 Proste ga me ber? Jovo bremi predamo tudi v tom, ki ti ogleda. Tudi, če ga me opravili, je veliko svopozrenjav. Dve kvalitve, kaj je hrana Sama? Nowe, lahko je. Odstaju, ki jaz sesem sve dve kvalitve, Cezko Dičičjo in Sama Gupta? Here unfortunately there is no sezio teller. And we missed the opportunity of being all together here. And so I'm going to talk about temperature inversion and then now clarifying what do I mean. And the claim is that this is a phenomenon that really is from astrophysical to atomic scale. So it's like the subtitle of the conference in the reverse order. Because I was starting from large scales and then going down to small scales. So what I will be talking about. And what do I mean with temperature inversion is that I'm looking at states of systems where you have anti-correlation between the density and the temperature in the system. So you have non-uniform density, non-uniform temperature. But they're anti-correlator. But the system is sparser, it is hotter. And when it is denser, it's colder. So it's sort of the opposite of what would. Clearly these are non-equilibrium states, because otherwise they would be isothermal. So they are non-equilibrium. But we have just learned from Julian lecture that these non-equilibrium states can spontaneously appear due to Vlasov evolution in long range interacting systems. And they live for very long times. And they are typically in the community, they are typically called quasi-stationary states. And the claim of the talk will be that this phenomena can be observed from astrophysical scales to atomic scales, even if the known examples are all at the astrophysical level. What I will be talking about is not making a. First I will review some phenomenology. And then I will not be concerned with a precise analysis of some particular case, but just for looking for which are the minimal ingredients and the basic physical mechanism that can give rise to this interesting phenomena. And it will, in my opinion, it comes out that really you just need inhomogeneous states and long range interactions plus some other ingredients that will be. So this is one example of universality in long range systems in the sense of Julian talk before. And OK, we'll see how it will be relevant, the concept that is called velocity filtration that will spend some time about for a while. Now, what do I mean more precisely with temperature inversion? Is a system like that. You have that your density is non-homogeneous. It decreases, for example, in a longer given direction. And then you have that your temperature rises along the same direction. I have to precise that since we are in a non-equilibrium setting, what do I mean with temperature? Just the usual definition, kinetic definition, that is the locally averaged kinetic energy, what in the astrophysical community you call the velocity dispersion. The most known example of a temperature inversion that is also the field from which I borrowed the name, temperature inversion, is the solar corona. So the outer atmosphere of the sun and also other stars. And here is the picture. And you have that the green curve is the density. The red curve is the temperature. And you see that while this part is very sparse, so the density falls down very rapidly, its temperature rises over three orders of magnitude. You pass from thousands, OK, at least to millions of kemins. There are other examples that are less known, probably, to the out of the community that studies it. And they are molecular clouds. This is a picture. Actually, the molecular cloud is not the pinkish region there. And I put that just because the picture is beautiful. But the molecular clouds are dark in this picture, because they clearly molecule. That's one is atomic emission. That's H alpha. And what happens there, a similar thing. Then you have, this is a plot, where in the horizontal axis you have something that you can measure and signal that is roughly proportional to the density of the system. And on the vertical line you have the velocity dispersion, what I would call from a statistical point of view temperature. And you see that they are clearly anti-correlated. Also for molecular clouds, there is this intriguing. There are these intriguing power laws discovered by Richard Larson many years ago. And you see that that seems to be really an anti-correlation between velocity dispersion and the density of these structures. Then there are many other examples that may be a bit more complicated. Also some particular kind of elliptical galaxies that sometimes show an anti-correlation between velocity dispersion and radius. And then density and also the hot gas in galaxy clusters. But for various reasons these other examples are less clean. So I will not concentrate on them. Now, for the moment I am at the astrophysical scale, because the non-examples are from here. If you look at what is commonly thought about in astrophysical literature, you see that these phenomena are not seen as examples of something of a more general phenomenon. Typically, a specific mechanism is invoked for each case to explain the phenomenology. The most studied phenomenon is also because we have a lot of accurate data, is the solar corona. And there are essentially two very different kinds of explanations of this phenomenon. One that has many, many sub-different explanations is the simple idea that there is some mechanism that takes energy and brings it towards the far regions. It could be done in many ways. And so there is really energy injection. You take energy from somewhere and you put it in the sparse region. There could be many mechanisms and I am not entering the detail here. Another one is called velocity filtration and we will talk about later on. So for the moment let's do that. Then, for instance, for molecular clouds, that generically is invoked the fact that there is turbulence. And so these are fluids and they could be turbulent. But there is not a real clear mechanism why you can observe these anti-correlations. Then for the other examples, there is more complicated. So let's keep them for the moment. Now what is this velocity filtration mechanism? I spent some minutes on this because I think that this is really important for, it is more or less the basic thing that could be unifying all these things. It's a mechanism that has been proposed in the 1990s to explain the problem of the heating of the solar corona. This was not very well received in the solar physics community for the reason that maybe we can discuss but not very relevant to what I'm going to talk about. And I will underline some limitations of this idea in the following. And the idea is really very simple. And the model that was proposed by Jack Scudder is very, very simple. You think of non-interacting particles that are in an external field, like a gravity field, for instance. So think of a one-dimensional problem. You have a gravity field, you have a ground. Things, for instance, like an atmosphere or something like that. You have a ground level, then you have a potential that brings you down. And you have particles in this situation. And suppose that at the ground level, you have a stationary boundary condition. So your distribution function is fixed. And this stationary doesn't depend on time. Then what happens in this system? Okay, since the particles are non-interacting, clearly you can write a collision and bonds equation when the force part is given by the external potential. But clearly in this case is just energy conservation for the particles, nothing more. But you can write it this way. And what does it mean that the term velocity filtration, the very simple thing that you can find also in Feynman lectures on physics, that only particles that at the ground level has sufficient kinetic energy can climb up to a given height. You need sufficient velocity to climb the potential well. That's the idea, the simple idea. Now, for instance, suppose you define the density this way. Suppose that you have a thermal boundary condition at the ground level. What happens? I skip the calculation, it is very easy. It happens this one. That the temperature stays constant. But clearly density decreases. Due to the effect of the external field. And this is what called the isothermal atmosphere. And again, this is an example that you find on textbooks. Okay? What Skander noticed, and was not surprisingly, was not noticed before, is that this result requires that you have a thermal boundary condition. Because if the boundary condition is not thermal, then this is no longer true. And this can be understood from a graphical argument. So suppose that you plot the log of your distribution function against the height, no, sorry, against, not velocity, but against kinetic energy. And just to have the symmetry, you plot it against signed kinetic energy. So you take positive place, velocity is negative velocity, but it's a function of p squared, not a function of p. And what happens? That clearly, since it's Maxwellian with the log, you get these straight lines. And you see that how this black line, the black dotted line is the ground level. So the boundary condition that you impose to the system. How can you build the distribution function at higher places? Well, you just get the, from here, you get the part of the particle that have sufficient velocity to climb there. And only those will go there, but their velocity will be reduced because you need your velocity to climb up in the potential well. And so what happens is that you get these parts of the distribution function in glue together at the beginning, so at the center. And this is the distribution function that you get at a higher point. At an even higher point, you get this one. You see they're all parallel here. Indeed, if you want to calculate the temperature, they have to be scaled by the density, and if you do that, the points collapse. So that's the reason why the system is isothermal because you have this scaling, this collapsing. But suppose that you start with a non-thermal, and in particular with something that is super thermal, that is that its tails are fatter than a Maxwellian. Then if you do the same, they will no longer collapse. And what happens by doing just this cutting of the distribution function and gluing together at the center, is that you will have the temperature increases because the tails are higher. So the higher you go, and the broader the distribution is, that's called the observation. Just this, but it's very simple, very effective. So what you get in the picture that I showed you at the very beginning. So you get an increasing temperature while rising, while the density is growing down. So it is, why this is very interesting because it is a simple and general mechanism that doesn't require any exotic ingredient or specific ingredient of a given system. This is something that could be completely general. Okay, clear that there are some problems. First, okay, to the first order, you can even consider your particles not interacting, but then there are interactions. And so, what about interactions? Then the other point, and this was the major criticism that has been done in the solar physics community to this idea, to solve the specific problem of corona heating, is that, okay, since you need a non-thermal boundary condition that should be some mechanism that keeps your boundary layer out of equilibrium. And nobody knows which could be. So it's a very strong condition and it's something that you have to impose. So again is a sort of an active, a dock ingredient that you have to put to your sips. Okay, my idea is now to show that this idea can work in a much broader context. And to do this, I will use a toy model or as Julian told, the toy model of long range interaction that is called the Hamiltonian mean field of HMF model. And here I show you a possible way to derive this model, just to show that it's really very general. Possibilities that you take a generic long range interacting system, then you restrict to one dimension with periodic boundary conditions and then you expand your potential in a Fourier series just taking the first contribution and you end up with the HMF, this one. And this can be interpreted in two ways, either as a model of particles moving on a ring or as Julian told, as a model of spins on a complete graph. So with mean field interaction is a matter of taste, which one you like. Now, we have just learned that at least for below a given time scale, the dynamics of this system is governed by the laws of equation that is written this way. And when you have the self-consistent part then we may have also an external field. And what happens is that you have what Lindembell called the violin relaxation. So in a short time scale, rearrange your initial condition, you end up on a quasi-stationary states and then eventually, but really eventually, you may go to thermal equilibrium. Now, observe this. Suppose that you are in a quasi-stationary states. So you can, as far as Blasov is the equation, you can say that your F is stationary. Suppose that the net effect of the possible external field plus the collective mean field is attractive, then this laws of equation is essentially the same equation that I showed you for the Scud and model, that are essentially the same. So this phenomenon velocity filtration may induce temperature inversion also in the HMF model. So without the need of a boundary condition, but just giving this as an initial condition. So you take an initial distribution function that is not thermal. And you let it evolve. If the taste of the distribution do survive the initial fast evolution, and this clearly nobody knows, but you look, and then you may expect that this happens. Actually this happens, and typically this kind of initial states do survive the vitalization. So you observe the temperature inversion. You see, this is temperature and this is density. You see where it is low, it is high and the other way. And that was the first observation we made on this, but this is not the most important. Now, so the first thing is this, that this mechanism is actually more general than Scud and thought, because it works also with long range interactions, and really does not need a stationary boundary condition. It can be applied also to isolated systems. So I think we made a step beyond, but still there is a problem. Who prepares the system with a given initial condition? Again, it is no longer needs something that takes a boundary layer of your system and takes it there with a given non-fermal boundary condition, but still you have to start from there. So you have someone that at certain point prepares your system. So the interesting fact is that you don't really need to prepare the system. You can do something that is a bit more natural than this. And this is related to a different question. So it appears that what is efficient is to disturb an equilibrium state and not just to prepare something special. So let's ask a simple and seemingly unrelated question. What happens when you perturb a thermal state of a system? Supose, you suddenly perturb it. So suppose that you apply a sudden perturbation to your system, a big one, not a small one that you can treat in a perturbative way, a big one, a kick. We call it a kick. Or maybe you can quench a field. This is more or less the same idea. So if your system is short range, then it will relax to another equilibrium. That's the difference between, usually when the perturbation is small, it will come back to the original. If it's big perturbation, maybe the system changes its energy. And I'm always thinking of isolated systems. So maybe it's in the acquires or loses some energy, but then it will go to another, again to a thermal state. Long range system, as we know it will not do that. It will go to a quasi stationary states. In general, as Julien told, we don't know which one. But there are some cases in which we know, and we'll have some talks, and the forthcoming calls will explain us by Fernanda Benetti and Jan Levins, that there will be, in some cases we know what happens, but only in some cases. But we may ask a simpler question, but how different this state is from equilibrium, and does it have some general features that seem typical? And the answer is, quite surprisingly, to me is yes. And the fact is that the quasi stationary state that you end up with in your system has a non-uniform temperature, if it is non-homogeneous, and displays and exhibits temperature inversion. And it looks like you can even understand why, even if still not at a very precise level. So first I convince you that it works. So you take an HMF system, and you prepare it in a cluster state, or magnetized in the magnetic point of view system, and it's thermal equilibrium. Thermal state, but collapsed one. Then you evolve it for a while, just to be sure that everything stays there, and then you, for example, you switch on an external field, for a very short time, and then let the system evolve. What happens? Okay, what happens is this. The system starts to, the mean field in the system starts to develop oscillations, and this oscillation damp out, okay? And, but the system is very, very far from equilibrium. In this particular case, when the system eventually goes to equilibrium, magnetization will be zero, but you see that magnetization is very far from being zero, and since we have 10 million particles, it is simulation that the time scale over which you observe equilibrium is very, very long, but you do observe that if you wait long enough. Now, and so when you are in a quasi-stationary state that is here, for long time here, how does the density profile and the temperature profile look like? Exactly, invert it this way. We have precise anti-correlation of, so this, so it works also in this case, and the interesting thing, you can also do the perturbation in a completely different way. You can quench one parameter of your system. For instance, you take an HMF model with an external field, and what you do, you quench the value of the field to another one suddenly, okay? There is a lot of business in these years about quenching even if they are typically in the quantum regime, but the idea is the same. What happens is the same, no? You see exactly the same picture as before, and the same, I don't show you the picture, but the same if you quench the coupling constant instead of the field. Again, same thing. So it looks like it is very general, this one, you don't need to do, there is a bonus in the case of quenching, and I think Andrea for the suggesting that it can be done, you could cool the system this way, because what you can do, you quench the system, and if you quench it the right way, the average temperature goes down, and at this point your hot particles are in the given positions, so you just get rid of those particles, you remove the hot particles from the system, and you quench again, and again it seems to cool down, and again the system stays in an inverted temperature situation, but cools down. So it's a bonus, I mean if you wanted to do something, you also do something else. Okay, so can we go beyond toy models? Yes, we can. Let's start again from astrophysical scales. One very interesting problem is that this inverted temperature profiles are observed not only generically in patches of molecular clouds, but in some structures that are very important in these molecular clouds, that are called filaments, they are nearly cylindrical structures, this is an infrared view in three different infrared bands from the Herschel space telescope, this is in the Taurus cloud, and you see you have these nearly cylindrical structures, they are very important because they play a relevant role in star formation, and a very basic model that is something more than a toy model that you can do of this structure is to assume a perfect cylindrical symmetry, that at least on a finite scale is something reasonable, there are some recent papers on this as a model, and then the problem becomes a 2D gravity problem because you're in perfect cylindrical symmetry. So what happens if you do the same thing? Suppose that you start with a thermal equilibrium state, the depth for 2D gravity is well known, for example, the striker solution that describes the thermal state, and you do the same as before and you apply radial perturbation to the system. What happens? Same thing. You observe the inverted temperature profile exactly as before. Another very interesting thing is you don't need to start with the thermal state. You can also do a cold collapse, and it works in the same way. Also you can put magnetic fields, so you can really go beyond pure toy models in this way. And then I promised you atomic scales, and here we are, and Julien told you that one of the examples where there are effective long range interactions that emerge is that of cold atoms in an optical cavity interacting with a standing electromagnetic wave. And this is a figure that I stole from the paper by Giovanni Amorigi. And the second thing I remind you is this one. You have atoms in one dimension that are trapped in an optical cavity and they interact with a single mode of the electromagnetic field. And if you take the semi-classical limit and if you assume that dissipative effects are negligible on the time scale that you look at, and this is a delicate issue, and I admit this is a delicate issue, and I suppose that you can do that, then it comes out that the model that describes this system is very close, is a mean field dynamics, is a conservative mean field dynamics, and is very close to the HMF. Actually it's half of an HMF. Instead of the full HMF, you just have half of the magnetization, and then it's very close. And I hope that we'll learn more, really from the absence maybe these two for coming calls we'll explain much more on systems like this. Okay, you can do the same again on this system. What happened? You prepare a system in a regime where the system is magnetized or inhomogeneous. At a certain point you quench the coupling constant. In this system quenching the coupling constant means that you change the intensity of the pumping transverse laser. You can increase it or decrease it, so in principle it can be done. I do not claim that this is a feasibility, this is an experiment that is feasible because my knowledge in this is not sufficient, but in principle it's something that you can do. It's not something that is impossible. What happens? The same. Very senior to what we have already observed. And again, as in the HMF case, not surprisingly, which is very close, you can cool the system using the same. I'm not claiming that this cooling is very efficient. Probably is not as efficient as the other. But in principle they do have a different way of cooling in principle is not bad since everyone wants to cool down systems, so... So, which is the physical picture that we have? This point is this. You take a long range interacting system in thermal equilibrium, but not really need that it is really thermal, because for the gravity you see also a cold non-equilibrium state works as well. You perturb it, quench it somehow. What really matters is that you bring your system thermal equilibrium in a way such that the mean field starts oscillating in your system. And this is what typically happens in this case, unless you perturb the system in a very specific way. Typically it does this way. And then what happens is that the system, if the system sets down in the inhomogeneous quasi-stationary state, so it doesn't go to homogeneous regime, then you observe the regime versions always. It's typical. It's not something special. It's the rule, not the exception. And so, this is a generic feature of long-range interaction systems, because it happens in the field, but also in more realistic potentials. Attractive repulsive works the same way provided in the repulsive case you have a confining field, otherwise clearly. Now it happens. For the moment we started just one DN2D, hopefully we can do something in 3D, but yet we don't. This is very robust by changing parameters, protocols, whatever. So it seems really something universal in the language set before. So, can we understand what's going on? Okay, now comes the, and I'm going to the conclusion. I think we can understand what's going on, even if we don't have a real theory of that. So we have a physical picture that is very different from a theory. So we can't calculate the profile out of given level, but I think we can understand what's going on in these systems. So what is going on? As I told you, the important thing is that your mean field starts oscillating and you always observe in these numerical experiments. So it is that there is a wave, that perturbation induces a wave in your system. So, but at a certain point the wave is damped and Julian told us how can you damp a wave in a non-collisional system, in a dissipative system, you have a very efficient way that is lambda damping. So, okay, lambda damping, as Julian told us, is a very complicated issue in the nonlinear regime or whatever, but we know even by papers by Julian's group that it happens also in cluster states and not only in the homogeneous states. And this kind of wave particle interactions have a feature that is interesting. They are selective in velocity. So the particles that strongly interact with the wave are those that have a velocity that is sufficiently close with the phase velocity of the wave. If you are completely off resonance, you don't feel the wave. It's only if you are not too far from resonating with the wave that you get. And this is what you find in all the books in plasma physics. If your particle, that is also the picture, if your particle has a velocity that is a bit smaller than the wave velocity of the train, then the wave gives velocity to the system. So the wave loses velocity to the particle system and the other wave out. So now, typically, the distribution function that you have are decreasing for positive velocities. So what happens? This is the simple way why this kind of interaction that leads to a damping of the wave because you have more slower particles than faster particles. So the net effect is that the wave loses energy to the particle system. But then, and this may be close to a question that Michael puts before, one of the signatures of something like that is that after, if something like this happens, then in the velocity distribution, you should see local changes. So something should change not, you change from a Gaussian to a different shape, but in some more or less localized region of your distribution, you see a shape. Typically, you develop a shoulder due to this. So does it happen in our system? Yes. You see what's going on. This is the initial velocity, this is for the HMF, okay? This is the case of the simpler case. But also for the gravity is closer to this. This is the initial masswellian. And then, you see that soon after the perturbation, immediately soon after, all these altars symbols are successive time, you see that you have developed the tails. Here is strong interaction here and also something here. And the central part is completely undisturbed, okay? But now we understand why you get the temperature inversion because you have created a suprafermal velocity distribution. And then since the system is non-homogeneous, then velocity filtration can work in this system and produces temperature inversion, you see. Because if you now plot the distribution, not the cumulative one, but the distribution at given spaces, you see that this one is the distribution, they are all non-thermal, but the tails are much larger when the system is sparse, when the density is low. This is exactly the effect of velocity filtration, nothing more, because you see, you just take this and you rise it. And exactly the same one, that is just reason. Okay, so I am at the end. So what's the take home message from this is that temperature inversion is a phenomenon that has been discovered at astrophysical scales. And okay, I have to be precise. The only in astrophysics, they call, they speak of temperature inversion using these words only in the solar corona problem. In molecular clouds, you will not find someone that says that this is a temperature inversion, because instead of temperature, they use velocity dispersion. And because in molecular clouds, you have another concept of temperature because you have molecules, no? So you have systems in internal degrees of freedom. What typically is called temperature in those systems is the temperature that you can measure from the energy level populations of the molecules. That clearly is completely different with that, because the others are gravitational. The molecules seen as particles that interact together and their velocity dispersion, clearly are completely disconnected with the thermal excitation of the molecule. Typically, the temperature is of the order of kelvins, while the effective temperature from velocity dispersion is much higher, because you have kilometers per second of velocity dispersions. Okay, this said my opinion is that the various examples that have been seen in astrophysics has not different phenomena, but different realization of a general phenomenon that at a very first approximation can be described by velocity filtration. Clearly then, every system needs, if you want to be quantitative, you need to introduce the specific properties of each system. So clearly, you can't explain quantitatively the schedule model that doesn't really work for the solar corona. There are some predictions that are good and some that are worse. But I think at a zero order, it's a good idea, then clearly you have to put something more to describe the details, but this idea of velocity filtration can be applied to any long-range interactive system. And there are minimal ingredients that you need. You need long-range interactions, so that the dynamics is Vlasov. You need a clustered steady non-equilibrium states, clearly in uniform states, you don't observe this. And you need that somehow fat-tailed velocity distribution can be created, but the way you can create it is very easy, because it's sufficient to disturb equilibrium. You don't need to do something, there is no fine-tuning in this. It's just, okay, take your system and kick it. And then regardless the way you do that, apart from specific case, you get this. So this is the end. My idea is that we can understand what's the fizzing behind, even if clearly I can't yet calculate the profile from the initial condition that would be very nice. So what next? Next would be first to study other examples, both in the astrophysical regime and in the down and earth regime. Especially these are, in my opinion, the 3D self-gravitating, because these will be relevant to molecular clouds that are the cleaner system in this way, and also to study this example of particular galaxies could be interesting. Another atomic scale systems could be traped ions. So you have ions in a trapping potential or particle beams, and they are respectively close to an anti-ferromagnetic HMF or to a 2D column system, so we expect that the same thing may happen. And then clearly that would be really, really nice to do experiments and see if we see something like that in atoms in a cavity, but also for, OK, I was, and I think I hope that for coming talks we can learn something more on how to do experiments in these systems. And now it would be very nice to turn our physical picture into a theory and I must admit that for the moment I didn't succeed in doing anything close to that, but who knows. And I think that there could be many hints from this conference, from the papers by Julian on land outdamping in clustered states, for instance, but also from the theories by Yanlevin's group and the possibility of predicting in special cases but not so special, the outcome of the quasi-stationary states from the initial conditions. So maybe with Tarsizio we discussed a lot of how the possibility of applying this approach is that for the moment we didn't succeed. I hope that maybe something can be done and that's it. Thank you for your attention.