 All right well first I'd like to thank the organizers for inviting me to give the stock and you for staying till the last day and waking up in the morning to come here. So I will talk about self-gravity in system so this kind of ties in with what Michael Joyce was talking to us about but I will talk about isolated system. So the idea is that the idea is that we want to starting with some kind of initial condition we would like to predict where the system will relax. So it's a system of stars and we want to evolve within time and see where it arrives. So this work was done with people in our department so Renato Pachter, Felipe Risata. So these two usually work on non-linear dynamics and plasmas. Tarsicio was a was a graduate student and then postdoc. Fernando Bineci who you heard the talk and the poster was the first day and then Bruno Marcos who visited us in Port Olegre for a year. So as I said the idea is to try to understand the structure of let's say an elliptical galaxy so often you find in this kind of system that you have a quite well-defined core structure and then there is a halo of surrounded stars and this is actually very similar to what is observed in plasma physics so if you take ions and you inject them into the accelerator so in general you can think about kind of lines of charge of propagating in in one direction and of course the magnetic field the solenoidal magnetic field provides a confinement and then if you eject the particles into the accelerator what you see often is that there is some kind of oscillations and relaxation process that goes in and then the system relaxes to stay which again has this characteristic kind of core halo structure there is a dense core and then there is a halo of highly energetic particles which surround the core. So this is there is certain similarity with the gravitational systems of course in Coulomb case you have to have a confinement while for gravity you don't need to confine at least if you are in two-dimensional or one-dimensional gravity in three-dimensional gravity things are much more complicated and I will just mention a little bit at the end what we have done with 3D gravity. So maybe before I get into the discussion I will show you a little movie that actually Bruno prepared. So this is a system of two-dimensional gravitational particles so the particles interact by the logarithmic potential I started them from some uniform distribution and what you're seeing there are this kind of density oscillations so what what actually happens is that the particles start here and they kind of cross through each other and expand so in the end it looks like the system collapses re-expands and you see this formation as as a time progresses of this kind of core structure here and you see this halo of surrounded particles so this process goes on and on and after a while we will see that this core you can already see that the oscillations of the core are becoming dumped out and there is a very well-defined halo which is that is being formed and what we want to see is given the initial condition that the initial water-bag distribution where the particles were uniformly distributed inside the disk and I gave them some velocities from some kind of distribution of velocities so this my initial condition I want to see can I predict what will be the final state of course if the system would be equilibrium this would be quite simple because we know the final state corresponds to the Maxwell to the Boltzmann equilibrium and then we could just solve both on Boltzmann type of equation and predict where the system would evolve for the self-gravitating systems this is not the case because what happens is that if we really want to study this kind of systems in the thermodynamic limit the collisional effects become irrelevant so the two body collisions are not important in the thermodynamic limit and everything is governed by the collective effect so you have the collective effect that one particle just feels the mean field produced by all the other particles and if we look at the evolution of the distribution function what we see is that the distribution function f evolves in the phase space according to what we have heard is a velocity equation so the velocity equation has this form here and one way to think about this velocity equation has been just a convective derivative over the phase space so and of course the convective derivative on the phase space tells us that if I look at one particle the density along the evolution of this particle the density of other particles in the neighborhood of this one particle will be preserved along the flow so this is a peculiarity of the velocity equation compared to the Boltzmann equation which would have a collisional kernel here which would drive them the system to equilibrium in the velocity cases is not the it's not the case so one of the one of the curious things of course of the velocity equation that it has an infinite number of invariants which are called as casumers basically any local function of the distribution function is a casumere invariant okay and in particular of course Boltzmann entropy is casumere invariant so if as the system evolves the Boltzmann entropy is preserved and you can say well but you showed me the simulation where I started with some uniform distribution which obviously had lower entropy and then it's evolved to something which was much more disordered so clearly entropy is growing so what's going on well to see what's going on in the system we really have to look at the phase space more carefully so so this is the HMF model famous HMF model that Stefano introduced to the community a long time ago so here what I'm doing is I'm just plotting the initial distribution so it's a little distorted because the way that we created the initial conditions but it doesn't make any difference so this is for HMF model so I give the momentum and the angle of the particle so this is initial distribution and then I just turn the equations of motion so it's just Hamilton's equation of motion Newton's equations of motion and then we see the evolution of this phase space and we see that the phase space gets distorted so it becomes kind of elliptical and then it's kind of starts folding on itself because I have periodic boundary condition so I have the stretching and folding of the phase space but if I look at the area of this it's exactly the same as area of this exactly the same as area of this and exactly the same as area of this so what's happening in this process of the dynamical evolution that the entropy is preserved so if I look at the microscopic entropy I see that it's completely conserved however as the process of evolution keeps going and the system keeps evolving what we see that this filamentation process keeps going and going on here I'm still having exactly the same area that I started here but if I look at this thing I lost all the memory because and this is not such a large time for the system so after some time we come to this state of course if I look microscopically I will see that this this area is full of holes so in principle it will be exactly the same as but on the coarse grain scale of the simulation or whatever measurement that I will be able to perform this thing here obviously has a larger entropy and if I calculate the entropy really for the using the Boltzmann definition of entropy I will see that this state here has larger entropy than this initial state all right so we kind of understand that but there is another peculiarity of this of this loss of dynamics and that is this conservation of density this incompressibility of the flow over the phase space and so to visualize this I mean if I take so imagine that this is my face space here and I start with some water back initial distribution so I just have a uniform distribution of velocities and positions of my particles so this is kind of what would go water back distribution so if I would be in two dimensions or three dimensions this would be the maximum radius until which I will distribute my particles and this would be the maximum velocities distributed to uniform and this would be then the maximum phase this is my face space density so this at this phase space dance so as the system evolves what we will see is that this different this levels will just get spread over the phase space and of course we can divide the phase space into the macrocells and microcells but the point is that this microcells cannot have two of the squares two covered squares sitting on top of each other because this would violate the incompressibility dynamics of the velocity flow so this is this this is really the fundamental observation and as far as I know Lyndon Bell that Fernanda mentioned that her talk was really the first one to make this observation that there is a syncompressibility dynamics of the velocity flow which if we want to construct some kind of series this is the most important ingredient of course there are infinite number of conserved quantities but this incompressibility is really the fundamental thing that we have to take into account so so the idea is that well as the system will evolve what has to happen is that there will be final distribution but for that the initial condition that I said the water bag initial condition what we see is that the distribution function always has to be less than or equal to the initial phase space density so it can never exceed this value because of the incompressibility of the flow so this is really a fundamental observation that would just that we make and which will be the basis for everything that I will tell you so for example going back to the gravity so here is a case of 1d gravity here so again I distribute my particles uniformly inside this water back initial distribution and so this is 1d so the same idea that the gravity in 1d satisfies the Poisson equation so particles interact linearly with separation so the potential is linear so I just turn the equations of motion the Newton's equation of motion evolve my system and I will arrive to the same kind of core halo structure but remember this is in the in the phase space now so this is a velocity and this is a position so I have this core in the in the phase space here and then I have this halo of particles so what we really need to do is to understand them where this core halo structure comes from and the series that I will show you then is going to be this solid curve here and the points are the molecular dynamics simulation so as you will see the series really works very well to predict starting with the initial condition we can predict where the system evolves and this initial this dotted lines here is my initial distribution so the system moves quite far away from the initial distribution so these are two different initial conditions so this is a density distribution this is velocity distributions so you can see for two different initial conditions that the system evolves exactly where our theory will predict that it should go very good so from now on I will concentrate on 2d systems so just to to to sketch the theory for you but then we will go back a little bit for 3d systems and but you will have the idea of how things work okay so so we'll start with a velocity equation here so the idea is that you have to solve velocity equation in conjunction with a Poisson equation so I have 2d gravity particles now interact by logarithmic potential so I have to solve the velocity equation and of course the potential here is the gravitational potential satisfied the Poisson equation this is the math of the particles here so this is the density distribution so to get the density distribution I integrate over the velocities plug it in here and I can do this process in principle iteratively to solve the Poisson equation the problem is Poisson equation is really hard to solve it's extremely unstable it's hard to discretize of course there are some experts here Tarsicio I don't know he's here so there are people who know how to do the things well but it's a lot of work and for gravitational system it's very hard to do you can do it for simple models like HMF and we did it but in general this is not something that we want to do so the idea is that I don't want to solve the velocity equation I want to start with the initial condition and predict where the system will evolve without going through all dynamics so so to understand what's going on we have to look at the particle evolution so let's let's take a simple case we can think about the first thing that we want to do is we saw that the fundamental thing that was happening in that movie that I showed you is that there were this huge density oscillation so I had this system of particles which clearly was oscillating so what we can do is we can define an envelope of this oscillation so I started with a water bag and then I just look at the root mean square displacement of my particles and I see how this thing will evolve okay and for the system actually we can write fairly simple equation which is which is an approximate equation which tells me how the envelope evolves so we can derive this equation it's not perfect obviously but it predicts us how this mass evolves so with this kind of equation we can predict that the envelope of this oscillations has this periodic structure and of course since this is there is no dumping in this thing so it just keeps going this oscillations if I look at the simulations we actually see that the periodicity of oscillations is captured very well by this equation so this is a maximum velocity this is my maximum radius of the initial distribution and then I just solve this for initial conditions so we see the periodicity of the oscillations is captured very well the other thing that we see is that in the simulations actually there is this dumping so where is this dumping comes from well when we see something oscillating and we have particles inside an oscillating potential we know that there is going to be some resonance so some particles enter in resonance with this oscillations and they can gain a lot of energy so once this particles gain a lot of energy they escape from this main course and go to the halo okay so the resonance drives the particles to the halo but of course the whole energy of the system has to be conserved this is just the Hamiltonian dynamics so the energy that the particles gain has to come from somewhere and it comes from this oscillations so this is kind of the process of land out dumping which we heard about already which dumpens this collective oscillations passing energy from the collective mode to the individual particles so to try to understand it more quantitatively what we can do is well let's look at the dynamics of one particle so now I know that I have a pretty accurate equation of motion for this collective oscillations of my envelope so let's suppose that the density inside the envelope is going to stay constant so I just finally rescale my density distribution so I compress expand but the distribution inside we're going to say stays uniform okay so with that I can solve for the potential produced by this uniform mass distribution so I know exactly what the potential is and this potential is going to oscillate well the since I know the boundary of my distributions and this is 2d gravity so what we see is that the force for the particle when it's outside this envelope is going to go like this so it's just derivative of the log so I have one over r so this is just a force in 2d and then when the particle is inside this uniform mass distribution then the force is going to be this so I take a test particle and I put it inside this oscillation oscillating potential and I want to see what's going to be the motion of course since I have oscillations I know exactly what is the periodicity of this oscillations so I can do a stroboscopic plot so I do like a punk area section of this thing and plot the dynamics of this one particle so each time the envelope is at its minimum upload the position and the velocity of a particle so what we see is when the oscillations are very small so if I start with some initial condition what where there are almost no oscillations then of course the dynamics is just going to be integrable dynamics of the system so I just have this normal orbits that I see if the oscillations are large what happens is exactly this resonance that I told you so we have the formation of this resonance island and some particles which were close to the border of the envelope suddenly gain a lot of energy and escape from the envelope and they go to up to this position so they can go up to this thing here so that's a huge energy gain for the particles and so this is a mechanism of the formation of the halo the particle gain energy from the oscillations they escape as they escape they dump on the oscillations and this process will just keep going of course if the system would be collisional what would happen is that well particles keep evaporating the rest of the system just keeps cooling down cooling down until you get to the minimum of the potential energy so you just collapse to the point but in lots of dynamics you cannot do that because you have that constraint that the maximum density that the system can evolve is that eta from the original distribution so it's kind of like a spin degeneracy in quantum systems you get to the maximum you fill up all the lowest energy levels of your system and there's nothing more that you can do so the oscillations will stop out when all the particles which could evaporate from the system and the core just collapses to this final maximum density state so with that we can really make an answer so here just to show you that that based on on this one particle test particle dynamics we can predict very accurately so here is just just this one particle dynamics with different initial conditions so we can just predict very accurately where is this resonant energy so how far the particles can go so this is just one particle dynamics here so we see this and if I look at the whole and body simulation problem with all the particles interacting gravitation we see that the maximum energy where the particle will go is exactly the same as this so it's really is a resonant process which which expels the particles from from the core and drive them to form the halo so the idea is the following okay so now we understood this one the stood this dumping process so let's just make an answer for the distribution function so what we make is an answer that's exactly what I said the process of evaporative cooling just cools down the core that means that all the low energy states up to the Fermi energy which we will define as a maximum energy will be occupied with the maximum distribution which is on the face space which is allowed by the velocity of dynamics which is this initial at that we started with so all the low energy states are occupied and then the dynamics and drive the halo particles towards the resonance so the resonance then the resonance energy we calculated from the from from this one particle dynamics so we have the number exactly for this thing here so we know that and we say that the the halo part here then it's going to be occupied with some density so the core is occupied with the maximum permitted density halo is occupied with some density chi which we don't know which we have to determine so we have in this thing we have two parameters we have this Fermi energy here which enters here and chi so we have two parameters which we don't know but we have also two equations for the conservation so I know that the dynamics is such that it has to preserve the total energy of the system because it's Hamiltonian dynamics and I have to preserve the norm so with those two conditions I can determine these two parameters which I have in my theory then I solve the Poisson equation with that density distribution on the face space that I define and we find the distribution of the particles over the face space here so this is our halo part this is the core part so these are the simulation points here and this is a series that I told you about so there is no no adjustable parameters in this here so we predict we calculate where the the resonance energy is and then we just solve with that unsung distribution so this is the velocity distributions and of course you can see that it's completely non-maxful Boltzmann there is this kind of halo part here which has this fat tail here different from just exponential decay so we can also look at the temperature distribution so this is what Slapo was talking about the other day so you see that of course since the system is out of equilibrium the temperature is not uniform inside so this this particle distribution didn't evolve to equilibrium it's in this funny quasi-stationary state which in principle will go somewhere then we'll see where it's gonna go but if I look at just big beams along along the radial direction what I see is that the temperature varies like this and this is exactly this temperature inversion that that Lapa was discussing the other day so you go so this is the core part and you have this huge jump here which is for the halo part but then of course it dies down because the velocities of particles are confined by the resonance this one yeah no but because I have to integrate over the velocities yes this is this is the phase space so this is the the distribution function and you see that it's going to be stationary solution of the velocity equation because it depends only on the one particle then energy so this epsilon is the energy of one particle which should have said and so the this is one so we have to when we get the density distribution in the field we have to integrate over the velocities and this is what I was plotting and if I want the the velocity distributions and I have to integrate over the positions okay so this is a temperature distribution so we see this inversion here so as I promised to I wanted to say a little bit about about 3d gravity so 3d gravity is really is a nightmare for anybody who tried to do 3d gravity knows how hard it is first of all there is even I mean you have to soften potential how you soften it makes difference then the particles there is no resonance like in 2d or 1d because the particles can actually gain sufficient energy to evaporate well this is an old problem I mean in three body problem you often find that you start with some state and the particle one of the particles just gains enough energy from the other particle and just goes away so this is the whole problem of stability of solar system it's gonna fly away at some point so it's really very difficult to do anything with 3d gravity I mean we tried for a long time to get somewhere with that what we manage so far it is to study to study 3d gravity for this special initial condition so this is the virial condition so in 3d you remember that there is a virial theorem which tells us that two times the energy has to be equal minus the potential energy if the system is in the stationary state so what we do is we prepare the distribution the water back so our distribution is still a water bag we know from the virial theorem that if the system will evolve to final stationary state if there is exist a final stationary state somehow it would have to satisfy this virial condition so what I say is I will produce a water back distribution but I will adjust the velocities in such a way that the virial theorems is satisfied okay this does not prevent this doesn't tell me that the distribution that I constructed is gonna be stationary distribution to be stationary distribution it has to be solution of the velocity equation this arbitrary distribution that I started with would just satisfy this virial condition will not be stationary and will have some kind of dynamics which will relax and go to the stationary condition however if I start with the virial condition what we know is that the oscillations will be small so different from this resonant oscillation that we saw in the other case if I start with virial condition what we see is that the particles will not oscillate so we will not see very strong oscillations so we don't have the resonances and the process of relaxation to equilibrium should be kind of adiabatic well if the process is kind of adiabatic then we can think well maybe the dynamics of the system is really not so far away from some kind of integrable dynamics so if I think about the particles in my initial distribution so exactly what Fernanda was telling us about in the case of the Hamiltonian mean field model if we start with with a particle it will feel the mean field and this mean field will change very little over the evolution because the system is doesn't have any strong resonances so the particles will just evolve under the action of the mean field so they will have this orbit if I look at some density shell what will happen is that the particles over the density shell will just get smeared out because it's nonlinear system so I have different frequencies so after some time I would expect that the dynamics will just smear out and the particles with a given energy will just be occupying the whole orbit which is permitted so even if I start with some initial condition which is I don't have in my initial distribution the whole orbit field the particles after some time will feel out this whole orbit so then we can think well okay so the dynamics is such that it's almost the integrable so it's kind of adiabatic dynamics so let's say so let's construct the initial district let's construct the distribution to which the system will evolve so as I said I started with some number of particles this particles will move under the action of some potential so I know exactly what how many particles I will have with a different with with a given energy so I can calculate and this will be preserved because the number of particles the particles there's no resonance interactions as the particle just moving a fixed potential so the energy of which particle is conserved right so this is a difference with the non with a resonant case where particles could gain and here's the particles just moving some mean field potential so if I know what is the initial energy it will be the same as a final energy of the distribution so this will be then the number of particles that I will have with a given energy here in my initial distribution and the difference here is that we really have to be much more careful because this are subtle effects so we also have to take into account that since my potential is spherical symmetric my angular momentum will also be conserved so what I do is I look at the number of particles in my initial distribution which have given energy and given initial angular momentum and I calculate how many particles I have with given energy and given initial angular momentum and this gives me this number here to get the distribution function what I have to do is to take this number of particles and divide it over this phase space that is permitted so what I do is I calculate the density of states that I have so I just calculate again for a given distribution function I just integrate over the phase space with a constraint that my energy has to be given to a certain value and angular momentum is given to a certain value so I can calculate this thing here so given an initial distribution we can calculate what will be the final distribution which has this kind of complicated so this is my initial distribution as a function of R and V and all I have to do is substitute this thing here this this is my energy this is a potential which in principle I do not know yet because this is a mean field potential under the under which the dynamics will evolve and so this is my distribution function so for a given energy and L and angular momentum I know how many particles I will have pure pure volume of the phase space okay so what what do we do we have the distribution function so now we just integrate over the soul remember that this is a particle energy here so we can just integrate over the velocities and this would give me the density distribution and then I plug it into the Poisson equation and solve that so this is the predictions of the theory so so this is my initial distribution would try different initial conditions different initial distribution so it doesn't have to be water bag anymore because we don't use this conservation of I mean everything is naturally preserved like in the velocity flow because basically the evolution of particles is non non-interactive so we just fix the potential which was undetermined self-consist so these are the the points the molecular dynamics simulations for 3d gravity and this is the theory this so this was the initial condition that was started when the system then relaxes to this so this is a density distribution this is a velocity distribution this is the initial condition so the system moves quite far away even so that we fixed the virial condition it does not fix the final distribution the system will evolve somewhere but without exciting the resonances so this is a different initial condition so we see that again the system follows exactly where the series and this is the case which is which is nice because we don't have why can we solve it because we do not have strong oscillations we do not excite the resonances and the particles just evolve into very controlled dynamics so with this kind of approach we can calculate the final distributions as soon as you excite any oscillation then you start producing this evaporations and life everything goes to hell so I still have a few minutes so what I want to tell you just this crossover so of course if we're gonna wait sufficient time the system if it's finite it should evolve to equilibrium at least for 2d gravity of 1d gravity the system should evolve to equilibrium so we can just look since we know the distribution of the particles in in the core state here we can just define a measure here how far away the system is from the final quasi stationary distribution here so what we see is that so I take this thing so this is what we calculate theoretically and this is and we measure in this in the simulations and then we just do this integral numerically so we start somewhere here at at time equals zero and the system very rapidly evolves to this quasi stationary state and then if we wait long enough then the system starts to go somewhere else so this depends on the number of particles so for five hundred seven hundred and fifty then you've got to twenty thousand twenty thousand we cannot wait sufficient amount of time because it just takes so long to evolve to this other state so the system just keeps staying in this quasi stationary state so as then goes to infinity exactly what we expect this quasi stationary state will just last forever so if we take takes this this time variable and we just rescale it with with this kind of dynamical time which depends on the on the number of particles we see that we can collapse all of these curves so there is a crossover time which scales as and to this one point three five four this this one this molecular dynamic simulation that we constructed and all these curves and collapse on the one which tells us that there is a characteristic crossover scale while the where the system moves out of this quasi stationary state and goes somewhere else where does it go well it goes to the Boltzmann equilibrium so as I said if the system goes to Boltzmann equilibrium then things are very simple because in this kind of thermodynamic limits that were taken the correlations between the particles are irrelevant and we can just solve the Poisson Boltzmann equation so I my distribution function is just going to be the usual distribution function for the for the Boltzmann equation so this is a velocity this is the self consistent gravitational potential I just plug it and integrate over the velocity the couple so it just gives me a constant so I solve this equation here and when the system has crossed into into the state here we wait a long time the system crossed into the state we look at the velocity distributions we look at the density distributions and we see that they perfectly satisfy for the 2d gravity case perfectly satisfy with the prediction of the Boltzmann Gibbs statistics so when the system moves out of the quasi stationary state it goes to equilibrium it doesn't go to funny generalized entropy statistics it goes to thermodynamic equilibrium as it should okay I don't think I will have much time to tell you but let me just show you the movie so the other interesting thing here in the systems is is that if you start with the initial condition so let's say you take 2d system and you start with the initial condition which is very hot or very cold so here we started with the initial condition which was quite hot so the particles had a lot of velocity so the system just kind of starts exploding and so this is 2d gravity again so we're looking at the x y coordinate so I started with a disk in which I put particles with some positions arbitrary distributed and given them some velocity from some initial velocity distribution but the velocity I put a lot of velocity in them so this is why we are seeing but of course particles cannot escape because in 2d potential is logarithmic so it would take an infinite amount of energy to take the particle to infinity so they go far but they come back and what we are starting to see is that along this evolution we're starting to break the spherical symmetry so there is this breaking of the spherical symmetry that happens here and if we wait we will see that we started with this kind of state this is x y coordinates this is the initial so we start with this kind of distribution and then if the virial number is not too far away from the virial condition so when it's close to 1 we go to the normal core halo state here if the virial number is too large or too small compared to the one then we break the symmetry and up with this kind of so this happens in 2d it happens in 3d but in 3d what will happen is that in this collapse and the re-expansion you would just have so many particles ejected that you will just you you don't have a very clear picture again of what's going on but it still will break the symmetry in the same way that happens in 2d system so I could just sketch very quickly for you how how we do the calculations let me just so what we do is we basically look at the distribution functions in arbitrary dimension so we can do this calculation in arbitrary dimensions we define the envelope here for the since we are in x y and z so this x1 x2 x3 will correspond to x y and z coordinates four dimensions we can go and this is the velocity and that what we do is we construct an effective equation for this velocity for this envelope evolution so this x y and z the average velocity position distributions this rms distributions if we allow for the symmetry breaking we have to construct this thing and there is well when we solve this equations here this dynamical equation what we see that there is actually symmetric and anti-symmetric mode and what happens is that the oscillations of the symmetric mode drive the oscillations of the anti-symmetric mode and then there is a certain parametric resonance if we are above certain value of the virial number which will then result in the symmetry break so unfortunately I don't have time to go through this but just to conclude so here this is the value that we find for 2d so if the virial number is less than this or larger than this we will have this instability in symmetry breaking in 3d we find this numbers here so let me just conclude so systems with long-range interactions of course don't have in the thermodynamic limit do not they do not have ergodicity or mixing but there is a certain universality that we saw in this core halo distribution then describes very well what happens with plasmas with HMF because we see it all the time so it's appears everywhere in plasmas HMF for gravitational systems so there is a certain degree of universality of course it's not the same as equilibrium the district the the final distribution will depend on the initial distribution that I started with but the final distribution always has this kind of core halo structure in the phase space so I think I'll stop here and then if you're interested so there is a review here as we wrote which has a lot of details about all the systems that I discussed