Thank you very much. So yeah, I think it works this great So of course my first words are for the organizers who invited me here and gave me both the honor and responsibility of giving these introductory lectures so I hope I'll be up to the task and not to To a boring for you because I have two hours. That's a good challenge So of course the title of my talk is a pretty obvious in connection to longer interactions and and of course I cannot claim that I have the Definition and the idea of what longer interactions are so that will be my view on this and I hope it can connect with yours as much as possible So may all I have all the things that go to many people have worked within this subject of course over the years and So you notice come from University of Nice. So it means that the The last ten years almost all my work was done in this so my first slide is a happy view of the prom that this angle Of course he has been so sadly in the headlines. So that's a happy view of from that is only which is a very beautiful place So okay, so so I start So I called it theoretical physicists view on longer interactions because I'm theoretical physicists And my main interest in this is that when you have a system with longer interactions Well, they can be very vastly different systems in very different physical fields Longest interaction in use at least may induce some common features for these very different systems So that's nice for a third set equal physicist that usually looks for Universality in different systems. So it really suits this natural tendency And and of course that's the reason for which there is this conference, right? So the idea is that we all have to learn from each other from our different we Approaches to different physical systems. So That's the that's the main reasons and actually that's of course the main message of my talk So I could almost stop here actually and enjoy the the the seaside, but I will go first a bit a bit longer into details okay, so I'm the third called physicists some sometimes mathematically inclined. So this is why There'll be some hints to mathematics and the idea is that of course when you have like atoms on one side and Stars on the other side, for instance, they don't share many things in common except sometimes some mathematical structure that will underlie some common phenomenon phenomenology Okay, and then last thing maybe as a for these introductory remarks So I'll try to keep emphasis in various physical systems so that as many of you as possible can connect to what I'm saying and maybe Disagree with what I'm saying that's still also of course, okay But I cannot claim to be competent today to have expertise in all the fields. I will be Giving a few words on so I hope I will not tell so much Wrong things and if I do please Interact me and tell me I am doing something telling something not so true so There is there's the first part of my lectures so of course being an introduction and You notice the first subsection will be on the definition of long-range interaction So that was already pointed out by Michael that's it's not it's not so true It's so easy to see to know what's longer interactions are So so that would be our first task be to a bit sort out between possible different definitions and pick up My mind because I have to do and then and then of course we can discuss on these Then we'll try to give a lot of examples so from different physical fields and as many as possible But of course I will not cover everything with longer interactions. That's not possible and then I'll go into a more a few Some remarks on on the structure of long range interactions and what what it implies So definition So preparing the lecture Like you've surveyed a little bit literature not so much because it's so it's huge But you can find several different definition the first one Well, I put it here in red because it's further on is the is the one essentially I will use So you can try to sort these different definitions by looking at the interaction potential between two particles in your system you imagine the many particle system and you can sort it as according to the Potential and how the potential decay was distance the interaction potential decays with distance and the okay The strongest possible measure like a meaning for longer interactions is the first one So the pen potential decay say as a polo or something looks like a polo One of distance to the power alpha with alpha smaller than dimension So if you think a little bit of it if you have one of our alpha smaller than dimension it means that if you have say homogeneous systems System than the potential energy of the system will grow More than linearly with the volume, okay So somehow sometimes people call this energy not extensive I would rather call it than the energy is not Additive and we see later on additive means I put two homogeneous places pieces of system together I glue them together then the energy of the total system is not in any sense Even if there are macroscopic systems subsystems the sum of the two subsets energies of the each subsystems, right? I will come to that later. So that's first definition of long-range interactions Then sometimes you find this so you hear but interaction potential decays like More rapidly than than than the one of the art of the dimension right my sigma is positive And yet I mean it makes sense to put give this sort of definition for instance in spin systems like in the 70s people were doing studying Renormalization group and critical exponents for these kind of systems and they found that if there is like usually some critical dimension that says that if the the potential energy decays slower than some critical power then the Critical exponents are changed and if it's more more than that and critical exponents are not unchanged with respect to the nearest neighbor case so this also Like a kind of a valid definition of language interaction. Let's use in literature Another one sometimes is okay anything slower than exponential is called Long-range and Something it makes sense for some sponder valves it makes sense because you can expect maybe that the correlation functions then There's a signature on this on the correlation functions. So that's also maybe physical physically valid definition and then I found a review on the In the literature, which is like a longer interaction in nanoscale science this maybe it's maybe even possible that some of the authors are in the audience and I found this this sentence so we can propose definition of long range on the nanoscale Starting with extending being us beyond a single bond. So now you can you have a just a Spin system for instance in condensed matter and then if you have next nearest neighbor interaction Then you can start thinking of it as long range So, okay, so that makes a quite really broad-scale for a long range interactions and In this very same review actually by French and Calabasters There's this conclusion what constitute the long range as opposed to short range Interaction depends primarily on the specific problem under investigation. So okay, so we are safe then for each problem We can decide if it's long range or not and and and that makes a life easy. Okay So for my purposes, I will concentrate on definition one, which is the strongest one So I would say a recall here one of our part of the alpha series alpha smaller than dimension and Yet, I hope that some of the ideas that I will develop and expose in this setting are somehow relevant in the For some other long range interactions for instance, we can imagine that they're relevant even if the if the interaction is not really Power low, but if say the interaction range whatever it means is the same order of money to the size of the system that would be essentially the the idea and Okay, I don't pretend that with this small discussion. I have like Settle the problem of definitions of long range interactions. You can for instance notice That I have used the potential in discriminating between the different Differentials of long range, but I could have used the force to write and say, okay Maybe it may be it's not important what if the interaction is integrable in the sense Maybe it's the force so we will come a little bit to that later on but that's okay to tell you that there are many issues So now we have essentially a definition of long range interactions. So we can start giving examples of on what what goes into the definition of what does not So first remark is that actually many fundamental interactions. They actually fall in this into this category apparently So of course have the Newtonian gravity with the attractive potential Problems, okay And and of course, that's the Extremely important interaction for astrophysics. So we'll come back here to that later I'm sure for in different lectures or and seminars during this conference well one difficulty from the Physicist point of view experimental physicist point of view, of course, it's not so easy to make control experiments on this Because you have observations not control experiments another fundamental interaction is cool interaction, of course So and I have divided it between say two categories So one non-neutral plasmas and neutral plasmas because so non-neutral plasmas are purely repulsive interactions and there is no real Screening in the usual sense in this system and neutral plasmas. There is of course screening so the you can already tell that it's almost in contradiction is what I've told you at the The first with my first introduction that long-distance interactions. Okay, they share Very their universal properties. So here, okay, there's already your hints of course, if you have a neutral plasma and if you have Newtonian interacting systems like your galaxy then okay completely different phenomenology. Okay, so so The my first remark had to be taken with a bit of caution, of course And if you think a little bit you probably know that if you have like a neutral matter Well, it's longer interacting and yet there is a some other make limits. That's that's for sure Okay, we had you will live in the some other man said on any climate in a sense So and and if you have a purely Newtonian galaxy, it's not in some other make limit that you would use describe it So they're really fundamental differences, but we'll come back to this later Okay, so that's for fundamental interactions Now of course many systems with long-range interactions. They're actually effective interactions So you have so in behind behind the scenes you have some more fundamental things but when you describe it to more microscopic level you have effective interactions and then that these may be long-range so a first example is Vortex-Vortex interaction in 2d fluids. So you have the Helmholtz Hamiltonian that described the interaction of vortex and it's In saying 2d you have a log r potential. So that's definitely alpha equals zero. So it's definitely long-range with my definition And you have many others another like broad categories the wave particles Interactions so you have sometimes particles that interact that sustain a wave and interact With this wave right so the waves is like a global degree of freedom in the whole system It's a like a mean field interactions between the particles and the waves and so here you have in plasma and fluid dynamics You have this to free electron laser We have free electron laser right a few kilometers from here at electrolyte and you have this kind of phenomena going on there called atoms in cavity also that will give Some details later on on this so they enter into this category okay, so few more details, so I've checked me some of the speakers of the the conference and the the whole week and So many people I didn't know many people I did know and So I'm not sure they will be talking about What's what I will present here, but most people of course are connected to one way or another To longer interactions and these are the kind of long-range interactions that I could I could guess more or less would connect with people that will be speaking so one one idea is Colloid that interface so with these people there so you have a you put you have a fluid interface and you have colloids and and if the colloids may deform a little bit the interface and because of this Deformation of the interface it induces interactions between the colloids Colloidal particles and then depending on length scale it can be a long-range one So they'd like you have a capillary length so distance between Colloids has to be smaller than capillary length and also not too close to one to another Otherwise, there are other effects coming to play in that case. There is something that looks like 2d gravity there is a logarithmic potential between them and Then one thing you can notice is is how you see it of course For these colloids particles the dynamic will not be at all Hamiltonian like it was for instance for vortices or like you can imagine it almost is for stars in a galaxy It will be more another than dynamics. So again, okay universality in some sense and we'll go to these universality universal features later on but Many differences between when you look at specific systems specific systems, of course Another one was this well this one it looks like the previous one. It's a Chemotaxis, right? So it's a specific so phenomenon which is really Really common in in biophysics. It's like how Cells can can move by sensing a gradient of a chemical quantity in the in the background So so this is the so okay So the the cells are for instance diffusing and this is they move up to the gradient of some chemical quantity C Right. This is just a drift term and this is dynamics of the chemical quantity So it's a diffuses itself and it may be degrades at some rates and this is created It's like the the cell acts as a source for this chemical quantity. They emit this chemical quantity, right? And so in some limit if for instance if the lambda is very small and maybe this dynamics here is very Is very quick very fast then you then just a little bit of algebra you find the same kind of Interactions as the one is the one in the previous slide and you have again something which is similar to over damped 2d gravity, right? So this is a Typical model for Chemotaxis and it is a huge related activity with this in also in mathematical biology Okay Was this one so cold atom so I think we'll have many Seminars and talks about cold atoms and longer interaction cold atoms It's just some some interactions, but there are several types of long range interactions called atoms so I don't claim it would be always like this and One important thing is that I'm rather Classical physicists as opposed to a quantum one So there will be very few quantum mechanics in what I'm going to say and probably more in the seminars So I apologize especially to organizers that are very fond of quantum mechanics So I count on on their further and later seminars to To fill this gap and fill this hole in my presentation anyway So here the the system will be seen as essentially classical for the atoms and even if quantum for for the light so the Which cold atoms usually you have the you have a wealth of interactions that are mediated by light essentially So you have photons that interacting with the atoms and sometimes So dynamics for the photons and the light is very fast and you can just integrate over this fuzzy group freedom And you end up with an effective interactions between the atoms, right? So this is the idea here and So imagine you have so this is what happens for instance in my topical trap You have atoms here and you have lasers and the lasers are tuned very close to resonance So the atoms absorb photons and then they reemit photon in a random fashion Spontaneous emission and maybe an atom here will again absorb the photon coming from the first atom that was heated right and of course the photons they carry momentum so once the This atom absorb the photon coming from this one. It takes a kick in the opposite direction Okay, and the kicks and means that it suffers a force a repulsive force from the the first atom, right? And if you average this over many cycles of absorbing a photon emitting a photon and the second atom absorb the photon Then if you average over these this is a fast time scale And if you average over many cycles like this then you end up with an average Repulsive force between the atoms and this repulsive force turned out to be like Decreasing like one of our squared so it corresponds to potential is one of our so it looks like a Coulomb force, right? and why one of our squared just because like the the Solid angle at which one atom see another one is Decreases with one of our squared with one of the distance squared Okay, so this is a cool like force. Of course, it's not the the atoms are neutral Then everything is neutral, but the effective force is cool. Okay, and this is okay This was introduced by what says co-worker with Wyman like 25 years ago. It's more as a standard model for this interaction Another type of interaction that also takes place in Magnet optical traps and that was introduced by Jean Dali bar So almost 30 years ago some type called shadow effect So imagine you have your atomic cloud here in light blue have lasers coming from one side and from the another and they Concentrate on this atoms on the left, right? So on this atoms on the left it receives photons from the left So did so then it's it's an average force towards this direction and photons on the right so an average force from this direction So on average, it's a equilibrated right except that the laser is coming from the left from the right Then has to cross almost the whole cloud before coming here and crossing the clouds. There are some photons are absorbed So the the intensity of the laser decreases While it's propagating on the cloud. So coming here intensity coming from the left is a larger than intensity coming from The right there's an imbalance and the average force goes towards the center. So this can be described as a collective interaction between inside the cloud and that's Can be modeled as an effective attraction long-range attraction between the atoms and Okay, train things. So so it looks like gravitation because the divergence is minus the density to a Constant but it does not derive from a potential theory. So it's not a potential force So it's okay. There's no contradiction because I'm not it's an effective force. I'm not saying it's So it's not conservative and it's not a contradiction, but still it's an interesting feature Okay, now again called atoms but in cavities So I have my cloud of a cold atomic cloud. I mean cavity here and I'm pumping them with a laser and Maybe now the laser is typically Further from resonance. So dynamic will be more conservative than in the previous slide on many optical traps and and the The atoms then the emits Coherently light so they absorb light from laser and they remitted in a coherent fashion in the cavity and then the so there is a standing wave in the cavity with some of course there necessary some losses and And In some in some limits, then you have then you have like a wave and particle system. So here you see In my picture the atom atoms are bunched so that is already Because there is a standing waves in the in the cavity and the atoms react to this wave and and they react to this wave by Bunching and they also since they were bunched and they can emit light coherently and then okay, and that's a positive feedback so this is kind of mean field Interaction and somehow attraction between the the atoms in this in this system So this is for instance the experiments and theory developed by Giovanna Morigy and her group In Germany Okay, and again, you can integrate over the cavity degree of freedom and have Interactions between the atoms.  It's not It complicated the force and you have at chose distances is dominated by you maybe one of us a tube and and if you if you but if you Put the laser in a clever way and you average in a clever way then then you can Make this one of our cube contribution vanish and you end up with a one of our interaction and That may be an isotropic depending on the your setting but anyway, you have something that looks like a gravity like force inside your your Bosenstein condensed cloud So you have so description would be maybe gross pitaevsky with a Long-range interaction. So That's another setting that was proposed like 15 years ago More recently you have active particles. So you probably have noticed that it's a very trendy topic in statistical physics in general and biological physics and In some cases you have may have long range interactions in these particles and here the So maybe not develop but if you have so one way to make particle move in active active ways to have them coated with some say on one side you have maybe Metal and then you shine them with laser and then the it creates a temperature gradient because the part with the Which is coated with metal absorbs more light and then creates more heat releases more heat and of course if you write an equation for the temperature then you have something like a diffusion equation for the Temperature and if you have the fusion equation for the temperature then you have it corresponds somehow to a long-range interactions, right? If the the the temperature Is fast it could be it's fast, right? So in some cases, for instance the goleztanian they're claiming that it one would see something like again Would like look like would look like gravity like dynamics Okay, so this is just for maybe for more for mathematicians and for More theoretically oriented. So even if you take eigenvalues of random matrices take for instance, so complex general ensemble is just a one of the simplest one you can think of you have complex matrix and and they're the entries of the matrix are independent their Gaussian and then you look at the eigenvalue Joint probability density of all the eigenvalues of the matrix, of course you have then there are complex eigenvalues and you have and eigenvalues typically And then you can see that the you have this kind of joint distribution So this is a kind of a harmonic trap here and there is an interaction in log so with a minus sign something that the particles typically eigenvalues in that case typically a repellent rather and It's called like a 2d Coulomb gas. Okay, and if you take all the random matrix ensemble, then you have some similar usually some similar things and I also Mentioned it because there is a huge activity in mathematics on these These systems and rated systems and there are sometimes called determinants all processes and so on so so again I'm not sure it will be irrelevant for What you what you're interested in but they there may be some mathematical tools to to take here Another one. Okay. Oh, I'm Also quantum mechanical here So if you take just a free failure of fermions in a harmonic trap one dimension and then you just Write the theory and use poly exclusion principle on your right So this is the ground state wave function in case of a squared the right state wave function square Then you you find exactly the same thing as before so you can see that there is a x square It doesn't work that well the thing. I think I think you're more. Let's see died more or less. Yeah. Yeah, okay, so how Okay, so then then you'll have to be really careful because I cannot really point out what what I'm saying about So it seems like you have to be really you cannot sleep too much Okay So it was saying so okay, so you so you have this this wave function and You can see that there's a oh great so thank you very much so you can sleep again So you have this this product here, so if you this product if you Instead of a product you say it's exponential of a sum then the sum you will find is is just a sum of a logarithm of Distance between xk and excel. So just the same as before so you again find like a Coulomb gas in one in this case It's one d. Okay, one d log gas essentially Okay, then you're free you have plenty more example. I cannot Dwell on all of them an interesting one. Maybe it's this one I think yeah, Tuma will give a seminary and I'm not sure if he'll be talking about this But anyway find it interesting. So you have Still dynamics around the massive black holes and if you look at so of course you have the interaction They the gravity is dominated by the presence of the black hole and but he and you have many Stars orbiting around the black hole So the first order you have just have say Keplerian orbits around the black hole But then if you look at on longer timescales these stars start interacting with one another So you can describe this as an interaction between orbits actually and you have an effective model of interaction between orbits And of course, it's kind of a strange long range interaction between these orbits, right? So another interesting example, I think Okay, and I will end up this section by a few words on toy models Since the underlying idea and then my message was there there are some universal properties that are Brought by the presence of long range interactions Of course, it's tempting instead of using very complicated systems since there's the properties We are after our universal then maybe it's a simple to the simplest model We have at hand and study and illustrate these Properties on the simplest possible model. So of course, so it's it's a reasonable idea indeed It has been used in many occasions Starting maybe with a tierings models to illustrate like peculiar statistical physics of long range interactions So tiering models, I think the end of the 60s or maybe beginning of the 70s And this one here is what I call the toy model because it has been used extremely extensively in the People interested with longer interacting systems in general. So HMF So I call the HMF because I'm not sure what the HMMF or for sometimes it's Heisenberg field Hamiltonian midfield Anyway, HMF usually people understand and The thing is it's a say take your XY interaction Mean field x y interaction plus a kinetic term And you've got this this is a kinetic term and this is the mean field x y interaction that TTI They're like in the angles or periodic boundary position with periodic boundary conditions And it's a one of the simplest non trivial model you could you could think of and indeed this has been very useful and Just to conclude this slide, of course the toy models are great You can do many things and indeed you can understand many new things, but of course it goes with all the caveats we're in toy models so it's Maybe there are only toy models and we have to make sure that what we are looking at is not just a specific feature of the very Very simple model we have and the last thing about this HMF So for a while it was really thought of three one toy model But the atom in cavity that I told you before like the experiments by German avant-garde's group Maybe they're not so far actually from this model So maybe not only toy ish after all HMF will see So before I go further on some caveats again, so I've Told you about maybe 10 examples and I'm far from An expert in all these fields, of course quite the contrary and almost ignorance most most of those And again, I will concentrate now essentially in classical physics. So apologize for People more interested in quantum physics. I hope the other lectures and seminars will make for this and I go now on some basic remarks like general remarks. So I will now I'll try to to like Substantiate a little bit the universality claim that I make made at the beginning So one thing is okay when you deal with long-range direction you have some specific difficulties and one specific difficulty that I will Develop a little bit later on that you cannot cut your system into pieces. Okay, so it's rather Obvious when you have this very long-range interactions. I've been talking about so if you cut your system into pieces Then even if it's macroscopic system macroscopic pieces, then the physics of the pieces would not be the same as the physics of the big big one so that may be a bit of a problem is related to the fact that time limit is not relevant for this and Another problem is the is numerical. So if you of course if you want to do numerical simulations as we often want to do Well, if you do an a use an a algorithm then each time step you have for each force Times that each force is order n squared So even if you have playful computer, then you quite often get stuck at rather small values of n. Maybe a problem and Of course longer interactions it also goes with some specific advantages and actually there they think they make For more than they are disadvantages specific difficulties. So I said You have one particle that interacts with many others So that was part of the specific difficulties and it's also part of the advantages because if you have one particles If I'm one particle and interact with many others, then it's a good approximation to say that I'm saying I'm seeing some mean Some mean interaction right plus maybe some fluctuations, but I have some good Some good idea how to control these fluctuations Okay, if you think of low of large numbers that would be on mean field and then I can look at the fluctuations That would be like central limit theorem and even further large deviations We have good theories to deal with this right and it's easier than if I have one particle And I interact with just a few others than the future for fluctuation I cannot describe this as a mean field for fluctuation, right or it's not as good. So the related idea is that somehow Mean field or mean field approximation. They should be a good approximation for the systems and we'll see in which respect But that's that's of course great advantage as it comes to a theoretical description, so I think essentially if I'm done with this introduction now I have two Main sections for this for my talk of the rest of the lectures equilibrium statistical mechanics and say out of equilibrium or the kinetic theory so I'll start with equilibrium statistical mechanics and probably To be coffee break before I'm done with this and that's that's perfect So since I'm in really Cartesian, so I'm my first part. I had three subsection my second part also have three subsections And so I'll be very Cartesian like this. Okay, so first Subsection is about scaling extensivity and non-additivity. So I already said some word about it Maybe it was too quick and not very detailed. So we're going to some details into this because it's actually maybe it's actually the Most important part of the heart of the matter then a few words on the mean field approximation, maybe and then I'll give a lot of that will do the Largest subsection will be about examples because they try to give examples and and discuss them Okay So just to Set the stage what I'll be interested in so you take long range interacting systems Or maybe a Hamiltonian like this or maybe a spin system like this you can imagine so that that's on the lattice That's maybe off lattice and then there's a potential here The decreases this long range decrease like one of the R to some power alpha here I have put the alpha here so alpha may be smaller than the dimension of the lattice and And I'm interested in this kind of quantity here. Of course. I could also do grand canonical But let's stop here. So it's micro canonical equilibrium. So I put a measure on my face space that is Concentrated like on the energy shell given energy E. Okay, I have this or Gives measure so exponential minus beta times the energy and I try to my goal of course is to study these different measures and understand what What they mean? What are their properties? So now scaling one thing which is rather obvious is that the usual thermodynamic limits Is not really adapted and here's a few more wordless. So imagine you have fixed density like medium with fixed density row and and and you in you try to Compute the energy of this fixed density medium Then if you decrease like one of our if the energy interaction decrease like one of our to the alpha and I'm integrating over a larger domain then the of course if alpha is smaller than the dimension it's not integrable, so it means that the energy Per particle will actually be more than than the constant will be decreasing will be increasing with the the size of the system, right? So That's that's a problem for for thermodynamic limit and it means in particular that if I did the potential energy Increases much faster than the number of particles. I keep putting imagine again I can I can I put particles and each particle interacts with all the other they have already put in my system, of course, I will end up with a sub super extensive energy say a potential energy now is thinking about entropy Well entropy I have n particles and typically the entropy of n particles. I mean whatever that happens Typically expect to have something like this if they have maybe say something like a fixed volume so it means that if I take something like a Thermodynamic limit that I what I would usually do that my potential energy always wins, right? So if I fix the temperature my potential energy always wins Then my system is always in the ground state whatever the temperature is so so it's not very satisfactory, right? And even if it's a if it may be well, right that if I fix the temperature and I increase The number of particles keeping the thermodynamics scaling then maybe my system will be in the ground state It's not it's not the right way to describe it, right? It's not what gives me my best My best information that system, okay? So this is this is what I mean by scaling For if you have a standard condensed matter system, then you have you know what the scaling is You know that you have to study your system in thermodynamic limit And that's what will give you the right physical regime and then best physical informations here in longer interacting systems So that's maybe a specific difficulty It's that we'll have to find what's the right scaling to give to get the right information, right? And it's not as given as in the usual condensed matter short-range systems and well the idea is very simple, right? So if the if you use the quantity I want to to compute if I If I know that the limit of you is zero, okay? It's a valid information But maybe it's not I have a system with a fixed number of particles and it's not very informative, right? If I can find another scaling where where the type of information I get is something like this, right? Maybe if I have this then this is true But this is more interesting and this gives me more information to my system. So this is okay This is really trivial things, but that's what's behind the the scaling we have to find so And this is another also trivial example that tell you that they can be several interesting scaling So if the quantity you have to compute is this imagine you don't know this and you know only the Kind of limits of this in some scalings it depends on two parameters So for instance if my a is fixed then I can compute maybe the limit of n times you and they have a Given given value and that's the first important piece of information But maybe if a is very small and all the 1 over n then that's not the relevant way to do and then I will Miss an important part of information if I use this scaling and I have rather I should rather use this kind of scale scaling And then I will pick up also in my limit this part here So it tells you that you and you have your system then you have some physics behind that Will have to decide you have to be taking turn this physics you have to decide between one or another scaling And I will give the examples like concrete physical examples later on Okay Nothing is just we see several examples. Okay, so first example this one So this is my spin system and and I want to Get some non-trivial statistical mechanics from this for this spin system right so then interact with one of our alpha and Say I'm in say one dimension here and of course if alpha smaller than one then I have a non extensive system and and if I use a Fixed temperature then we'll find that my system is in the ground state. So maybe if it's a electromagnetic like this everybody's Point in the same direction. So if I want more information I can renormalize I mean Tony and here by this quantity here here and of course if you I really want to stress Usually then when I talk about this people say me, okay, come on But your Hamiltonian does not depend on and that's stupid to make this kind of rescaling that that's not what I would I say I'm thinking, okay, I have a my system has a fixed n has a fixed alpha and nothing depends on n Maybe but I have a fixed and I want to understand what happens then I cannot do computation with fixed n I have to do computations. I mean analytical computation with infinite n, but I can choose What do I keep finite or not in this infinite and limit? That's my scaling and that I in this by making this this is the choice I make and Making this choice is a smarter than than making the choice that Without this crusade rescaling just in smarter just by the same thing that it's better to know that n times u Tends to a constant when n goes to infinity that just knowing that you tends to zero when n tends to infinity, right? Okay, so of course here's the scale the energy you could also scale the temperature in this the same same remarks Well, no example is severe irritating system. So people doing like statistical mechanics for severe irritating systems They they keep this quantity fixed, right? So there's the volume of their systems with the energy There's a total mass. This is G the gravitation constant And of course then there are several ways to to understand this can you you can enforce the scaling by maybe keeping the total mass constant and E constant or or E that goes like m squared so you you okay you have several ways to enforce to enforce this and Okay, just a small caveat if you are doing a segue dating systems then you have also short-range singularity and that makes a problem for statistical mechanics, but I won't enter into this and Okay, neutral class plasma is interesting because you have the same short-range singularity as in Severe editing systems and but okay, then you can regularize it for instance if you're in condense matter Then maybe you have an exclusion principle or something like this So you forget about the singularity and then it is proven as theorems that there is a well defined thermodynamic limit I mean really thermodynamic limits not another fancy scaling limit, and that's okay. That's a usual matter, right? So it means that even if you have long range interactions you in some case you can have some some anemic limit that is relevant. So this Again caveat against the universality. I was advertising So now about Extensivity and additivity. So I hope I have convinced you that maybe Telling that the energy is extensive in the sense that is a proportional to number of particles to proportional to a Number of particles to some other power. Maybe it's it's not that a relevant Concept because actually Usually you have not and the number of particles not the only Parameter you have other parameters and when you send n to infinity you have to choose a some scaling limit So the fact that it is extensive or not depends on the scaling limit you choose So it's not maybe it's not a very well-defined concept. What's what extensive is or at least in my view Maybe you will disagree with this Yet additivity. That's well-defined. So I have to here have one big system Microscopic one say with longer-range interactions and I cut it into two pieces and I have the energy of The whole system is something which is well-defined the energy of each subsystem alone also And of course I had the total energy which is different from the sum of the energy of the subsystems Just because there is of course like the interface energy here is also the same order of magnitude as the others So that's a well-defined concept and if I scale the energy by Whatever quantity this will remain true. Okay that I cannot Decompose my system like this. So non additivity. I think that's that's rather what characterizes this system rather than non extensivity Anyway, so one thing that you may Directly infer from this kind of remarks is that No face there is no face separation at least in the usual sense that is possible for this system What's face separation? So you your systems spontaneously divide into two pieces and the the two pieces essentially are interact very So just through an interface and the energy of the interface is small and and and you have face separation here That's not possible to have something like this. So it only tells you that the thermodynamics will be strange for instance one thing that probably you'll have Details on it means you go to chat later on during the week is this if you have face separation Then it implies that your entropy and function of energy will be a common cave function Because so this is the energy of my face separated system if I have no interaction energy between the two then I know that my system will be in a maximum entropy so place so and this is the entropy of my face separated system so If my entropy is not concave then I can have it made concave by using face separation, right and Then if you go further in your reasoning then, you know, the free energy is the Legendre transform of the entropy that's under some dynamics, right? And and you can go from one when you use really legends transform You can go from one function to it and transform and back with the inverse version transform Only if you're dealing with convex function or concave where then you have to just to play with the science, right? So if your function or not concave anymore, it means that you cannot go back and forth between these two representations So it that's that's a hint that you may have some problems going back and forth between canonical and micro canonical This is the possibility of non equivalence between these ensembles. So this is one Really here I have used just a regeneric things about longer interaction. So this is an example of universality I was meaning at the beginning, right? I'm sure you'll have details on this with you go. Okay So now About about main field. So you what I said before so you have each particle is interacting with many others So really a mean field description should be very good, right? So that's that's the main idea and for instance I mean, this is correct And if you choose a well you have once you have sorted out this scaling limit problems Then then then you're in good shape One one consequence is this so for all the system that we're talking about they really Exponent critical exponents are classical. So maybe find it boring, but okay. Maybe that's also a nicer universal properties for this system On the technical level then we have a nice mathematical tools large deviation theory that we will able enable us to do many nice computation on this and then of course Once I have told you about these are universal things so universal Validity of mean field in a sense to be a pretty more precise universal tools like large division theory then caveats Of course, you still have second order Transitions with big fluctuations. So we still have to be a bit cautious when you are close to second-order phase transitions Sometimes you have short range singularities too and these short range singularities the maybe dangerous for mean field because it means that the force a troll change May dominate. Okay, you also have sometimes short-range interaction plus long-range interactions so then it brings additional correlations and the one that will be a bit also a Detail they give a bit detail on later and that you have you may have made more than one scaling limit That is relevant. Okay, and if you have more than one scaling admit which is relevant Maybe in one scaling that's will be like the mean field scaling and you will be able to do some things using mean field And the other one not mean field. Maybe I'll give example on this later Okay, so now I turn to the last part for equilibrium statistical mechanics, this is Examples so the first the chief examples that are self-gravitating systems. Of course, it's a really exciting to have to take just a Particles that interact with Newton and gravity and and look at what the statistical mechanics statistical mechanics of this Right, so people have tried to do this for a long time because it's a very natural problem. Maybe a bit a big at economical because Even galaxy or they're not really pure self-gravitating systems like this, but it's really interesting system and and Okay, then the incentive was still in into the in the observations and this is a very naive remark And this is actually wrong, but that could be an incentive to study statistical mechanics statistical mechanics They will give you access to some large scale structure large large scale regularities of your your systems And since we can observe some large scale regularities in the galaxies You can and you can more or less classify them then it's natural to try for statistical mechanics description Okay So then there are specific difficulties So of course one thing is I have defined usually a volume here You have no volume here of course and and and the interaction is not confining It's the potential is one of the are so if you particles to go far away they can escape this potential So this is one difficulty to define statistical mechanics another difficulty is short range singularity because so of course if you take two two stars or two points infinity close one to another then the gravitational energy is minus infinity so it It Releases a whole lot of energy for the rest of the system and the rest of the system can become extremely hot and the two more while There is just one binary that becomes close to only Two particles of the same points. There's another singularity that somehow you want to avoid To define a statistical mechanics for the systems. Anyway, if you manage to take care of these difficulties Then you find this main features And this was done by Antonov I think maybe so a long time ago in the 60s probably tells you that you have a well-defined equilibrium or at least a metastable equilibrium Until the central density reaches some limit which is well defined Depending on the parameters and once you have the past this limit then there is no Equilibrium even metastable and and you have what's called Gravitable Catastrophe and and and your system will tend to collapse So that's That's really a beautiful theory. I mean you have a well-defined. I mean Nice computation you have a well prediction that is really nice but Okay, as I said in the beginning it's difficult to have controlled experiments on astrophysical scales, of course And it seems difficult to find really clear situation where this is applicable. So I Heard sometimes people say that Globular clusters, so maybe you know so you have galaxies and global clusters. They're clusters of stars was a dense cluster of stars that are In the after that are a bit isolated from the rest of the galaxy and they're so they can be considered as as more or less More or less clean systems of self-gravitating systems. Maybe in some cases. It's a bit relevant for for them, but I don't know exactly So difficult to find the real applications of this beautiful theory and this is why actually this this models of interacting orbits I was talking about at the beginning is interesting because Somehow maybe it's an example cleaner example of applications of statistical mechanics for Systems for self-gravitating systems, so we call you it's not directly self-gravitating systems But they're so you have massive black holes and orbiting stars around the black holes And you have an effective model of interactions between the orbits and this this you can then this you can do statistical mechanics on this interaction between the orbits and this may be relevant for this Maybe I don't know maybe we should ask to have two more or it is on this so Okay, maybe maybe I'll do this and then it's time for for a break So another very important example of longer interacting systems or vortices like this Well, what I say is very important because okay may say vortices. Maybe you really we don't find any Clean systems are interacting vortices, but this is a this is a model for Interacting vortices is a model for two-dimensional turbulence So you can you can see two-dimensional turbulence as if you say discretize two-dimensional turbulence by interacting vortices then you have Model for interacting turbulence and and 2d turbulence you may say okay 2d. We don't care about 2d We have we live in 3d, but if you have rotating systems then it tends to confine the dynamics in 2d and while the rotating system we had we had this Picture of the earth and earth earth science and the climate change science of course When you look at climate you always you most often have two-dimensional models So maybe interacting between different shells, but you have two-dimensional models, right? because of the strong rotation so so this and this is the Say the basic thing about about this model So you have interaction between vortices and you can do statistical mechanics of this system, right? Of course, there is it's not so easy because there is singularity. Well, not so strong. So maybe it's okay and So this was not by on Zagheur, I think I'm probably so Quite a long time ago in the in the 40s I think and the main feature was this was that if you take this System and you say fix the energy and you vary the energy then for some range of energies then the system So it tends to maximize its entropy I recall in maximizing its entropy means it will be creating large-scale structures So this is the the great finding by on Zagheur in that case is that it's Entropically favorable to make to form large-scale structures So when you intuitively one would think that okay, now you have large-scale structures I mean that your flow is getting more ordered in that case Maybe it's getting more ordered, but yet it's entropically favorable and I remember we had a talk at a stat fees for those who were stat fees by Dan Frankel actually he was making the point that if you have heart's fear then the Ordered the crystalline states of heart's fears at high enough density would be actually the more entropically Entropically favored like the most probable one. So entropy is not always So maximizing entropy is not always maximizing disorder at least in a what would be called disorder in intuitive sense So here the same thing happened happen you maximize entropy and you actually create some order and these large-scale structures What's what's amazing? That's indeed. That's what's seen when you do Experiments in 2d turbulence if you start from very disordered states and you will be you will see like large vortices that form you will see like Well, of course, I'm not claiming at all now that that this model is relevant for what's happening in the atmosphere But yet in the atmosphere you see that the the the flows is organizing to large-scale structures like jets and large-scale vortices and so on okay, and then this one this is about statistical mechanics and effective wave and particle models, so this these are plasma physicists and The the idea is that you when you have plasma You probably know that you have a probability of long rear waves and the damping of long mirror waves It's called under damping and then a question was Does the wave Damn completely or that's would be called a complete nonlinear non-damping or do you have something that the waves maybe stays at some non-thermal level and this these these people they have So if it's gets in my case in people they have tried to give a statistical mechanics answer And I'm not sure it's relevant or not really but but it's quite beautiful because the so they could find a threshold by statistical mechanics So if you have in some range of parameter you have some kind of phase transition and in your way It does not damn completely and some other range of parameters than your wave dams completely So it's a kind of statistical mechanics answer to an old All the dynamical questions. I will come to that later on. Well, I think it's a 10 30 So it's time for coffee and I will give more examples and kinetic theory after the break. Thank you