the units to be one like mass is one frequency is one but then there is dimension less nonlinearity epsilon and then we can compare this motion so this is say x of t for you know original nonlinear oscillators and we see that of course we accumulate a mistake so but now we can actually develop perturbation theory I'll talk a bit later about it so you many of you probably know about stationary perturbation theory in quantum mechanics but there is nothing quantum about perturbation theory you can develop it for classical systems so you can get a better Hamiltonian and if you do this perturbation theory you'll see much better motion as I said I will mention where this expression comes from so now if you correct for epsilon for the Hamiltonian so the Hamiltonian as you see is still extremely simple it has same orbits so it's a function of same H naught so it's basically harmonic oscillating harmonic oscillator squared which means that you can still write all analytic expressions for how the particle moves and so on and then if you look into better motion the blue line you see much better description of actual dynamics but now if you wait for the long time it's the same plot but now time is much longer than even this will break down just because our period is not exactly the same and if you you see mass but basically after a while all lines become everywhere so you cannot really predict accurately using approximations where the particle is so but then of course you can do even better approximation better approximation and then you can improve improve and improve so but after a long time even for this deterministic picture actually we can predict much better not precise position of the particle but it's stationary distribution so where the particle is so if you think about this imagine that I have a classical description of say electron and atom and it moves extremely fast so and when I do measurements I basically take say photograph of this electron but because it moves so fast I essentially perform it in a random time so the stationary distribution will be obtained if I do like many many photographs at random times and see where electron is and then the only information I can get that it's somewhere on this orbit right then uh suppose I measure only coordinate then I can ask what is the coordinate distribution and if you think about the intuitively uh it's more likely to find electron or particle when it's slow right because when particle moves fast it goes through this point very quickly so you there are small chances you will detect it on your camera but when particle is slow so it's near turning point then there is much higher chance that you will see it right so and if you think about this probability is proportional to time which is spent by the particle in a given point in space and this is inversely proportional to velocity so then the stationary probability distribution immediately is one over square root of e minus v right because this is velocity so but if you think about this this is nothing but micro canonical distribution statistical mechanics so this is again the simple example there is no chaos and we only do time averaging so we deal with particle at a fixed energy so the relevant distribution is micro canonical right the particle always have the same and I assume all of you uh know this that micro canonical distribution tell us that particles equi probable everywhere on the constant energy surface right so let's see how it works so uh probability of coordinate is a marginal right so we need to take micro canonical distribution and integrate over momentum and then we will find what's the probability of coordinate but you all know that if I integrate delta function I need to divide by derivative and derivative of p square it is p which is basically it's p over m because p squared over 2m is when you differentiate p over m which is velocity so you get one over velocity absolute value of velocity have to be more precise so we see that actually this one desystems irgodic and then we can ask the same question what will be mistake in my stationary distribution so now I compare harmonic motion non-linear potential and this is like a his two histograms p of x for harmonic motion and non-linear potential for the same non-linearity I showed before point one and you barely see the difference so again I just want to stress that the perturbation theory in quantum mechanics is more robust not because it's quantum but because we deal with stationary states analog of stationary state is not a phase space point it's it's a probability distribution I'll come to that micro canonical probability distribution so for these distributions they are much more robust so we see that if we abandon the idea that we need to know precisely where the particle is and we only want to know what's it what it is doing after a long time then we have much more stable situation even in these systems which are not chaotic so and then you can convince yourself it's not just time uncertainty it's uncertainty in anything because usually yeah so I'll come to this so in one dimension so yes you get ergodicity just for one particle which is not chaotic but in higher dimensions I'll come to that you will need chaos yes so now you can convince yourself that it's not only uncertainty in time which leads to the same result usually we deal with ensembles of particles right we again if we study maybe one macroscopic pendulum it's not the case but if we study you know many small systems like atoms even if they are classical so the quantum mechanics is not important we very often deal at least with ensembles of particle and these particles have slightly different masses slightly different Hamiltonians right so we can say how some particles have slightly different magnetic field and some other particles right and all these things lead to loss of a precise position of the particle so this is the word decaherence which many of you here has actually a workshop on the open systems so decaherence does not necessarily come from the fact that you have environment but simply from the fact that you have many particles and there's somewhat different different Hamiltonians different masses so you can have different clocks different initial conditions so anything different and if you go through the same line of arguments you will see that stationary distributions are much more robust so precise dynamics is very fragile and this is by the way the reason why you know quantum computers are much less stable at least in my opinion not because they're quantum but because they try to deal with time-dependent information it's much harder to control okay so now let's try to actually coming back to the question let's not try to increase the complexity and now I will have either two particles so one particle in two dimensions it's how you treat it but these particles are not interacting in sense they don't know anything about each other so the Hamiltonian is some of the Hamiltonians and of course there are two separate conserved quantities and those who like experts you know that conservation laws are related to symmetries I wonder if anyone can tell me what the symmetry is there is an extra symmetry in the system I am not assuming that v1 and v2 are related so it's not like rotation of x and y and so on but there is still an extra symmetry which leads to conservation sorry I cannot hear yes but you have two energies anyway it's a bit subtle you have two time translational symmetries because you can introduce a separate time for x Hamiltonian and y Hamiltonian anyway so and two Hamiltonians you know Hamiltonian is is related to time translation anyway you have two energies which are separately conserved and then if you go through the same argument which I gave if you do time averaging actually you will not end up in thermal equilibrium right so I will get in the product of micro canonical distributions for x coordinate and y coordinate so we'll still get a stationary state which is still will be pretty robust against fluctuations which keep Hamiltonian separable now I want to to be more careful but this is not a correct statistical ensemble it often happens you start from something simple everything works you're happier you make an x-tap and suddenly everything fails so and this is actually a generic situation in integrable systems so systems with extensive with many conservation laws usually as many as number of degrees of freedom so we basically have like each degree of freedom separately formalizes and this is why actually after a long time the systems are less predictable than ergodic you need more information to describe them it's not just enough to tell what the energy of the system is you really need to say what's the separate energies so in some sense the systems are much simpler we can solve equations of motion at the same time they're much harder because we need much more information to describe even their long time state okay so now let's try to make situation even harder even more complex and let's imagine that our potential includes interaction so it's some function of x and y so now what do we do well energy conservation allows us to find magnitude of momentum I will assume that masses isotropic just to simplify things but not the direction so direction we cannot solve from this and there are no like other conservation laws if we say arbitrary which will help us so we cannot really say what's the momentum is as a vector without solving equations of motion so and this is a big difference with one dimension so what do we do and here it's comes as actually very old philosophical principle of indifference which I think according to what I read originates to Bernoulli but I'm pretty sure it has all the history maybe not mathematically formulated it's like there is a standard joke about some stupid people you ask them what's the probability there is an elephant outside and you say it's 50% why because it's either there or not there but it turns out that this stupid principle is actually well mathematically formulated and this this is example of principle of indifference which tells us that if we know nothing about our system that's what we should apply but then we can try to bias this principle a little bit so like with the elephant you can ask like where you are if you're in India it's one ends if you are somewhere north pole and another ends right so same here you start if you apply this principle to physics you're just saying that without any extra knowledge we just assign equal probabilities to all outcomes and this is actually the principle of maximal entropy and we know that it works so in this particular situation it actually this principle leads to micro canonical ensemble because we know that energy is conserved so we know that for a given particle if we look into time average probability distribution it should always have the same energy and then we'll say well let's assign probabilities randomly within this energy shell and this is your micro canonical ensemble so it sounds like a stupid principle but it actually works very well so let's see it through more like refined argument why this is expected answer and I learned it first when I was a student myself from the book of Landau-Liffchitz where it was kind of written as obvious but you know all subtle things there I stated is obvious so anyway so I'll try to go through the proof and maybe I'll ask someone of course students on there to say where the mistake in this proof is so let me not mistake subtlety I mean it's not the wrong place mathematically it's wrong of course but physically it's it's not entirely wrong so let's imagine that we start from a given initial condition and then we evolve our particle and time according to say Newton's or Hamiltonian's equations of motion right and then we formally define stationary distribution as time average of my instantaneous distribution at any given moment my distribution is a delta function because particle always has a well-defined position right so then I do time average of course I can introduce some broadening into delta function so on but this is not not the issue now this is the argument which I learned first from Landau-Liffchitz so the stationary distribution and this is indeed very easy to show that if you do this time averaging this probability becomes stationary since it stops changing in time you can imagine it if you have bounded motion and you start doing photographs after a while and then you average over these photographs after a while you'll go to some stationary profile it's very easy to show when t goes to infinity I think I forgot to divide by t it's anyway time average if time goes to infinity this distribution is stationary it stops changing in time which mathematically means that it has vanishing Poisson bracket with Hamiltonian right in quantum language vanishing commutative that's what stationary means well because it has a vanishing Poisson bracket the argument says that it should be function only of conserved quantities right but in this example I'm assuming there are no conservation laws no symmetries except for energy so it should be a function of h but h is the same as energy because I consider particle at one point right it's always the same so the only distribution I can get is micro canonical right because energy is fixed so from this argument I derived this micro canonical distribution right at the particle as I already mentioned several times a zipple probability to be at any point and this is like principle of maximum ignorance and so on and then I ask questions only to students not to postdocs or faculty so where's what's wrong with this argument as if if if it would be right and I want to say it's not completely wrong it's actually works in many many instances but there is something missing now step number one we just say let's consider time average yes yes yes I understand but so far you see I didn't talk about chaos or not chaos I just the argument is that I start at one point and I do time average of my distribution I don't really know what equations of motion are with a delete to chaos or not of course at the end it's going to be important but for now I'm just asking where's the mistake in the argument so I do time average so basically I think about this I do many many photographs and plot a histogram where I see the particle right so this is mathematically time average of this delta function right then I'm saying that this distribution if time goes to infinity it stops changing in time and this you can prove mathematically for any bounded motion the proof is it actually very simple so in the sense that if you take this distribution and evolve x and p according to time this distribution will be the same it's time transition because it doesn't change in time it should be a function of only conserved quantities and the only conserved quantities h and then we arrived at this is a micro canonical distribution yeah yes yes you're right yes this can happen but suppose I'm in simpler suppose I have situation like this yes there could be subtle situations indeed that I can never go to the other side but then at least this argument will tell you that you'll be micro canonical within connected phase space right so you talk about the situation when your phase space is disconnected so but even this is still not correct not always correct let me correct myself I'm just assuming that energy is conserved and I'm arriving to the result that it should be with the same energy it's actually very subtle I I know like many people say that this is a proof uh so I actually I'll get to this question don't worry if you have any more ideas just let me know so now you kind of have two if you want conflicting ideas I loosely saying Newton and then Laplace but of course there was no real argument between these people about this issue but this is sort of an argument between I would say textbooks or like our you know colloquial understanding of what's going on so on the one side we have determinism right so whatever the potential is I will show some pictures for this particular potential so you have x squared y squared non-linearity so we can solve equations of motion and they have unique solution for a given initial condition right so and then we can find what exactly particle is doing and then we can basically determine so our distribution time averaged or not is actually always unique right on the other hand we have another argument just ignore equations of motion they're complicated they're hard to solve just use energy conservation and assign probability distribution so let's try to do like numerical simulation so we take this potential and we'll see what's going on so while you see a picture I'll explain what's plotted so this is an equal energy surface because I have two particles my phase space is four-dimensional I don't know I don't have a software which makes four-dimensional pictures so it's a two-dimensional motion and what's plotted is equal energy surface so it's basically a surface which particle can never cross right it's where momentum becomes zero both x and y right so then I initialize the particle with zero velocity at one of these points and then well it's just computer solves the equations of motion and this is for sorry for a small font it's like a small non-linearity 0.5 and if you look very carefully well my distribution is not becoming micro canonical you might say well maybe I don't wait long enough but this is happening for as long as you can simulate sure there are good reasons to believe that this will continue forever so it looks like you know wins and then you can imagine you can do the same game I was doing in 1d you can solve first linear equations of motion for harmonic oscillator then correct them then maybe correct them more and so on now let's try to increase non-linearity it's the same system but now non-linearity 4 and then that's exactly something which mentioned what you start to see that the motion becomes totally unpredictable so just from this picture you know that there is no way some analytic function will tell us what's going on the motion is completely crazy moreover you just see that this motion starts looking like ergodic right my particles kind of go everywhere so yeah I forgot to say these two points are exactly two nearby initial conditions the conditions which are displaced very little and moreover you can see that even if you go to machine precision you will get the same story it just will be delayed by very little in time so and this is the answer to the question which I asked so in the left picture well it's a partial answer it's the story is much more interesting more complicated but the partial answer is that the left picture is constrained by emergent conservation law basically perturbatively addressed you can think about this that at epsilon equals to zero there are two conservation laws so say angular momentum you can say two energies so you can say total energy and angular momentum right the system is rotational and then if you try to perturbatively correct you will see that your second conservation law will not disappear completely it will be modified yes yes I will come to that yes yes yes yeah exactly but let's see how it works so and I'll try to develop stationary perturbation theory for classical systems so let me take I'll start from one dimensional example again so and then let me do the following trick I will introduce complex phase space variables instead of x and p it's a kind of suggestive from this picture I will say that p is like imaginary part of some complex number and x is a real part of course if you remember a quantum mechanics these are analogs of creation annihilation operators but there is no h bar and they're not operators they're just complex numbers a and a star so I need to introduce these factors square root of a omega and square root one over m omega just to make dimensions the same right because if I went to want to add x and ip I need to make sure they have the same dimensionality and factors of two are just from convenience again apart from h bar these are your creation annihilation operators but these are my complex variables and then if I do reverse transformation then essentially I will say that x will be real part of a and p is imaginary part of it right so obviously I don't lose any information I have same number of variables now I can ask what are the Poisson brackets using these variables I know hopefully I assume that everyone knows it so in xp it's like d by dx d by dp minus opposite so now it's a very simple calculation I'm not doing it and by the way I want to apologize I will skip some derivations because otherwise I won't go too far so but for those who see it for the first time it's it's actually a good exercise so you can check that Poisson bracket in terms of these complex variables looks almost the same it's like this um it's q d symplectic derivative with respect to a in this star but there is extra factor of i and now if you are quick you can actually see that Poisson bracket between x and p in this way will be exactly one as you want right because for example dx da will be one over square root db dp da star will be plus i times square root of omega over 2 m omega will cancel you'll get one half and from this you'll get another one half so you'll get one so now uh hopefully remember that equations of motion for any function of x and p that d any function dt is Poisson bracket of this function with h it's again comes from if you have function of x and p not of time in a sense that I always have at each moment of time I'm looking into the same function x squared x plus ip whatever x power four then you can say it's like defunction dx dx dt plus defunction dp dp dt and then dx dt is dh dp and so on so you get the Poisson bracket so now I can ask uh what happens uh with my Hamiltonian and then uh again you probably all took quantum mechanics you know that Hamiltonian in terms of these variables is very simple omega star a there is no h bar because it doesn't appear in definition of a but it's the same thing so now if you uh uh check what equations of motion for a we'll see d ad t is omega a so my solution is just rotation and this is exactly rotation in the phase space so but now with this coordinates for harmonic oscillators I brought it to a circle so basically renormalized my axis that my motion is in circle spacing now let me go to my non-linear oscillator and then I look into non-linear term remember it's a plus a star with some factors and then if you take a plus a star power of four then you will get this right a star power of four plus this binomial coefficients and so on so so far I didn't do anything but now let me do what we do always in perturbation so we'll go to rotating frame so essentially I know that due to this first term my a rotates quickly so I will do this transformation a goes to a times e to the i omega t and again it's a very simple exercise to see that if you plug now this into your Hamiltonian equations of motion the first term will simply disappear but there is a price to pay and the price to pay that my a becomes now time dependent and now if we look into this perturbation we will see that all the terms oscillate in time except for this one because this contains same number of a star and a again if you're familiar with quantum harmonic oscillator it's number conserving term or if you want angular momentum conserving term and then we will say that okay so if these terms are small and they oscillate in time we can just ignore them in the first approximation and then actually you recovered this Hamiltonian which I mentioned in the earlier slides so it's really see it's one line calculation in this language so and this is a Hamiltonian obtained with in rotating wave approximation then you can actually go beyond that using like floccus theory I'm not going to talk about this I don't know if there are talks this but you can if you develop a proper appropriate floccus theory you will recover exactly perturbation theory for energies and so on and then this I'm saying without the proof but you can sort of believe me that in each order you will see that this rotating approximations if it can be written as a function of h naught a function of n so there is emergent emergent integrals of motion n which is a or h naught which is a star a and h rotate but here of course they are not independent because h rotating is itself the function of n so I well I just corrected my equations of motion but now let's try to do it for two dimensionals later which is harder so I do the same thing but now I have two pairs of complex variables for x1 and x2 so I used x1 x2 and here xy I apologize I was copying from different parts anyway I have x and y as an earlier slide and then if you do rotating wave approximation you will arrive to it's the same idea you will get to the Hamiltonian which of course has this nonperturbed part but also has a half perturbed part which only contains same number of a's and a's stars they could be a x and y for the same reason because everything else is lost later never and now if we stare this expression we will realize that it still has a conserved operate obviously h what Hamiltonian itself is conserved but also n total number of particles is conserved but now this Hamiltonian h rotation is not a function of n so this term is not a function of this number because it contains like a x star e y and so on so this is now a new conservation law which we obtained in first order of perturbation theory and now we can do simulations this is some epsilon 0.3 and we can see that indeed it works so if we look into nx and y we see they are not and without if the system is linear both an x and an y separately conserved right this is like energy x energy y but the moment we introduced non-linearity we see this conservation law breaks down actually very quickly so they start oscillating but now if I look into the sum of them it's actually has it's much better conserved it has very very small oscillations and actually I can keep going I can improve this if I use floc a theory I can improve this conservation law even more and I will get smaller oscillations so and again you can show that in any fixed order of perturbation theory you get two conservation laws yes exactly it's a canonical transformation yes using these variables exactly yeah there is nothing quantum about Bogolubov transformation absolutely which is number conserving in terms of new variables yeah Bogolubov transformation mixes a and a star but you still have canonical so Bogolubov transformation is canonical transformation which preserves now Poisson bracket between this a and a star which has to be i and this is actually unitary rotations so there is no quantum mechanics but all unitary rotations preserve this Poisson bracket yeah you're right yeah so I'm coming there so each order of perturbation theory does not care about non-linearity so in each fixed order you actually get a second conservation law and I like this example because we know we just discussed if you have two conservation laws you cannot have chaos right so and then we learned something very interesting from this example it's it's because we just saw a picture that chaos can happen right we know it can happen and it turns out that this expansion to almost always is symptotic in a sense they are not convergent have zero strictly zero radius of convergence and I don't know exactly about this problem but most certainly it also has a zero radius of convergence yes yes so then I do the following trick I I remove this by going to time dependent ax and ay so my ax and ay oscillate in time and then you can see when you differentiate maybe we'll just write down it's very easy to see I don't see any chalk so suppose you introduce a tilde which is like a times e to the i omega t I hope my signs are right if they're wrong then maybe minus i omega t it should be and then I have i d a tilde dt right so I want to find what it is and then it will be i d a dt thanks e to the minus i omega t plus omega d a tilde dt right so sorry but omega a tilde I apologize so but this term will appear also in the Hamiltonian when they have d h d a star right so this term will cancel and if you remember your quantum perturbation theory that's exactly what you're doing you're just doing transformation such that without perturbation your well there it's coefficient of expansion of wave function doesn't oscillate so once you do it you see that you effectively have Hamiltonian without the first term right I reinserted it back because it corrects the energy but sort of it doesn't appear on equations of motion there are some subtleties between frame transformations but let's say it's not there just pretend that it's not there so now you have a Hamiltonian which is purely non-linear right it's quartic and a star and it's small in a sense it's proportional to epsilon but on top of that some of the terms are oscillating so you have and they're fast oscillating right because now in this rotating frame dynamics is set by epsilon omega is gone right so this omega is actually fast frequency it's much faster than dynamics I know it's much bigger than epsilon right so then what you can say in here I'm using words but there is flock your theory periodic theory of periodically driven systems behind so there is actually a very nice problem about capitsa pendulum in in Landau Lyft its motion fast oscillating field so they go go through this in detail so what you can do in the first approximation just ignore fast oscillations and if you ignore it only this term remains if you do next order of expansion you will kind of get various commutators between remaining terms but you will in each order you will see that number of a stars and a's will be always the same if you do correct high frequency expansion so this n should be conserved but there are some caveats this n you have to correct for your variables so this anyway n will be modified as well let me put it this way so it's not that a star a plus whatever this n is conserved to you'll get corrections like you're getting correction to h rotating but you also get corrections to n but you will get conservation law in each order so all terms which you see appear as a function of epsilon or omega squared actually there's a good exercise if you do quantum perturbation theory so you just start from some state with some energy and just find leading order correction to this term you'll actually recover this expression so this is literally identical to stationary perturbation theory in quantum mechanics but you are not doing it for individual eigenstates but for the whole Hamiltonian answer your question okay so in other standard example of chaos is kick throtter so and this is known as standard map so we basically have just free motion of the particle and periodically turn on gravitational field because i find potential so in the equations of motion extremely simple so between the periods i don't have any potential so i have a free motion so it means that p is a constant and phi gets is increased proportionally to its velocity or momentum in this case and at this moment because you have a delta function kick it so fast and coordinate doesn't have time to change but momentum get a kick right it's very easy to see and this is called like a standard map and this is what's going to happen according to scolopedia so someone else a long time ago did these calculations and these are the phase space portraits for the system for different values of k or different values of the period and then you see a very similar story so if kick strength is small so basically each time you add momentum a little bit it's almost like a continuum for system so instead of discrete equations of motion you can write continuous equations of motion the energy conserving and so on and then you just see you have very very nice orbits and as you increase k then you start to get a mess and eventually it's again becomes basically ergodic but you actually see that expansion which i was sort of advocating or when i applied to this system cannot be exact because if you have exact conservation law we cannot get chaos but even here for the small kick you start to see like a bit of noise here the systems become chaotic and here you can see it even more visibly it's not ergodic but yet it's chaotic and this tells us that this construction cannot be exact so we can maybe get some approximate conservation laws but not exact conservation law and there is a famous theorem in mathematics which i honestly i don't really understand so the proof is far too complicated for me but this is sort of the statement it's kamogorov-arnold mozer a very famous which basically tells us that in classical finite-dimensional classical systems actually ergodicity under some conditions that is no degeneracy so on actually my previous example has degeneracy so it doesn't belong to the theorem but still it says that under some mathematical conditions ergodicity doesn't happen right away so basically a big perturbation preserves almost all tory so i just showed in integrable systems you have the motion along the tory so and this is like actually video i found this is vadimir arnold like and i like to tell the story so this is kapitsa pendulum and the story is that he was in in in committee for a olympiad high school olympiad in in soviet union and the head of this committee was kapitsa and kapitsa was yeah i had and he was deputy of course arnold was responsible for mass and then kapitsa told him about this funny story that for you know his experiment he had to solve this problem that if you have an oscillating start shaking it like in this picture then you can stabilize upside down motion and arnold said that he didn't believe kapitsa because he saw that there's no theorem in mathematics which tells you that this motion should be stable and according to his words on the way to home he figured out that actually their own theorem which they developed might apply to this and then he worked it out and actually found that indeed the theorem applied and they can prove the stability of this motion and then he was very challenged by kapitsa kapitsa used swiss you sorry uh this zinger and suing machine to to to uh demonstrated and then arnold made his own experiment with a razor he even says that first time it failed and he redid all these calculations and figured out that his pencil was too long and then he cut it by four centimeters and then it worked anyway but coming back to what i was saying so you see the same story you start to see like if you start shaking the face space portrait becomes really mixed so now back to the studio example we see that uh if you look into probability distribution just by counting again how many times point visits a particular particle visits a particular point in face space now it's a coordinate representation you can see that at this large non-linearity i guess the same number four which i showed before you can show that it approaches ergodic distribution so it's very easy to check that micro canonical ensemble in 2d is very special distribution uh of of coordinates as a sata function in a sense it's always the same where the motion is allowed it's kind of property of two dimensions so we see that chaos leads to ergodicity okay so now after this long introduction let me go to quantum systems so what do we do so there is no notion of trajectories so we cannot define in a lepunov exponents of how sensitive trajectories are and it's actually a long quest how you properly define chaos and there is also long history longer term so let me mention uh some not in historical order but some of them so one possibility is that let's try to define an object which is sensitive to lepunov exponents in the classical limit so but this object should be defined quantum mechanically so i cannot say distance between trajectories right because i don't know what trajectory is and then the ideas came actually from early work of larkin and afchinnikov and they go under the name of otoc out of time order correlation function and the original idea was yeah and then basically there was a lot of activity uh starting from kitai from all the centerworks and so on i'm not really going into that direction but let me still mention what these objects are so they had an idea that let's try to look into a strange object of commutator squared of some observable at time t and time zero i just chose momentum randomly it could be any observable so and then if you open the brackets commutator squared then it's it's an easy exercise you'll get four terms and they just want to highlight that these terms appear in kind of strange order in time so that's the name like in particular get t then zero t and zero so and if you try to write on a five-man pass integral approach you'll see that time increases decreases increases decreases and there is no way that can time can increase so you cannot really rearrange your uh quantum and because of this this correlation function don't have causal representation so they don't appear anywhere in linear response so you know that usually correlation functions appear as dissipation kubos acceptability non-linear susceptibility and so on so this type of fun of correlation functions don't appear but they are sensitive to liapon of exponent why because in the classical limit we know that commutator becomes a Poisson bracket up to factor of h bar and you remember what's the Poisson bracket between two functions if one of them is p it's just d first dx d second dp which is zero minus opposite yes oh this is some initial state yeah you can so if you want to have classical analogy actually this is an important thing you want psi zero to be localized in phase space point so this could be say coherent state if you want to be it's like analog of phase space point right so basically here the logic that you take an object which is which has well defined classical limit and which is sensitive to this liapon of exponent and then you see why it should be sensitive to this liapon of exponents because what is dp dt over dx zero this is precisely how momentum at time t will change if you have infinitesimal change in coordinate at time zero right so we know that in the classical limit it should diverge with twice the liapon of exponent twice because it's squared sorry can you speak oh yeah that's a good point so this is average of initial probability distribution so if you want a semi-classical limit it would be Wigner function corresponding to psi naught this is the best classical analog right so normally for classical systems we take one trajectory and there is no average but if I want to take here classical limit I have to put a bar because what will happen is average of a distribution Wigner function corresponds to psi naught and if I have coherent state it's almost phase space point right okay so yeah oh because if you take a linear it's a good question if you don't square the commutator on average you'll get zero because when you start usual commutator will appear in kubos susceptibility and kubos susceptibility is a never big and the reason is that if you start from even narrow probability distribution you'll get lots of constellations you'll have some trajectories which will go one way some another and when you average there will be nothing exponential square means that you always have positive numbers you average over positive numbers yes that's a very good question oh I think I missed something yeah thanks for asking questions so it turns out and this was figured out by Boris Fine first that actually you can get similar sensitivity to what you see an actually mathematical equivalent object if you kind of study echo so basically if you start in his work which I'll show briefly you start from some say magnetized state so state with magnetization you evolve forward in time you do small perturbation evolve backward in time like it reverse Hamiltonian and then you compare magnetization you got with initial magnetization and it's intuitively clear if you are chaotic you should it should disappear you start from magnet magnetized state in situation where in equilibrium there is no magnetization anyway so what you can show after some work is that this echo is equivalent to to this Poisson-Berkin square so this is physical but it also shows it's a bit weird right so you need in order to measure it you need to reverse time and this is an example from a paper by Viktor Belitsky group 2017 who studied same kick throtter and they introduced very small h bar and they basically studied this object and compared with classical there are many curves I'm not really going into details but let me just say so what's plotted a Lyapunov exponents one obtained from this out-of-time order correlation function one in a standard way so these are basically red circles and triangles there is a small delay between them but this is not even quantum effect because this is like averaging of exponent is not the same as exponent of the average there is small difference between them anyway so they agree in the small h bar limit so everything is good so maybe we can use this as a definition of quantum chaos but there is a problem and I think again in the same paper I mentioned by Boris Fine they to be big surprise to themselves found that if you consider truly quantum model so not the model which doesn't have classical limits so I'll introduce those later but think about spin one half spin chain right so spin one half is always quantum it has just two states even if you have 10 spin one half so 100 spin one half it's still not a classical system because there is no obvious microscopic classical description right so and then if you take these two systems he didn't see some sensitivity so here I am showing precisely it's it's from their plot precisely this deviation of magnetization from initial value which is this echo which should also go with twice the level of exponent so it's basically ot oc and this is for classical spins and then you just see as a function of time it's exponential so you just see you have linear scale and time and logarithmic or exponential scale in in this deviation but now they did the same numerical experiment for a quantum system and this is nothing exponential it's not just curve but look into numbers so numbers barely change right so here you have many orders of magnitude and here well let's just say one order but over much longer time so and then later there was a proof actually by by Tamash here in his group that if you have local systems with local interactions and local Hilbert space dimension so it's like spin one half so then you have at most polynomial growth so this idea just doesn't work and I don't think people found a way around no it's even not that so Holstein you can use Holstein Primakov that the problem is that you cannot replace commutators with Poisson brackets in your equations of motion if you take large s-limit then it will work for the time which scales as log s so actually you need very very big s to see exponential behavior because this time it's called Erenfest time is actually very small so but beyond that you just cannot there is not enough room because at short times you always have some perturbation and loosely speaking short times you already have large quantum fluctuations and at long times you kind of reach one so there is no room for exponential growth and classically there is a room because you can start from something very very localized yes so there are exceptions I'm not going like syk model and so on but at least my understanding even for the it's it still has large n but my at least we tested it for like infinite temperature states it's still the same so it seems that it is I don't know a single counter example when you have exponential behavior which is not described by saddle point and saddle point equations always have Hamiltonian structure so basically write any pass integral you have some large n parameter large n large s small h power whatever saddle point parameter so if you take saddle point equations you will get classical equations in a sense you'll get Poisson brackets and they always describe this exponential growth I don't know counter examples but I also don't know proof maybe Tamash knows I don't know proof which mathematically says that it should be the case in in space I guess Tamash is here but nearest neighbors are local so I don't know how fast they should decay exponentially decaying is probably local you need some Lib Robinson bound so I think people argue that Lib Robinson apply for power power-law interactions with power bigger than something but depending on dimensionality but I don't remember yeah sorry can you speak about Lojmitecho it's also like in short that people tried it but somehow it it cannot be used to distinguish chaos there are many other probes I will talk also a little bit about operator growth maybe but it's also didn't work I don't know if I'll have time so people try it I'll come to that because of course I hope we have at least some positive answer to the question how you can unite quantum and classical chaos in a single framework so hopefully I will reach maybe tomorrow at that point yeah all right or talk where correlation function yes well childish formula is just pass integral representation of this correlation function but yes formally just think about it so you start from some state it's actually better to be the disigons this is eigenstate of your operator which you are measure say monetization for various reasons close to eigenstate then you evolve it in time up to time T and then do you do infinitesimal unitary with say perturbation B then you reverse Hamiltonian and come back and now you have two operators what you measure and with what you perturb and it turns out that in this case when your unitary rotation is small so you can expand it so suppose epsilon is rotation angle so if you expand in epsilon then this will map exactly two out of time order correlation of between these operators M and B with what you measure and what you perturb yeah and then you can do kelvish contour and so on okay so we I would say as community generally failed with chaos quantum chaos I just wrote one example but how about thermalization or ergodicity so and they are actually the progress was much better so in starting from 90s so let me again start from basic so now I give up the idea of looking into trajectories how chaotic and so on but I want to ask after a long time will my system be described by thermal equilibrium or not and how I can reconcile it with quantum mechanics and actually there were many years of frustration when people thought about this problem from the early days of quantum mechanics like von Neumann was probably one of the first people who started thinking about this and the apparent paradox is here so our evolution quote-unquote is linear I kind of really don't like this language because it's linear because we use Schrodinger representation but let me not go into these details so in a sense I can take any wave function initial state expand in the eigen states of the Hamiltonian and then my motion is very simple just the oscillating time right so I have many harmonic oscillators now let's look into some observable and look into expectation value of this observable and then well I will get the double sum and then I will get of course this oscillating terms and matrix elements of the variable and this is called density matrix this object and now I will ask you know in weak sense I will ask whether my time average will formalize exactly I was asking in the beginning and then while time average I was assumed that I have generic system with no degeneracies and so on then all oscillating terms will average to zero right and then what I will see that out of this sum I will get only diagonal elements of Rovich remain and then it's actually for Neumann as I mentioned was first to realize that if you want to reconcile this language quantum language with which I can describe also classical systems right they have complicated states what many then somehow ergodicity should be encoded in the structure of eigen states because you see if you don't say it there is like a problem because these diagonal elements of Rovich are just basically mod c and squared they don't depend on time so if I want my distribution to appear Gibbs distribution there is some problem or like micro canonical because these probabilities these are probabilities to occupy energy states they are time independent and there was a long history behind I'll mention some of it but essentially the answer is that the eigen states encode all this information so let me again start from some simple examples to to to illustrate how it works so if you go to same one-dimensional oscillate I started from then I know that my stationary states are described by WKB so this is basically their structure and if you remember WKB if you average over this oscillations this p of x remember there is one over square root of p appearing in square root so one over p momentum so this is exactly this micro canonical ensemble so and this is a comparison between micro canonical ensemble and quantum state so if we average a little bit of oscillations again for many reasons like we have more than one state or we have slightly different masses slightly different anything these oscillations will average it will just reproduce micro canonical so and again this situation with stationary states is kind of simple in in all integrable models we can basically develop WKB in h direction but in chaotic systems we have nasty orbits so from this we kind of conclude there should be nasty eigen states so let's just see where the problems are so suppose we have like two-dimensional potential now and we we need to solve a simple Laplace equation say if we have a billiard then my wave function should vanish at the boundary right so and actually people for many years tried to do it and they were failing except for some special situations and then actually it's it's serve Michael Berry who came up with a famous conjecture that the Wigner function which is basically a wave function squared in some sense this is an analog of probability distribution p of xp for quantum eigen states approaches basically random superposition of plane waves so the first random appeared here so and this is like numerical experiment also take anything from Wikipedia so this is one of chaotic cardio billiards and this is exact eigen state so this is I forgot is a real part or absolute value of squared probably real part of wave function and this is random superposition of plane waves roughly with the same energy so you pick up plane waves from the micro canonical shell and of course the patterns are not the same because left is random this is not random but if you look visually they look pretty similar so separately the word developments by by Wigner Dyson who sought about totally different systems not classical chaos at all but but some spectrum of nuclei and they like did from some experimental observations they eventually came up with is an idea that maybe this nuclei described by random matrices so if you look into levels there is also a long interesting story behind but I think I will not cover anything if I go into details let me just say in words that the main feature of this random matrix theory or at least one of the main features is that probability of level spacing being small is approaches zero and the reason is that you can sort of see it from this picture if you have two by two block and if these guys are random then in order to energy difference to be zero you need most real diagonal part to be zero and off diagonal part to be zero and moreover if off diagonal part is complex then it's even harder because you want both real part and imaginary to be zero so you get more repulsion if you have complex Hamiltonians and this should be contrasted with sort of naive expectation that we have random energy levels if you have random energy levels it's sort of like mosquitoes in the tent you get Poisson distribution and which means that probability of level spacing is exponential so and there are two powerful I mean people at the end connected all these ideas and then there were two powerful conjectures one it has a name of Berry-Tubber conjecture about generic integrable systems who say that our distributions precisely Poisson so if I take basically generic means I don't have special degeneracies between energies so they're integrable but they don't have some extra symmetries so in the conjecture that your energy states described by Poisson statistics and this BGS by Higa, Giannone and Schmidt conjecture who kind of generalized this Berry conjecture and they said that for chaotic systems probably should say ergodic systems the energy levels are described by random matrix statistics basically show how it works so here's an example so we have just square roll potential but these incommensurate walls so there is no rational number relation between x and y lengths and then if you look into level spacing distribution so basically you measure your numerically measure your energy levels you look into what's the distance between levels and then you plot the histogram right you actually get very good approximation to the Poisson statistics and if you consider a billiard in this case Sinai billiard this is original work by by Bajigas then you will see a random matrix ensemble so you see that probability of having very small level spacing is actually suppressed and then of course numerics got better so people studied various examples and they found that this random matrix statistics for chaotic billiards works extremely well so the better numerics the better the results so even though this is a conjecture but I think it's now completely accepted so and this is kind of surprising result at least on the first site because your original Laplace equation it doesn't have any randomness so and then I guess I will finish with some overview for examples finish this lecture so there were many many tests of this conjecture so this is a regional heavy nuclear experiment so these are level spacings between different nuclei I forgot exactly which but there are like more almost 2000 spacings and they actually this Wigner-Dyson distribution worked very well then there is a hydrogen in a strong magnetic field we all know hydrogen atom if you have small magnetic field there is a magnet effect and so on but once we introduce strong magnetic field actually the system becomes chaotic ergodic and you approach if you go to high and higher in this unit you go basically to high and high energies closer and closer to unbound state you actually get better and better description in terms of this Wigner-Dyson distribution this is a funny example which I'll skip to save time and then actually and after like you know 2008 people started checking many particle quantum spin systems and this is one example from work of Lies Santens and Gubin who considered purely quantum system so it's a spin chain in this case it's integral x-axis chain with a magnetic field in the center and then they looked into level statistics as they increase magnetic field in the center which makes the model chaotic or ergodic and they also found that if this field is zero model is integrable you're close to Poisson statistics just look into the same quantity level spacing but if it gets stronger you are closer to Wigner-Dyson statistics so this probe seems to work very well and there were many many other tests and other numerical systems and mostly of course one-dimensional spin chains because that's what one can do numerically and this is this always works at least when integrability breaking is strong and and this is considered this emergence of random matrix statistics considered a standard definition of quantum chaos but again I would rather say it's quantum ergodicity there is also very interesting relation to prime numbers and zeroes of Riemann zeta function but again because you have five minutes left I probably skip this it's very interesting story behind but anyway so prime numbers have something to do with random matrix statistics let me put it this way so another manifestation of random matrix same thing instead of level spacing you use so-called spectral form factor and again like Tamash is an expert he used this measure and many other people too for other things so this looks a bit so it doesn't the advantage of this that and it's it only uses energies doesn't really use eigenstates and there is some measure accumulative measure of energies and basically we define sort of a partition function sum of all energy states of this exponents and then you take a square of this function you can introduce imaginary part if you want to weigh your energies with some temperature so some low energy states could be the weight and so on but you can say that this bit is zero so you just look into z squared and then if you carefully go through random matrix prediction so you will see that if you average this again over ensemble of random matrices it will have this function will increase which is a bit surprising because if you think about it your average the z of t and you get many oscillating terms and usually you think that if time gets bigger they oscillate more and more and then they deface but actually random matrices have this long range correlations between energies and because of this long range correlations this logic is not entirely true and so what happens is that after this initial intuitive decay of this you'll get a linear ramp and this linear ramp is also taking as manifestation of random matrix statistics so oh these are different times yeah this uh this is particularly mean time scales and usually let me put it there is always a short time which depends on the model so this I just stole picture from internet it's from my syk model and intentionally didn't want to go into details it will bring me very far but essentially you have a short time scale uh at which everything decays but then there is a typically it's called tau less time it's tau less time it's basically when if you think about frequency space or energy difference space your Hamiltonian starts behaving as a random matrix and after that time so for this model it's of the order of one because it's not local but for local models this number this time this I will come to that so this is typically a diffusive time so it's l squared over d it's time when you start feeling boundary so and after this time you have basically unlimited ramp until you reach what's known as Heisenberg time which is inverse level spacing so if in thermodynamic limit uh if this time is finite or at least this time at most polynomial you can say it lasts forever but you have to be careful so this value is still much smaller than this value so it's not like experimental probe but it's good numerical probe okay see I have two minutes left yeah I think it's probably a good point to stop here so that we are not late for the break questions yeah it's a Hilbert space and you it depends how we define it it's whether you normalize it to one it equals to zero d squared so in this way it's normalized to one when time is equal to zero so according to this different distributions of levels you can see that your quantum system is q-take or it is non-q-take like maybe localized if you take say a mean field approximation to your quantum system this now gets back to a classical system will you be getting the signature of classical cues in some way um not necessarily because uh you know mean field approximation can be for example you know one-dimensional right if your system is symmetric and so on uh and uh overall I think this is very interesting question and to which level so mean field approximations kind of lose information and they can lose information about chaos so it now depends so mean field again we are talking more about dynamics so mean field is uh usually classical approximation so uh in a sense that again you you you have few degrees of freedom which are described by Sanon or Poisson brackets and so on but the situation could be even more subtle like for example you can take integrable spin chain and then you take this classical limit several point approximation and then you will get essentially equations describing classical spin chains with the same interactions and those turn out to be non-integral so and opposite is also true I think so this um transition between between quantum and classical chaos is more subtle so some models when you approach classical limit it's bounded to be the same right because if you say send h bar to zero your saddle point approximation uh becomes classical but also quantum chaos or authenticity is only defining the limit when h bar goes to zero because in a way this is a sympathetic statement so finite two by two matrix you cannot say it's chaotic or not right so you need more and more levels to say that as you increase Hilbert space size you approach say vignodizing distribution and if you approach classical limit at the same time when you increase Hilbert space size typically this equivalent statements so you get chaos in both so ergodicity in both so not causing uh but in other situations when I guess mean field is less justified uh this is not the case but overall it could be very subtle actually well here ballistic I I wouldn't say so I don't know this model that well so ballistic is the order of one when you feel the boundary but this is like completely non-local model right so in in local models ballistic will be like L and uh so if you want this is time when information spreads so you know entropy spread but then they also call it diffusive so here I actually don't know was it and so in local systems it's n squared and usually uh this is uh uh when uh I know your conserved quantities start to feel boundary again what is it in syk model maybe there are experts in the room who know so these regions are specific to this syk model like yeah for this for normal systems you will see this so ballistic will be like of the order of n and and diffusive will be of the order of n squared if our system is not ergo did uh are we allowed to use renormalization group to find the fixed point of our system renormalization group for what for equilibrium to find the fixed points of our system fixed points in what sense so very often we talk about fixed points in equilibrium so if you talk about yeah in equilibrium actually chaos is much less important because you just assume you have equilibrium assume that there is a partition function uh and so on if you talk about dynamical fixed points then it's sort of an open field so people talk about resermalization as sort of fixed points like generalized Gibbs ensembles and so on so in in this sense yes uh you can use this notion but I don't think it's defined in in our g sense like vigorously so there's some words behind that um your first papers by by burgers and others that your first flow to like a stable distribution which is fixed point and numerically always you often see like it's fixed point if you look at the Fermi-Pasta-Ulam problem after long time it goes to some strange distribution which is localized in time basically forever and eventually people believe it's normalizes and it's very particular distribution I have no idea why it's stable and maybe there are experts who know it but I'm not aware of any you know mathematical renormalization group ideas which will predict what this way you can say Kolmogorov turbulence is an example of this point but again there are some considerations why this is stable I am not aware of like rg type equations which will tell us that you you go to this no I'll try to get there I mean there were many confusions uh among us including myself like when we started and there was like very ignorant I don't want to use bad words but maybe ignorant approach like when you started looking into quantum chaos or good DCT just thinking that uh we can ignore like previous knowledge so there is uh I'm not going to talk about many body localization I don't know if there are lectures or about this but this is like an example of where like community failed so there were many irrespective of whether people believe it's localization not basically there was so so many mistakes like so many statements about that and part of the reason is because people were confusing chaos and their good DCT these are not the same things what happens that usually in thermodynamic limit you will always ergodic but if you look into finite system and come to that and you ask how you go from integrable to ergodic regimes you will go still through this phase which is chaotic but not ergodic and that's kind of the reason why I spent so much time discussing like you know chaos and ergodicity in classical systems or ergodicity no level level spacing distribution does not tell you about chaos it tells you about ergodicity but ergodicity comes with chaos so in this sense you can say yes it's a diagnostic of chaos and very often in literature people define chaos like this so if it's a definition then of course you can say level spacing distribution defines chaos so therefore systems are chaotic which have this but it's a pathology if you define chaos and I'll come to that as unpredictability then you can have perfect person statistics and yet you can be completely ecologic you know as I said it doesn't work I'll come to that either after the break or tomorrow and I don't want to use definition but this is basically the work we were doing for a long time and I'll try to say through basically adiabatic transformations and long time response so there classical or quantum again you can define this complexity and then you can distinguish chaos ergodicity and integrability but again I don't want to use its definition because it's not really accepted so some people might have different opinions but at least I will try to say how you can distinguish these three notions very well yeah but this conjecture I am BGS conjecture in a way is wrong about chaos because this is like a long problem if you take not billiards billiards are very special just take this model which I showed to you and look into level statistics you can break all the symmetries you can say it's symmetric and so on there is no way you can get vignetized not even close so BGS conjecture or eth is is sufficient condition for like chaos because you are unpredictable you can say it's ergodicity but it's also unpredictability you have word random you cannot predict eigenstates you can say yes this is sufficient condition for chaos but it's not necessary so billiards as I said they are very special so once you have fun on this mixed phase space or whatever you you can never get vignetized statistics and then I will mention that in the systems which are chaotic in any sense of this word you cannot really predict anything even if they are many body with weak integrability breaking they still don't have vignetized statistics so vignetized statistics it's really that's why I spent so much time on and I will talk a bit later about relation of this random matrix statistics to thermodynamics it turns out that once we take this as a conjecture and I will introduce eth very soon it's a generalization of BGS then we can recover all thermodynamics but already I mentioned that this stationary eigenstates appear because of time averaging right so like stationary they are analogs of time average trajectories if you think about this they're just quantized so when we talk about stationary eigenstates we talk about long time behavior so we talk about ergodicity and indeed from this BGS conjecture you can predict emerging thermodynamics but when you talk about chaos you have to talk about unpredictability this is not the same thing right should we continue or well I don't know well I guess yeah I mean we can post like we can start or maybe in private maybe so perhaps I suggest that shall we maybe start later okay so that they have a half an hour break okay and the break is like outside the terrace and the coffee break okay thank you anatoly