 Turbulent Mixing and Beyond 6 International Conference 10th University Program. The program was supported by the US National Science Foundation, the US Department of Energy, the Abdul Salam International Center for Thoretical Physics, the New York University as well as the University of Western Australia. Financial support was provided. So Turbulent Mixing and Beyond Program is a scientific program that has been organized with a certain goals in mind. And the goal that we had in mind when we were organizing it 10 years ago was to expose the generic problem of non-equilibrium. Those days we were saying Turbulent, these days we're just saying non-equilibrium and Turbulent processes to a broad scientific community, to promote the development of new ideas and tackle the fundamental aspects of this problem, to assist in applications of novel approaches in a broad range of phenomena where these processes occur and to have a potential impact on technology. From the very beginning, the TMB program or Turbulent Mixing and Beyond program was a program that was established for scientists and by scientists that was established as a solely merit-based program and that is shaped by requirements of academic credentials, novelty and information quality. In fact, actually we do know that we study very complex processes and we would like to set certain standards of quality and quality criteria. TMB program was founded back in 2017 years ago with the support of international scientific community as well as national and international funding agencies and institutions. Today our community unites thousands of researchers from academia, national laboratories, corporations, industry at both early and advanced stages of their careers. If I would like to give some numbers actually for the TMB program, since 2007, as I mentioned, there was a substantial support from the direct sponsorship of the TMB program from funding agencies such as the National Science Foundation, Air Force, Office of Naval Research Department of Energy as well as institutions including the University of Chicago, NYU, Carnegie Mellon University, International Center for theoretical physics is our basic primary sponsors, CEA in France, UWA, ILE, Russian Academy of Sciences as well as there is an estimate for the participant contributions because usually the funding that is being provided by sponsors we apply for publications and for support of those who cannot support their travel to the conference where there is a significant part of people who are coming just with support of their own institutions. TMB conferences were organized in 2007, 9, 11, 13 and 14. They were organized as ACTP regular and hostage activities and as well as it was once organized as a school back in 2009. It also was once invited mini-conference of the American Physical Society meeting in the plasma physics division in 2013 and it also invited as a symposium on interfaces and mixes by the US National Academy of Sciences which will be held in November 2017. Since in these 10 years we published essentially 16 books and the 16th book that we have with us is this book and I kindly point your attention that even though it looks like very innocent it's actually the regular publication because it has an ISBN number and we placed this ISBN number very purposefully on this book because quite often the information, especially high quality information that is presented at the conference is sensitive. However, as long as we have an ISBN number this information is a formal citation record and as well as the record. In addition to these publications, of course, which have been published by the ACTP, we also published a number of topical issues, focus issues, invited issues in physical script, we published with the philosophical transactions of Royal Society and Royal Society Publisher which included theme issues and edited research books as well as we are preparing now the special feature issue in the precedence of the National Academy of Sciences in the US. So, if we will Google, even though it's not kind of really strict criteria but it's still a number, if we will Google today that for Turbulent Mix and MV-On we will find approximately 1,500,000 results or so. I believe that this number will be dropping out sometime in the fall but still it's a good estimate. So, TMB program, TMB conference, essentially they designed to build the bridges because the conference itself provides the opportunity to bring together scientists from many different areas including high energy density physics, plasmas, fluid dynamics, Turbulent, geophysics, astrophysics, optics and telecommunications and we did have these presentations at some earlier stages. Mathematics, applied mathematics, probability statistics, data processing and computations focus their attention on the long-standing scientific problem and as well as on the connection to reality and why we are talking about connection to reality because in fact we do tackle fundamental problems which play important role in a broad range of phenomena and for which actually there are no well-designed methods so we do need to understand how to formulate the standards of quality. The TMB community members, they are leading experts and researchers at advanced and early stages of their careers. There are also graduate students from developed and developing countries and this mixture of senior people and younger people is quite important for the success of the program. What is important for the program is that it's based, it's actually a science-based program. In this conference we found that in our communications that TMB related problems haven't come as a set of outstanding research issues. The solution can significantly advance a variety of disciplines in science, technology and mathematics and that our community itself involves participants who conduct highly innovative research and their interactions strengthen the community more. Non-equilibrium processes, these are processes that play a broad, a key role in a broad range of phenomena from astrophysical to atomic scales with conditions of high and low energy density including inertial confinement, fusion, magnetic fusion, light-material interaction, non-equilibrium heat transport, material transformation under impact, shocks, explosions, blast waves, supernova, accretion disks, early universe formation, stellar, planetary and non-business convection, planetary interior, mountain leader spirtectonics, premixed combustion, oceanography, atmospheric flows, non-canonical, well-bounded flows of supersonics and supersonics, as well as in many processes in cutting-edge technologies such as laser micromachining, nanoelectronics, free space, telecommunications, applications in aeronautics and aerodynamics, as well as in oil and gas, gas and oil production industry. So, in fact, a reliable quantification of these non-equilibrium and turbulent processes is a highly formidable task because the theoretical description is one of the most challenging problems in theoretical physics and the addressing complexity of this problem may expand the horizon of modern theory of partial differential equation, may encourage the development of novel perturbative integral and stochastic approaches, is actually calling for connections, for new connections to establish between kinetic processes at microscopic scales and processes at continuous scales. It also suggests and actually encourages the development of new methods for predictive numerical modeling for the error-estimated uncertainty quantifications. If you will be discussing non-equilibrium processes on the experimental side, these are very challenging problems because they are hard to systematically study, implement, and control in a well-controlled laboratory environment due to the high sensitivity and transient character of the dynamics and this sensitivity and transient character of the dynamics impose higher than usual requirements on the accuracy and special temporal resolution of the measurements as well as on the data acquisition rate. And in fact, we are now returning back probably in terms of the science development to the questions that have been asked by scientists maybe a few hundred years ago. How can we ensure that essentially we are getting the opinion-independent result? How can we ensure which is the scientific result is supposed to be? How can we assure that we might eliminate the influence of an observer on the observation results as well as on the data interpretation? So, from this perspective, non-equilibrium processes is an intellectually leech and highly fascinating problem. Its exploration may have a transformative effect on our understanding of a broad variety of physical phenomena on fundamental principles of mathematical modeling as well as on the technology development. And with this idea in mind, we have been working pretty hard. I hope we have been productive, efficient, I believe, in our studies on non-equilibrium dynamics. We have obtained and reported solid and fundamental results and have made and reported discoveries. And about this, we will be discussing actually during the conference sessions as well as during the roundtable discussions and, you know, the public lectures, et cetera. So, we have organized now the six International Conference Tense University Program. We have certain objectives in mind which are listed here. It's actually, and they are tightly connected to the goals of the problem. Our TMB conference actually consists of invited lectures that are usually, you know, 30 to 35 minutes long. Contributed talks that are from 25 to 30 minutes long. From posters, there will be one roundtable session on Thursday as well as open discussion. What the roundtable is, it's an informal discussion under which, at the end of which we are going to elaborate a set of action items to proceed further. And in addition to the roundtable posters, we will have also the two competitions, which are the competitions for the young poster, for the young scientist award, which is here, as well as the competition for the best poster. So, as I mentioned, you know, as a community, we are a part of a TMB community that unites currently about 2,000 researchers. Our mailing list contains over 5,000 emailed addresses and our community is growing while it still actually remains pretty, I would say condensed because its natural bounds are set by the requirements of academic credentials and quality of research. Now, at TMB 2017, we have about 150 people. We will have 195 contributions of 375 others, and it includes about 50 to 60 invited lecturers. So, all our participants include students and young researchers, and 30 of them will be competing with the oral presentations and 30 of the poster presentations for the best poster and actually TMB for your award. We have seasoned scientists, members of the leading scientific institutions, as well as the National Academy of Sciences, industry and high tech, as well as worldwide. And then we have such a broad international community, one of the important words that actually drives the behavior within this community is a key word for us, is essential respect, because, you know, we studied challenging problems and we would like to be respectful to our problems, to our results, to our work, and to one another. Here's a list of the organizers of the TMB program, which includes members of the Organizing Committee, Young Scientist Award, Best Poster Award, as well as Financial Support Committee. There is a member of the Local Organizing Committee, Ken, which is Marcelle. You have... Yes, I'm nervous so much, but actually, my point was that if you have any questions, then follow the members of the scientific advisory committee because of course we keep everyone who has been ever involved in this. And I can't ask you to applaud him, because here, while, you know, he formally will not be associated with the team. Okay, thank you so much. And also, we break the logistic systems from the ACTP because this is a really unique place worldwide and it's essentially always was and probably will be the point of East-West communication, especially these days, because the, you know, overall political situation becomes more complex, but ACTP in this situation remains the place where people really focus on scientific development, regardless the country of origin, gender, age, you know, and other sense. And we would like to express our gratitude to the Conference Support Office and Ms. Federica Del Conte, who did enormous really work because organizing, you know, 150 people, it's a huge work, usually standard ACTP activities are smaller in size, it's probably one of the biggest. To financial office, who usually very accurately handles all the finance that is provided by the sponsors as well as the ACTP. The Visa Office and Mr. Adriano Maggi, who helped many of you to get the visa on time. Housing Office and Ms. Titzana Batazian, Ms. Elizabeta Capella, as well as Publishing Office and Ms. Sabrina Wiesentyn and Ada Arden Ateli, Computer Office and Johannes Grasbenberg, who works at the ACTP for a very, very long time and, you know, many of you might just, you know, have an internet access to his help, as well as to the Science Dissemination Unit and Dr. Enrique Canezza, Carla Fonda and Marca Zanara. And I kindly point your attention that all the ACTP, all the presentations that will be handled here as well as in the earlier lecture hall, they are video recorded and they will be available online essentially one hour after the presentation. So we essentially have a live broadcast. We also gratefully acknowledge the support of the National Science Foundation, Department of Energy in the US, the International Center for theoretical Physics in Italy, the New York University and the University of Western Australia. So, TMB 2017 is organized to advance the state of the art in our understanding of fundamental physical properties of non-equilibrium processes, to have a conspicuous, positive impact on their predictive modeling capability and physical descriptions, and overall to have a better control and understanding of these complex processes. As usually, the success of the TMB program, because it's a program organized by scientists, for scientists and by scientists, consists just from the successful work of all of us. And I strongly encourage, and you are strongly encouraged by the organizing committee to highlight the strongest parts of your work when you are giving your presentations and when you are discussing your work with your colleagues as well as the students. So, welcome to the TMB 2017. I would like to actually also return maybe to remind one of the first letters which has been sent to me back in 2007 by Eugenie Meshkov. Yes, and I'm very glad that now in 2017 this will be the first time when he attends this conference. So, yes. Regarding the formal structure, I kindly ask you to open now the program and to proceed with me such that I could explain which is which, where and how. I do not have questions anymore. The program, this is the booklet. The program usually lists only the title of the talk and the name of the presenter. The booklet, there is a complete list of the contributors as well as the institutions. The booklet, there is a list of authors which you can find in appropriate presentations. The presentations here are scheduled as separated by the TMB scenes. They go by the last name of the structure before our standard routine. Actually, we have our sessions running from 9 to 10. 10 to 10, 30, there is a coffee break. And from 10 to 12, 30, there is a two-hour session. Then there is one hour and a half for lunch. From 2 to 4, 30, there is a coffee break. And then there is an afternoon session from 4.30 to 6. There are actually a few exceptions because while we try to keep the term of time of the coffee breaks and time of the lunch, the more or less set, they are not set in stone. In fact, we do prefer for people to spend time of the coffee break maybe as short as 15 minutes. If the coffee is available, it will go to the left-hand side to the first floor, there is a bar and also over there, there is a coffee machine. Now, we will have a parallel session. We will have parallel sessions, unfortunately, even though we will try our best not to have them. Yes, I will do it. Thank you so much. We will have parallel sessions. The parallel sessions will not be run, so there will be the sessions that don't have parallel sessions like this session, and there will be the sessions that will have a parallel session. So the parallel sessions, they will be scheduled, they are scheduled in earlier lecture halls. So in order to go there, we just need to go from there, from this hole to turn the right and follow the signs. It's a smaller lecture hall, however, and the presentations that are scheduled there, as a rule, there are several sessions which are dedicated to the TMB for use award, as well as there are other sessions where people present longer 25-minute talks as regular contributions. I strongly ask you, especially our invited lecturer, strongly encourage you to attend the TMB for you sessions. They are marked separately in this program, and please let us know your opinion about the works because the students and young researchers, they take the TMB for you competition very seriously. There will be the poster session, which will be in the poster hall around the main lecture hall on Tuesday at 5.30. There will be one public lecture, which will be given by Dr. Srinivasan, and it will be not only for us, but also ICTP-Wide, which is scheduled on Wednesday from 2.00 to 3.00. And there will be also the roundtable discussions, which will be from 5.30 until 7.00 p.m. in the Oppenheimer room. It's those who have attended the TMB conference in the past. You know what it is. It's actually the room that has a real roundtable. So we will be sitting around this roundtable. And we will have a banquet from 7.00 to 9.00 on Wednesday. And we will also have a reception, farewell reception on Friday from 7.00 to 9.00. So I think we are ready to start. And I would like, as a continuity of the session, to invite Professor David Campbell to start his talk. And I kindly point your attention that Professor David Campbell was one of the at our 2007 conference. Better movements. You're reading my title? Yes. I'm reading your title. Because these days he will be discussed, Professor Campbell from Boston University will be discussing intermittent mind-body dynamics at equilibrium. Thank you, Susnana. Thank you for all here. So good morning. Good morning. Good morning. Et cetera, et cetera. It's a real pleasure to be here, to give a talk at this turbulent mixing and beyond. As Susnana said, I gave a talk at the very first conference. And I actually talked on the same subject, the Fermi-Postulon problem. Although it's not apparent from the title. What that indicates is not that I'm stuck in one area, rather that that problem is so important and has been so influential that it's influenced the study of nonlinear dynamics and non-equilibrium phenomena since it first started. So let me first mention that this work was done in collaboration with Carlo Daniele and Sergio Sege Flach, and has already been published in Fisbeve. And at the end of the talk, there'll be a short quiz and you're supposed to tell me what those two fingers mean. So I want to talk about the context, because the equipartition and the thermalization in many body systems, and it was focused, I want to focus on the first study that really showed that this might not be a trivial problem, the pioneering work of Fermi-Postulon and the failure to observe equipartition. From those original simulations, there are four questions that arise immediately. I will have only time to focus on only one today, and this is what are the origin of these Fermi-Postulon recurrences, which I will tell you about. What happens if you crank up the energy density, that is the energy in the lattice per particle? Will the original data go to equilibrium? And finally, and this is a subject which is relevant to our non-equilibrium dynamics, how do you find equilibrium and what are the dynamics or what is the dynamics at equilibrium? Since I want to make sure I get to the results, independent of time constraints, what we find is that even in equilibrium, the FPU dynamics exhibit very large fluctuations, you can call it intermittent dynamics, which are driven by some sort of stickiness due to coherent structures that exist in normal mode space, so-called cube breathers, which I will explain briefly. And these cube breathers also explain the recurrences. If the coherent is not trivial, it's quite complicated. Thanks to a referee who challenged whether our results were general, we're able to show that in the Klein-Gordon system, which has a different form of non-linearity, again, these coherent structures cause sticky dynamics and these are intrinsic localized modes, so this is what I hope to cover in the next 35 minutes. So here are the three gentlemen, Fermi, Pasta, and Ulam. In 53, working at Los Alamos with a machine called the Maniac for mathematical analyzer, numerical integrator, and computer, they tried to study the question of how equal partition and thermalization occur in a very simple classical non-linear dynamical systems. Now, for those of you who have seen the movie Hidden Figures, or know the book Hidden Figures, this is the hidden figure in the FPU problem, Mary Singleton, she was Mary Singleton, she actually wrote the code that they ran, and there's a great article by Thélie Dachois in Physics Today describing her role. The Hidden Figures, of course, were the computers, the African American women who worked in the space program, but she was a hidden figure in the FPU problem. So this is a picture of the original L.A. U.R. preprint, the last from Los Alamos National Laboratory. I actually had the last paper copy that they distributed. Fortunately, I made copies of it before somebody took it. So here's the original preprint. Studies of non-linear problems won, you'll notice that. Here are the authors, and you'll notice that there are Fermi, Pasta, and Ulam, and there is Mary Singleton. She did the work, but she didn't write the report out, so it's the FPU problem. This is the abstract from that paper, and the important thing here is to notice that the key conclusion is the results show very little, if any, tendency toward equal partition of energy and freedom. This system did not go to thermal equilibrium, did not go to equal partition, and the question was why? And of course, the fact there were no further studies is sadly led to the fact that Fermi died in 54, and therefore the work was not continued. I won't go through the whole preprint, but I want to call out a couple of things. Here, notice the aim of establishing experimentally the rate of approach to equal partition of energy among the various degrees of freedom. This is experimental mathematics. This is the first time, really, that a computer was used to study an analytic problem that was untouchable, sorry, a problem that was untouchable analytically, and let me read you, if I can hear, what von Neumann said. Our present analytic methods seem unsuitable for the solution of important problems arising in connection with nonlinear partial differential equations, and in fact, with virtually all problems in pure mathematics. Really efficient high-speed computing devices may, in the field of nonlinear partial differential equations, as well as many other fields which we now are difficult or entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress. So von Neumann wrote that in 1947, he knew the purpose of computing was not numbers, but insight. And one final point here, notice that at least Mary Tzingo got an acknowledgement for the work she did in coding the problem. Okay, so how many people actually know in detail what the FPU problem was? Wow, two hands. Okay, I'm glad I have, I am very glad I put in this introduction. So this is the FPU problem. It's a nonlinear chain, so I should say sorry, it's a chain of masses coupled by nonlinear springs. So this is a, so we have 1, 10 fixed walls and they studied n equals 32 and 64, mass m coupled by potential v, and of course v, the simple term here is Hooke's law, linear restoring force, but these cubic and quartic terms are nonlinear terms. Now FPU knew, it's easy to prove, that if you have only the linear term, the system could never go to equilibrium, you know that, you analyze, Fourier analyze the initial state, the Fourier coefficients stay the same, they just rotate with phases, so you'll never go to equilibrium. They knew that, but they thought that adding this nonlinearity would lead to immediate, almost immediate, equilibration. And this is a quote that Ulam wrote in Fermi's selected works. I'll take you, you can read it yourself, but the point is, what they found was a very surprising result to Fermi and he called it a little discovery in providing limitations in the belief that any kind of mixing and thermalization would happen as soon as you put in nonlinearity and this did not happen. So what actually did happen, how we study it? Well typically, we would study a system starting from its linear limit, so the two coefficients from the cubic and quartic terms are said to be equal to zero. We then get a simple equation of motion, which we all record linear, this is the linear phonon dispersion relation from solid state physics and if we look at assuming that yn has this form, we put a lot of spacing for convenience, then provided that we have this dispersion relation relating omega, the frequencies, to k, the normal modes or momentum, if we have this dispersion relation this problem can be solved. So now, if we have a weak nonlinearity, alpha, let's look at only alpha, beta is zero, and we expand in terms of normal modes, then this separates the problem into weakly coupled harmonic oscillators. Here are the harmonic oscillators, omega k has that sine dependence, so these are uncoupled, and here's alpha and there's a coupling coefficients which can be calculated, but coupled with normal modes. So for stronger nonlinearity, FPU expected that whatever the initial conditions, the normal modes would share the energy equally over time. And so the question is, what actually happened? So for n equals 32, I'm going to show you some results of the so-called alpha model excited in the lowest normal mode. Only the few lowest modes were excited. In fact, in the FPU problem, only modes one to five were excited. The original study, hand plotted, reprinted in Fermi's collected works, and it's a little hard to see, especially from the back, so let me walk you through it. This is the initial state all the energy is in the lowest mode, the simple sine mode that goes to zero at both ends. It starts there, precipitously this falls down, mode two starts up, mode three, mode five, looks like mode four comes biggest here, then mode three, then mode two, and then mode three, mode four, and then we come back to where mode one comes almost back to its initial state. This is in a very short period of time, you can't read this, but this has n equals 32 parameters, alpha is a quarter, and the delta t squared for the numerics was one-eighth. So you see what we have are recurrences in the FPU problem. This is the same problem, so it's a stronger non-linearity, you would expect this therefore would go more quickly to equal partition equilibrium. In fact, the repeat cycle, the referendum on recurrence is much closer in time, it comes sooner. These recurrences are to 97% the initial energy is back in the initial state. Now, in case anybody is wondering whether this is a Poincare recurrence in a non-linear dynamical system, Poincare recurrence is somewhere over in Slovenia. So this is a very, very surprising result, a shocking result. It really is if if you show the movie of a glass of water having been poured out, reassembling in the glass, going back to the initial state. Now maybe they didn't run it long enough. So in fact, later, and now Mary Menzel, Mary and single Menzel published the result themselves, they ran the FPU Alpha problem much longer and what they found was not an approach to equilibrium, but a super recurrence. So instead of having 97% of the energy back in the initial state in a finite time, much shorter than the Poincare recurrence time, they went to almost 99%. So these FPU recurrences were a huge surprise. They actually led to the whole development of nuclear science, to the development of the concept of solitons and Hamiltonian chaos from this. I don't have time to go into that in detail, but I will say that the FPU dynamics were well. So the point is they didn't absorb your partition, the energy stays localized. These unfortunately, thanks to Bill Gates, my simulations here are not active in the actual PowerPoint presentation. So what I'll do if I have time later on is show you some actual time evolutions of the motion in normal mode space and the motion in real space and you'll see that these things stay localized in normal mode space. So here are the three questions, or three of the four that I mentioned before. What is the origin of the FPU recurrences? Most people are familiar with the ideas of Zabowski and Kruskal, who in 1965, actually developing the word soliton, showed that the FPU problem was closely related, if you took a continuum limit, to the Kordaveg-DeVries equation, which is completely integrable. So in that sense, the Fermi-Postulon problem was in some sense for low energies close to an integrable system, and that's why it didn't go to equilibrium. Actually, a more precise explanation, due to Flock and his collaborators in 2005, called cube breathers, these are linearly stable, coherent structures in the FPU system in momentum space, in normal mode space. So periodic motions in normal mode space that are linearly stable, and they actually explain the recurrences quantitatively in terms of times, when they occur. A good question is, what happens as you increase energy? It turns out that's well studied. This was the original work by Izraelov and Chirikov in 65, which led to using the Chirikov overlap criterion led to the understanding of when you should have stochasticity and approach to equal equal partition in FPU. And then there have been many, many studies taking the original FPU data, not increasing the energy. Original FPU data will the Fermi-Postulon problem reach equilibrium. And there are a huge number of studies. The answer seems to be yes, but on an exceptionally long time scale. So now let me give you the warning because it's easy to confuse. These so-called cube breathers have nothing to do with quantum mechanics. They are coherent structures that are localized, typically exponentially, in normal mode space around some particular normal mode. So there are exact solutions, periodic solutions linearly stable around that involve some modes around the normal mode but are exponentially localized around that mode. Okay. So now let's do some serious work trying to see how we define equilibrium in the FPU problem and how we would approach it. Well equal partition means that all the modes must have the same energy on the time average, obviously not instantaneously, but over time the average would be the same. So here is the energy in a normal mode its momentum and its frequency times its location our frequencies have the same the same structure as before. This is the energy of an individual mode compared to the total nu k. So we will track two variables the participation ratio which just is the inverse of the sum of these squares. This tells you how the energy is divided up among the different modes if they're all equal and the participation ratio takes a simple sign simple structure. And then a logarithmic like quantity sorry an entropy like quantity which is defined by s of t which is nu log nu and we'll look at the normalized version of that which is s of t minus s max divided by that, eta is limited between 0 and 1 by its definition. So these two things are going to be our measure of whether we're at equilibrium. Now what do we mean by equilibrium? What we mean is that the averages exist, this is our definition of it you can argue about this, but this is a standard definition, that the averages exist and that they can be computed from the Gibbs distribution. So I won't go through the details, I have it on slides if people want to see the gruesome details, but if you calculate these quantities p and eta for Gibbs distribution then you find that eta is 1 minus the Euler constant over the log of the number of particles in the system minus s0 and the inverse participation ratio is 2 over n, where n is the number of particles in the system. Question of course, sorry I couldn't hear yes whereas what? The temperature? So this is a confusion this beta is 1 over kT, the other beta it doesn't have to appear it doesn't have to appear you have to do the calculation I can show you the calculation but you know the fact that a system can go to equilibrium at any temperature at any temperature you can define equilibrium and if you define equilibrium in terms of the values of these two variables in a Gibbs ensemble you get those results beta is 0, that's infinite temperature infinite temperature may be an exception, but I show you the calculation, it's surprising but if you think about it at any temperature a system can go to equilibrium that's the point and this is the result for the FPU problem independent of temperature sorry? for the selected? I mean you can define the same thing for 5 fourth, the point is if you define these quantities this is a participation ratio of the individual energies relative to the total and this is an entropy which was first used in the 90s by Cassetti and Poinot and the Italian groups you will find just by straightforward calculation that those are the values I admit it's surprising but if you think about the fact you know a system can go to equilibrium at any temperature then it's not so surprising I'll show you the gory details in private if it were so assuming that's true let's consider the manifolds on which eta and p, the averages over time achieve their equilibrium values which we'll call fn f of eta and f of p now these manifolds co-dimension 1 manifolds so they're generalized ergodic Poincare sections and what happens as a given dynamical trajectory moves through this very high dimensional phase space and equals 32 64 dimensional phase space it will pierce these manifolds an infinite number of times as the trajectory goes to infinity the time intervals between the piercings will contain information on whether and when the dynamics visits a sticky region of phase space if you have a very long time integral interval between two passages you know you've gotten stuck somewhere away from equilibrium there's something sticky out there in the original fpu problem n equals 32 eta should have this value 0.1218 and 1 over p is n over 2 so it's 16 so these are the values that we should see if we run an fpu simulation to equilibrium if it had equilibrium right? everybody clear on this? because otherwise I've lost you right? okay oh yes the next slide takes a long time to come up because the it's printed in details so these are actual from our runs of the values of eta instantaneously and p instantaneous values are shown in black which is why that slide took so long to come up because it's a bitmap the average values over a time window are shown in red and you see the eta you can show it starts at 1 and p starts at 1 comes down over time and the average eventually goes to this value which is in fact this dotted line is 0.12 0.1218 value okay p goes up to 16 you can see that the different colors are for different values of epsilon remember epsilon is the energy density the energy you put in the total system divided by the number of particles and fpu energy is was very small for larger epsilon you put more energy in the system it goes faster and faster to the equilibrium value right? that's the blue curve the red curve and the green curve you see small fluctuations around equilibrium and the magenta curve this is log 10 in time scale so this is 10 to the 10th units and the fpu epsilon is 4 times smaller than this epsilon here in magenta so it's way out here when it will finally approach equilibrium here's actually a calculation of the time to the equal partition using the t eta this is the eta variable going to its equilibrium value versus epsilon so here's epsilon in log of epsilon the open circles are data from paper of casetti at all I'll show you the reference in a minute they sort of predicted that fpu would be 10 to the 11th if you extrapolate this curve you get 10 to the 11th the black field data black field circles are our data again we find our guess is that tfpu is 10 to the 15th which actually would be actually to get to that value would require depending on how you do it between 14 and 80 years on our cpu or 3 months to a year on a gpu cluster these are very very very long time scales and that's of course assuming that there isn't another see this is the vertical this is the value of the fpu value of epsilon what they actually study is out here and you see it's going to be at least 10 to the 14th and notice that there is a kink in the slope that casetti at all did not see right here at about point 0.01 which makes our data go at a rise faster and we can't say for sure that there isn't another kink there so the ecopartition time is well beyond the scope of current calculations to make the value that fpu used now of course going back to this picture it'll come back yeah I'll come on back well I'll ask you to remember the picture where we saw that as you increased epsilon you went faster equilibrium this is the casetti at all reference it's a central reference in this field anybody wants to copy it down I'll leave it up for a couple more seconds so the message is you could tend to equilibrium but one of the initial important calculations grossly underestimated the equilibrium time and it looks like it's a lot longer so this is probably one of the two most important slides in my talk for the new results what we see here are the time intervals these are the time intervals between subsequent piercings of the manifold remember I said you have this manifold around and it pierces it these are the time intervals this is the interval number so t1, t2, t3, t4 and you can see that there are long excursions in phase space so here's an interval this is in time, this is the value of eta and you see that here's the equilibrium value for this whole long time period which is of order 10 to the 6th 10 to the 5th the system is out of equilibrium question? okay so if you plot the distribution functions for the return times of eta as a function of r of t you find that you find a power law tail for this distribution this is a fit of the power law which depends on epsilon for large values of epsilon it falls off rather rapidly so you don't go away from equilibrium very long and the first moment is finite but the second moment diverges if gamma is too small this is also true in the correlation functions now the difference between the red curve which is where you have eta above equilibrium value and the blue curve where eta is below its equilibrium value is pretty obvious physically namely it's much more likely the system will be away from equilibrium than more equilibrated this is more equilibrated than the value of eta you expect naive the thing that Kazeti did they assumed eta equals 0 was what they were looking for and that's not correct it's the 0.1 to 1a value so what is the origin of this stickiness? here's an example of the in normal mode space so there are 32 normal modes 1 to 32 here's the time and here's what you see excited you see strong localization in normal mode space around a high value of k this is for epsilon fairly reasonable and there's a definite localization in normal mode space the scale is such that the light colors are way up here at the top indicating that the energy is highly localized in one of these cube readers there are similar results for 5 fourth classical 5 fourth field theory on a lattice obeys this equation notice that the non-linearity here is what we call on-site it doesn't depend on the difference between two sites but it's on-site so it's a different kind of non-linearity so it's a different system in this system this supports so-called generalized modes or discrete readers here's a picture from an article that Sergei Flaouf, Yuri Kipscher and I wrote for physics today and I'll go through it in some detail what you see here is the 5 fourth equation for a certain value of delta x I think it's actually delta x equals 10 so it's a very large value I should say obviously as delta x goes to 0 this becomes a continuum equation this is just the discrete version of the second derivative so it's a partial differential equation but for large delta x this is a couple lattice problem like the FPU problem and here is the region as n goes to infinity this region becomes dense with the phonon spectrum and this is a phonon spectrum it has a finite mass gap from about the square root of 2 to the square root of 2 plus 1 over 10 squared square root so there's a finite region in which the phonons can exist is everybody happy with that? these are delocalized linear excitations and these arrows here are what's called omega breather these are the breather frequencies and what you see here are two kinds of breathers stare at these for a minute and tell me what you think the characteristic of this breather is and the characteristic of that breather so in this one what do I mean by that? this is the amplitude of the solution at this n equals 0 here is the next neighbor and then 0 beyond that here's another one amplitude of two sides what you see here is an optical like alternating discrete breather which has a frequency above the continuum here and an acoustic breather which has a frequency below the continuum these are discrete systems they are localized exponentially localized on the lattice now notice that if this frequency which is about 1.3 is below the continuum and all harmonics of it are above the continuum there's no way it can couple and it can't decay there's no localized mode in 5 fourth similarly this state is already above the continuum so all its harmonics are above it so it can't decay so these are stable modes and exist actually in any spatial dimension if we actually do the same simulations like as in the FPU case for the 5 fourth case we see the strong organization now in real space so this is now lattice sites 1 to 32 and we see that in this case at about n equals 22 it has most of the energy and it exists there for quite a long time and again the pdf of the distribution is power law this time as epsilon increases it gets smaller but the result is the same in terms of localization that there are these localized excitations either in normal mode space the cube breathers they continue the 5 fourth breathers which account for the stickiness and the deviation from simple dynamics at equilibrium so let me summarize I'm going to finish on time maybe a little ahead I introduced the idea maybe a little too fast of the famous Fermi-Pasta-Ulam recurrences and the FPU paradox namely that the system did not go to equilibrium we gave a natural definition of equilibrium in terms of the observables and whose behavior can be followed in time both eta and p we can calculate them at any time on any trajectory so we know their values and we found that they go to an equilibrium value but when they go to that equilibrium value there are non-trivial intermediate dynamics where the system deviates goes off equilibrium in a matter of time and that's due to stickiness of orbits that are related to locally stable coherent structures in both the Fermi-Pasta-Ulam case and 5 fourth we've actually tested this method with other observables and can be extended to other models and so what we have actually is a new quantitative way to probe out of equilibrium dynamics and the relaxation to equilibrium in high dimensional systems here again is the reference and with that I'll thank you and thank my collaborators Carlo and Sergio, Sergei thank you