 Okay so I would like to thank the organizers for having invited me to talk about this series of exercises initiated by our concern with the stellar dynamics that is dominated by a central massive body. That dynamics is relevant to centers of galaxies which is the set of systems that initially motivated my interest as well as the interest of my colleagues but it certainly applies to other systems in fact much of the thinking about these systems started around the solar system itself where the Sun would play the role of the dominant body or in the case of planets in terms of ring systems systems of particles around planets and their ultimate equilibrium evolution with caveats as to the processes that may be deriving a thermal evolution in the different systems. So in this talk I will focus on stellar systems which are the systems that have provided me with the motivation and then ultimately the various formulations which I feel are extremely natural to the context of this meeting. I must say that the stay here has been a thoroughly aesthetically engaging and intellectually engaging one. I mean what with oscillating state between my balcony and the Adriatico guest house. This is one of the views I was able to collect and beautiful quenched atom states I found myself in by coming to the Leonardo auditorium. So I couldn't have asked for more beauty. The physics I must say on the cold end seems particularly exciting though if I gathered correctly from colleagues still quite a bit messy and farmed from the desired objective. So I'll be looking forward to thinking about some of these systems with friends and students back home. So what is in the title of my presentation? I will be talking about the dynamics and thermodynamics of systems of particles whose mass remains a significantly smaller than a central body which dominates the dynamics of that systems of particles. The relevance of this system for our discussion as you might have heard and it started with Julianne's presentation and moved on to this morning's talk by Jan Levin gravitating self-gravitating systems are notoriously pathological and toy models the interesting models even some physically motivated models have all tried to deal with these pathologies in one way or the other. Normally what you do with pathologies even in social systems is you confine them. So what people have been doing to gravitating system is just confining them one way or the other. You put them in boxes, you put them in spheres, you add boundaries, you consider 2D systems in which you're dealing with lines, you take point particles, you think of lines of sheets. This has been the story so far but that story has provided statistical mechanics with very interesting examples albeit one remains with the question what does this have anything to do with physical reality. So what I'm proposing here is an exciting route that is provided by nature itself. Now that we know that black holes at least I hope you agree that we know that are a physical reality and LIGO definitely dispelled many suspicions. We can take it as a natural source of cut-off source of boundary in some sense in self-gravitating systems provided by a central black hole. So the problem that concerns me goes from the dynamics which I explored thoroughly with students and colleagues exploring in some sense instabilities in these stellar systems as well as collisionless relaxation to saturated states such as the kind you might have heard Jan talking about this morning. So on the other end irrespective of the process of relaxation in the system and we know that relaxation in these systems can be quite fast much faster than is expected from let's say relaxation in our galaxy or in elliptical galaxies or even in clusters, sun clusters relaxation is fast enough that it becomes relevant to the ultimate evolution of these systems. So within a matter of 10 to the 8, 10 to the 9 years you can expect significant relaxation in these systems. So you can ask irrespective of the process, okay, what is the maximum entropy state that you can expect in this setting? And so we developed a theory associated with these maximum entropy state for a model which is motivated by physical consideration, albeit, you know, rather restricted to allow us to make progress and the progress was very interesting as I shall show you. Finally, now that we've looked at collisionless relaxation associated with instabilities and looked at thermal evolution associated with relaxation processes in these systems, you can ask, okay, I need a theory that can take me from an equilibrium or a series of quasi equilibria, perhaps through instabilities down to the fully relaxed state. Such a theory we were able to put together very recently. And in that sense, we have a complete story and a system that is now just beginning to look reasonably exciting from a theoretical point of view. So this is ultimately 10 years of work. And I can very confidently say that the work is just beginning. Okay, so let's look at the astrophysical motivation. Okay, so again, the systems that I'm dealing with always involve a massive body, which is controlling a much lighter system of bodies around it. Okay, so the natural one, the motivation for my work had had to do with supermassive black holes, which after a series of very interesting and truly remarkable physical arguments starting way back and with Lindembell, the figure behind much of our thinking in self gravitating system thermodynamics having introduced the suggestion that one of the most powerful engines in the world may be powered by supermassive black holes. So now after a series of arguments and exquisite observational feats, we are pretty confident that most galaxies harbor supermassive black holes in their centers. And they tend to come in the range of 10 to the six 10 to the 10 solar masses with people cribbing about the limits of both ends of that spectrum. What is interesting for us here is that these supermassive black holes come dressed with nuclear clusters, stellar clusters. And these clusters can be on the order of a 10th of the mass of the black hole, they may be more massive than that. And they tell a story of mergers. Okay, they are sort of the tell tale of mergers, perhaps of clusters that have found their way to the center and got disrupted. Perhaps also a story of star formation, as we shall see in the center of our galaxies, as well as the mergers of black holes, which is also a potential route for formation of massive black holes for less from less massive ones. Close by, we have the puzzling galactic center, which harbors supermassive black hole, and the black hole in the nucleus of the sister galaxy, the Andromeda galaxy. So let's look at these. Okay. All right. So the galaxy has been the object, objective of very fine observations by two groups of astronomers, one based in UCLA, one based in Germany. And these two groups have managed to put together a very, very clear picture of what is happening in the closest regions of to the central black hole, indications of which have already manifested in various observational campaign radio astronomy being the earliest one. So what we know is that we've now managed to observe stars that are tightly tucked very close to the black hole, and whose motion is dominated essentially in a Keplerian, Keplerian sense by that black hole. So this is the result of actual detailed observations of stellar orbits in the center of the galaxy. Now, I would like you to note that these orbits don't look anything like what we expect from our solar system, where things are much colder, really nearly circular, they may compare to some cometary orbits in the solar system. And in fact, this is a useful analog to keep in mind, the cloud of comets, the Earth cloud, maybe a useful analog to keep in mind. So these orbits are so close that they give us a very tight bond on what sort of material structure could be generated, generating that gravitational field. And pretty much all alternatives to black holes are ruled out by the degree to which orbits are tucked close to that central concentration of mass, whose mass to rise ratio leaves no doubt that we're dealing with a supermassive black hole, whose mass is on the order of four million solar masses. That's a useful number to keep in mind. If you look at the region which that black hole would control, it will be on the order of two parsecs. That we would think of as the sphere of influence of the black hole. Okay, you can ask a similar question for the Earth, what would be the sphere of influence of the Earth, i.e. the region around the Earth, which will be largely dominated by the Earth for getting the contribution of the sun. And you get a number that is associated with what is known as a hill sphere. Here we're dealing with a collection of stars. So that number here is determined by the mass of the black hole, but also by the velocity dispersions of stars in that sphere of influence. For a rough estimate of what that sphere of influence looks like, you can think of it as the radius with which it enclose a mass in stars, which is equal to the mass of the black hole. So that is the length scale that is introduced by the black hole, which is the starting point of our cutoff. Okay, now the black hole is isolating the system from its surroundings in a natural and very interesting way. Okay, within that sphere of influence and very close in, there is a kinematically hot disk of stars, hot in the sense that orbits are distributed widely in inclinations and eccentricities. And they're young stars, so that created what is known as the paradox of youth. Okay, how can you form stars so close to a black hole, which is notorious for its tidal ability to tidally shred anything inside? Okay, so that's one thing. So people thought maybe they migrated in and were disrupted, but now we also know that you can actually under certain conditions form stars reasonably efficiently close to a black hole. So that's a paradox within the community. And then you are left with the trying to attempt to answer how these stars actually are so hot. So all I'm trying to suggest is we have a system whose kinematics requires explanation and whose dynamics and thermodynamics may be ultimately controlled by instabilities on one hand and strong relaxation processes on the other. Okay, that is as far as this I see I have to exit here. Sorry. Okay. So this is one proposition for the structure of stars in the center of the galaxy. And people have suggested that maybe a counter rotating population of stars is there. Other group, the other group, the California group prefers to think of it as a thick population of stars. So it's one disk, which is a rather thick and gives a sense of having counter rotating populations. Counter rotation is significant to the kind of dynamics I like to think about. Okay. Now to the triple nucleus of M 31. Okay, so this is the system that got me started. And this is a system that shows a very unusual feature. So what we have learned by careful studies is that the black hole is stuck somewhere here in that blue part, which also constitutes one of the nuclei known as P3. It's a population of stars, a disk of young stars, probably associated with a starburst, perhaps not unlike the one we're talking about in the center of our galaxy. So the black hole is here. You have a disk like structure here of young stars. And then the curious thing is that the light distribution around that black hole does not show any axisymmetric feature, rather, it seems to indicate a lopsided feature with most of the light coming at the slowest part of the disk. So if you look at kinematics, stars farthest away are slowest and seem to be lingering longest in their orbits and shining more than they would shine. And the other part, which is the anti pole of that distribution close to the black hole where they're moving fastest. So farther slower, closer, faster, it suggests a Keplerian situation. Okay, in an eccentric Keplerian situation, you move out conservation of angular momentum would want you to slow down, you move in, you want to be moving much faster. And so the suggestion is that this is perhaps an indication of a traffic jam. If you manage to get orbits to align, Keplerian orbits to align in terms of their orientation, eccentric Keplerian orbits to align, then you would naturally get an over light distribution on the slowest part, which is at the Apple apps and slightly smaller, a much smaller peak in the light and the brightness on the peri apps and where things are moving much faster. So that picture will be the subject of much a dynamical discussion to follow. So in that system, one has three nuclei to explain. And the question is, how are these nuclei related to each other? How do we explain the double nucleus? Is it the consequence of a dynamical instability? Is it the consequence of purely relaxation processes in the system? What is at work? And within the Milky Way system, one is trying to explain discs, hot discs of stars, young stars very close to the black hole. This is as far as the motivation is concerned. It's useful to keep time scales in mind. So if for an outside audience, this is I think what is most relevant as takeaway. So I mentioned, I mentioned this length scale, which is the sphere of influence that is in which the black hole controls dynamics essentially into Keplerian motion, which is slowly modified by the mean field potential of the cluster that is enclosed within that sphere. Within that sphere of influence, there is a hierarchy of time scales. And it is that hierarchy of time scales which we will employ to our advantage. So one has the orbital time scale, which is the usual Keplerian time scale. So it's useful to think, let's say for the galactic centers of time scales on the order of tens of years, two thousands of years in terms of orbital time scales. Then you have the secular time scale. This is the time scale which people like Laplace and Lagrange would have computed for the solar system. And this time scale is on the order of the mass of the black hole to the mass of the cluster multiplied by the Keplerian time scale. So that could be on the order of a factor of 10 to 100 in terms of slowness of time scale. This time scale is not present in the actual Keplerian system in which orbits are naturally fixed in space. This is the famous degeneracy of the Coulomb problem. And that degeneracy is at the heart of much of the dynamics that characterizes stellar clusters around black holes. So this time scale will be on the order of tens of thousands to hundreds of thousands of years typically. Okay, now we come to relaxation processes. The fact that orbits, Keplerian orbits are fixed in space, all right, essentially, means that they have a long time to interact with each other and that time is controlled by that precessional time scale, which is not true for stars that are whizzing by each other within the field of the galaxy. There they're moving along dynamical time scales and the two body relaxation time is notoriously long. It's on the order of what is known as a Hubble time. So it can be so long it is irrelevant for actual systems. In centers of galaxies, the fact that orbits are degenerate means that they have a much longer time to see each other and so they interact with each other much more efficiently and that has been introduced by Rauch and Trimaine in 1966 as a form of what is now known as resonant relaxation. That relaxation does not affect energies, so the semi-major axis of the orbits is not affected by this relaxation, but it affects their eccentricities and their orientation, so it controls angular momentum and not energy. Finally, the energy evolution which has been the subject of numerous studies starting with work of people like Spitzer and on to the 70s with a series of calculations associated with two body relaxations around nuclei. That is the longest time scale and that time scale will not be relevant for our calculations. So this is the hierarchy on the order of hundreds to thousands, tens of thousands to hundreds of thousands and then this could go on to 10 to the seven, 10 to the nine years for the galactic center. This becomes very interesting and this is orders of magnitude longer still. So this is the two regimes within which we will be working. Now within these two regimes one has within the sphere of influence of the black hole within that hierarchy of time scale you have the fastest orbital time scale now and that orbital time scale is so fast that we can appeal to an idea already introduced by Gauss namely to replace point particles by a smear distribution of mass in which the density would be proportional to the time spent on the orbit. Okay, so that is Gauss's idea which is ultimately the first application of what Arnold would like to describe as the averaging principle within Hamiltonian system. This is the first instance of that use of that principle explicit use of that principle it was used implicitly by people before him. So that would imply through this averaging that the momentum conjugate through the degree of freedom on the orbit will be conserved and that is precisely the semi-major axis of the orbit. So orbits in the sphere of influence largely satisfy a sort of invariant okay which is associated with their semi-major axis so they can do whatever they want the distribution of semi-major axis will largely be invariant. Now you might say averaging breaks down around resonances resonances mean motion resonances in these systems are extremely weak and that affects thus exponentially small when it comes to affecting the evolution of the semi-major axis. So we drop that. So now we average a particle is replaced by a ring and ring-ring dynamics now has very interesting properties rings now relax resonantly and that's pretty fast so their semi-major axis doesn't change but their inclinations could change and their eccentricities could change and then these systems could be studied for dynamical evolution so you can consider a distribution of particles now turned rings so you can ask I have a distribution of rings all preserving their semi-major axis distribution and then you find that they are prone to secular instabilities instabilities that evolve over the slow time scale the precession time scale so instabilities for instance in the center of m31 the andromeda galaxy would evolve over time scales on the order of a few hundred thousand years that is extremely fast when thinking of centers of galaxies so a series of calculations proved instabilities in systems particularly involving counter-rotating streams of stars which may be unnatural and planetary systems which are highly collisional but there is no problem with counter-streaming stars within galaxies as the earlier picture suggested so back to m31 here is the system that we're trying to explain we're trying to explain this over density over brightness on one end of the black hole on the slower end of the black hole and the model here was associated with tremaine when 1995 proposed exactly this aligned Keplerian orbit picture that i had suggested earlier so he proposed that this traffic jam would provide the answer to many of the observed properties of that systems brightness kinematics everything you want in fact it proved extremely successful and next generation models did were significantly correlated with observations so by now we believe that this is indeed what is happening in the center of m31 the question remains for a dynamicist how do you actually contrive to bring orbits in alignment like that okay so there are various propositions i will talk about ours naturally so ours which is one of two or three propositions in the literature considers a counter-rotating stream of orbits and then it had already identified such streams as unstable streams and the instabilities we believed or suggested will actually relax into a lopsided configuration by arguments that had to do with conservation of angular momentum in the presence of an instability so here just to show you that we're dealing with things that are closer to reality than our theoretical work nowadays people in papers that do not accept theory you need to have numerics otherwise i can't take your word for granted so here is the numerics before the theory so we considered a 10 to the 8 solar mass black hole and carried out i think at the time some of the most serious tests of n body calculations undertaken to date with our meager numerical computational resources so we considered a 10 to the 8 solar mass black hole a prograde population of 10 million stars a counter-rotating population of a million stars and we allowed them to relax in some suitable fashion to eliminate any source of fast instabilities in the system and let them evolve so this is what happens so this is when it comes to the dynamical part this is the prograde on the more massive part of the distribution and they go unstable over a time scale of the order of 200 thousand years in the course of the instability the massive population settles into this lopsided uniformly processing configuration which is exactly what it remain like model would like to see and the retrograde population is this first in a fashion that is reminiscent of what Yan was describing in terms of resonant pumping of populations in fact we did look very closely to that resonant pumping in a fashion that is similar to what he described this morning so a result of that is that the density the mass density of that distribution evolves into a lopsided configuration whose properties as I will show very shortly are compared very favorably with those that are observed in the center of m31 so we have a counter streaming population which goes unstable long range interactions settling into a saturated state collisionless saturated state I might add okay so here there is no collision evolution to speak of it didn't have enough time to operate within that collisionless evolution we get this beautiful lopsided processing distribution what symmetry breaking bifurcation of sort and this is indeed what is going on now this is an account of what the saturation looks like in that system so if you look at the modes this is the evolution of the m equal to zero mode in the system which is the axis symmetric mode but this is the interesting one okay the m equal one mode associated with the massive population is the one that is giving the signature to that processing disk and this is the evolution of the precession rate as the system settles down into its quasi equilibrium yes it is a quasi equilibrium so this is in some sense a first look at what can happen to stellar systems within spheres of influence this is the numerics of it now if you want to look at face space this is the structure of the mean field associated with this distribution and this is what the face space look like when I look at eccentricity dynamics the prograde population now is here the retrograde population is being pumped by the prograde population the massive population to occupy a very wide and much hotter region of face space as it saturates into its equilibrium not unlike what Yan described for his cylindrical systems this morning okay this is the before and after and then you may wonder how does this look when it comes to observation well we do extremely well I would say given that we didn't tune hardly any of the parameters in the system so if you look at velocity dispersion which is a key kinematic signature around the black hole this is what the observations are like and this is our solution and our solution in addition to recovering things like the peak and the tail on one end recovers the tail on the other end which has proven extremely difficult for alternative models to obtain so that's as far as dynamics so one would like now to develop theory okay and okay so the numerics here is trying to provide a Monte Carlo a dynamical evolution of a Monte Carlo sample of this face space distribution which is evolving according to the you know collisionless Boltzmann equation about which you've heard a lot this week the potential here involves the self potential of the cluster okay and the external potential which in our case involves the black hole so we have a black hole we have the self potential in the cluster and you can allow for additional perturbations and we do allow for these when we need to since the dynamics is dominated by the black hole we have nearly Keplerian motion so the natural suggestion was to okay let us orbit average okay orbit averaging already provides a significant reduction of face space size so I don't need to do things to the system in order to have a properly manageable system already with orbit averaging I have essentially a four-dimensional face space for a triaxial cluster if I'm dealing for a two-dimensional cluster a disk I have a two-dimensional face space with a distribution of semi-major axes which is invariant so this is extremely natural and a consequence of the averaging as I pointed out is the conservation of semi-major axes so this is an exact with quotes invariant in the system all right now that you average you go from particle to wire you have this hierarchy of time scales that you can exploit in the averaging so you end up with fast orbital motion over which you average and then slower motion associated with the mean field of the cluster that is a slower time scale then you may have sources of variability that obtain in binary systems which are extremely relevant to binary black hole systems and finally you have relaxation processes and these will be responsible for taking the system to its thermal state okay we've talked about that we have turned particles into wires and now we would like to generalize all of galactic dynamics okay to systems of wires instead of systems of particles and that has been our program for the last 10 to 15 years when it comes to thinking about these systems so Vlasov was turned into a Vlasov over systems of rings over a reduced face space thermal dynamics was reduced over a system of rings instead of systems of particles and kinetic theory was developed for a system of rings instead of a system of particles and everything works through beautifully consistently and without any of the pathologies that are familiar from other settings so just in case time finds me overboard let me give you the progress report and then we'll take it leisurely from there assuming I have time to be leisurely about okay so the progress report over what our more than a decade of work is that counter rotating nearly Keplerian stellar disks okay which are some of the disks I've shown you in the simulations are violently unstable okay they are violently unstable and they evolve into lopsided uniformly processing configurations over a rather fast okay time scale which is a collisionless saturation time scale okay now the actual process of saturation something we can talk about there is no full theory for that saturation yet okay finally when we look at the thermal evolution of these systems of rings okay now we've taken particles and turned them into rings so we're dealing with a lower dimensional of face space compact face space I might add with no pathologies associated with infinities in the velocities because velocities disappear from the picture velocity associated with fast motion on the ring has been absorbed into the Keplerian energy which is a conserved quantity so no more velocities in the face space face space is purely configuration space if that rings familiar to people who look at vortex dynamics this is precisely what we have here okay rings in our systems have a lot in common with the vortices that have occupied on zager in fact much of my thinking on the thermodynamics is motivated by on zager's work even though the conclusions are very different all right so finally and just in the last three years and I have to thank Jerome for inviting us to Paris for this myself and Shridhar it was an extremely productive time in which we put together a first principles theory of resonant relaxation okay something that has been lacking for the last 20 years now we have the foundations for thinking about resonance relaxation alarak and trimane in a grounded coherent fashion so indeed the fun is just beginning okay having gone over this progress report I'll just go very quickly over the system okay that has motivated much of our calculations insights it's a rather it's a system of reduced complexity as you must have but it's certainly not a toy model in any sense it has all the physical justification that goes into providing us with a nicely truncated dynamical system whose physics we can study nearly completely okay so what is the system in question all right so the system in question involves counter-rodating discs of stars around the supermassive black hole and dynamics is dominated by the black hole so we can do the orbit averaging I was talking about semi-major axis is conserved okay it's conserved in fact we take all particles having the same semi-major axis all right so and they all have the same mass they can only have the freedom of going one way or the other but otherwise their eccentricity is absolutely free so they can go prograde retrograde they all have the same semi-major axis the coordinates are naturally the eccentricity and the orientation of the Keplerian ring or the geometric equivalent of those in terms of a polar representation the phase space has suddenly become compact okay they all have the same semi-major axis and it's natural to think of the lens vector on the Keplerian orbit as the vector of interest and that vector the dynamic of that vector is best represented on a sphere okay where the co-latitude provides a measure of the angular momentum and the longitude provides the orientation of a ring in the plane this is the prograde side this is the retrograde side and they're coupled to each other looking at the coupling okay one would like to recover interaction of rings which finally reduces to a function involving differences between their lens vectors okay so this is a prograde ring this is a retrograde ring this is the lens vector associated with these eccentric ring ring coupled ring system so the question is how do I get the interaction between them well we're going to do the orbit averaging or the usual Newtonian interactions between two particles now when you do it over systems of rings that have the same semi-major axis you can show and it was already shown in the 80s by Goldreich and Tremaine when looking at ring systems that the interaction is essentially logarithmic and the difference of the orientation of the lens vectors all right now this may sound familiar logarithmic interactions in terms of position vectors are natural in the context of point vortices well the analogy is not exact it's almost there but it's not exact the dynamics that we have is natural for point vortices on a sphere okay so if you take a Helmholtz-like treatment for point vortices on the sphere you get something that is quite close to what we recover okay so now that we have the ring-ring interaction reduced and the dynamics regularized through the orbit averaging that I described compact phase space velocities out you can't ask for anything better now we're doing essentially statistical mechanics with very clean long-range interactions involving vectors okay which are continuous rather than the discrete configurations you may find in quantum mechanics so the question that we ask if we take a distribution of prograde and retrograde rings and let them evolve what happens if you do the statistical mechanics of these systems okay do they manage to contrive into aligned configurations of the sort the answer is yes okay thermal equilibria for counter-rotating systems more often than not want to be in these beautiful lopsided configurations dynamical instabilities take you there but also thermal evolution pure thermal relaxation in these systems will want to favor a conserved angular momentum and energy lopsided configurations so the program is straightforward you pass to the continuum limit okay which is now a function of what canonical version of this eccentricity vector I was talking about known as a Poincare variable you introduce an entropy okay this is the Boltzmann entropy over prograde and retrograde populations and you maximize that constant n, l and u where l is the angular momentum and u is the total energy and note that the energy right now is just mean field energy this is the Hamiltonian okay there is no kinetic energy term left in the problem just the average potential okay so you go through the procedure you end up with a non-linear integral equation its solution is quite complicated Harry you try various techniques I'll just tell you something about the program and some of our results so the program is to solve for axisymmetric equilibria we did that ask whether they're thermally stable we find that a lot of them actually are at entropy saddles okay so they are thermally unstable are they dynamically stable well we find whenever they're thermally unstable that they're dynamically unstable okay and if thermally unstable what are the global entropy maxima we search for these with various techniques and then finally if dynamically unstable what are the saturated states and how do these states compare to the maximum entropy states okay so the complete story discussed in various guises in various lectures in the course of this very interesting conference all right so here is the report axisymmetric equilibria are prone to lopsided m equal one deformations over a broad range of energy and anger momentum okay this is maximum entropy statement equilibria are symmetry breaking naturally over a broad range of the physical parameter space resident relaxation tends to drive nearly capillary in this to these lopsided states so m31 had it had long enough time would find itself in this lopsided state so this thing would suggest with some variations over how planar the distribution is okay so you have to take it with a few caveats all thermally unstable disks are dynamically unstable this does not follow generally okay thermal stability implies dynamical stability thermal instability need not okay now it turns out when you look at the linearized stability analysis all unstable configurations from a thermal point of view are dynamically unstable and the dynamically uninstability drives systems to lopsided uniformly processing equilibria so the same picture I showed you in the numerics of the full n body calculation I showed earlier the full story involves the complimentary action of both collisionless saturation and collisional relaxation okay so this is a sample of what the equilibria would look like in the axisymmetric regime all right this is the mean eccentricity associated with the population the disks okay as a function of angular momentum and color and energy all right so the high the lower the energy in the system the greater the concentration in the central regions now let me show you how an equilibrium a global equilibrium is found in this situation so this is a micron canonical Monte Carlo simulation of a distribution of rings where the points represents orientations of the eccentricity vectors okay so we start with a nearly axisymmetric configuration and we let it evolve through a Monte Carlo search for the global equilibrium and lo and behold it wants to find itself into a fairly highly eccentric and orbitally aligned configuration of the kind I have been talking about so far all right how much time do I have no time okay we play it a Mediterranean style okay so when you ask for the stability of the axisymmetric systems I have argued that they are thermally unstable and here we're showing in dashed the fact that for every angular momentum this is zero angular momentum and this is higher and higher angular momentum as the color goes down for every angular momentum there is an energy you know below which okay systems are thermally unstable now you ask if you look for global equilibria okay you're no longer restricted to axisymmetric configuration you find that the equilibria follow lopsided configurations such as the one I will show you here so as a function of energy at zero angular momentum you will start at high enough energy with an axisymmetric configuration and you will turn increasingly lopsided in terms of symmetry breaking bifurcation at the thermal equilibrium if you look now this was the phase space density mind you this is what it looks like if you look at the actual configuration space density associated with such a ring or a distribution then you will get this hot lopsided configuration which is typical of what I have shown you for the m31 system and which we can suggest is natural to obtain for stellar systems in the sphere of influence of black holes okay I will jump over dynamical stability if you ask me a question about how it compares to the thermal equilibrium I can try to tell you more about that and I'll just take a minute to finish with the last bit of work it's just one slide that has to do with the route to thermalization okay so this is now having characterized collisionless instabilities and their saturations and having characterized thermal equilibrium in these systems we want actually the route that will take us gradually from one equilibrium to another equilibrium on the road to thermal equilibration so here is a recent work with my colleague Shridhar of the Raman Research Institute in India so what we do is we generalize a beautiful work by Gilbert which does a 1 over n treatment of a BBJKY hierarchy and actually finds a closure for that hierarchy which is natural in 1 over n expansions okay this is one of the cleanest treatments of collisional evolutions in stellar dynamics so we generalize that to the realm of Gaussian rings okay instead of particles deal with Gaussian rings around black holes then that allows us to get the first principle theory of resonant relaxation which is the first formally grounded approach to rock and remain which is an ad hoc treatment of resonant relaxation finally we look at the interactions between resonant populations which is ultimately the source of driving of resonant relaxation okay and it involves the feeding of what Gilbert describes as a wake now it involves the wakes of 2 rings instead of 2 particles which then feed the orbit average Boltzmann equation with collisional flux that is the right hand side of the Boltzmann equation we recover a Fokker Planck equation with explicit terms involving both the dissipative and the fluctuating part for the evolution of a distribution of rings and we are now just waiting to start looking at realistic systems with whatever resources are at our disposal I thank you for your attention