 Professor Gover, dear colleagues and friends, I am very moved by your kind words, it's a great honor which you have bestowed on me. I feel even somewhat embarrassed, especially with so many eminent colleagues here. I don't know if I deserve this honor, perhaps my merit has been to be interested in irreversible processes already at an early stage, when this subject was not very popular. I'd like to remember a conference which I had been organized in Benavez-Stilberian in 46, which was dealing with irreversible processes. And at the end of my lecture, some of the most famous thermodynamics of this time stood up and said, I really don't understand why this young man is interested in irreversible processes. After all, irreversible processes are transients, and after some time you come to equilibrium. Why, therefore, not be directly interested in equilibrium? I was so astonished that I had not really the presence of mind to say, well, we are all transients. Therefore, it's rather natural to be interested in our human condition. At this stage, to use the expression used by Professor Feigenbaum, everybody was convinced that there was one universality, and this universality was equilibrium. If you perturb the system, well, like let's say a pendulum with friction, it goes back to its equilibrium position. If you perturb a liquid, a gas, it goes back to equilibrium. So, we were very astonished that when we studied non-equilibrium processes, it came to the conclusion that at the well-defined distance from equilibrium, some say quite different happen. And as was mentioned, it is different. It is new things which happen. Thus, as Professor Gover already mentioned, that there is a possibility of new coherent structures. And this is generally the case if there is some non-linear feedback. For example, x becomes y, y becomes x, proteins give nucleic acids, nucleic acids give proteins. Then, the type of laws to which we are used generally are no more valid, and you come to some very unexpected phenomena, such as we see now today with the Jabotinsky-Belouzov reaction of chemical oscillations that we can see also, space structure, spirals, waves of all kind. And we are only at the beginning. And it is very interesting that this really has shattered our ideas about, that is for example, the chemical reaction. Generally, we had the idea that a chemical reaction is a random encounter of molecules, that you produce organized patterns, coherent time series. Of course, this is not adequate. There must exist long range correlations which are absent at equilibrium. I like to say in a slightly, I would say, anthropomorphic way that matter at equilibrium is blind, but far from equilibrium it begins to see. And if we push the system further away from equilibrium by pushing matters who's a reactor more rapidly, or by heating some part and cooling some other, we have some other type, more complex type of processes which appear, and chaos is one of them. Well, personally, I have some hesitation to emphasize the word chaos, because chaos has a little negative connotation. It is the difficulty of prediction which is emphasized by this connotation. I like to speak more about self-organization, about the possibility that for given boundary conditions and for a given mechanism, you may have more and more possible structures, these broken time symmetries. When I was young, my teachers were very proud, they could prove that for given boundary conditions and a given mechanism, you had only one solution. Now we are very proud when we can show that for a given mechanism and given boundary conditions, there are many. When I was invited to give a talk here, I had to make a difficult choice. I could either speak to you about non-equilibrium structures and the relation with chaos, and this is today a very fashionable subject, you know, these ideas have been applied in many fields. And now the possibility which was open must try to give you an idea about the work in which I am involved with my co-workers and two of my closest co-workers are here, Professor Petrovsky and Hazegaba, and to speak to you more about things which make me believe that indeed chaos may be a new science. And as a subject of the conference is really chaos is a new science, I decided to try to do this. However, this is not an easy task. Let me start with the following remark. After all you have seen these beautiful structures described by Mandelbrot yesterday. You have heard about these non-linear equations and the road to chaos beautifully described by Feigenbaum. But you can ask the question, what is the relation between all these interesting and remarkable phenomena and the basic description of nature as embedded in Newton's equations or in quantum mechanics. And here I would like to emphasize that the type of processes described yesterday by Feigenbaum occur only when there is a broken time symmetry. It is then that you have the Stokes Navier equation, then that you have a reaction diffusion equation. Therefore irreversibility is very important that I already just mentioned that is when you go away from equilibrium that you have all these new phenomena. What means irreversibility? Irreversibility means that there is a natural time ordering. It is obvious that you can say that industrial revolution came after, let's say, a Neolithic time. It is obvious that we can make a distinction between young men and an older one. But is this time ordering basic in nature or is this time ordering simply in our mind? And that is really a question which has been discussed since centuries. And the reason why it has been discussed is that if you take the simplest phenomena such as a pendulum, it can start here, go there, or go from here to there. And you cannot say that really earlier. It seems that once you go to the basic description, then you essentially lose the direction of time. And this is also true in quantum mechanics. Therefore the question is really time real, time is irreversibility. Now this question was really openly posed with the discovery of thermodynamics in the 19th century in 1865 when Closier made his famous statement that the entropy of the universe is increasing and therefore brought in a kind of evolutionary view of the universe. But you know the great mathematicians of this time, the great physicist, didn't pay any attention to this. This was a science produced by engineers in physical chemists which are very low on the scale, you see, of, I would say, of hierarchy in natural sciences. Therefore one of the merits of Boltzmann was, who was certainly a great physicist, was to ask, what is after all essential about that? Is there something really fundamental? He was very much influenced by Darwin and he tried to make a kind of evolutionary picture involving populations, not of living beings, but population of molecules. He came out with this kinetic theory. But people were rapidly pointing out to him that this was incompatible with the basic time symmetry of Newton's equation. And he became gradually more and more unhappy about the situation. He was really like a man who had two loves, two women he loved at the same time. On one side he knew very well the basic equation of physics and the properties. On the other hand he believed in an evolutionary world and he could not have made up his mind. That gave really a kind of tragic flavor to his life. And the defeat of Boltzmann was considered by his hair, to some extent, as a triumph. As a triumph of the idea that time is in our mind, where Einstein, much closer to us, has written time as irreversibility is an illusion. I always found curious that his conclusion was so readily accepted. Because if you really think that time is an illusion, well human life is an illusion and with it all the attempts we can make to find more about the world. That is a remark which Karl Popper has made and which is completely correct. And I think if there was no crisis due to the defeat of Boltzmann, this came from the fact that the physicists were convinced that the aim of physics was to express the description of the nature in terms of universal time reversible laws. And this comes back really to the ideology of the period in which science as it exists even today was conceived in the 17th century. At this time life was emphasized very much. The fact that for God nothing can be hidden that essentially for a well-informed observer is what will ever happen is already present in the present. Therefore, in a sense, the time has to be eliminated. And this idea to describe everything in terms of universal laws was really the goal, the ambition of physics. And often it seemed that it was physics was very close to discover really that it was right that we were at the point where we could reduce the universe to this universal time reversible laws. And every time something went wrong in the sense quantum mechanics had to introduce the observer, the strange role of the observer of measurement and so on to which I shall come back later. And general relativity came finally to the idea of the Big Bang, which is certainly a temporal event. Last year's conference in Minnesota at a very provocative title the end of science. I don't believe that we can speak about the end of science, but we can certainly speak about the end of certain form of rationality associated to classical ideology. And I think it is building up of a new rationality, of a new view of nature, cause as it is generally understood certainly plays an important role. Then, as I already mentioned, the question is this all in our mind or not? As today a different has today different aspects than it had at the time of Goldsmith. Because we know precisely that going away from equilibrium, going away from situations which are timeless, nature takes new forms. You have all this non-equilibrium structures which I call dissipative structures, which come in, which present a lot of broken symmetries which are maybe temporal sequences or space space problems and so on. Therefore the question is where to look? How to incorporate time inside dynamics? And let me give you a very simple example where you can see that this can be incorporated in type of systems which one calls unstable dynamical system. And the discovery of unstable dynamical systems is really one of the great discoveries of the end of last century and of this century. It's quite remarkable. This century is associated to two great conceptual revolutions, quantum mechanics and relativity. But classical sciences like thermodynamics or classical mechanics have also undergone quite a bit of change. And this change, as I shall show at the end of my lecture, is likely to change quantum mechanics and relativity. Let me explain you what an unstable dynamical system is. And I don't know if this is very clear. And you see, it is a geometrical transformation in which you take a square, you squash it, you put the right hand above the left, and you obtain more and more this type of stripes. You see, you fragmented more and more. And you can immediately understand the two points which were closed in the beginning will show up in different stripes there at the end. And these systems are characterized by instability, which means, which has been mentioned yesterday by Feigenbaum, and the Apunov exponent. That means that if I take two points very close, initially they will go away exponentially. And the lambda here is a kind of inverse of a temporal horizon. And in this case, this takes a very simple value. And this is an unstable dynamical system. It's a chaotic system because you can no more associate to it at all the regularity of periodic motions, like the motion of the Earth around the Sun. It is a highly unstable system. Particles go away, and it's very, very stochastic. It is an example of what's called a Kolmogorov flow. And this gives me the opportunity to mention the enormous, the extraordinary work of Kolmogorov, because case stands for Kolmogorov. And to say also how much I regret that in the popular literature about his subject, his work is not sufficiently mentioned. Because Kolmogorov really was the first to understand that two concepts, which seem to be of opposite, can in fact be closely related. And this is very important. And one is the idea of regularity, which we generally associate to dynamic motions, like periodic motions, and the idea of stochasticity. This system is a highly random system. In fact, when you follow, I have no time to show it, but if you follow the trajectory of points, you can show that the next position of the point is as random as you would play on a roulette. That is the reason why this system is also called a Bernouy system, Bernouy referring, of course, to games of chess. So this is already an example of an unstable system. Now the important point is that for such unstable system, we can introduce a new concept of time. In addition to the idea of time as motion, as changing from this first picture to the second to the third and so on, we can introduce a time which is not related to the motion, but which is related to the topology, to the number of stripes, to the type of conformation, which is a global judgment. It's like when you look on somebody and you want to guess what is his age, you should not look only on his hair or on his fingers or that. It's a global judgment. And here, the vein which the red and the white distribute is a global judgment. And in fact, as my colleague, Misra, has shown, you can introduce a time operator that's of no importance which measures the number of fragmentations which you have to do to obtain a given picture. So there appears a second time, a time which is no more related to motion, but which is related to something more qualitative. And I would like to show you now that in most systems which we meet in nature, we can also introduce this type of second time and this is, can be introduced for unstable dynamical systems and that is a vein which time enters now into the fundamental description of nature. And here I come to the second giant of unstable dynamical system and that is certainly Poincaré. And in 1892, Poincaré asked a very interesting question. He didn't ask it in this general form, but it is equivalent. He asked the question, can we eliminate interactions in dynamical systems? As you know, a dynamical system is described by the energy which physicists call Hamiltonian and which contains momenta kinetic energy of the particles and interaction, potential energy. And he asked, can we make a transformation, a canonical transformation, a one which keeps the usual form of physics such that you can eliminate the potential energy? And his answer was no. And that is very fortunate because in a sense if the answer would be yes, that would mean that the universe would be identical, isomorphic to a system of non-interacting particles and this would mean that there would be no coherence. If there would be no coherence then there would be no chemistry, there would be no life and there would be no noble conferences. So it has some good and bad aspects. But anyway, Poincaré pointed out that we cannot do it and he showed something much more. Poincaré showed why you cannot do it and you cannot do it not because of the forces, but because of resonances. Every child knows what a resonance is when he puts in motion a swing. And here I've drawn two harmoniques, which are in resonance. Then the mathematical definition is that you have two objects or two degrees of freedom, of course you can have many more, which have frequencies omega one, omega two. And if there is some integers and one and two such that the sum is equal to zero and one of the integers have to be different from zero, then we have resonance. Now the existence of this resonance of the strong coupling because of resonance, as I mentioned, is not related to the existence or non-existence of forces. That's a different problem and this needs to, the famous problem which Poincaré called the problem of dynamics, it is the problem of small denominators. Why? Because if you consider a dynamical system and suppose the dynamical system is made up by two parts, one which is integrated, you have eliminated the interactions and then you put a small perturbation which contains the interaction and you want now to eliminate these interactions. Then when you make the perturbation calculation to try to eliminate it, you come always to situations in which this small which is a resonance condition appears in the denominator. Now if this is zero in the denominator, that means then it is infinite. The expression is infinite. That leads to the problem of small denominators, therefore to infinities and that is, it is infinities which make a system non-integral. Of course I have no time, it's always on the point to make this more precise, but let me simply say that if there are enough resonances then the system becomes non-integral. What does it mean? It means that you can no more eliminate interactions by methods in which you would use perturbation and expand everything in powers of the perturbation constant. So the system, the situation, remains there till the fundamental work again by Kolmogorov in 1954. In 1954 Kolmogorov, soon followed by Arnold and Mozart, created what's called now the Kamm theory, which is one of the most beautiful theories of physics and mathematics, and which was the first positive answer to Poincaré's impossibility theorem. Because what Kamm have shown is that if the coupling is small, then in spite of Poincaré's non-integrability, most motion will remain periodic, like in celestial dynamics. Only a small variety would become erratic, chaotic. So in a sense this permits to solve now this non-linear non-integrable system to some extent. But there is something extremely new in the work of Kolmogorov, and what is so new is that essentially you had the problem of the small denominators, which as I mentioned is what's called the fundamental problem of mechanics by Poincaré, then you had the Kamm theory, and as a result of this you have now most motions become regular, if the interaction is small. But there are always now in addition random trajectories, trajectories which wonder seemingly erratically in the system, and then of course then the appearance of stochasticity, of probability, of chaos through the Lyapunov exponent, through this divergence of trajectories. And this is of course therefore a fundamental discovery, and when you look on the type of diagrams which have been shown already yesterday by Feigenbaum, you see for example here you see what's called a Poincaré section, you see these points are all the intersections of a trajectory with a plane coming from a single trajectory. You find some islands which corresponds to more regular motion, but most of the points seem to be distributed just at random on the surface. So essentially the great discovery of Kamm was stochasticity, that stochasticity and dynamics they're not opposed. Now the type of problem which my colleagues and I have been considering is just the opposite of the Kamm problem. In the Kamm problem you can still say something when the interaction is weak, when most of the motions remain periodic. But what happens if there are more erratic motions, well you can follow this on the computer, but you cannot say very much analytically. But therefore it's interesting to consider another extreme case. It is extreme case is the case in which all motions are erratic. And that happens in a class of systems which is extremely, I would say popular, which is extremely present, which we find nearly always in physics. And then there are the situations in which there are so many resonances that each resonance is in between other resonances. Well let me give you an example. A particle, a harmonic oscillator in a field, but if I consider a harmonic oscillator in a field, the field is a continuum and there will be frequencies of the field as close as I want from the frequency of the harmonic oscillator. And therefore I cannot more speak about periodic motions at all. In mathematical terms you have to say the spectrum is continuous and you have continuous sets of resonances that are of no importance. The physical idea is that no resonance appears everywhere. And this appears, I would say, in most problems of classical quantum mechanics. For example if you consider a quantum transition, an unstable state which is destroyed by emission of light. Then you have a resonance between the energy of the state and the energy of the photon which is emitted. And to calculate the lifetime you have to make an integration. I will go into the details about and you have to make an integration over the frequencies of the field. And the same is also true in kinetic theory. Kinetic theory is essentially a theory of resonances in which particles come together, collide, come together, collide, go away. And when you analyze the mechanism, and I shall come back to this, it's a resonant mechanism, a resonant transfer of energy between the particles. So these so-called large Poincaré systems are prevalent in nature. And now it appears something very curious and very interesting. And that is a fact that essentially I can speak about integrable systems. That is most of classical physics, classical or quantum mechanics. And then we have this breaking point, Poincaré's non-integrability theorem. Out of this breaking point comes then the theory of come, 54 and later, which leads to dynamical chaos. And then comes the theory in which we have been interested. And the same, which I immediately to emphasize, this is really a collective work in which Professor Petrosky has played a very important role. And we come to the large Poincaré systems. And what is the interesting point which I try now to explain in the rest of my lecture in a not too technical way, is that for this limiting case, we can find new methods of integration. We can exactly solve again this problem. And the solution of this problem is quite different from the solution of the classical problem. Many things which are true for integrable systems, like for example, variational principles, that the particle takes the shortest path and such kind of thing, are no more true for this type of system. But on the contrary, this type of systems are more stochastic. They are more probabilistic. You cannot say so much as you can say about integrable systems. Not only they are more probabilistic and more stochastic, but they are also in a sense, of course, corresponding to much more irregular motions. And they have, and that is the main point for us, they have a direction of time. Therefore, a large Poincaré system lead to the incorporation of dynamics in the frame of incorporation of time in the frame of dynamics. Now, how to make this integration? What do I mean by these new methods of integration? And here I would like to give you first a small example and then present the general recipe. The small example which I want to take will be taken from quantum mechanics. Now, you know quantum mechanics is really a revolution in the way of thinking, in the sense that in quantum mechanics we associate to each physical quantity, to each observable, as one says, an operator. Instead of being simply a number, it is an abstract quantity called an operator. You should not be afraid by the word operator. Operator is simply a multiplication and derivation, something like that. And then the main problem of quantum mechanics becomes an eigenvalue problem. The operator acts on some function, derivative of some kind of thing like that, and then for some function you regenerate simply the initial function. Then it's called an eigenfunction. It's regenerated apart a multiplicative constant, and this constant is then called an eigenvalue. And what do I mean by speaking about the spectrum which is formed by the eigenvalues which may be discrete or continuous? And interestingly enough, Poincare's non-integrability criterion applies to quantum mechanics for continuous spectrum. Precisely in the case in which each, for example, a harmonic oscillator is embedded in the field of seators and you cannot really disentangle one from the other. And you have exactly the same problem. Suppose the Hamiltonian is the sum of an integrable Hamiltonian plus some perturbation, then can you solve this problem? And in general you cannot solve it. And in fact this is one of our difficulties in modern field theory. Modern field theory can solve all problems of free fields, but it was already mentioned yesterday. If you come to the problem of interacting fields, then it's extremely difficult. In general you cannot do anything. Therefore the problem is, and this is related to Poincare, because you can no more expand the solution in powers of the coupling constant and you cannot obtain a conversion, this expansion. Now the point in which we have been interested always was the problem of time. And so we came to this idea which seems at first a little strange, but not so when you remember the example of the baker which I gave you before, can we not solve the problem by incorporating the time, a kind of internal time, a kind of functional time into the problem of integration. To explain you what I have in mind, let us think about a very simple problem. We can have a stone falling and outgoing waves. We can, in principle, have incoming waves which would make the stone going away, but that is not what we generally observe. What we observe is stone falling, outgoing waves. Therefore what we observe is a very defined natural order, natural order. Stone first, waves later, not waves first and stone later. When you now think about a quantum jump, radiation, you expect also a natural order. You expect that you have first have an unstable state and then you will have the emission of the wave. That was the idea of war, the wave would be a kind of retarded wave going away. Now when you write this problem in perturbation you come out to the small denominator and therefore to the Poincare divergence. Here the denominator is the difference between the frequency associated with the particle and the frequencies associated to the field. Therefore the question arises how can you cure Poincare divergence? Well, as I said, you can cure it and what we have shown, you can cure it exactly, rigorously, into all orders by modifying the small denominators, modifying the small denominators by associating to each denominator a physical process. Then you go from one to k, you go to the future, then you go from the field to the particle, you go to the excited city, you go to the past, and this then immediately leads to ideas about mathematical methods for introducing time or here I cannot go into details, analytic continuation, forget about it if you don't know it. So essentially you can give a meaning to Poincare's denominator and the remarkable feature is that this then eliminates all Poincare divergence. You come out now with eigenstates which are complex, which include damping, which means it includes no dissipation and have a broken time symmetry. What have you done? You have essentially extended the type of transformation which Poincare had permitted. Poincare had permitted only let's say transformations which would keep the eigenvalue real by permitting now a larger type of transformations. We can solve problems which you could not solve before and of course we recover but everybody knows, but in addition we recover the existence of an entropy or an h function which decreases monotonously in time when the particle is decaying into the continuum and emitting its weight. Now this is a very interesting point even from the point of view of quantum mechanics because in quantum mechanics there's the question which was very much angering Schrödinger, are there quantum jumps? At first everybody, I mean both theories are quantum jumps but when you think about Schrödinger's equation which is a fundamental equation of quantum mechanics it's a continuous equation, there are the jumps and this has given rise to a lot of discussion between famous discussion between Schrödinger and Bohr and they never agreed and Heisenberg wrote somewhere well it's clear that they didn't agree because obviously the jumps come from our observation and this is very difficult to accept because if quantum jumps are due to our observation we ourselves are the cause of our existence because quantum jumps are the mechanism of autochemical reaction, of all chemical reaction which are the origin of life. To say that is because we observe reactions that the quantum jumps is that because we observe nature that we exist that I think is a relatively strange metaphysics and I can hardly believe in it. John Bell has written a book which he called very nicely speakable and unspeakable problems in quantum mechanics and he makes a lot of fun out of these ideas and I think quite rightly. So in a sense by making a more general transformation into the complex plane we can now say in this formulation there is a quantum jump irreversibly the particle will decay and you cannot restore the initial situation after a sufficiently long time. But this is a very simple example in which I can give a simple formulation to the time sequence. I have an unstable state which is decaying. However in general it is impossible to find such kind of very easy time order and we have to turn to the statistical description. What means is a statistical description that something which Gibbs has introduced many years ago instead of considering one system you consider many many of them you are not interested in the fate of one example that you are in the fate of the totality. I did like what I explained yesterday about voting and so on you're not interested or every a single individual will vote that you're interested to know if there will be 30 percent of yes 70 percent of no or what so ever and this then leads to the idea of a large phase space and in this phase space many coordinates many momenta and it's described by density function and this density function depends on the particle positions under the momenta and you can apply similar ideas to classical mechanics and to quantum mechanics. Now how time enters into the statistical description in addition to the Newtonian time and times very easy to see. I spoke about collisions but collisions create correlations two particles which have interacted go away but they keep I would like to say the memory of their interaction it's like two people where the conversation one of them goes to Australia and the other goes to Washington but they keep the memory of their interaction and they carry it around if I make a velocity inversion then they will come again back together and now the two people meet a third one it becomes a ternary interaction a ternary correlation the ternary correlation becomes a fourth correlation and so on and you can see this very well on the computer calculation how the correlations are building up in a computer you see that binary correlations are building up very rapidly for short distances for longer distances it takes already much more time then ternary correlation takes more time and so on if I would have here a glass of water then I could say that this water is still aging I can make now a distinction between young water and old water and old water there are correlations between millions and millions of particles which have appeared in young water when the molecules have been brought together they have no correlation so in a sense the time here it is the analog of the internal time which I introduced for the Baker transformation the time here is the time not in the units not in the motion but in their relations if you take for example a primitive society a Neolithic society the people they are about the same as we are today from all that you can guess however the relations they're very different the correlations they are very different we are living today in a time in a period in which time is accelerating and this acceleration of time is related to the fact that we create more and more correlations we have the mass media time is related to the global view we can have on the events which are going on in the universe therefore there is a way of introducing an internal time in all this dynamical system and that is the flow of correlation the flow of correlation which is flowing this Newtonian time that introduces a new dimension a new aspect of this time exactly like in the Baker you have the Newtonian time coming from the Baker transformation that in addition you have this topological time related to the way in which the stripes are distributed and this of course without any much mathematics you can understand that this will destroy the symmetry between past and future if i start with an initial state which had no correlations a young state then i go to a state b in which there are a lot of correlations between molecule my old glass of water you see it's not at all the same as if i start with the old glass of water and i try to rejuvenate it by suppressing the correlation i would need to have an enormous control on all these correlations which i never have in fact the best correlations i can to some extent control today are three body correlations in fact this glass of water about which i spoke has an hour of time which if i leave it alone and prevent prevent strictly evaporation has an hour of time which will go on till the end of the universe because this hour of time will go on because the glass of water contains 10 exponents 23 particles and before i will have the right correlations of billions of billions of particles that will really take time which goes beyond every imagination so in a sense the existence of this time correlate this this correlations introduces a new time element and again once i know that this is the existence new time element i can of course introduce this into the mathematics i can solve the equation for the evolution of the system for unstable dynamical systems for Poincaré systems i can solve them exactly taking into account a new way of treating Poincaré's denominators again putting a small i epsilon plus a small i epsilon minus according to the fact that i go from a binary correlation to a ternary or from a ternary to a second order and so on so essentially we can now solve rigorously the evolution of ensembles and we can solve this evolution of ensembles because we have essentially this idea that we can introduce into unstable dynamical system a supplementary element and that is the idea of temporal succession expressing the relation between the elements and non-Newtonian non-Einsteinian time in fact this goes further first of all as i can only introduce this time element this time ordering on the level of statistical ensembles i can no more go back to trajectories i can no more go back to wave functions therefore i obtained an irreducible description which is statistical which can no more be reduced neither to Schrodinger neither to Newton it is a stochastic description which is irreducible the basic description of unstable dynamical system is a probabilistic statistical description that is the new integration as i have shown you a few minutes ago there is a box the box of classical integrable systems of Schrodinger equation Newton's equation Lagrange and so on and then i have now a new box and this new box deals with highly unstable large Poickery systems and this new box is can be again integrated as rigorously as Newton and Schrodinger have done it however the result of this integration is quite different the result of this integration is that it remains a stochastic system it remains a stochastic system and you cannot reduce it because if you would like to reduce it you would lose an information which only refers to the statistical ensemble which does not refer to trajectories or to refers to to to wave function now there's also another element and this other element is that essentially the description of what happens in such a system is quite different from the usual description in Newtonian physics or Schrodinger physics because here now you see the resonance plays an essential role but what is resonance resonance is coming together of two spacetime events which are now correlated by resonance then somebody for some of those of you who know a little statistical physics they can remember a little the form of diffusion equation for Kaplan equation they are equations which contain the second derivative not the first derivative as Hamilton's equation with the same vice second because you have in fact two at least two elementary events which are linked by resonance therefore you can no more decompose the history of the world into independent events like Feynman diagrams you can decompose into events which are non-linearly strongly coupled by resonance conditions and this of course is a big change in respect to classical description therefore essentially let me summarize we have on one side we have integrable systems with trajectories and wave functions and on the other hand we have non-integrable systems large Poincare systems for which you can introduce new integrations which lead to new solutions of the eigenvalue problem and doing a reducible description in terms of distribution function and to dissipation and entropy therefore you can say that the paradox which I started has to some extent been solved in the old problem in the old situation you had on one side the macroscopic level the level in which there is time ordering in which people become older in which the thermodynamic equilibrium is established in which if I introduce a difference of concentration it is leveled out and you had a time a description a fundamental description a microscopic description which was time reversible now I must say that was my driving force over the many years I thought about this problem I never could understand how time can arise from no time how this time breaking symmetry breaking can arise from equations which have which have no symmetry breaking in time which are time reversible that I could never understood and that was really the paradox and the answer of some people was to say oh the macroscopic level is all in our mind it's all our imagination the basic level is a level of eternity it's not the level of becoming it's the level of being that is the fundamental level that Einstein's point of view however now we come to a different situation we come to a situation in which you can say we have three levels we have one level which is the statistical level which is a reducible which has a broken time symmetry which is still dynamical which applies to large chaotic systems and by averaging over this you you obtain the macroscopic level with a broken time symmetry if you would start with trajectories or as close as you want from trajectories of this wave function they are destroyed by the chrystochastic character and you come here therefore this is the statistical level which now becomes the fundamental level of the description and here let me therefore a little try to situate the problem of chaos now to avoid some misunderstanding in a sense I think it is very important to make a distinction between dynamical or Hamiltonian chaos and dissipative chaos dynamical chaos as dealt with the scum theory and based on Poincare's theorem and on the other hand dissipative chaos as I've been explained beautifully in some examples yesterday by Feigenbaum so in a sense we see now that we have a two level chaos that the chaos is a concept which contains two levels one level is a dynamical chaos as induced by resonance the basic mechanism in nature is resonance because that is present everywhere I mean it's present from the three-body problem up to the end-body problem of fields of kinetic equations and so on even if you take the three-body problem my colleague in Austin professor Zevehi gave me a very beautiful example if you take a three-body problem you have three particles going around one of the other whatever the initial condition after some time one of the body will go away and you remain with the two-body problem therefore you have already a kind of arrow of time even incorporated in so simple problems as a three-body problem and not to speak about the end-body problem once you have this dynamical chaos you can from the dynamical chaos generate all the beautiful equations which were mentioned yesterday the Fourier equation the Stokes Navier equation the chemical equation and then for them you have different type of behavior according you are close to equilibrium or far from equilibrium but the first condition is that you need to generate the distance from equilibrium and that is generated through the dynamical chaos therefore the chaos the dissipative chaos is chaos mapped over chaos it is the second level of chaos there are two fundamentally different types of chaos one is the fundamental chaos which is related to Poincaré's theorem and with the existence of resonances or instabilities on the fundamental dynamical description and the other chaos is the outcome of the first chaos on the macroscopic level and this chaos as i said is not a good name because the chaos as you see it in macroscopic system is in fact order because when you see for example the period doubling which was beautifully described yesterday it is already a highly ordered process highly ordered process i would like to call it more self-organization and coherent structures space patterns and so on which we now in the last few months remarkable progress has been realized in the discovery both when computers end on in the laboratory of space space structures beautiful space structures all the natural only beautiful time structures but you can call all this what emerges out of equilibrium is as a process in self-organization even the chaos even is when you think about very chaotic phenomena they are much less chaotic than thermal noise in other words they are already organized phenomena and therefore therefore what you see is that order is generated out of chaos in other words here you have really chaos and out of this dynamical chaos by putting in the hour of time you generate self-organizing structures therefore you have strange universe in which at all levels we see the appearance of organization out of a microscopic background which is essentially chaotic now you see what is also important is that we see now that chaos is closely related to the problem of time and time is related to the problem of control irreversibility we cannot avoid an unstable particle to disappear we cannot avoid a three particle correlation to become at a later time earlier or later a four-hour correlation or fifth-hour correlation i like always the same sentence by nambokov nambokov is written what is real cannot be controlled and what can be controlled cannot be real and i think that is exactly the situation which to which we come but what is also interesting is that this changes of course some of the basic features of quantum mechanics and of of relativity i have no time to go into this let me simply mention that in this introduction to the new physics paul davis has written very correctly as the rock bottom quantum mechanics provides a highly successful procedure for predicting predicting the results of observation on micro systems but when we ask what actually happens when an observation takes place we get nonsense after half a century of arguments quantum observation debate remains as lively as ever the problems of the physics of the very small and the very large are formidable but it may be that this frontier the interface of mind and matter will turn out to be the most challenging legacy of the new physics then this is very closely related to what i discussed because here now the basic level is on terms of probabilities it's no more on terms of amplitudes like in ordinary quantum mechanics therefore the question how to go from amplitude to probabilities is automatically solved for unstable chaotic systems and therefore you can now give i am going very rapidly you can now give an exact recipe that is a measurement apparatus then it is a chaotic system it has to be a chaotic system it has to be let's say a large point array system in which the description is in terms of probabilities it cannot be reduced to quantum mechanics to wave functions and you have no more this famous problem of the reduction of the wave packet and all the difficult and the role of subjective observers and so on which was really a kind of strange story disturbing scientific community since a few decades and similarly when you think about a relativity relativity was always thinking in terms of points in space time instantaneous events but when you have collisions many of resonances there are no points they are connected points which are connected by the strong resonance conditions and therefore there also there is a change now therefore let me conclude by saying our world view had been essentially dominated both in classical and quantum mechanics by the theory of integrable systems by the theory of relatively simple systems in which resonances play no role which are relatively regular which in which we could associate to the ideas of determinism of time reversibility of universal laws and so on what we see now is that this extremely extremely important window I would say is still only a window it's only part of of our physical universe that a much larger part is described by non-integral systems non-integral systems which need to different properties you can integrate them and that is very nice but you lose some other properties as I said you have no more Lagrangians no more variational principles you have a different type of physics but you have it's more unified physics because it incorporates thermodynamics second law probability and dynamics and therefore is certainly certainly a great progress in the direction of unification of our physical world well I don't know if I've succeeded but what I've tried to do is to show to you that indeed chaos may be the starting point of a new science thank you very much