 So, just a microphone, okay. So, I'm going to talk today about a work that I have done with Ana Carolina Hibero Teixeira, myself, Renato Pacted and Ian Levine, and we're all from the Federal University of Rio Grande do Sul in Brazil. And the focus of this work is on quasi-stationary states. So, as has been talked about in the previous lectures, if we have a system with long-range interactions, for example, if we have N stars or N ions, and they start off in an initially perturbed distribution, so not in a stationary state, they'll undergo a process of violent relaxation and reach quickly this quasi-stationary state due to their collisionless dynamics. So, the objective of our work is to look at some approaches that exist for describing these quasi-stationary states to predict what their density profiles will be, what their velocity profiles will be. And so, we're going to look at the HMF model, specifically in this paper. So, the HMF has already been introduced several times today. I don't think I need to introduce it again, just to point out the equations of motion of this model. So, each spin, the equation of motion for the spin is given by minus M times sine theta, where M is the mean field. So, M is just the average value of cosine theta, and it's the magnetization of the system. And in the thermodynamic limit, when N goes to infinity, then the system should be described by the Vlasov equation, the collisionless dynamics, and so it should reach a non-equilibrium stationary state. So, the magnetization, this order parameter, that's the mean field, the average value of cosine theta, it distinguishes between two phases of the system. So, here I have, just to show two examples of these quasi-stationary states and the two phases, I show a phase space, the distribution in phase space. So, here are initial distributions. On the y-axis is the momentum, and on the x-axis is the angle. And we have these initial particle distributions, and depending on the initial condition, we can reach two different phases, a paramagnetic phase, where the spins are distributed homogeneously, or a ferromagnetic phase, where the magnetization is greater than zero, and we have some mean alignment between spins. So, in this case here, we see that the spins are clustered around zero, and they exhibit this characteristic core halo formation, which is a dense, low-energy core, a cold core, surrounded by a hot, less dense, to diffuse halo of particles. And so, for this kind of configuration, the core halo configuration, we have a theory that can describe this kind of distribution, which Yan Levine will talk about on Friday. But what we're interested in this work is what happens if we don't have a core halo formation. So, how do we describe the quasi-stationary distributions of systems that don't form this core halo? And the important thing is that this kind of formation, it happens, it occurs, when the initial distribution undergoes strong mean field oscillations. So, if the initial distribution starts off in a configuration that is very far from the stationary state, it will oscillate strongly, and this will lead to this kind of formation. So, in order for us to determine when the core halo formation will occur and when it will not occur, we can use the virial theorem, because this will show us, this will decide when the initial condition will have these strong mean field oscillations and when it won't. So, the virial theorem, it states that in a stationary state, so in the quasi-stationary states that we're looking at, this relation should be obeyed. So, the time average of the sum over all particles of the scalar product of the force times the position coordinate should be equal to two times the kinetic energy. And for example, for 3D gravity, this leads to the famous relation that two times the kinetic energy is just minus the system's potential energy. And what we have seen in previous works is that if the initial condition does not satisfy this relation, so if the initial distribution doesn't have this relation between the kinetic energy and the potential energy, then it will oscillate strongly until it reaches a configuration in which the virial condition is satisfied. So, if the initial conditions don't satisfy the virial theorem, we have strong mean field oscillations and a core halo distribution. So, what we're interested in is therefore when the initial conditions do satisfy the virial theorem. So, there'll be close, the initial distribution will be close to a stationary state, but it won't necessarily be a Vlasov stable, a stationary state, there will be some evolution. And our goal is to find out to which stationary state the system will evolve to under these conditions. So, in this case we'll have minimal mean field oscillations and a quasi-stationary potential because if the distribution doesn't oscillate in phase space, the potential, the mean field potential depends on the distribution, so therefore the mean field potential also won't have these strong oscillations. And what we've shown in previous works is that under these conditions for initially virialized conditions, Lindenbell statistics gives reasonable results. So, Lindenbell statistics, as Julien mentioned earlier, is a very elegant approach to describing the relaxation of long-range systems. And the idea behind it is the incompressibility of the Vlasov flow, so the preservation of the volume of the density levels of the distribution. So here, this figure, it's a schematic representation of phase space and the gray area is a one-level initial distribution, so it's a uniform density inside this gray area. And what happens under the Vlasov dynamics is that the initial distribution will start to evolve in this filamentation process. So it will evolve on finer and finer length scales indefinitely since the volume of each density level is preserved, the entropy is also preserved, and so this dynamics continues indefinitely. But on a coarse-grain scale, we can say that the system reaches a stationary distribution because at some point, the length scale will be so small that we won't have enough resolution to see this evolution continue. Is the microphone good? To see this evolution continue. But on a coarse-grain scale, it seems to have reached a stationary state because we don't have enough resolution to see the dynamics continue. So this is the idea behind Lindenbell statistics. What he did is divide phase space into these two scales, which are represented by macro-cells and micro-cells. So each macro-cell is divided into a certain number of micro-cells. And since the volume of each density level is preserved, the volume occupied by this initial distribution has to be the same as the volume occupied in all of these occupied micro-cells here. So if we sum over all of these occupied micro-cells, the gray micro-cells, it has to be the same as in the initial distribution. And Lindenbell's idea is just to do a Boltzmann counting of the number of ways that you can distribute these phase elements that are occupying the micro-cells between all the micro-cells. So it's the number of possible configurations of the system. And so doing this, you have the number of possible configurations of the system, W, and therefore you can define a coarse-grained entropy, which is proportional to the logarithm of W, and maximizing this entropy, given certain constraints, the constraints that you're interested in for your system, you find the coarse-grained distribution, the Lindenbell distribution, which in this case, for a one-level initial distribution, is like the Fermi-Dirac distribution. And so this distribution is Lindenbell's idea of the distribution that describes the quasi-stationary state. So here is a theory that describes, that aims to describe the distribution of the quasi-stationary states. But an important thing to keep in mind is that this process of Boltzmann counting, it requires a certain assumption. So it requires the assumption that each micro-cell has an equal probability of being occupied or not. So this means that any phase element here, for example, a very low-energy phase element, can occupy any of the micro-cells. So it's really an ergodic process, and there is this phase mixing that Julian mentioned as well. And the reasoning behind this is that the potential will oscillate strongly, so there will be this violent relaxation, that will mix the distribution in phase space. However, from what we argued previously, we said that if we have initial conditions that oscillate strongly, there can be a core halo formation. And in that case, Lindenbell statistics does not work. And that we have shown in previous works, that Lindenbell statistics works reasonably well, if there aren't strong mean-field oscillations. And so this is a bit curious, how can we reconcile these two visions? Because if there are no strong mean-field oscillations and the potential is quasi-stationary, for a long-range system, inter-particles correlations don't matter very much, and what really matters is the interaction of each particles with the mean field. And so it's almost as if the dynamics could be described by uncoupled particles. So particles that are just evolving in a quasi-static mean field without exchanging energy. And this is the opposite of the idea of ergodicity and mixing that Lindenbell proposes. So this is more of an integrable dynamics idea. And this is exactly the idea behind the second approach that we're going to look at in this work, which is the uncoupled pendulum model. So this model was proposed by Pierre de Buil, David Mukamell, and Stefan Rufu as an analogous model for the HMF. And basically the idea is very simple, is you take uncoupled particles, so the particles have no interactions between each other, but they follow, they have the same equation of motion of the HMF, but instead of having the magnetization, the mean field here, they have an external field H. So this external field is static. And in some way, in order to make an analogy with the HMF, later on I'll explain how, but we'll have to associate this external field H with the mean field magnetization of the HMF model. But under this dynamics, the uncoupled particles, since the particles don't exchange energy, the external field is static, then the energy distribution is conserved because each particle is just going to evolve along its constant energy orbit. And so since this dynamics is integrable, we refer to this model as the integrable models of the abbreviation IM that will show up in future slides is because of this name that we give it. So to better illustrate this idea, here we have once again a schematic representation of phase space, so momentum P, angle theta, and the solid lines here represent constant energy orbits, so they're the orbits along which the uncoupled particles will evolve. And the blue dotted rectangle here represents the limits in phase space of a given initial distribution F0, so this F0 here. So inside this blue dotted rectangle, we have particles. And this shows that these red lines show the initially occupied regions of the energy orbits. So since at first we only have particles inside the blue rectangle, there will be energy orbits that are fully occupied by particles, and some energy orbits that are only partially occupied by particles. But after a transient time, these particles will evolve under this external field, and they will occupy homogeneously the entire energy orbit in a process of phase mixing. And so after this transient time, we have a homogeneous distribution of particles along each energy orbit. And therefore, in order to find the coarse-grained stationary distribution, all of the stationary marginal distribution, sorry, all we have to do, for example, to find the distribution in theta, is integrate over this energy distribution, over the energy orbits, over momentum. So we have a conserved energy distribution given a certain external field H, and given an initial distribution F0, to find the energy distribution, we just have to integrate F0 over the constant orbits. And then to find the distribution in theta, we just have to integrate this energy distribution over the momentum. And then to do the association with the HMF, we have to impose self-consistently that this external field H be equal to the magnetization of this stationary distribution. So we just have to integrate cosine theta times this distribution. And this gives us a closed system of equations. It gives us all the information of the stationary state of this uncoupled pendulum model. So we applied this to initial conditions for the HMF model that satisfy a generalized virial condition that we developed in a previous work. And we used it in initial conditions that have one density level in theta, but multiple density levels in momentum. So it's like an overlap of several different water bag distributions. So each color here represents a different density level. The blue is a higher density. The green is less high density. And the red is the lowest density level. And the number of density levels of these initial distributions we call L. So it's an L-level, multi-level momentum distribution. And then we have a closed equation for the distribution function in energy since we have a given initial distribution. And we have all the information we need for the integrable model. And then we compare both approaches, Lind and Bell statistics, and the integrable model with molecular dynamics. So here are results for the angle of the density distribution on the left and momentum distribution on the right of molecular dynamics. So there are the black squares. The red lines, which are kind of thin here, but I think you can see them, are the integrable model results. And the blue dotted lines are Lind and Bell results. And this is for a one-level, so just a normal water bag initial condition. And we see that both work quite well. So both the molecular dynamics agrees well with both the integrable model and Lind and Bell statistics in this case. You can see a slight deviation here from Lind and Bell statistics and the integrable model, but both are still very, very good approaches. However, as we increase the number of levels, so if we start with an initial distribution that has three density levels in momentum, then we start to see that the integrable model works better than Lind and Bell statistics. And this is because due to the statistical approach of the mixing of the distribution function in the Boltzmann counting and Lind and Bell statistics, you lose the information of these initial density levels. And so here we can see that the integrable model, it maintains well the information of the different density levels, so these humps or bumps here, while Lind and Bell just gives just this smooth distribution. And so these are for different initial conditions that we have tested, different initial distributions with different energies and different initial magnetizations. And it works quite well. And this becomes even clearer when we look at the energy distributions. So here, we have the distribution in energy for several different initial conditions. On the top, just for one level water bags, so normal water bags, and on the bottom for two level distributions. And for the one level especially, we see that the integrable model works very well. So what that means is, since in the integrable model we're just conserving the energy distribution, this means that the molecular dynamics is also conserving very well its energy distribution. So we have this plateau here of the cold core that the integrable model and molecular dynamics matches very well, while Lind and Bell predicts a warmer core, so it decays here more smoothly. And as we increase the number of levels, so here we have two levels, we see that the integrable model still gives better results than Lind and Bell statistics. But it starts to deviate a little bit in the tails, the integrable model, but it still preserves very well the information of these different density levels. And finally, just a last comparison between Lind and Bell and the integrable model, here we show the root mean square deviation of the energy distributions. So the triangles here, these colored triangles, represent the root mean square deviation of the energy distributions from the integrable model with molecular dynamics. And the circles here represent the root mean square deviation of Lind and Bell and molecular dynamics. And each color represents a different initial condition, so we tried several initial distributions. And what we see is, we just confirm what we saw in the last figures, that really Lind and Bell has a greater deviation from the molecular dynamics results than the integrable model for all levels of the initial distribution function. But as we increase the number of levels of the initial distribution, both Lind and Bell and the integrable model also increase the deviation from molecular dynamics. And we believe this is due to the internal mixing between the density levels, that are virial condition that we use, it doesn't fix well enough that there won't be mixing between these different density levels. And so this leads to a discrepancy between the integrable model and the molecular dynamics, but still works very well. So just to summarize, what we've done is to compare these two different approaches to the quasi-stationary states. And they're based on very different fundamental assumptions. So one considers that the dynamics is very ergodic and mixing, and that if you have a phase element of very low energy, in the Lind and Bell case, I mean, if you have particles of very low energy, they can go to very high energy and vice versa in any way they want, that it's ergodic and mixing. And the integrable model is the opposite. You say that particles conserve their energy. And we've shown that for the HMF model, the integrable model works better than Lind and Bell statistics. So it's an indication that this dynamics really is more integrable than ergodic and mixing. And, well, the agreement decreases as we increase the number of density levels in the initial distribution. But we have also applied this model, this integrable model, so it can also be used for other long-range systems. So not just the HMF toy model, but we've also applied it to its 3D self-gravitating systems in this work here, which I think Yan Living will talk about a little more also on Friday. And so the 3D self-gravitating system is an even harder problem to solve than the HMF model because of all these problems of short-range divergences. And it's a more complicated problem, but we successfully applied it. And it worked very well to predict the quasi-stationary states of this system. So the uncoupled particle approach is a good approximation, at least to guess the quasi-stationary states of these long-range systems. And that's it. Thank you.