 So, I go again through a few examples of equilibrium statistical mechanics whose long range interactions. So, another typical example you take say ions in a trap and you have so just that's a potential energy. So, you have the interaction here and maybe the trap energy here and there is a balance between the trap that tends to a trap of course the atoms and this one is repulsive interaction. So, then your systems acquire a typical size maybe and something like this and if you really think a little bit you can look at the ground state at the mean field level and mean field level if you look at the ground state and mean field level that means you describe your system just by one body density. What do you find? You find that the ground state is something like this. You have a mean density over say a sphere or a disk if you're in 2D and if you look really at the ground state of this beyond mean field. So, looking at the position of each particle not only at the mean density then you find the next order. So, you have here correlations between the positions of particles. So, overall they are homogeneously distributed on this disk, but then they may organize inside the disk. So, I give this example because in the example where you will have several different scaling that are relevant right. So, if you fix first this is first scaling that would be say the mean field scaling. So, just don't give details but if you then if you do this you will find that you can describe your system with a mean field theory and that's good because describe the cloud shape and if you change beta in this scaling then you will at a large temperature you will have your system is almost Gaussian no interactions and if you decrease the temperature then your system will become closer and closer to this uniformly charged sphere right and that's nice and there's no special phase transition that's the mean field description of your system. But it's not the only interesting thing so if now you take another scaling with beta fixed then you can see beta fixed then this term will become more and more important as n goes to infinity and then you will approach say the absolute ground state and you will be probing now the the correlations between the particles right inside your uniformly charged sphere and then phase transitions are possible between say more as Poissonian distribution of the particles inside the sphere and all the configurations like say lattice configurations at small enough temperature. So, the message here that here you have two different scaling that are interesting so and and and that's that's physics here your own experiments own system that will tell you what's really beta is and what's if it's better for you to describe your system to use one system one scaling or another right so this is and by the way they like understanding the absolute ground states of this kind of systems really it's and also it's beta dependence it's a long-standing mathematical problem and interesting one okay so I now turn to some some conclusions to about long-range statistical mechanics equilibrium statistical mechanics with long-range interactions so there I hope I've convinced you that there are many universal features related to this so say non-additivity and the and the say non-equivalence between the ensembles we've had the questions about negative specific heat and so peculiar effect transition and so on so that's a good part and then in my opinion so I love this theory and there are many interesting things it's very beautiful but there don't think there are that many experimentally meaningful meaningful applications of this equilibrium statistical mechanics with long-range interactions theory and so I tried to give to give some and then there are some I mean qualitative features that are definitely very interesting but quantitatively it's not so easy to find really applications and maybe the orbiting the orbits around the black hole or maybe one example of that so then I'm interested in in learning more about this but okay that's a general of its statement and actually there is a good reason why it's not so easy to find the experimental realizations of these equilibrium statistical mechanics and the good reason for this is the considerations about time scales and relaxation time scales and so you start from some out of equilibrium out of equilibrium configurations what is the time scales needed to reach equilibrium and typically as we'll see they tend to be slow for long-range interacting systems so that's a reason why equilibrium statistical mechanics is not always the relevant theory and this is a well worked transition to the next part which is of course now kinetic theory to try to understand these time scales and what can we do if we don't have equilibrium statistical mechanics so only two subsections on this third part so I'm sorry it's not exactly three sections and three subsections in each sections so the first part will be on the Hamiltonian case so I will use a so take a Hamiltonian systems long-range interactions and try to understand its kinetic theory and here there are two against paragraph and that's it will be essentially for two very different time scales so that's good and then a few words maybe if I have time on non Hamiltonian systems because you have noticed that in many examples the interactions between the particles or part of the systems in effective interactions that often goes with some non-conservative part maybe friction or maybe more than that forcing okay so kinetic theory start with the Hamiltonian setting so the Boltzmann picture of course which is appropriate typically for short-range interactions you have you have maybe hard spheres and they jiggle around and once in a while they have a very strong interactions between two hard sphere that is typically a collision so of course if they have a collision their velocity and their momentum changes very rapidly from this direction say to this direction right but the collisions so they have a strong impact on each particles each collisions have a strong impact but collisions are rare right so particles typically have a long mean free pass and then once in a while they have a collision and if you describe your system with just a one particle distribution function that so depends on the position velocity and time then I have written here Boltzmann equation just hiding the complicated part into this collision operator he see here and this type of equation as you probably know can be obtained from microscopic setting in a specific scaling again specific scaling she's called usually Boltzmann grad scaling she's precisely the scaling in which the interactions are rare right the collisions are rare or otherwise too much correlation build and at least in mathematical sense it's not possible to derive these kind of equations of course now we'll talk about longer interactions and the picture is completely different so I'm longer interactions so I have one particle here I still feel another particle here and of course there's no that there could be something like a collision but collision will be a would be like driven by short and the short range part of the interactions and on top of this you always have like this long range part of the interactions so when you have longer interaction what is the picture picture is completely different so I have one particle here that interacts with all the others and as I have already mentioned the introduction then you will have something like a low of large number one particle here I feel the force say coming from many other particles so at leading order I will see something which is a low of large number I will feel a mean field force created by all the others so that will be my first like the dominant contribution in many cases and then what happens on top of that of course I have a lot of large numbers but I'm not interacting with an infinite number of particles so I also have fluctuations around this low of large number that are like finite size fluctuations so that will determine another time scale which will be the this time scale of these fluctuations so I have this natural difference between like my mean field time scale and my fluctuation time scales right that will show up in what follows and that's that's again that there's that has some universality for these long range interacting systems and that happens in plasma physics as well as an astrophysics and other cases as well so first part now so I have this divide between the mean field time scales and the fluctuation time scale then I will concentrate at first concentrate first on this mean field time scale there's already plenty to say about this mean field time scale right which is the short time scale the one which is the shortest and in this case what you obtain so you obtain now a kinetic equation that will replace somehow the Boltzmann equation right but just using now the the mean field force so here is so he's still my question for the one particle distribution function he's just a transport term and this is my mean field force phi and this is the transport in velocity space which is with this mean field force and if I have lots of question equation this is my personal equation for the potential that depends of course on all the density of particles right so this is called okay this is a lot of question equation right and there is no collision I have put zero here because the collisions what would be called collision in that case that would be the related to the fluctuations around this mean field force and I don't consider them for the moment and then this is so this slide is meant as a another example of a universality of the concept for a different long range interacting systems of course if I now I'm not in astrophysics and I look at point masses then they have something like he like this now I have a not minus sign because now it's a attractive interaction here but I have this say blast of Newton equation astrophysics they sometimes call it collisionless Boltzmann equation because there's no collision here I've been talking about point vortices so if I try to describe point vortices it's slightly different I don't have a function f that depends on time on space and velocity rather I have only a vorticity which is the mean density of the vortex vorticity that depends on space and time this is my omega depends on x and time and this equation here you may recognize it at the Euler to the Euler equation in in vorticity formulation and of course if I start with the particles and wave then I can also write kind of a collision less kinetic equation like this which would be a Vlasov plus waves right so all these equations are called collision less because they're there's zero here on the right hand side there's no collisions there's only the effect of the main field force and because they are collision less kinetic equation so some some would not even call them kinetic equation because there are no collisions they do share an important number of properties that are really related to the fact that they are collision less so they're okay Vlasov type equation to the Euler equation there are others there they share many many properties oh I have another example here actually when you have lights propagating in a nonlinear nonlocal medium you describe it with a nonlinear Schrodinger equation and you have regimes where indeed you end up with a Vlasov equation for the local density of of lights okay so that's the fact that all these long-range interacting systems they can be described over some time scale by this same type of kinetic equations that's another say example of the universality induced by the long long-range interactions so a few have a few words now on the on mathematics because I was asked by Michael to to talk about everybody in the in the audience and there are a few mathematical physicists I guess on mathematicians so a few words on mathematics and I think actually I think it's it's important it's important to to understand what this collision less equation means I mean not deep in the technical mathematics but a little at the precise level mathematically and that helps really the physicists actually so I will spend a few words on this so the idea is okay the formal derivation is very easy okay mean field approximation should be okay so then I just write the mean field equation from this from this and I obtained Vlasov equation and that's fine and okay I think Vlasov equation is good but then that's not very precise okay we would like to say maybe something like this that that I start from particles and then Vlasov equation becomes should become exact in a sense I mean exact limit in some some end going to infinity scaling limit right that's something like this I would like to say and and in which sense it becomes exact and maybe on which time scales and which end do I need which size of the system do I need for this to be correct this kind of thing we would like to to understand right to be more precise so to be more precise okay first introduce this quantity that's the central quantity is called empirical density so I put I have particles Xi VI position Xi velocity VI and I put delta functions each time I have a particle right normalized by and so that they have a probability density and and that's really what what you do when you do numerical simulations right you observe that's what you observe and I mean that I mean you you you you may average over several initial conditions but if you do run-run then then you have just one realizations of these particles and that's that's the kind of thing you observe and then the question we ask so how is this quantity there empirical density is close to my function F which is solution of the equation when you have you have the initial conditions of F on one side and my particle system on the other side that are close enough okay so this is slightly more precise way to formulate my problem and now there are two key observations that were made in the 70s I think by no insert and then bone hep and the machine so that the empirical density itself is a solution of glass of equation so a lot of equation you can of course you have derivatives so you think that maybe you have you need to have a regular functions to solve of glass of equation but you can define slightly more generalized solution and you can see that in reasonable sense empirical density is a solution of glass of equation so now we're in good shape because we have to compare two solutions of glass of equation one is the empirical density and the other one is my approximate version of the empirical density maybe when I have course grain my initial condition right this is what I have to do and the second key ingredient is that if I take two solutions of the glass of equation then they do not that are close one to another at some point then maybe they of course they diverge it's not they go to different places but not too fast and then at most exponentially in time and this see here of course it's there's no end here it's independent on the number of particles right so I so they have to take two solution of glass of equation they may separate exponentially but not more than exponentially so then if I put together these two these two key ingredients then I have my theorem right I have on one on one solution of that equation is my empirical density one other solution is say my solution of glass of equation when I have course grain my empirical density at t equals 0 I evolve both of them of course they separate a little bit but not too far not too fast and since they were very close at the beginning then at some finite time they remain fairly close so that's the content of this theorem yes of course yeah I was trying to bury this then I'm comparing two functions or two measures so I have to to it's not like Euclidean distance I have to define a distance so there are several ways to define this and so one way you can use transport distances between so you have two distances that can compare two different functions typically there I mean they're they're weak distances so you not I mean you're not comparing for instance like the the supremum of f1 minus f2 or even the integral of f1 minus f2 squared that that would not make real sense because then I cannot apply it for something irregular like this you are comparing measures so you were typically integrating against test function and you want you take the supremum against some class of test function typically say bounded and lipschitz test function and then then this give you bounded lipschitz distance or transport distance if you know if someone is familiar with Wasserstein distances you can use this too but use some kind of weak distance here but of course that's an important part of the theorem that I was trying to to bury thank you or so here's now the theorem again with this distance that you have to precisely define of course and what the theorem says theorem says that I take a finite time t and I fix it right then I increase my number of particles right and so I take an initial f condition f0 and then I can take initial conditions such that with particles such that the empirical density of particles become closer and closer to my target f0 right and if n goes to infinity then I can be as close as I like and then I can evolve up to time t and I remained smaller than some predetermined quantity epsilon right but then I have fixed the time scale time horizon t and then I have sent the number of particles to infinity so this is very important because and I'm in remark two here that I fix the time horizon and then I send n to infinity and then I came that rather of equation is a perfect description of my system but does not mean that if I have of course a finite n systems and I evolve it over time after some point rather of equation may be totally irrelevant and in fact you really twill right so you cannot exchange these two limits and this so this is one reason I think then you understand well the okay here the mathematics help you to understand what's going on and another thing is what is this number three then you can ask okay how if I have a given n how far can how far can I go in time the thing is you cannot go very far because the time you can go with a lot of equation from the proof it appears to be proportional essentially to log n so it is not much even for n large so so maybe soon in trouble okay and that's actually an interesting mathematical question is really can we go further than log n or and that the answer is yes but then which condition to where okay what should I tell again yes I mean there's no average needle neither though yeah so you just take and that that's good so in the theorem there's no average so you have one initial condition you evolve it and then your theorem tells you that okay you're close to a solution of lots of equations so that's one and another thing is is this hypothesis here okay that's at the force and it's assumed bounded so of course if you were doing plasma physics or astrophysics then you are in trouble because it's not at all bounded and this nice theorem breaks down so what can I tell you about singular interactions so then maybe I'll go fast on this then it the thing is it's been a huge mathematical question to try to go further than the standard theorem I gave you before and go to a singular interactions there have been several here contribution free with so the forces were getting more and more singular when you go up to more recent contributions and there have been very recent contribution by people and collaborators that I mean really an improvement and that go up to including the Coulomb case so which is the one really we're happy to have it included so making use of let's say new new techniques like using probability degrees of freedom that were not really used or not as well using previous attempts so I am hoping that this is this will make you for a teasing for I think the the seminar by pico hoping he will talk about this I don't know actually so now a few words on on this blast of equations okay so we have understood that on the short time scale we have blast of equations so first thing to understand kinetic theory of these longer interacting systems is trying to understand blast of equation right so this is a very cartoon about blast of equation in paradigm boundary condition I start with this so it's a transport equation so here I have even even if I have no force I have these particles that go this direction this direction and then they wrap around the the periodic boundary conditions so it thanks for kind of filamentation in this in the velocity direction and what's important to understand is that here I have a so here I have a two level initial conditions say one and zero and the surface the area here which is occupied is always the same and the and the density is always one or zero but the same always same surface occupied but then filamenting right so that's a basic thing so what that's what's happened maybe if I have an external field external potential so that's not like this and I have particles that are doing oscillations in the potential and they will unless there is some special feature with the frequencies of the particles they will filamentate in this in this way again keeping the the area of one and zero constants so here you can understand that even if there's no even if there's no relaxation to equilibrium no friction in this no collisions in these equations then if I say maybe it's easier to see here if I average over the velocity and I look at the density maybe here I have a bump in the density here and here it's almost homogeneous right so there is a this phase mixing mechanism is irresponsible for some homogenization and some pseudo relaxation and this is the the the heart of a lambda damping phenomenon lambda damping phenomenon which I will say a little few words up on it after so this is the first the picture now this is 2d Euler you can open let's do this so this is a perturbation of 2d Euler equation this is a simulation by Yotoshi Morita and you can see maybe I play it again so you you can see that there is really also a filamentation so it's not a lot of equations it's a vorticity so a vorticity perturbation you can see they're like a really filamentation in this direction which is very very clear so so I hope I have a that you you will more or less believe me then they'll say you that a lot of equations and 2d Euler they do they do share some common phenomenology okay so there are some properties of Vlasov equation that maybe again the idea is that all these properties they will be shared by all these collisionless Boltzmann equations or collisionless equations kinetic equations so that will be important universal features for all these long range interacting systems on short time scales so you have conserved quantities this tells you that so some are inherited from the particles and some are not directly inherited from the particle system there are extra conservation laws that are induced by the fact that you go from particles to a continuum descriptions and all these extra conservation laws there are sometimes called scatter mirrors infinite number of them incantable infinite number of them so they're they will be responsible for some special features okay so this sentence here about the fact that all these conserved quantities and this the fact that each level set of s the volume of each level sets as was clear in the previous trend slides the volume of each level sets is conserved this is actually the same the same statement okay so what does Vlasov dynamics actually do it's actually a so the volume of each level set is conserved but it it will filament each level sets and mix them and this makes for very complicated dynamics that involves fine and finite state scale I mean special scales as time evolve so what happens then for a lot of dynamics so because I have my long range interacting systems I know on on some time scale it's relevant to consider Vlasov equation so an important thing to understand to understand my long-range interacting systems in this thing what's the asymptotic behavior of my level of equation right what's this asymptotic behavior so it seems reasonable to look at stationary solution and what happens I have many many stationary solutions I have an infinite number of them and uncountably infinite number of them right so this is well known for instance in astrophysics you have many say models of galaxies that that could be constructed right so so there are several ways to construct these stationary solutions so whether homogeneous it's very easy otherwise you can construct them starting from conserved quantities and looking at say critical point of conserved quantities that that gives you some stationary states and okay a consequence of this you have that many stationary states that chances are when you evolve that you will get stuck in one of these stationary states and you will not evolve to stash to full statistical equilibrium statistical equilibrium is a stationary state of level of equation but it's one of the out of many of them and there is no reason that you will precisely end up at this state when you use a lot of equation so that that's what makes a lot of equation a complicated object right there it's asymptotic behavior is not at all clear whereas Boltzmann equation you really expect that the asymptotic behavior of Boltzmann equation is statistical equilibrium so it's a complicated object too but at least you have some idea of what's the asymptotic behavior okay so then if you have that many stationary states what can you do well one of course then it's the difficult problem and then you can have several approaches I think this is my my next transparency is then you have several approaches then you can look at the stationary states and at least try to see which one are stable so at least then you can maybe discard some that are unstable look at which one are stable and this is a this has been of course an endeavor that has been undertaken since the early 19th century 20th century it's for a lot of equations say and and it turns out a very interesting theory so you can you have a stationary solution you can linearize and try to understand what's going on and remember I have started from a Hamiltonian system so I have started from a Hamiltonian system it it translates for Vlasov equation into special properties of their stationary states their stationary states so the the spectrum of the linearized dynamics around the stationary states cannot never be asymptotically stable each time you have a negative eigenvalue and the eigenvalue is negative part by symmetry you have an eigenvalue with positive real part and then you are unstable so the the most you can hope for is a neutrally stable state right and these neutrally stable states around this neutrally stable state you can have the phase mixing dynamics that I was showing you before and that will entail lambda damping right so this is this is understood since the work of Landau in in in and then and then the experiments in the 60s in plasma physics but the important thing is that the this landau damping phenomenon is really another universal feature of these long range interacting systems once you have long interaction you see you've got a good chance to have a relevant collisionless kinetic equation and when you have this type of collisionless kinetic equations then somewhere you've got a good chance to find landau damping right so it's it has been discovered in plasma physics by landau but then it's like a standard concept in astrophysics too and also in 2d fluids mentioned order equation and it's okay it's known since maybe one century more than one century and also it has been discovered in many other instances and also even non-hubbinitonian systems like a synchronization model like a Kuramoto model some of you may know it there is some kind of landau damping there other models so again this is a kind of universal concept that that may be relevant to many long range interacting systems okay so uh landau damping now what what can I do to understand the asymptotic behavior of blast off equation so remember so the reformulate yeah yes in the experiment you have to to discriminate between a really landau damping on one side and and more easier friction so um well you can compute what would be landau damping and then look at how should landau damping evolve as a function of parameter and then change your parameters and then see if it evolves in a coherent manner uh like in plasma physics there they also have been they did some plasma eco experiments so it's not exactly landau damping but then it's it's kind of a way to to see that your evolution is really actually reversible despite the fact that it seems to uh to filamentate so you so if you're able to do some kind of eco but it entails a really good control on your system so it's not easy otherwise um yeah you can think of this good question okay um actually uh maybe uh maybe it's even possible to see to see a difference uh with the regularity of initial conditions because that would be important for landau damping if you can if you can see a difference with this um typically a normal friction the rates of the of the exponential rate of decrease to equilibrium would not depend say on the regularity of initial condition so may okay regularity is very uh mathematical concept but yet if you change really the shape of your perturbation if you manage to change the shape of the perturbation it should have an effect on landau damping i don't know just some guesses i don't know okay um so what can we do to understand the uh uh asymptotically behavior of landau equation so this is the the question say you start you have an initial condition you have a collision less equation you have to understand what's going on at last time and it's has been a very active topic in in mathematics actually so say uh you can have say a dynamical system approach so that was uh say landau damping is part of this so you have a stationary states you look at um stability of the stationary states maybe you look at um then bifurcations and non-linear dynamics close to the stationary states like using tools from dynamical system theory okay of course the uh so you have a plenty of tools a priori but then okay it's not so easy because it's an infinite dimensional and also one drawback is that the results you obtain that typically they're valid close to a stationary states or in a weakly non-linear regime in a sense then you have statistical mechanics approach the idea is that um so your level of equation it mixes really it seems to mixes the your levels the your levels of the distribution function so you can say okay maybe I will not try to understand really the the fine dynamic fine scale dynamics of this or just use statistical arguments and say in a sense it will mixes the the most possibly these levels and I will use a statistical mechanics approach for a lot of equation this was a really nice idea by Lyndon Bell by Lyndon Bell in the stroke physics and it was also done then in 2d fluids trying this to to um make this statistical mechanic approach works maybe I have a few words on that later and then there are also other ideas because um obviously what happens is that neither approach one nor approach two is completely satisfactory so uh then you can maybe try to mix both of them and then use uh dynamical arguments uh as most as you can and then when you cannot do anything more than then you you use statistical arguments to conclude or something like this so um have just made just a few words on the different dynamical approaches uh so these dynamical approaches I think there I can now again divide this dynamical system approach in two branches one branches would be uh the search for for stability criteria and this was really driven by astrophysics because in astrophysics it's not so clear so you have plenty of models for say galaxies um that are stationary solution of your browser of equation or collisionless Boltzmann equation in their astrophysics terminology then you want you want to know if it's stable and that's not easy to know if it's stable or not right and so starting actually Antonov was one of the pioneers again uh he used a variational approach to understand which one was stable or not and to get stability criteria so the idea is that you have if you have take a conserved quantity of your any dynamical system and you take say the say the minimum of a conserved quantity then it is a stationary state and it is stable in a sense right if you think a little bit uh if you have your your like a Hamiltonian you have the minimum of a Hamiltonian it's a conserved quantity then minimum Hamiltonian is a stable stationary state right so the same kind of things in an infinite dimensional setting you've got um a criteria for stability okay so um and you have a say powerful criteria say uh tells you that for instance if you have a solution of the browser of equation that is uh expressed like this so you just have oh hey so you have parentheses that are missing here so um the uh distribution function only depends on the one particle energy then it's enough that my function phi is uh then function f here is decreasing so that it's enough for that so that my system is stable my solution is stable that's a kind of a powerful criteria of course then um this kind of thing gives you no information the dynamics you have forgotten about the filamentation process I told you and you don't have landau damping for instance okay and uh if you were uh like me a bit mathematically inclined what happens is that here when you look about stability again then you have the same question as before about the distance when you were looking at the distance between two uh um solution you have I mean you're in an infinite dimensional system so you you have to uh define what the distance is and it matters a lot here you have to define what stability means I mean okay I remain close to when I start close I remain close but what does close mean and here in these systems close mean something without derivative right because if you have derivative then you have your um your um uh filamentation that that creates higher and higher derivatives uh values for derivative and then you're not close anymore so you here you forget about filamentation now the dynamical system approach two is really you try to be much more precise you you really try to describe the dynamics around uh around the stationary states and you try to define to to uh then capture also landau damping so um so this is like my uh linearized so in blue it's a linearized result of equation and red is a nonlinear part and you can see that the so the nonlinear part you have a velocity derivative here and I shown you that there there was this filamentation so when you have velocity derivative and and it's filamentates in a v direction so these terms become larger and larger so really uh linear the fact that linearization works it's not not clear and this this was known even to uh landau I think that okay you can it's it's a good idea to study linearized other equation but it's not clear at all what does it mean for the full system for a full nonlinear system and the um again here this mathematical remark is that I'm looking for uh stability in a sense so we what does stable what does close what does small means isn't mathematically important so uh just a few words on this uh because you probably um have heard of this or maybe not maybe yes so I said it's a very active uh topic in mathematics and it was really revived for example by the theorem by Muon Villani a few years ago so really they've shown that if you start close enough to a stable stationary state of load of equation homogeneous setting right you start close enough then landau damping wins and and you really landau damp and the potential will damp to zero in sense right and but these these guys here they have somehow um um converse that uh if you and they tells you and this is again um uh thing that that tells you that really mathematics can be important so I landau damp if you believe Muon Villani he said okay if I start close enough to a stationary state then landau damping wins and I go to a stationary states of lot of equation no problem but then these guys say okay but if you measure your distance in a different way and in a less demanding way right then you can start as close as you want to your uh stationary solution and and and yet go go elsewhere and and not landau damp your your perturbation so it's it's a bit um counterintuitive maybe for physicists but really what the way that the the the way you measure the stability and the closeness between your uh your solution is important so this is again now this is a uh uh old simulation like 20 years ago by Giovanni Manfredi I took it took in the internet so it tells you it's okay so this is the initial condition and you have landau damping there and you see it's not complete it's not completely under damping then it seems to oscillate forever so it tells you the dynamic can be much more complicated than just uh complete landau damping and understanding all this it's it's actually not a trivial task to go a bit further okay um so then uh not dwelling into details but just um a few words that's why if I want to continue on this dynamical system path right dynamical system approach to random equation uh so I have stationary states stable stationary states so where does it go when it's stable is it's when it's linearly stable is it non-linearly stable then the next step is really um um say a weekly non-linear theory so bifurcation theory so for instance I have a I I vary until the parameter my system sits in a stationary state stable one a very little bit of parameter and it becomes slightly unstable then my system destabilizes it goes somewhere where does it go and does it go far does it go close does it saturate close what happens that's that's really a bifurcation problems and that was especially important in in plasma physics and it's also relevant in astrophysics and I guess in several other uh longer interacting systems and and I think this is uh I mean even so at the mathematics level it's uh yeah it's very difficult I don't think there's there's so there's not much even at the physics level there's plenty of things to do there it was an interesting contribution by these people I would not have time to develop but the uh the big question that okay classify all these bifurcations really um in more um comprehensive way I mean it's still a very interesting question okay uh so uh I would just conclude on this that um so the organizer told me that I may I could have saved like a half an hour to talk about my research activities and of course I will not have time to do that so you can relax but um so one of my recent research activity with Yoshie Maguchi and David Metivier was precisely this so this dynamical system approach and this understanding the bifurcations of these systems of the Vlasov systems right so uh this is in the pgc of David Metivier and there is a poster by David I think where he explains this so you even if you're very disappointed that I cannot dwell into the details you can go and see the poster and there are also other theories related to this say dynamical systems approach but anyway my conclusion on this is I think it's a very rich problem and there is still plenty to do in this in this area then there's this statistical mechanics approach so again I don't have time to really enter into details but this so this was pointed by Lyndon Bell and I recall you the basic basic idea is that okay I have a Vlasov equation I know on a time scale maybe reasonable for my experiment I have to look at Vlasov equation then I have to look at statistical mechanics of Vlasov equation so these statistical mechanics that takes into account all the consents quantities of Vlasov equation right and there are many more than for the complete statistical equilibrium so that was the idea of Lyndon Bell and it gives rise to a I think a beautiful theory but I go to the comments on that so it's a beautiful idea and sometimes qualitatively it gives some really valid clue of what's going on but the the assumption of like you really have to go to a maximum entropy it's not so well in many cases not so well verified so okay I would say that Lyndon Bell the problem is difficult so in some cases it gives a very good result some cases it's misleading so that's a problem you have to then it the problem becomes to decide when it's interesting theory or not so okay so it's not it's not the whole the whole thing and again problem with Vlasov equation clearly in many cases it does not go to the its maximum entropy as defined by Lyndon Bell so then you have a kind of mixed approaches and there maybe Bernanda or maybe Jan Levin in their seminars they will give more details on this so again the idea is okay dynamical system approach is too restricted full statistical equilibrium of Vlasov equation this is too demanding I mean you don't have I mean dynamics is still important so you can try to do something in between and use as much dynamic as you can and then when dynamics becomes too complicated then you close in the sense your argument with statistical mechanics so say this this is a more or less the idea in this kind of approach and maybe Jan will give say a few words on this no no okay so now in a few minutes they have left I'll go beyond Vlasov equation so of course so remember I start from my particles and they have one first time scale which is Vlasov equation given by the the evolution is given by Vlasov equation then they have a second time scale much much longer in which the system will evolve according to well on the following the effects of the fluctuations of this mean field right and this is what I will call this collisional time scale much longer than the Vlasov time scale and so to point out again the the universal feature if you have so the fact that there is this separation between the say the mean field time scale and the fluctuation time scale this is a universal feature of long-range interacting system right at least for this conservative one Hamiltonian ones okay so this is something that one could look for instance look for in to in experiments with cold atoms like the in cavity which is not far from being Hamiltonian in some cases maybe you could look you could look for this phenomenology because this phenomenology with two different timescales is well known for instance in astrophysics of course and well known in plasma physics and what I'm telling is that it's relevant in many other fields as well okay so what can you do when you have to take into account these fluctuations then you have so Vlasov equation you just have the mean field dynamics in the particle dynamics you have mean field fluctuations so you have to take into account these fluctuations and you can formally analyze these fluctuations because again you have the separation of timescales between the two so that helps I mean we're in a better shape than with a short range interactions with this respect and you obtain the equation that well you can obtain several types of equation depending on the approximation you make but say of ballet school and our type ballet school and other plasma physicists so this was first obtained in plasma physics and okay maybe it's not important the details on this but you can see here there is a one over n so it means that the my timescales it's so this is for plasma physics say my timescales is much longer it's a small term time space is much longer and depends on n and this is a collision term because so it looks like a little bit like a Boltzmann if you see that you have a quadratic in f like f times f okay so actually rather than Boltzmann it looks like a diffusion equation so one term would be friction here and then would be a friction that would be diffusion okay so it looks like maybe a friction nonlinear focal point equation and this term would drive the system to statistical equilibrium over very long timescales so we're still safe with if we believe ballet school no equation then we do over long timescales we go to a statistical equilibrium and okay for this one there is no mathematical mathematical proof it's much more involved so I have to tell you so the basic physical mechanism here to relaxation is resonances so take for instance stars in galaxies so you so you have so the stars at the vlas of order they orbit in the galaxy right they have a almost like keplerian or not keplerian because it's not a keplerian potential but they orbit in the potential of the galaxy and you have other stars that have another orbits and maybe these two orbits have the same frequency or at least a rational ratio of frequencies and then they will interact strongly and these stronger these resonances they drive this linear ballet school evolution I said linear ballet school was for plasma physics but then it has been extended in astrophysics of course then it's much more complicated because you have a non-homogeneous background so you have to look at the evolution say also in space as well as in velocity space so it's much more complicated but still people are doing this and maybe we'll have some seminars showing us where what is the state of the art on this I don't know anyway state of the art is that people are trying really to do this to use this kind of of tools to understand astrophysical problems okay may I skip the last question I don't have time to this so this is important I this I have this summary here for dynamic evolution so I start on initial condition I tell told you on a time scale which is I called meanfield time scale or vlasov time scale sometimes sometimes it's called dynamical time scale my system evolves according to vlasov dynamics up to an asymptotic state of vlasov dynamics presumably this asymptotic state is not easy to go to compute to predict and it's sometimes called quasi stationary state because it's stationary for vlasov dynamics but not stationary when you look at the fluctuations around vlasov dynamics so this one is called quasi stationary state here and then over much longer time scale you have evolution towards statistical equilibrium and this is sometimes called collisional dynamics but I stress again that it's not really collisions right there rather fluctuations of the meanfield that drive the dynamics rather than really collisions are as we would be used to in a Boltzmann case okay so maybe maybe it's best this if I stop here to leave time for free questions so because just a few words on if I have non-amrytonian systems maybe so the simplest non-amrytonian systems I have is I have also a friction time scale so in the previous picture it so my friction time scales maybe drive my system towards say in a simplest case towards statistical equilibrium in other cases I have friction and maybe I have forcing so it drives me towards another state and then I have so I have another time scale in my problem so if I go here then I have to understand where is this new time scale is it is it here very short is it here very very slow is it somewhere in between so when I have my non-amrytonian systems then I have to take care of these all the time scales to see if there is something of this picture which remains or nothing for instance if my non-amrytonian time scales and maybe that's related to Michael questions if I have a system which is really over damped then I don't see a blast of equation at all means that my time scale is really here my non-amrytonian friction time scale is here and I just see this time scale and I don't see this phenomenology here if I have really a under damped system that maybe I can still see some blasts of dynamics here and then friction takes over and drives my system towards equilibrium and if my non-amrytonian effects are very very small then maybe I can see this whole thing and this is actually relevant for instance in in astrophysics they have all I think these different situations arise so a galaxy it's really not as I understand I'm not astrophysicist as I understand it's not really reasonable to model a galaxy's isolated systems so you have so they have external influences and so on and and and so so this this external influence forces that act on the galaxy they there are like this non-amrytonian time scale and and and you have to understand what's what if is this is the more important thing most important thing or if maybe the the the collisional linear balescu like time scale is important right for galaxies I guess the the collisional one is not important but for a global of clusters that are smaller systems if they may be important actually I think I will stop here maybe I have some conclusion here so to stop with this conclusion so okay so that was my view or part of my view on long-range interactions actually it's probably not yours and I really hope actually have many different views on long-range interactions in this this conference and again that was a starting idea it's also the concluding idea so there are many common features due to long-range interactions there are universal features so really I think we have to I mean we have a lot to learn by talking to each one to another and that includes also mathematicians really I think they have plenty I mean understanding really the some mathematics of it it really can help understand the physics too so okay I'm now excited to learn a lot more hearing at the the seminars and lectures to come thank you