 Good afternoon, everybody. I'd like to welcome everybody to this afternoon's College of Science and Engineering Colloquium. We're honored to have with us today one of the most famous physicists, not only in Latin America, but in the world, in the area of statistical physics. It gives us great pleasure this afternoon to welcome Dr. Constantino Salas to the American University of Armenia. Professor Salas is a pioneer in applying statistical physics to fields in which it's not usually applied. Complex systems, critical phenomena, biogenesis, chaos, nonlinear dynamics, immunology, economics, cognitive psychology, population evolution, and many other fields. Fields where if you open up a statistical physics textbook, you won't necessarily find things about these fields. And I'll let him go into why and what he's doing differently. Dr. Salas is based in Rio de Janeiro, where he's head of the Department of Theoretical Physics at the Brazilian Center for Physics Research. He's also the head of the National Institute of Science and Technology for Complex Systems, which has researchers spread throughout Brazil. He's also co-director of the International School of Complexity in Erije, in Sicily. He was born in Athens, Greece, and grew up in Argentina. He attended university in Argentina, Bariloche. He did his graduate work at Sorbonne University of Paris, where he obtained the title of Dr. Deta de Science Physique in 1974. He then moved to Brazil, where he lives today. But he also, during that time, held post-doctoral positions at Oxford University, at Boston University, and at Cornell University. He has received numerous international and national distinctions. He's a member of the National Academy of Sciences of Brazil, as well as the Academy of Economic, Political, and Social Sciences of Brazil. He's also the main editor of the El Sevier Journal, Physica A. I'm actually an assistant editor of that journal, so he's my boss in that respect. He has received four titles of Dr. Honoris Causa by universities, four different universities in Brazil, in Argentina, and in Greece. He is the recipient of the Mexico Prize of Science and Technology, which is one of the most prestigious scientific honors, including scientists in all of Latin America, as well as Spain and Portugal. The president of Brazil also awarded Dr. Salas the National Order of Scientific Merit, the contribution for which he's best known, and which he will discuss today, is the generalization of statistical physics, namely the Boltzmann-Hiebs theory of statistical physics. And the goal here is to extend statistical physics to non-equilibrium systems, to systems that are typically non-urgotic living systems with correlations, as opposed to just inert matters, opposed to the study and the application of statistical physics just to solids, liquids, and gases, as is usually the case. This generalization of statistical physics is being actively studied throughout the world. There are somewhere between 3,000 and 4,000 articles on Professor Salas's generalization of statistical physics by well over 5,000 scientists from around the world, and well over 10,000 ISI citations as well. Over 2,300 citations alone were for his original paper on the subject. He has spent much time traveling, speaking, and engaging in many research institutions throughout the world. He has coauthored a number of papers from his association with the Santa Fe Institute in New Mexico, where he has collaborated with, for example, Nobel laureate Marie Gellemann. The volume of his accomplishments is such that I could probably fill up the whole hour if I keep going through them. So I think that I won't do that. And instead, I'll just introduce his talk today. It's entitled Complexity in Natural, Artificial, and Social Systems. And he will describe the general idea that complexity in such systems is conveniently characterized by the nature and the strength of the correlations between the different parts of the system. Afterwards, we will have a brief question and answer session here. But then you are all welcome to join us up in the Akyan Gallery on the fourth floor in the west wing of this building. There'll be food there, too, and lots of opportunity for an informal discussion with Professor Salas and with each other. So without further ado, please join me in welcoming our distinguished guest, Dr. Konstantino Salas. I'd say it's a great pleasure to be here in Armenia, making some sounds. So it's a big pleasure for our variety of reasons, first of all, because when I was a child in Argentina in my family, we had a very good friend from my parents. They were together in practically every week. And they were Armenian. They spoke Greek, but they were Armenian. And I remember very well the face of the lady and her face I recognize in many Armenian women. Complex faces. Complex faces. Beautiful girl. And another reason is that since 16 years, I know the president, Bruce, or Hossian. So it's a pleasure to meet him here. Another reason is my father used to tell me stories about the sick road, about Armenia, about the Samarkand. So as you see, I have many reasons to be happy to be here. So let me start with this. And let me tell you the main idea of what I'm going to talk to you. First of all, I know you are from various fields. So I do not expect everybody to understand everything that I'm going to say. A few of them, a few of you will understand everything or almost anything that I would say. But some of you will understand only part. But it doesn't matter. Anyhow, we never understand everything. So if at the end of this talk, you have the feeling that you didn't lose completely your time, I will be very happy. So let me start with this concept. You see this table? So it has a surface, the surface of the table. And if I ask you, what is the volume of the surface? The surface is a planar object. What is the volume of the surface? Can anybody say what's the volume of the surface? Zero, because it has no height. So it's a kind of silly question with a kind of trivial answer. So it's not fun. Now I will change the question. I will ask what is the length of the surface, the length. Can anybody guess what's the length that feeds the surface? It's a field. Another silly question with a rather trivial answer. So let me now make the intelligent question. What is the intelligent question? What is the area of this? The area of this is, for example, two square meters. Not zero, not infinity. Zero and infinity are dreams. Two square meters is not a dream, is this? And it is different from the surface of that table that it has possibly three square meters instead of two. So if I go to a shop and I want to buy mosaic for the bathroom of my house, I cannot tell him. I want mosaic for zero cubic meters of bathroom, because he would say this guy is crazy. I have to tell him I want mosaic for 20 square meters, square meters, not cubic meters of mosaic. And then he goes and he brings the box. So who is telling us what is the intelligent question to put here? Is it us? No, it is the system. It is the geometry of the system that tells you. If you want to, you can put all kinds of questions. You can put stupid questions. But if you want to put an intelligent question, ask the area. That's the intelligent question for a surface. Well, what I'm going to argue, let me give you another example here. You have this object, which is a fractal. And you see it has, say, 10 centimeters. In the next hierarchy, you eliminate this. I think it's mine, but it's making a noise. But you know, these companies never stop. It's the Brazilian company just making me messages to buy things. As you see, they never stop. So here, you eliminate the center of third, and you have this hierarchy. Then you go further, you eliminate the center of third. You eliminate the center of third here and here. And you go down forever. And at the bottom, you have a fractal. It's a lot of points. It's not an empty set. It's a very fragile set, but it is not the empty set. So if you want to ask the size of these, are you going to say how many meters or square meters or what? The intelligent question for this thing is what is the measure in not centimeters, not square centimeters, but centimeters to the power 0.6309, because this number is the fractal dimension. So for this object, the intelligent question demands a funny measure, which is a fractal measure in this case. So what I'm going to tell you that the entropy, I'm going to talk a lot about entropy, the entropy of a system, which is a measure of ignorance on the system, the entropy, you are not going to decide what is the entropy. The system will tell you what is the entropy that you must use. System tells you. The geometry and the dynamics of the system says, if you want to talk with me with an informational tool, like Shannon entropy, you must use this informational tool. So it depends on the system, not on you, not on the commandments of God. It depends on the system. So this fractal thing, it has to do with what? With symmetries. So there is no translational symmetry. An example here, you take one piece of this and you translate it and you recover the entire work, the entire work. You have reflection, you have an axis, that's a different symmetry. You have rotation, like 60 degrees, a different symmetry. And you have a funny symmetry here, which is called scaling, scaling, scaling symmetry. So you take this, you amplify, you have this, and then you take a piece of it, and the piece of it has a similar shape to the whole of it. And then you take a piece of a piece, and it has, again, the same shape as whole of it. So there you have a lovely subject for the next time you're going to a part, to here, because we are accustomed that the parts are inside the whole. But I'm telling you here that the whole is inside the part. So what about that? The whole is contained into the part, to do as I want to understand better, these kinds of things. So there you know how these kinds of things work. And for example, here you have a fractal, and you take a piece of it, which is this, and you take a piece of the piece, which is this, and they all have basically the same shape. This scaling symmetry, subtle one, it is behind many things that I will tell you. It is behind many things that have to do with complexity against simplicity. And I will come back onto that. And first of all, talking about entropy, I know some of you have never heard about entropy. All of you have heard about energy. So because we hear many times the word energy, at the end we have the impression that we understand that's a human characteristic. You hear it a lot of times, and you think you understand it. So not necessarily all of you understand what is entropy. Not necessarily anybody understand what is entropy or energy. But energy is more common than entropy. But let us have a look what and reconfirming, Nobel Prize, one of the most distinguished physicists in the 20th century, what he says in his book, Thermodynamics, from 1936. He says, the entropy of a system composed of several parts is very often equal to the sum of the entities of the parts. He says, very often, an intelligent man like a reconfirming does not write very often just like that. It's because in his mind, he has a counter example. He has some subtlety. He has something in mind. This is why he writes very often. The not always is to what my talk is dedicated to. Have a look on this table. Those that know a little bit of mathematics, those who don't, it's a pity, because mathematics can be very useful. So those who don't, just study a little bit of mathematics. So this line is Boltzmann gives Shannon von Neumann entropy. It's the same mathematical object, which is minus the sum of probability, logarithmic probability. Are all equal in the central cemetery. In the center, you have also Johann Strauss, Schubert, and you have a study of Mozart, Beethoven, Beethoven. The central cemetery of the past in Switzerland. But it is double fun. The military is more fun than the Soviet. Although he has a lot of history. Yeah, sorry. Salon, like people don't do that. Exactly. I was telling right now to the president and to Michel that there is a passage, which is extremely interesting, in Faust. And that passage, Mestofeles, is explaining to Faust how the humans are. And Mestofeles knows a lot about humans, because he is the devil. No. And he says to Faust, the humans are like this. When they do not understand something, they put a name, and they are all satisfied. So you don't need to understand strictly, but essentially, because it has a name, and we are all satisfied, and we are talking about that. I'm going to refer, especially to this line, where you see immediately there is an extra letter, q. If in this line, you take q for 1, this line becomes this line. And therefore, I am not talking about an alternative to Boltzmann Gibbs for Norman Shannon Entity. I'm talking about a generalization of it, because it has the other one as a particular case. This line has been able to explain 140 years of physical phenomena. So this is a great line. We don't want to change this. We want to make it more powerful, to be able to address other systems in which that one fails. But so here, you have a function p log p, and this is a generalization of this function. And given a function, there is a trillion of possible mathematical generalizations. And for those who know a little bit of mathematics, it's very easy to generalize a function. So a general mathematician can generalize a function very quickly. So why this particular generalization, among one trillion of possibilities, why that one? That's a long story. But let me tell you that this one is a kind of minimalistic generalization of that one. It's like a zen generalization of the other one. You know, you touch just a little bit. So these two, they share a lot of important problems, like concurrency, I think, a lot of it. These two are very risky. These appeared in 2011 by a group from Japan, and these appear this year by a group from Hungary. But of course, you must violate something, because the only thing that violates nothing is p log p. So if you want to generalize, you must violate, but at least one solution. We're going to violate additivity. These expressions, additive, these expressions, non-additive, and I will come back on to this one. So violating additivity is, and I think Bruce Bogosian made many years ago a comment along that line, is similar. If you take Euclid axioms for geometry, there are five postulates. The fifth one has been an intriguing postulate during the century. The fifth one says, from a point outside a straight line, there is one parallel and only one parallel, otherwise. This postulate, if you violate it, if you abandon it, like here we are abandoning additivity, you get the so-called curved geometries, Lobacevsky, Riemann. And with the curved geometries, you can do a lot of things. And in particular, you can invent general relativity, which is what Einstein did in 1950. And by inventing general relativity, he made an intellectual, very interesting breakthrough. But for not saying that it is useless, general relativity has a practical application also, only one, which is in your cell phones. The GPS that you have in your cell phones uses corrections coming from general relativity. So if your cell phones detects a car here, the car is here and not there, because the relativistic the general relativistic correction is being incorporated through talented engineers in your cell phone. This for not saying that it is completely useless. So let me go on. And let me talk about this word and this word, additivity. So this is the one we're going to violate. So I'm taking the definition by Oliver Penrose, Foundations of Statistical Mechanics, 1970. So he says, an entropy is additive if, for any two probabilistically independent systems, the entropy of the sum equals the sum of the entropy. Remember the sentence by Enrico Fermi? If it is like this, it is additive. If it's not like this, it is not additive. SQ, the entropy I was talking here, this entropy, is contributed in three lines that it satisfies this. And therefore, SQ is non-additive, accepting Q equals 1, where this disappears. And you go back here, and you go back to Boltzmann Gibbs channel for Neumann, which is additive. So this is additivity. What is it, extensivity? Extensivity is much more subtle. Extensivity is the entropy of n particles, or the entropy of n degrees of freedom is proportional to n. If it is so, we will say that the entropy is extensive. If it is not so, we will say that the entropy is non-extensive. Now what is this? As you know, the science of nutritionists is based on the fact that if you eat an ice cream of 200 grams, I'm punching you because you don't have that problem. So if you eat an ice cream of 200 grams, you get inside the double of calories than if you eat an ice cream of 200. Sorry, I said 200. An ice cream of 200, you get inside the double of calories than if you eat an ice cream of 100 grams, the double. Not three times, not square root of 2, in fact square root of 2 is a very complicated concept for most nutritionists. It is exactly the double. So this is the basis of their sciences. Now you know what is the basis of the science of nutritionists. The entropy is proportional to n. Now, if you invent an entropy and you give it to me and you ask me, is it additive? I can answer to you in 20 minutes if it is additive or not. But then if you ask me, this entropy that I invented, for this system, for this system, because the professor also can be seen as a system. So this entropy for that system is extensive. I might be unable to give the answer in 20 years. This is a simple thing that you can answer right away. This is a very complex thing, because it has to do with the correlation inside the system of the elements of the system. This is subtle. So I would like that you understand the difference between these two words. And let me try to succeed to that with two examples. So this is the total amount of possibilities. This W is the W that is on the grave of Boltzmann. Boltzmann entropy is logarithm of W. This W is this. So this is the amount of possibilities. If you have n coins, this is 2 to the power n. Because it can be head and tail, head and tail, head and tail. So the whole is 2 to the power n possibilities. Now in such a system, arrives a new coin. And the new coin says, multiply your possibility, which is 2 to the power n. By the two possibilities that I like, which is head and tail, multiply. And the others say, no problem. This is a free country. We multiply 2 to the power n times 2, which gives you 2 to the power n plus 1. So they are all very satisfied. And they are very long. Because independence goes together with loneliness. Don't forget that. People like to say, I want to be independent. OK, but do you want to be lonely? Because they go together. Now a different system. Instead of being exponential with n, it is a power law of n, n say n squared. So here the coin arrives and says, multiply your possibilities by the two that I like, head and tail. And the others say, no problem. Here we have strong correlations. We have language. We have music. We have poetry. Emmy Dickens, John Milton. We have a grammar. Grammar is tentative correlations. We have Wall Street. We have living organisms. Living organisms are very complex. And they have big correlations. If I cut here the air like that, I cut it the air. And now, OK, it's the same. If I cut her arm, she goes to hospital and I go to jail. So she's not like the air. The air is a simple system. She is a complex system. So this case is very different from this. If you have this case, what entropy should you use? Boltzmann. Because Boltzmann entropy is the logarithm of w. And if you take the logarithm of this, the n goes in front. And it is proportional to n. And that's extensive. That's proportional. That's Clausius, 1850s. And that's OK. You say it's proportional. Here, if you use Boltzmann entropy, you apply log onto this. It's going to be proportional to logarithm of n. But thermodynamics says it is not to be proportional to the logarithm of n. It's to be proportional to n. So Boltzmann entropy for such a system by y. But if you take sq with q equal 1 minus 1 divided by rho, the same rho that you have there, if you use this that entropy is proportional to n. It's OK with thermodynamics. It is extensive. It's OK with Clausius. We are back. And you have another example, but it's not so important. So here, you use a non-adiddy entropy and non-adiddy entropy, because I told you sq is non-adiddy, in order that for that system to be extensive, so you take it non-adiddy in order to be extensive. Because if you take it additive, which is Boltzmann, it is non-adiddy. I mean, it's not what you want. So you see those two words are different concepts. Let me tell you more. This entropy, which is Boltzmann, is additive. sq is non-adiddy. I put here another entropy, which is also non-adiddy. If you have this class, you should use Boltzmann entropy, because it is extensive. And you should not use these or these, because they are non-adiddy. If you have this class, you must use sq. If you have this class, you must use s delta. So if you married Boltzmann, Gibbs, Shannon, von Neumann, you will stay here until the death separates you. But if you married thermodynamics and clauses, and let me tell you that it is a better match, you will stay here until the death separates you. So what am I talking? I'm talking exactly the kind of thing that happened 3,500 years ago, even before the first Armenian historical notices, we said today 3,000. Maybe not. But that one is 3,500 years ago. So there was a big pharaoh whose name was Tupmusi, the first. And there was a big pharaoh like Rancesi, the second. And not like Tutankhamun, who was not a big pharaoh. But it just happened that they didn't know the things. So Tutankhamun has become very famous, because it's around everywhere because it's so beautiful. But Tutankhamun was not a big pharaoh. Dalman was a big pharaoh. So here, you have the Nye, Egypt. And they had good scientists, the Egyptians, in particular, good astronomers. And the north was said, along the river. What river? The Nye, the only river in Europe. So north is along the river. South is against the river. And they were all happy. But one day, Tutankhamun took his army and he invaded Mesopotamia, the Euphrates. And there you have the Mount Ararat, you see, here. So he invaded here. And then they found the Euphrates. And the Euphrates runs the other way around. He runs from north to south. Here you have the Mediterranean Sea. There you have Cyprus. And the Euphrates runs like this. It goes to the Persian Gulf. And then, not only the astronomers were amazed, even the soldiers were amazed. What's going on here? The stars go wrong. Why? Because the Nye was a sacred river. So the Nye, it would go north. It would come inside the island sea, which was infinity. And then it would go up in the stars and make the motion of stars. The river was moving the stars. And it would go back down here to Black Africa and would run again. So everything was cycling and beautiful and perfect. But here, what's going on? So they did not understand. And when they came back to Egypt, they made this. There was a fashion at the time. They made an obelisk. And they wrote here that strange river, the Euphrates, in which when you go along the river, you go against the river. So what about that? When you go along the river, you go against the river. Incredible. Because they didn't know the Amazon. The Amazon does not run like this, nor like this. The Amazon in Brazil runs like this. Every child knows that. In fact, today, every child around the world knows that the flow of the river has nothing to do with the motion of the stars. But they didn't know. They thought it was one concept. The flow of the river and the motion of the star. But we know today those are two concepts. So when you distinguish the two concepts, you make an intellectual progress. This is extensivity. They are two different concepts. But if you have worked during 140 years with the air, you think they are serious, and they are not. Let me mention quickly a very beautiful result that was obtained one year ago by an Italian that works in Madrid. He established a connection between this entropy as q and the Riemann zeta function. The Riemann zeta function is the most important function of the theory of numbers because it is directly related to climb numbers, 2, 3, 5, 7, 11, 13, et cetera. So we still do not know what are the consequences of this. But this is a very beautiful connection between information theory and the theory of numbers. Let me change a little bit the subject that's still in mathematics. If you take the Boltzmann entropy, which is a sum of p log p, and you optimize it, you make what in mathematics is called an extremization procedure, you get a Gaussian, which is like this. That Gaussian happens to be the attractor of the central limit theorem. I will tell you more about that. The central limit theorem is called central because it is the most important theorem in theory of probabilities. So the function, the distribution which optimizes Boltzmann entropy, happens to be the attractor of the famous central limit theorem. If you optimize SQ instead of optimizing Boltzmann entropy, you get this function that I will call Q Gaussian. And if Q equals 1, there you have the Gaussian. Q equals 2, you have a famous function which is called the Cauchy-Lorentz function. And changing Q, you have all these things. Wouldn't it be nice if it was possible to generalize the central limit theorem in such a way that these functions are the attractors of the generalized central limit? We started suspecting that there was something like that in 2000, but we succeeded proving that it was like that seven years later. And these are the two theorems that I am referring to. That one was done with the collaboration of two mathematicians in the United States. That one worked during several years in the same Tufts University where Bruce Boborsen was the head of the department. And the second theorem is when the same people plus the mythic Marweger Man, who won the Nobel Prize and who predicted the existence of quarks. So we have inside plenty of quarks. So these two theorems are quite complicated to explain, but it's not so difficult to understand the logic of it. So the logic is the following. If Q equals 1, which means independence, if the variance, which is a statistical object, is finite, the attractor is a Gaussian. If it is diverging, the attractor is a so-called David distribution. Why this particular theorem is important? Because in nature, there are lots of Gaussians distribution. Lots. Why? We believe it is because of this theorem. Because the attractor does not depend on the details. So it's diverging. For example, if a lady comes in with Chanel's sink, a little bit later, we smell it. That's convection plus diffusion. Diffusion is Bromian motion. That's a Gaussian. If instead of Chanel's sink, it's another perfume, Christian Bjork, we also smell, again, another Gaussian. So many Gaussians. Another example, if you take the distribution of the darkness of the skin, like the concentration of melanin, it roughly follows a Gaussian also. If you take the weight of people, it roughly follows a Gaussian. And have you ever observed a miracle? If you take a coffee, it can be an Armenian coffee. It can be a Turkish coffee, a Greek coffee, or an American coffee. And you put sugar here. And you do a couple of times like this. And a miracle happens. The entire coffee becomes sweet. You put coffee here. And then 10 seconds later, entire coffee is sweet. That's incredible how this happened. Convection plus Gaussian diffusion, Bromian motion. So Gaussians are everywhere because of this view. If q is different from 1, you have strong correlations. And we prove here that the q Gaussians are the attractors. And therefore, there is going to be many q Gaussians in nature. I'm going to show you some of them. There are very many of them. I just selected a few of them. And probably I will have to cut half of the selections because of time. But I will show to you that this prediction says, you are going to find q Gaussians everywhere. And remember, q Gaussians are directly related to the entropy sq. But before showing you that, I would like to show a very interesting paper produced by the president of the American University of Romania. 16 years ago, I gave a talk at the Boston University. And there was an audience, like you. And there was Bruce Bogosian sitting in the audience. And he heard what I said. And you might think he's not quick because sometimes presidents are not quick, but he is quick. He went out of the seminar. And a few weeks later, he had produced this paper, which is the first experimental connection of the theory I'm talking about with experiments, with physics. The first one was done by him. It is this electron plasma. And there you have the experimental voids. And you see a black curve. But the black curve, he proved, corresponds to q equal to 1 half, which is not voids. Voids is q equal to 1. So it's a pleasure to present to you his result here, and his presence. Let me go back. I want to show to you some q gaussians in nature, and elsewhere. So this was done by Eric Ruebs in 2003. And he said that if you have cold atoms in dissipative optical lattice with a lot of lasers, the distribution of velocities is not going to be Maxwellian. It's not going to be a gaussian. He said the distribution of velocities is going to be a q gaussian. And he said the formula for q. So he did that in 2003. In 2006, that was verified in London by these people. So they verified the prediction by Eric Ruebs in quantum Monte Carlo, and even better in the laboratory with cesium atoms. Why did I say even better? Because I am a physicist. Maybe a professor who works with computational sciences maybe he considers even better value. I consider it even better here. Why? Because here, you control. You do what you want. And the computer does what you say. Physics is different. Physics is like this. You put cesium atoms and you do not control it. Nature controls it. So you're just waiting to see whether nature is going to like what you said or not. Most of the time, she doesn't like it. But when she likes it, that's a big day to say, I discovered a secret of nature. This is why I said even better. So they verified the prediction of Eric Ruebs, Eric Ruebs, three years earlier, even in the laboratory. Let me show to you this work that appeared a few weeks ago by Gilmar Gris from Spain and Tassos Buntis from Greece. They happened to be together visiting my group simultaneously. And so we did this little paper. And what is the paper? The paper is here. You have a map. You put x, y at time n. And so this little formula, you get x, y one time later. And you put it back. And you do it again, and again, and again. So you have a map. It goes somewhere in the space x, y. This map is nonlinear if mu is non-zero. So this map has a very special property. It is conservative. It is area preserving, which means for those that are familiar with the word Hamiltonian, this is like a little Hamiltonian with two dimensions, x and y. So we can see what the system is doing in the computer. It probably has only x and y. So you put upon it, and you see what happens. The air in this room, the phase space, has not two dimensions. It has the Avogadro number of dimensions. The Avogadro number is 10 to the power 23. It's a huge number. You will never see, we will never see, what makes a point in the phase space with the Avogadro number of dimensions. But we can see that one, because it has only two dimensions. We saw it. That's the paper. So we'll do a region where the Lyapunov x form is almost zero. And the area of x and y and the geomark is one point here, near the corner, near the center. And then she runs two to the power nine iterations, and it appears a kind of eight. And she continues a kind of eight. And she continues a kind of eight. And she continues this, and she continues this, and this. So up to the moment when we stopped the computer, this was not fully black. It was like an eight. If it was fully black, that would mean that the system is ergodic. And therefore, it follows Boltzmann Gibbs from Neumann Shannon Entry. If it is not all black, that means that the system is non-ergodic. And that's not a problem. So maybe, because this Lyapunov x form is not exactly zero, very close, but not exactly. Maybe one day, after the end of the universe, this will become all black. But we leave that problem for the scientists after the end of the universe. For the scientists before the end of the universe, like black, like that, this is non-ergodic. So why ergodicity is important? Because each of us is a non-ergodic system. Here is ergodic. And she is non-ergodic. And I wish you 100 years of non-ergodicity. You see, if you open that door, that window, the molecules that are inside, they go outside, they come in, they come out. They use all the possibilities. Because they are independent, standing alone. But if she goes there, she doesn't jump outside. She does not use all the possibilities. She only uses the possibilities that are consistent with her remaining as a living organism. So she selects very much what she's going to do. Do you remember the second class that I presented to you? They have stronger relations that they select. This is why languages exist. This is why Wall Street exists. So while this system does not like to be ergodic, what happens with the distribution of probabilities with the functions? That in this air, they give you a Gaussian. What happens? Well, here you have it. A Gaussian, it's a parabola here. This thing is not a parabola. So because the system is non-ergodic, it does not like Gaussians. I said a parabola because it is long linear. And therefore, a Gaussian here appears on something like that. And you see, this is not like that. It has an infection point. It follows roughly this black line. And this black line is a Q Gaussian with a Q for 1.6. All of them are interested by the worry that we do about time. We believe that clock doesn't work, but the time does work. So here you have the famous LHC. So here you have Geneva. And there you have the accelerator, the large carbon accelerator, which has 27 kilometers of perimeter. And in this, you have four big detectors, one, two, three, four. One of those four is the CMS detector, which you see here. You see a human being there? Or you see the size of the detector? This is the biggest machine humanity has constructed until now, the biggest machine in thousands of years. This CMS collaboration has 2,500 scientists from all over the world. And they have published several papers in technology. In general, why? Because when you have 27 kilometers of electronics, vacuum, and all kinds of things, you can publish papers in technology. You, your children, your grandchildren, there is work for all of them, publishing thousands of papers. But I'm going to show to you the first two papers in physics, not in technology. So the two first papers in physics are these two curves on this scale. So what is that you see here in log linear scale? In this experiment, at 7 tererelectron volt, which is a tremendous energy, they make collides, two protons. So two protons like this, they collide. To make them collide is a big mess, because the protons are so little that they go like this. And nothing happens. You want them to go like that. So you must have great precision to make them collide. When they collide, there is a terrible explosion. And it appears quarks and gluons. And those quarks are the quarks of Marie Guilman. And gluons and something happens there. And then suddenly, this soup of quarks and gluons explodes. And it sends out atoms. And the atoms go like four or five jets. And they measure the jet. And they measure the longitudinal motion and the transverse motion. And I'm showing to you the distribution of the transverse momentum. And those are the experimental points. And the black line here is a q exponential. It is the function that comes out from the q entropy. And q is 1.15. If it was q equal to 1, it would be a straight line here. But you see, it doesn't like q equal to 1.15. And it likes that in 1, 2, 3, 4, 5, almost six decades. Six decades is a lot of precision. But these people are very talented. And they went down here. They continued, measured it. And they obtained this. This is a very recent result. It appeared a few weeks ago. Here you have 15 decades. And this black line, this black curve, is the same q exponential you saw before. You saw it here in the beginning. Now you have it all. The same value of q, it's the same function. And it fits the data in 15 decades. 15 decades means a precision of 0.00015 zeros on the point. So here you see two absolutely amazing things. First of all, how can these people can measure something in 27 kilometers that has a precision of 15 decades? That's the first amazing thing that's incredible. The second amazing thing is how can this little function that works here, it works down there. So when I saw this figure, I saw it in a couple of months ago, presented by these two people who are from Poland and he is from the United States. I was absolutely amazed, my god, 15 decades. So we have not much intuition with humans. Maybe Mephistopheles has, but we don't have, about so many zeros. If I tell you 10%, you say, OK, I understand. If I tell you 1%, you say, OK, I understand. If I tell you 0, 0, 0, 0, 0, 0, 0, 0%, you don't understand. You don't see it. Yeah, you say 0, but it's not 0. There is a 1 later. So I decided to have some feeling about what is in 15 decades and what I did. I represented in log log HL Newton for the energy of Newton is p squared divided 2 times the mass. But Newton for is OK here, but here it starts being wrong. And there you have the Einstein connection, correction. This is Einstein for 1905. So this is Einstein for which shows that Newton was a genius, that he was not perfect. Einstein was another genius, that he was not perfect. And one day, when people arrived very high energies, Einstein would also fail. Why? Because any human intellectual construction at some point would fail. So where are we now? There you have Einstein's expression of the energy. If you take p equals 0, you have the only formula that reporters know, e equal mc squared. And what is the highest controllable energy that we have today? 7-terra electron volt. So there you have the curve for an electron, which is the same for a positron. And they can do collisions, electron positron if they wish. And everything is controlled up to 7-terra electron volts. So Einstein is correct up to here, after we don't know when they will fail. So up to here is how many decades? 7 decades. 1, 2, 3, from that point, 7 decades. So Einstein, the famous Einstein formula, is experimentally controlled up to 7 decades. What you saw here is 15. So when I made this comparison, you go to this equation. There you have a technological application. It's all rounds of women. And they, using the entropy SQ, they invented a procedure to process very kindly the image. So there you have a monogram. This is the breast of a woman. And there you see nothing. But then they process. And you see those little points? Those are micro-pacifications. And there you have another example. A micro-pacification like that one is half millimeter. So not only the origin ecologies, but the hypocrites could not detect a half millimeter micro-pacification. But with this image, you can't. So it's very timely that it's cancer. So it's doing its work. And it will kill the woman. So this is an important contribution they did. Because with this processing, they went from 80% of truth. They could go to 97% of true positives. So 17 women more that you can save. Even one woman is important. What about 17 in 100? And not only that, simultaneously, they decreased the false positives from 8% to almost 0, 0.4%. The false positives. What is a false positive? A false positive is when the doctor says, I'm very sorry, but I have to give you very bad news. You have cancer. So the woman said, ah, and she goes home and the whole family, ah, and they cry during 15 days. And then they receive a phone call from the doctor saying, I have good news. It was a false positive. You have no cancer. No, that's good news. But nevertheless, 15 days, the entire family was very, very bad. With their processing, you can increase the positive, the true positive, and you can decrease the false positives. That's a great contribution. Let me give you another example. This was done by four Chinese engineers. It was produced a few months ago, 2020. Here is for guided surgery. This, the surgeon, is not looking where he is cutting. He's looking the monitor. He's looking where he's cutting here. And in the monitor, he sees the tool. And do you know what is this? A blood vessel is inside. A small kind of thing. And look, you see everything. As if you were traveling inside there, you see this thing. You see everything. Here is color, inside. And here is bronchus in the line. Very tiny. And you see inside. Without cutting the person, you see inside. This is a line of somebody who does not smoke. If you smoke, this is very fuzzy. So don't smoke. Stop smoking. Very easy. Let me give you another example. Darman was done by 1, 2, 3, 4, 5 resilience. So this is a study of multiple sclerosis. I'm very happy. This is what Stephen Hawking has. So there you have a cut of the brain in the original image. And here you have the processing with the Boltzmann method, the Shannon method, the Q in column 1. And there you have the processing using Q different form. So you see much more details here than here. And one day Watson asked Sherlock Holmes, how do you find who is the murderer? And Sherlock Holmes answered, the relevant detainees. This is how I find the murderer. The relevant detail for making the diagnostic of these multiple sclerosis was this one here. Here you would see it, and here you would see it. Of course, the famous, maybe I can talk to the presenter or something, but I don't know. Ah, the cable. Oh, maybe. Oh, there you have the cable. Don't talk to him. Don't do nothing. So there you have the famous porcelain of Mimori, the famous dude. And one year ago, it appeared on this paper by these two French people who worked at the Centre Européandre la Ceramique, at Mimori. And they studied experimentally, seasonally planted zilconia. And this enters into that kind of porcelain. And here, by using the normal procedure, classical simulated maniading against generalized simulated maniading, this is Q equal 1. This is Q different from 1. So here you have the depth of the atoms. Here you have 100 samples, and they are slightly different. And here you have 100 samples. And a good restaurant is not where sometimes the food is good and sometimes the food is bad. Yesterday, we went to an Italian restaurant and Professor of Bohossian, he told me that here, they have a great steak, always. He didn't say, let us see, maybe today is a great steak, maybe tomorrow it's nothing. So a good restaurant is where always the food is good. So here, they are always composed, and they can sell it more expensive. And this is what they want, because they like maniades. So I went myself to March many years ago, and they have so beautiful dishes and things and porcelain and maniades. And there were some dishes, and they asked, how much is this? And they told me, I don't know, 50 French francs. And this, 120. And I looked at them, but they had to say, no, they are not the same. Put many of them on top of the other. So you put here, and it is like that. You put here, and I just, this is why this is 120, and this is only 50. So you choose. So this is why they are interested in this. Instead of this, because they can drive the car. How are we doing in time? How many minutes more? So this is a contribution by two physicists into astronomy. So this appeared a few months ago, 2012. They found two new laws in astronomy. And I learned today in the Academy of Sciences of Armenia that one of the great scientists of Armenia was an astronomer, an astrophysicist. So they found two new laws, which are these. They measured the, you know, the asteroids, their origins in the planetary system, which is plenty of stones, asteroids, thousands of stones. So they analyzed the statistics of those stones. And here you have the distribution of periods. Love, love. You have seen in the movies where there is an asteroid that is going on to the Earth. It turns around, because it turns around, it goes slowly, and it's going down. OK, this turning around has a period. And another asteroid has another period. And another asteroid has another period. So you have a distribution of probabilities of periods. And this is the black dots. And the red line here is a Q Gaussian with Q equal to 2.6. So this is a new law of astronomy. This journal, in fact, it has one of the highest impact factors in astronomy. And they found another law here besides. You can have an asteroid this size, this size, this size. And they made, they constructed the distribution of sizes of the asteroids. And those are the black dots. And the red line here is a Q exponential with Q equal to 1.6. So in physics, we are happy with these copies, because I can tell you, I don't control, and they don't control, what the asteroids do. Nature controls. So when you find that nature likes a Q exponential, I say, oh, I found that secret, nature. Another example, these appear this year, 2012. These people make physical approach to complex systems from fault. And I want to show to you one of their applications. This is the cumulative distribution of returns of the biggest 100 American combats. The distribution as a function of returns for different times. You know these people in Wall Street, and as we are long on stock exchange, they make buying and selling every 10 seconds, or maybe every 10 hours, or maybe every 10 days. So it depends on the time in which you act. By doing this, they have different times here. Four minutes, eight minutes, 16 minutes. All of them are contractions, with Q, which is given here. For example, that one is Q for 1.34. That one is Q for 1.25. The longest, the longest time, Q is the closest to 1. Why? Because when it is very long times, it loses correlation. And when it loses correlation, it becomes lonely. And when it becomes lonely and independent, it gives you another example. There was the doctor of thesis of Archita of Pagia in Notre Dame University. She studied this, which is about a 1 millimeter thick. So they were in a plate there, a lot of them. And she was feeding them a disgusting thing she was putting there. But they either really did something like this. And she was filming. And she was analyzing the motion of the cells of these things. And they do not move like the molecules in this room. Because the molecules in this room have Q for 1. It's a Gaussian, it's a Maxwell, which is Q for 1. That one does not like that. It likes Q for 1.5. You see, we experiment on data, and that line is a Q Gaussian with Q for 1.5. She looks for food. The same way, you look for the keys of the car when you lost them. How do you look for the keys? You say, I left them in the kitchen. And you go to the kitchen. No, it's not in the kitchen. I left them in the garage. And you make a big jump from the kitchen. You go to the garage. And you say, no, it's not in the garage. What is it? Or maybe in my room, and I put another jump. This is how you look for the keys that you lost. This is how she looks for food. You say, what about the Hydroville Decimale that had Q for 1? They all died. Darwin killed them. We only survivors are Q for 1.5. Because these, they find food. So you say, why don't they use more Q for 1.8, 2.3? Why not more? Because they are lazy, like all of us, like all living organisms. So even with Q for 1.5, she finds food. She's like, OK, why am I going to make so big jumps from the kitchen to the kitchen of the neighbor? Why? To the garage, it's enough. Let me tell you a little bit about black holes. So this is a famous paper of Stephen Hawking, the one which is Cambridge University, at the Department of Applied Mathematics and Physical Physics. And I gave a seminar there exactly eight days ago I was in Cambridge and I gave a seminar here. So because I became interested in the black holes and the thermodynamics. And what is the abstract of that paper by Hawking? Here, he denies Boltzmann use. He says, this means that the standard statistical mechanical-canonical ensemble cannot be applied when gravitational interactions are important. And he's right, but not. And Gibbs knew that one century ago. So he's right. But look what he says here. The number of configurations is that he denies Boltzmann denies use. And he uses Boltzmann's use there. So the conclusion is that not only students, people make mistakes, but also make mistakes. It's a very abstract, there isn't inconsistency because the distribution he does not want to use comes from here and he uses that. This problem has been tackled by many people during decades, like all of these people. This is another advice. No, that one, of the Michael Daff, he's fellow of the Royal Society of London. That one won a few months ago, $3 million, the price of that Russian that gave a lot of money to scientists. Well, and here you have a letter in nature whose title is, when entropy does not seem extensive. And it says, why the entropy is proportioned to the surface rather to the volume? If it is a three-dimensional object, it should be proportional to the volume. That is proportional to the surface, and therefore it's non-extensive. So during decades, people have been accepting this. And they say, well, thermodynamics, it's not appropriate for black holes. And you see here a paper in 2006 by these people. The area, as opposed to volume proportionality of black hole entropy, has been an intriguing issue for decades. So you don't understand why. And you have one more. One year ago, he's a great specialist at Manabara. And he writes, thus, the extensive property of entropy no longer holds. He's violating thermodynamics. But if you ask to 100 physicists, what is the science in physics that we live forever? In 100, probably 100, we tell you thermodynamics. Everything will change. So he's violating thermodynamics. Nevertheless, he sleeps at night. That's all. How long? So I attacked this problem because I became a deep myself. How do these people really violate thermodynamics? So we found a solution. I will not tell you the details. But the entropy to be used is not that important. It is this entropy as delta with delta equal to 0.5. And when you have this, you have a power of the famous Peckenschein-Corking entropy for power 3 half. This is proportional to the area. Power 3 half is proportional to the volume. And it is extensive. Now we can see that. So let me tell you the very last application of these ideas. That was done with two friends and colleagues in real life that appeared one year ago. And what we did, there are a lot of formulas. But I guess some of you have some contact with quantum mechanics. Did you take in this equation q equal to 1? Here it becomes 1. Feel 0, cancel 0. And you have the Schrodinger equation, the famous most basic equation of quantum mechanics, that you will find in the next one. So this is a generalization of the Schrodinger equation and nonlinear generalization of the Schrodinger equation. And for a three-part group, we found the exact solution to be not a plane wave, which is the exponential of i k x minus only a plane wave, but the q exponential, q plane wave. And this is the exact answer. I was very happy about this. Why? Because you know what is a plane wave? Something like this. It's wider than the universe. And it is older than the universe, a single plane wave. So no human being can have a plane wave in this box. This I do like. It was the time I was a student. Well, the q plane waves are like this. They are localized in space and in time. And therefore, you can have a q plane wave in your box. So I say that's the quantum-insert way of thinking. Yeah, not the quantum-insert way. It's a plane wave. The solution for a four-tube is not the quantum-insert way. Go on with that. So this is the exact solution. The energy comes out, the formula of Newton, because this is not relativistic. Planck relation is satisfied. The Broglie relation is satisfied for all values of q, not only q equal to 1, which is the case that you will find in the technical case. Then we said, let us make it relativistic. So Klein-Grover equation. This is a nonlinear Klein-Grover equation. We put inside a q-plane wave. Again, it is the exact solution. And what energy comes out? Einstein's problem. For all values of q, we said, let us take the q. Who is the q? We did that equation. Here, the equation is relativistic. It has spin. It has matter, antimatter, electron, positron. Here, the equation has everything. So let us take the q. So we took the q. Here, you have a nonlinear theory equation. The solution is, once again, the exact solution is a q-plane wave. And the energy is, again, the formula of Einstein, because it is Lorentz invariant. It is relativistic for all values of q, not only q equal to 1, as it is in the technical case. And I will stop here with the examples. If you want to know more, there are all these books. And if you want to know papers, you go to this address. You will find thousands of articles dedicated to this business. And let me end by telling you how this happened, how such things happened. And I'd like to end in homage of the process with a text in literature. But not the literature he dominates, which is English, but Italian literature, which is as good as the English literature. Mamma mia. So this was written by Marco Bersanelli, who is an astrophysicist from Italy in Milan. And this is a true story. Sophia is his daughter. So with his family, they were in Austria, and they were walking through a path and going out. And they found a lot of interesting things. And then at certain moments of fear, la picola di tre anni, si vuol only three years old, she said, mamma, mamma, una fragola, a story. She saw a story there. The other two children run, but it was too late, and the soirena had already cooked the strawberries. So they started looking, and they found another. The documents of what I've reached in the sonoracle three, quattro, the hunting was open. We were finding strawberries everywhere. So when they came back, they returned with my sincere surprise, doing the same path that we had done going out. We found a lot of them, zero strawberries going out, and 100 strawberries coming down, an effect statistically incredible. What had changed? Big stir of mathematical concepts and entertaining, nonetheless. Are there questions? We can take a few questions here, and then I suggest we go where the refreshments are. Could you tell us a little bit, sir, about what did you do? Yes. How far away are we at? How far away are you? Oh, by the law. You will not use it. I think he will not use it. She will not use it, but our grandchildren, grand-grandchildren, they will use it. How far away are we at? About the cube. So all the examples were larger than one, but not more than like two points something. What can you comment on that? Or about two, less than one? Yes. Well, first of all, I skipped an example, which has to do with quantum computers. I have there, but I skipped it because of mine. That example has cubed below one, so there is no problem if it can be below one. Now, as you know, this cube, many examples are going 1.5, but in those are q exponential, the maximum value of q. If they are q gaussians, the maximum value of q is 3. This is why you solve 1.5, 2.1. So the maximum is 3? For q gaussians. Is it because you cannot normalize? Because you cannot normalize. They are so slow going down that the area diverges. And if the area diverges, it is not q. The sum of probabilities must be 100. If you use infinity, that's something else. This is why there is an upper value of q, but there is no lower value of q. It can go down to minus q. It's an interesting question. Could you show the side of the bibliography? Yes. Would the presentation be available? I can give a copy to Professor Bokozian. There you have it. Yes. I was going to talk about channel centrality where you are brought to the infinity coordinates. Yes. Yes. And we are not switching them on. The value is distributed, which are taken by minus infinity. Yes. Provides the highest value of certain zero energy. Yes. Among both of these energies. Do we have similar phenomena in the ocean? Yes, absolutely. Very, very similar. And the same for this delta k? Awesome. This delta is not so fragile. This q is always concave. This delta is concave only when the number of degrees of freedom diverges. It's less fragile. It's still. And it seems to be the case of black holes. But of course black holes. So you don't care about if they are friends with monks. Other questions? Of course. Maybe a question. And we can see also other degrees. For example, you go normal since you have a number of physical functions. Yes. Yes. Yes, you can consider all types of distributions. But I will show you here the classes that optimize and handle but those that optimize the entropy. Because those classes that optimize the entropy they satisfy the generalized central new theorem and they are going to appear everywhere as you saw. They do. But the people have used other distributions. The q-fibre has been used by some people. Q-scratch explanations has been used. I have not seen nobody using the q-log normal but there is no problem. Excepting you cannot derive it from an entropy and possibly you can prove a central new theorem. So maybe there are a few but not many. It's about a little bit more cosmopolitan. It's about the sense of the time. The entropy, the notion of the tension, in many cases is related to the direction of the time. Yes. With the q different than one. It's the same. The flow is the same. It's a pity because if we could inverse that we would be all the younger, all the younger. But we cannot. I understand that there is no universe now, no problem about that. However, when you will be 80 years old you will start having some problems. But today I understand you have no problems. But with this question, if the time is kind of independent variable or it is derivative of a really mechanical motion, if it is related to the entropy in some sense if there is no motion there is no time. If there is no increase of entropy there is no time. If the q is smaller it kind of depends that it is possible that there are systems that there is no increase of entropy and then there is no time. For all value of q the second principle of thermodynamics is satisfied. There are maybe 20 pages in the literature showing that. But you put an interesting point there and maybe Bruce knows Michel Barranger. You know, at MIT. So I will give you an image that Michel Barranger gave to his professor from MIT today that he gave in a colloquium in Texas. So he was giving a colloquium and he said, you have a system that lives in a phase space. I showed to you a two dimensional phase space for your mother who is business and I told you the phase space of the air here is three years of dimension. So you have a box and you have a phase space and your system is here. And as a function of time because of mechanics, you don't have mechanics, quantum mechanics, you have business mechanics, we don't care. Because of mechanics it does like this. And as long as you follow it in box with the laws of mechanics you know where it is at the end of the disease and it goes and the end is always because you know where it is. But one day you become tired and you say to my brother, I don't want to follow this all day long and you say, I will change my description. I will say it is in the box. I will not say where. I will say it is in the box somewhere. At that moment you increase the energy. You increase the energy. The little thing does not care. Because the little thing only cares about nature, about physics, from Aristotle. But you decide to change the description and then the entropy increased because you lost information on the system. Because you don't care where it is. Do you care where these molecules are or do you care only that they are here so you can breathe? Not too hard, not too cold, that's what you care. You don't want to know this molecule this year. You don't care. So the engineer who designed the air conditioning doesn't care about the positions. He cares about thermodynamics to make an air conditioning like that or like that. So at the moment in which you decide to abandon the description and change it to a less informative description you increase the energy. But in reality it stays zero or a second? What is the energy? I'm telling you, if you want to follow it it is zero for you. But if one day you say I'm sick of this thing I will only say it's inside the room, that's all. Then you change the description, the entropy changed. The reality is... We were talking with the professor there about John Newton. Maybe he can say it better than me. He says, the one whose mind are humans have their own space and time and they can make from heavens, a hell and a hell, heavens that's in paradise lost, John Newton. So you decide whether you want to follow this in all details or just to say what it is in the box. You decide. When you decide, the entropy accommodates to that information pool. You will find, just to be fair, you will find physicists that will tell you this is immense. Entropy has nothing to do with me. Entropy has to do with the system. Well, you discuss with them. I mean, my answer is what I thought. It depends on you. But do you want to know on the system? I think what I'll suggest now is that we continue the discussion but up in the Akyan Gallery. As I said, there's refreshments up there. We can have informal conversation with Professor Tsala. Before we do, I'd like to go back to the comment that he made earlier about the talk that he gave at the University in 1996 which is when I first met him and as I was sitting in the audience I saw that what he was proposing solved a problem that I couldn't figure out how to solve and it was a system that turned out to have a queue different from one and as he spoke I could see that it applied. So I sent him an email afterwards Oh, you sent him an email? I think so, yeah. I don't remember that. So what we may or may not remember was that you were very, very encouraging and helpful. And I think one of the wonderful things about people who are scientists and researchers one of the best qualities that they can have is when they're willing to spend time helping their colleagues and younger colleagues in particular work something out and just devote time to it devote energy to it. You and I exchanged quite a few emails the subsequent days and I thought that it was wonderful but you were encouraging like that and it depended to that paper. So likewise, I was very pleased to welcome you here at AUA This, by the way, is the first time that you visited the former Soviet Union was the first latest in the form of Soviet Union you visited and what I'd like to do is present you with this book so you will remember. Okay, I wouldn't remember anyhow without the book also. Thank you very much. What I'm going to suggest is now we thank Professor Salas and we go up to the room and we enjoy the refreshments and continue the discussion.