 OK. Now, I have the pleasure to introduce Professor Domokos Chash from the Hungarian Academy of Science. And he will talk about stochastic dynamics issued from a deterministic one. Thank you very much. And thank you very much for the invitation to this very interesting OK. So, thank you very much for this very interesting event. I'm enjoying both the school and the conference. So, this is the title of my tokens, my co-authors are there. So, the goal is to derive the heat equation from microscopic principles. So, this is the well-known heat equation, where there is a constancy, which is a specific heat. And kappa is the thermal conductivity. And kappa depends on the temperature. And for different materials, the dependence on the temperature is different. For a wide class of models, the dependence is square root of the temperature. For example, there are some examples, insulating materials, the guzzes of weakly or rarely interacting particles. Now, Bonneto Lebovic and Ray Biele wrote a survey about the progress in the derivation of the heat equation. And they were emphasizing that it would be necessary to add interaction between the moving particles, instead of points make them little balls. In fact, there are two competing types of models where one can, where people, mathematicians and physicists try to derive the heat equation from microscopic principles, billiard-like models and nonlinear oscillators, interacting nonlinear oscillators. In the mathematical theory for nonlinear oscillators, to treat the model, one always has to introduce some small stochastic noise. For billiard models, there is a hope that one can avoid this, one can really address the deterministic evolution. Now, there have been results where one could obtain the diffusion equation, the Brownian motion from microscopic principles, well-known in the Lorentz process, Bunimovich and Sinai and later Bunimovich and Chernov Sinai. But there was always only one particle and no interacting particles. So, Gaspar and Gilbert, two badger physicists in 2008, were suggesting a model of localized hard disk or hard spheres, and my talk will be very strongly related to their method. So, what they were doing, that they introduced this model of localized hard disks, I will of course explain what they had in mind, and the approach was the following. So, you start from the microscopic kinetic equation, the full description of the system, this is Hamiltonian model of course, and then from the microscopic model, you take a rare interaction limit, and in this rare but strong interaction limit, they obtain a stochastic process, which is a generator of a Markov jump process. The kinetic equation is related to the full phase description, all the positions and velocities of the particles. In the Markov jump process, you lose a lot of information. The equation, the stochastic process, is only related to the energies of the particles, that's a mesoscopic picture. And then, in the second step, from this Markov jump process, they hope to derive the microscopic heat equation. Now, the model is the following, the Gazpar-Gilbert model is the following. Well, on this picture, you see a chain of lengths too, but of course, you can imagine arbitrary lengths. This is a quasi linear model, so a chain of particles. So, the shaded domains, they are fixed periodic scatterers. And the white disks, white circles, they are the moving particles, and I will explain it later more precisely, parameters of the model are determined in such a way that these disks can interact with each other, neighboring disks, but they cannot leave their cells, so in this sense, they are localized. This simplifies the situation. Well, the model has even physically some meaning, so in this physical paper, they were just using a model, so a situation where there was no mass transport, only energy transport. Okay, so this model is a semi dispersing billiard, I will be more precise again, so just a very brief introduction into the notions, which I am going to use. So this is the well-known Sinai billiard, so you are on the torus, in this case, you have one scatterer, and the dynamics is uniform motion, and specular elastic reflection, and then, of course, you take into account the geometry of the torus. So this is a two-dimensional Sinai billiard. I will talk about the billiard flow and the so-called billiard ball map, so the billiard flow is the continuous time evolution, it has an invariant measure, the Ljubel measure, so the phase space is the billiard table in dimension d. It's a subset of the d-dimensional torus on the picture, it's two-dimensional, and for the velocities, you have the d minus one sphere, and the discrete map can be obtained in the following way, so there the phase space is the boundary of the billiard table, so in this case, the boundary of this scatterer, and again, the velocity space is the same. So well-known definitions, so a billiard is called dispersing or Sinai billiard, so if these scatterers, the fixed scatterers are strictly convex and semi dispersing if they are just convex. Now, the parameter choice of Gaspard and Gilbert was the following, so the box size is one, chain legs n on my picture it was two, then along the vertical axis, you take periodic boundary conditions for simplicity, and this is a notation, the radius of the fixed scatterers, which were the shaded circles, rho f of the moving this rho m, and there is a condition of confinement, very simple geometry, that the two radii add up more than one half, and also there is a condition that the two neighboring disc can interact at all, so there is a critical value when they can only touch each other, there is no interaction, and if the radius of the moving scatterer is a bit higher, then you already have really interaction. It's natural to introduce a small parameter, which is how much you are above the critical value for the radius of the moving disc, and what Gaspard and Gilbert were doing, they were keeping the sum of the two radii fixed, and then essentially the configuration spaces of this neighboring disc essentially depend only on rho, on the sum, and if you are just the critical, then you have a non-interacting billiard balls, so that's their model. Now, this is a sort of physical formulation of the kinetic equation, it was already derived, it's very simple, it was derived already in 1989 by Ernstendorfman, so this is the complete time evolution of the density in terms of the configuration points and the velocities of the particles, and then the time derivative is the sum of three terms. First of all, there is free motion, there is the wall collisions of each particle, and there are the binary collisions of neighboring particles. I took this equation because I will need the notation for the wall collision rate, which I denote by new wall epsilon, and the binary collision rate, which I denote by new bin epsilon, and oops, yeah. So, and when this epsilon, my small parameter tends to zero, there is a natural scale separation, namely the wall collision rate converges to some positive constant, of course, because you can imagine that the particles, this collide with the wall a lot of time before two neighboring walls collide, and if epsilon tends to zero, then the binary collision rate tends to zero. So, there are two scales, one for the wall collision and one for the binary collisions. Now, what Gaspar and Gilbert did in this limit transition, they derived a so-called master equation, so the generator of this Markov jump process, which I will be talking about, which was, as I said, a equation for the energies, the time evolution of the energies, and then they realized the hydrodynamic limits, so from this master equation, they obtained the coefficient, the heat equation, and they also obtained the coefficient of heat conductivity for this model, as I said, it was constant times square root of t. So, this is just a repetition, what I already said, that the mesoscopic equation is a Markov jump process. Just a small remark that, as to the second part, so the first part is dynamical, dynamical system problem. The second part is a probability problem from the Markov jump process, one has to derive the heat equation, taking the hydrodynamic limit in the sense of Varadan. The first starting step we made with Alex Grigu and Kostya Hanyin, we proved some bound for the spectral gap as n, the chain length increases, Sassada improved our result by having found optimal conditions for this spectral gap. Sorry. OK. Don't bother about it. And, OK, so now I return to the actual problem for part one from the dynamical system. So, that's of course a challenging question and there were at least two interesting results for that. First Dolgokat and Liverani introduced a model where the particles had each particle, it was a chain again, each particle had a nose of dynamics and the interaction was through a potential which was very weak. So weak interaction acting permanently and as a result they could derive the mesoscopic equation which was again a Markov process but it was a system of interacting differential equations not jump process because here the interaction is permanent. Another approach was by Keller and Liverani who were taking really already the rare interaction limit for a coupled map lattice of interval maps and they derived, they introduced the interaction in such a way that it was similar to this kind of hardcore interaction so that when the two particles had an interaction it was drastic, it was very strong and for this model they could derive the uniqueness of the SRB state and they could also prove exponential spacetime leak K. Now, what I would like to do I would like to give some idea of our approach what we have in mind. So the first is a reference to two earlier results so there is a classical result by Hirata Sossol and Vienti which is a rather general abstract result if you have a dynamical system which is mixing in a controlled way for example, alpha mixing and you have a sequence of nice subsets to avoid neighbors of periodic points and so on such that the measures tend to zero then the successive entrance times of the dynamics into this small set forms a Poisson process in the limit of course in the appropriate scaling so that's I think intuitively very clear a more quantitative formulation of this kind of theorem was two years ago given by Shazot and Collet so this is but I said it's an intuitive background now I return to the model itself and now I will only talk about the 2n equals 2k so this is a 4-dimensional billiard model the two centers are 2-dimensional so all together it's 4-dimensional model is a billiard it's easy to see and this model is isomorphic actually to a 4-dimensional semi dispersing billiard semi dispersing because the scatterers in the 4-dimensional space are cylinders with 2-dimensional bases and 2-dimensional generator spaces now of course it would be very nice to apply the theory of hyperbolic billiards if there were such a theory for this case so what Libertani Pellegrini so approved in 1992 that this model and the end chain and also a general class of these models are ergodic mixing Kolmogorov mixing even but no mixing rate was known so I remind you that in Hirata Sosova enti first condition was a sort of controlled rate of mixing this is what is definitely missing here now Peter Balint and Peter Todt also called Modjoro in 2008 they proved exponential correlation decay for higher-dimensional billiards for those you may know that for 2-dimensional case exponential decay of correlation was proved by Lai Sang Young but no results exist for higher-dimensional case so this result is the only one but it's only valid for the finite horizon case specific class I don't go into detail dispersing billiards and this theorem uses a so-called complexity hypothesis which is certainly a fundamental hypothesis in billiards theory again I will not use it in this now our approach is based on the following idea so there is a very efficient method for using some mark of property of billiards by Chernov and Dolgopiak and this method is called the standard bare method well where you want to show any statistical or stochistic property of billiards you use a sort of mark of method sometimes completed with some martingale ideas but the essential is the some mark of it there were mark of partitions mark of seas, mark of towers and in this evolution standard bare method which is probably not so widely known so I will just spend one slide about showing how this mark ofness appears in the standard method and what is the standard pair so the standard pair so this is a two-dimensional case so I am considering the billiard ball map and the standard pair is an unstable curve and a nice probability density on this unstable curve so that is the standard pair now what you have to imagine that you have this two-dimensional phase space which is one this kind of cylinder or several of them and you cover your whole phase space with a family of course uncountable family of standard pairs this is a proper family now if you take a standard pair from this family and you apply the dynamics I said that the standard pair first component is an unstable curve so for the dynamic axis that it increases there is a sort of exponential increase and also cutting so from this standard pair you get images a lot of other unstable curves and densities on it so this was our original and you get a lot now the first property which I show here that if you take a function on your phase space and you want to evaluate the expected value of this function after n iteration of the map then one learns at school probability so this expected value can be calculated in the following way you take all the images of this given standard pair so all the unstable curves and densities on them and calculate on all this L alpha n standard pair the expected value of your function and there are weights that are given by how the original weight decomposes into the smaller pieces so this is the low total probability and then the Markovity of the whole thing this is reflected in the growth lemma this says that there are some constants such that for every epsilon if so w this is the length of your unstable curve so if you weight sufficiently long for every n larger than this thing then the sum of weights corresponding to short standard pairs in the image oops, what happened? here short and this is very good because those which are short for the time being uniglect and and you know how much, what is the total weight of those uniglect so believe me that this is the main important the most important part of Markovity and then equation you have newborn standard pair forgetting about the past this is the Markovity actually now I want to show you what is the difference between Chernov dogopiat model where they applied this method very successfully the standard pairs and this problem which I am talking about so in Chernov in dogopiat case you have a disk of mass heavy disk this red one and you have scatterers again the shaded circle is on the torus it is a fixed scatterer and you also have a point particle and the two particles move and they have elastic reflection and this is a billiard what was helpful for them that if the mass of this moving disk is very large then the unstable the cones in general of the complete system are very close to the cones, invariant cones of the two dimensional billiard so for a short time this heavy disk almost does not move so for the small particle this is almost a senai billiard with two scatterers on the picture the blue one is the fixed scatterer and for a short time the red one is also a scatterer and you can use the nice stochastic properties of two dimensional billiards now in the Gaspar-Zirber model the problem is that the effect of collisions is very strong so that you cannot approximate the unstable cones of the Gaspar-Zirber model they are far from those of two dimensional senai billiard there is an additional problem which I will not address so this is I repeat that I am considering two moving disks in this Gaspar-Zirber model and we would like to do something now since there is no theory for four dimensional semi dispersing billiards therefore one has to do something different and in first step we simplified the model by taking a piston model and if I succeed I will show you piston so this is a piston model on the plane in our case we will only be interested in a quasi one dimensional chain you see that there are these piston particles which move on their interval here and there and there are these particles which occasionally interact with actually this movie was done by Thomas Gilbert one of the creators of this Gaspar-Zirber model so this is our model and it's a three dimensional billiard also semi dispersing and now this is a small technical problem that in this model the two pieces of the boundaries make a corner and if one wants then one can replace this model with another model where instead of this diamond you take the hyperbolic space and then octagon in it and take a billiard in it and then you can replace it also with a piston I don't go into detail later it will be clear why I mentioned this so this is again just one cell of this piston model so one diamond and one piston coupled to it so this is the diamond billiard phase space complete phase space for the diamond and the piston now so here I formulate the CRM I don't want to get lost in all the notations I just tell you that I am interested in this two particle model in the energy of the first particle of the billiard particle and this is what I denote by e then dynamics is phi corresponding to epsilon so this is just the energy of the billiard particle at time t under the flow and of course I have to rescale it because as epsilon 10 to 0 the interactions are more and more rare and of course since this region where there is interaction it has area epsilon proper scaling time is t over epsilon square and then the CRM says that for a wide class of initial lows the energy of the billiard particle in the appropriate scaling converges to a Markov jump process with some kernel and here is the kernel so it is calculated actually just for a short remark that this piston model so the diamond and it is actually a billiard in this three dimensional domain so you have this cylinder and for the interaction you cut out this lower corner which has height epsilon so this is the billiard and now I would like to talk about one part of the proof and this part of the proof tells about the following so we have standard pair and so Chernov had the following CRM for two dimensional billiards Lawrence process billiards for the same you take two functions which are nice so called generalized further functions one of them has mean zero then this correlation decays stretch exponentially with some constants which depends on so this is the stretch exponential decay now since we are applying we are applying standard pairs method so I want to emphasize that here the average is taken with respect to the invariant measure the Liouville measure since we are using standard per method we need a similar statement when the initial measure is singular is given on a standard pair so it's a curve probability density on it so this is our formulation of the CRM so you take a standard pair phi tilde is the measure given by the standard pair so it's in the complete phase based measure determined by this density on the standard pair and then you take the average of your function time t with respect to this measure and you have again a stretch exponential bound so this is our CRM and well I will tell you some words about the proof of the CRM to show that I was listening to the school part of the of the meeting that the method we are applying is actually Margulis sickening method and to show it how it works is the following that our phase space for the flow is three dimensional and here is the standard pair w u it's the curve and you have to imagine the probability density on it now since this is a singular measure you have to somehow make it smooth and this is what one does in the following way I mean you have the curve first you take for example in the center of the unstable curve a disc which is orthogonal for simplicity to this unstable curve in the center so this is the disc D and then first it's easy to shift your unstable curve to every point of D and then you get this nice picture which is on the on the figure however you have also to do something you also need in hyperbolic theory you need to work with the central stable manifolds they are two dimensional central stable stable manifolds and the flow direction this is the central stable and these are surfaces and they behave like they are on the figure so over since this is a singular dynamical system some central stable manifolds are very short close to the singularity they can be very short nevertheless we want to use this structure and we use this structure in the following way that well let me so this I said that D is a disc of radius epsilon yes and this I already said that these are the shifted variance and now let me focus on this so called Cs foliation first I take those points on my original unstable curve so the picture is like that that I had this original unstable curve in the center I took the disc of radius epsilon then I shifted this unstable in every direction and there are other central stable manifolds they can be short so my first job is to worry about the measure of the sufficiently long unstable curve so each is the set of those points of my original unstable curve where the central stable manifold crosses completely this U, U is the product of my unstable so of my unstable curve is this D and from classical theory we know that the measure of sufficiently long central stable curves is no sorry what I so so each is the set of good points and then the measure of the bad points is at most epsilon so this we know from classical billiard theory then the similar bound will hold on every on shifted versions of the unstable curve so that's very nice and all together we will know that if we denote by U note the set covered with good central stable so good points are in H in every point of H we take the corresponding central stable curve X should be downstairs sorry but I am saying in words then the bad set in U in this sausage is small and well I think when I started seven minutes oh that's perfect, thank you very much so well now I do the following so in this sausage so I have this sausage it may have some caps there are sufficiently long central stable many folds and there are also others and what I want to take I want to take the the product so I want to take the yes I think I so mu tilde is the real measure restricted to U U is this sausage and this measure can be decomposed as the product of two things the factor measure on every central stable many fold I remind you that central stable many folds there are shorter one and there are nice one which cross completely my sausage so J is the index set for all the central stable many folds and H is the index set for the nice one so we can forget about the other one so I just I will use this for H and the essential thing is the following that first of so what I will do is that on this original W I have a probability density phi moreover on the disk D so this is the disk in the center D I take a very beautiful probability density and essentially what I want to do I want to take the product of this density phi and the density given on this disk D on D the radius epsilon don't forget so the density will be something like that so this is to epsilon and I want it to have a probability density so it's natural that I have these bounds for the supremum of the probability density and it's derivative and I assume that both q and the derivative vanishes outside D and from these two things I create the measure in this the smooth measure in this sausage so this is the definition that I introduce a measure by using phi and by using the density q and the density of this created measure will be D rho divided by D mu mu is the removal measure and what I want to say that in the construction of course I cannot go into the detail but there is one important element that if I take a fixed sufficiently long x belongs to h if I take a sufficiently long a nice central stable manifold then the integral of this g-node density with respect to the conditional density I have to go back so oh sorry so in this the composition so this was the factor density and you was the conditional measure on every central stable manifold and this is what I have to use that on every nice central stable manifold integrating with respect to the conditional density on this central stable manifold integrating this g-node which I created as I said you before it's a very nice constant phi is the density given originally on the standard pair phi and beta is the derivative of the factor density with respect to the arc lengths and this formula makes it possible to evaluate the integrals which figure in the theorem and this makes it possible to apply this kind of marvelous thickening method while this is a technical thing so I think I finished by some remarks so I did about one part of the proof about how to prove this correlation decay for standard pairs of course there are other parts of the proof because I just mentioned one piece so you have this model and you have the piston particle so interaction occurs when the billiard particle and the piston particle meet they can only meet in this critical region now for the composed model the energy of the billiard particle on an unstable curve is not constant because the composed model is a composed model and even if it's constant the energy after one collision the energy will change along the outgoing curve because there is an interaction and it's a drastic interaction it changes so one has to handle also this question but probably another time we can talk also about it to close I want to say that once one can prove the result for the model which I mentioned it's almost automatic to get it of n cells so it's not a real difficulty the real difficulty is of course to get to the original Gaspar-Zilber model two-dimensional interakting I don't know so thank you very much