 nonconcave entropies and also about recording in progress how this is at the origin of ensemble in equivalence and I just had a slide on this micro canonical solution of the Blument-Capelle model. I remind you that the Blument-Capelle model is a model of spin one and there is a Q-revised coupling all to all with cut-series scaling in order to make the Q-revised thermal extensive in N and then there is a local term which goes like SI square. Okay so and delta is a positive coupling parameter to SI square which of course is ineffective if SI is equal to zero but gives a positive contribution is if SI is either plus or minus one. How to count the states of this model? You take for instance typical configuration which is a configuration of pluses minuses and zeros and you can swap the spins all the spins and this gives an n factorial for the permutation of the spins but then of course you get the same configuration if you swap the pluses if you swap the zeros and if you swap the minuses so you have to divide by this. This is very similar to how you count states in for instance in the ising model or let's say even how you count the result of a coin toss experiment to know where there are pluses and minuses or head and tails and you have a throw of say n throws and you want to know in how many ways you realize that throw and this will be n factorial divided the number of tails and the factorial and the number of heads factorial is the same binomial type this is trinomial because there are three states distribution then what you do you express so you use the stealing approximation so which says that log n factorial is n log n minus n it's only the leading order of the of the stealing approximation as Matteo was mentioning yesterday it's enough for this case and then you take kb log omega and what you get is this very complicated expression you can check that when q equals zero you get back the sort of q revised type of entropy but if it's not it's a more complicated one aspect which is important you will see later on is that you see that there are logarithms of quantities that can be negative so of course that loses any meaning and you will see that this is very important for a phenomenon that I've not mentioned and it's a godicity breaking I will I will do a comment on this in this case and then another in a relation which is important if you go back to the energy of the model you see that the energy is delta time times some s i square which is this q parameter which is called the quadrupole moment okay because it's clear that the quadrupole moments which counts the number of pluses plus the number of minuses is a maximum when when all are pluses or all are all are minuses okay so q measures the percentage of pluses so you see that the energy divided by n can be written as delta times q and then if you factor delta with respect to q also in the second term which is k m square is the usual q revised term k m square k is j over 2 delta okay so why we want to factor delta it will be clear later on because by changing k which is the ratio of q and m if you want the relative ratio of q and m you will go from first order to second order phase transition by kick mean delta constant so it's a way of crossing the phase transition in in a in a controllable way okay varying one parameter only only k okay so this is the entropy as a function of the quadrupole moment and of the of the magnetization okay it's a function of of two quantities so what you have to do and it's sort of lengthy you can do it as an exercise i didn't put it on the slides you can express the one of the two for instance q as a function of the energy and of the magnetization you just invert the relation epsilon equal delta or q minus k m square you write epsilon over delta q is epsilon over delta plus k m square and you substitute q in all the terms of where q appears in the in the entropy okay then in this way you get an entropy which is a function of energy per particle energy per spin and magnetization okay it's a more complicated expression i i have not written it down okay just simply replace q for epsilon over delta plus k m square okay in this expression and you get an expression which is s of epsilon and m okay so this is if you want is the equivalent of the landau entropy of the landau free energy in the context of the micro canonical ensemble because you are writing the entropy like you write the landau free energy as a function in the case of the landau free energy as a function of beta of the inverse temperature and of the order parameter m and in this case you are writing the entropy as a function of the energy which is not beta you know that energy and beta are related by the legion transform one to each other so you have now an entropy as a function of epsilon and the order parameter this is already an interesting quantity but it contains a lot of information and i want to cancel a little bit of this information by taking the maximum of this entropy with respect to m and this will give you the entropy as a function only of energy okay so and how is the entropy as a function only of energy in this model you can try to draw it so if you haven't i i suggest that you do a sort of a sort of plot okay i can tell you how more or less it is okay but then then you will check okay if you if you try to draw this entropy as a function of epsilon okay it will be something like this okay as a function of epsilon so it results as i've said from substituting q okay in this expression for epsilon and m and then maximizing with respect to m so if you don't want to spend too much time in coding maximizing you just sample m between minus one and one or between zero and one would be the same because the expression is symmetric in m and you take the maximum the maximum value so you know how to write a code you do it okay if i encounter and a value i start from some value okay which is my maximum if i encounter a value which is even larger i take this new value and and then i i i sample all the values of m i take thousand values of m and and then i get this entropy so for us when i first draw this it was very deceiving because we were looking for an ensemble in equivalence for non-concave entropies and what we got was a sort of a a graph that was perfectly concave so i went to bed that night i was very deceived it was 20 years ago so we were looking for non-concave entropies and then the morning after i i went back to this curve and i i tried to zoom part of it and at some point appears on the screen a sort of darker part that was very very small you will see that the numbers are really small and so that's where it was hidden the the non-concavity so as usual when you take a model at random and you look for something if you are lucky it can be the right model if it's not if you are not lucky it can be in some this is a story that i felt to everybody so don't don't be deceived it seemed that everything goes wrong so insist that if you are convinced that there is something you should insist okay okay this is the story but i i ask you to repeat this and to to zoom so first of all another important feature that i was telling is the fact that you have in fact negative temperature here okay which is expected if then if you go towards higher energies the number of states goes down and and so you get negative temperature you are in the part of positive temperatures and here you get a non-concave part now but i have to tell you how it works so you see that the numbers here are really really small so point so you see my parameter delta over j is so i'm varying by by very small quantities delta over j as i've told you k is j over delta is my my parameter and i'm plot i'm what i'm plotting here is maybe i need the okay okay so what i'm plotting here is is already some it contains already some information on the phase transition is the relation between the temperature and the energy okay so i'm computing beta through the micro canonical temperature so now beta is dsd okay and t is one over beta and what i'm varying i'm varying delta over j in order to cross the tri critical my the canonical tri critical point i i remember to you that the phase diagram is something like this is here t over j and there is here uh delta over j delta over j and i have a tri critical point here this is second order phase transition this is first order phase transition and what i'm doing in this plot i'm i'm crossing this line with with straight lines crossing the phase transition line with straight lines you see that i have the parameter is delta over j uh in such a way to reach the tri critical point okay and the tri critical point is delta over j equal log four over three you remember from the yesterday and what happens at the tri critical point you see that you can prove it but okay it's clear from the graph that the slope here is zero so this is typical of tri criticality that the derivative of the temperature with respect to uh energy which is the inverse of the specific it is infinite at the tri critical point uh okay so and then what happens it happens if you cross the tri critical point and you enter the first order phase transition region then you see that the temperature goes down you see that the numbers here are very small so that's why the so temperature goes down i will try to draw now the the the entropy i cannot do it because it's so small the region that i i would have to zoom in very small numbers but i will make you a drawing but it's clear from what happens here that there is a region of negative specific heat because if you go up in energy temperature goes down so here the on top okay and then it goes down okay so there is a region of negative specific heat so it's not self-crediting system is not plasmas where it's not vortices is a extremely simple model spin model uh and it can be solved exactly and it gives you a negative specific heat so negative specific heat is not due to any other feature that were present in such systems like singularities as short distances collapse non-equilibrium we are perfectly at equilibrium we are drawing the entropy of the of this system and we discover that there is a negative specific heat okay and then the relation here is continuous at some point and what happens in the in the canonical ensemble in the canonical ensemble you that you rather draw fix the temperature no so you go along this axis here you start from low temperature you increase the temperature and as you increase the temperature the average energy will go through this curve until you meet this point okay and at this point you see there is a slight dotted line here is the Maxwell construction and then you jump to this value of the energy and then you again you start so this is the drawing the average energy as a function of the of the temperature okay so while in the canonical in the micro canonical ensemble you have a system where there is no no no phase transition okay because there is a continuous relation between t and epsilon in the in the canonical ensemble you have a first order phase transition okay so the two ensembles don't give equivalent results so one ensemble gives you a continuous curve between a continuous relation between temperature and energy and the other ensemble gives you a first order phase transition so that the the two ensembles are not equivalent so so this was the first example of of in equivalent ensembles maybe I can quote the paper is a paper in in in prl in 2001 that started the long story of of study of systems with long range interactions that give uh in equivalent ensembles so in order to get in equivalence you have to break some principle of additivity this model as I've told as a Curie Weister which gives a non-additive energy and this is the origin of the of the equivalence of ensembles and if you go on okay and you uh uh you go from four six two to six six five you go smaller and you see that these become steeper and steeper and at some point this branch become vertical and at this point there is a a discontinuity that we have called the micro canonical first order transition because there is a jump in temperature at the given energy and if I go on the region of negative specific heat disappears and I roll in the jump and finally delta equal j equal one alpha so also uh i which is uh this point I I get a continuous relation between energy energy and temperature this is a full phase diagram for for this model it's okay are there questions just to try to I will try to draw now the answer be since there's a question uh from uh zoom uh let me read it so can something be done about the marker okay we don't see a reference pointer on zoom okay ah the pointer yes it's true uh I can point like this okay okay I will do it with my finger it's better that was the question okay okay okay just uh I will draw the the entropy for for for for this case which is the most complete one there is a jump there is negative specific heat so so what happens for the entrance suppose I would like to draw this entropy okay as a function of epsilon uh in fact when I do the maximum I will discover that the maximum is satisfied by two branches this is the m equal zero branch and it's valid high energy and then there is an m different from zero branch uh here and uh this branch also is convex is concave but then at some point it starts to bend and becomes convex okay and then it touches this branch here so here the derivative which is the temperature is not the same as here so the two derivatives are not the same and this gives the jump here the jump in temperature the jump in temperature this region where the temperature decreases with the energy is the region here uh is this region here where the convexity is up okay and this gives the region where temperature decreases when one increases the energy and this is the region where with concave entropy which gives the growing temperature with energy region okay so this is and then of course at some point this will go down okay okay and uh okay this region is here was this little dot in my in my curve because you see that the numbers are very close to each other and so you have to really zoom in this region now there are other models in which this part is larger so I have chosen this one because historically is the simplest one and there are but there are others in which the differences are order one so it's easier to see this uh is uh region this depends on the model on the parameters okay so this is essentially okay so what I can do if I want to be quantitative and I suggest that you try this so I expand the the entropy as a function of m okay this is the constant term this is the term proportional to m square the entropy is you can check is symmetric in in m uh this is the the the term in front of m square and this is the term in in front of m four you can try to derive them okay do the exercise it's a Taylor series expansion it's not too heavy and then you can check if you have done the exercise of yesterday of computing the free energy of this model you can check that the condition of criticality so the critical line in the canonical ensemble is the same as the critical line in the micro canonical ensemble so if you take this a and you put it at zero and then you you express the energy in terms of the temperature you will find the critical line that you had found yesterday solving the model in the in the in the canonical ensemble while the condition for tri-criticality is not the same so the condition for a and b to be zero is not the same and gives two different tri-critical points so what happens in this in this region of the you see that the numbers are very close but i will expand so what happens in this region is that you have this the you have the second order line and you reach the canonical tri-critical point okay then the black line is the continuation with the first order critical critical line in the canonical ensemble in the micro canonical ensemble the critical line continues okay you see this dashed curve here the the the critical line continues to reach the micro canonical tri-critical point is the point in which at some point at a given energy there is a temperature jump so the this the the plot of temperatures versus energies opens up and you get a jump in temperature this is this point at this point since there is a jump in temperature there is an upper temperature and there is a lower temperature so i have to follow the value of the upper temperature which is this one and the value of the lower temperature okay so and the two join when delta over j is on the on the t equal zero line okay okay so so the if you want the the phenomenon of temperature jump in the micro canonical ensemble is nothing but the phenomenon of energy jump in the canonical ensemble in the canonical ensemble is the dual of that phenomenon so in the canonical ensemble at a given temperature i have a jump in energy and the latent heat and in the micro canonical ensemble at a given energy i have a jump in temperature which is if you want to call it a latent temperature you are not used to this thermal okay but uh it's exactly the same thing uh it's a bit less i mean uh you are not used to the fact that temperature can be discontinuous and in fact uh that was a mystery for us and we went around telling this story people were not convinced how can it be that after given energy you have a system that has two temperatures is the system out of equilibrium because at some point it should reach the same temperature but you should consider that in the micro canonical ensemble temperature is derived quantities not temperature is the derived quantities not the parameter that you vary with continuity so and in fact finally we we have a done also a study of this phenomenon even numerically to convince ourselves to see what happens for instance we have considered my system i will not show it to you of of with a kinetic energy just to have a direct access to temperature and what we have discovered is that at this point where the temperature jump appears the probability distribution of the temperature is by model so in some sense that the system uh is swapping from one temperature to the other temperature and the distribution is by model uh so the kinetic energy is by model at at this uh at this point uh it's interesting because in some simulations of self-gravity system people have observed the same phenomenon that the system is is swapping in two different regions of of kinetic energy from the high kinetic energy to smaller kinetic energy which is exactly the same phenomenon that we observe in these uh in these uh simple uh in these simple models so these are simulations done in the group of of of scottreman in in in princeton is a self-gavitating system they discovered also in several gravitating system this this led me during the years so now i don't want to speak too much about that to interact with guys these guys in self-gavitating system area and there was a very interesting period of cross fertilization uh where we were trying to classify what we were observing in these simple models and to find the equivalent in the self-gavitating system case in particular i did this together with the pyrrhanishavan is from to lose who is an expert in in self-gavitating systems and uh it was very interesting because for instance they didn't have the same names the the the different transitions that they were observing so they were calling i call it try critically it was calling it differently because they were finding the thing it was very very nice period where we are written papers together and okay just to give you a a graph for the complexity i don't want you to understand all these okay but this is uh this is a classification that was done by julien barret who was my my PhD student and freddy boucher in a paper published in j start fees using sarnold singularity theory they were able to classify all uh uh this is uh co-dimension one singularity so you co-dimension one it means you vary only one parameter and uh and these are the shapes of entrophies and these are the shapes of uh uh inverse energy with respect to temperature some of them you have seen in my plot unfortunately since inverse energy and the different cases you you can have okay so the the different shapes of entropies and what happens at the phase transition with the with the entropy what i can tell you that uh several years later so the point is that this is classification so based on singularities types of singularities that can happen in these so these are functions okay s equal kb log q m and so on okay so you know in mathematics you can put question what happens if i vary a parameter in this function okay where is the maximum where is the minimum and so this is the type of questions okay and these questions are solved in in a field which is called singularity theory uh okay and they work together with the math math teacher and they found the classification for so that but then you don't have the model so so you have the what can happen but you don't have the physical model where this happens so we have tried by going around to find if there are these singularities and i can tell you that i found the other people found the different ones i found this one as a entropy in a model and it's very interesting because physically as a entropy you find as a entropy in binary mixtures you take the book of landau and if you are patient enough you reach a certain chapter 10 or 11 i don't remember there is a chapter on binary fluids which where they discuss it discusses different different phase transitions in binary in ternary fluids and for binary fluids there is a case where where is there is a reentrance in the canonical ensemble reentrance means that you vary the temperature you enter a phase and then you continue to increase the temperature and you go back to the phase where you were at lower temperature it's a bit strange you know is so suppose i have water okay i increase the temperature i go to gas then i increase the temperature i go back to to fluid it's very strange okay it's called reentrance but in binary fluids you can have these reentrance in the canonical ensemble so i fixed the fixed temperature what we have found is that if you go to the micro canonical ensemble this is co-dimension one perturbation i have two solutions for this case with two entropies that are both concave that are both concave and then when i vary this single parameter it's not temperature i move the entropy which is below up until it crosses the other okay in the micro canonical ensemble and so you get two first order phase transitions no because here you have two slopes okay one slope here and one slope there so when you observe reentrance in the canonical ensemble we observe azeotropy in the micro in the micro canonical ensemble so it's a it's a sort of of strange behavior and of course one can amuse oneself to find other cases in different models and to check ensemble in equivalence in in various different models because i put this slide just to give you an idea where the field is gone and for instance last year we have published a paper with uh with uh uh uh with david mukamel and uh uh on on a fourth order critical points and what is the phase diagram in the canonical in the canonical and micro canonical at fourth order critical point fourth order critical points in points where are points where b a was zero b zero at three critical c is zero so it can happen you but but you have to play with parameters more more carefully i don't want to enter in this part because it's it's a very complicated phase diagrams with it's not something that but it gives of course this type of procedure can give rise to too many interesting physics and study of different situations so this is a question what does invisible concavification signify does it mean entropy concavity does not exist in this in this a on honestly invisible means that these are metastable branches that are not seen if you maximize the entropy so they are invisible they could correspond the two metastable states for instance and unstable states and they are almost invisible i don't know how they could see it but they are below the maximum of entropy but you can reach those those branches by by by metastability the paper is published in j start fees i don't remember when around 2010 maybe i can give you the reference for that paper to the to the person who is asking okay okay so what we have learned from this first lecture so this is the end of the first lecture so this is the material that i had prepared to for the first lecture we are at the third unfortunately so so i was a bit slower so so the thermodynamic limit of long range interactions is well defined provided the interaction is rescaled using the cuts factor thermodynamic formulas can be adapted to allocate long range interactions through the legend French transform for non-concave functions which is an extension of of a non-concave function theory ensembles are not necessarily equivalent they can be equivalent but they are not necessarily equivalent phase diagrams differ in canonical and micro canonical ensembles but you can have similar non equivalences and we have found them between constant pressure ensembles and constant volume ensembles from grand canonical and canonical so so this we have examples so that this happens when passing from one ensembles and some to the other so i will give you a plot of the different ensembles at some point and the entropy can be non-concave and specific heat and other response functions you will see that also susceptibility can become and this finishes the first lecture okay so now i will stop sharing and i will share the file of the second lecture five minutes to introduce the second lecture and then you will get a break now it doesn't work ah yes because i didn't go full screen okay so just a brief now plan of the of the second part of the third lecture and then i think it's time to have a break so i will consider and i will go now out of the regime of because what we have seen is is in this model there are all-to-all interactions and local interactions so uh in order to to find models in which uh short-range interaction can compete with long-range interactions with all-to-all interactions we go get back to this model it was introduced by negal and kardar in the in the eighties and we have solved this model in the micro canonical ensemble and uh and we find similar phase diagram this is just to be sure that when you add nearest neighbor interaction the system the the the ensemble in equivalence is stable in some sense so so you you continue to find the ensemble in equivalent then i will discuss ergodicity breaking and metastability and i will introduce a method which allows to solve the micro canonical ensemble without counting so this is another path of of soluble models so you of course counting is the best way to solve the entropy but there are systems for which you cannot count because the the phase space variables are continuous or because you are unable to count so this method allows to solve the the micro canonical to get non-concave entropies and then i will apply the method if i can maybe this will go to tomorrow to to a model which is in the class of the negal negal kardar and then okay i will give you i will let you solve the negal kardar method using the using the transfer matrix uh okay this is the plan okay so maybe five minutes to introduce the model so the model is uh is again it's it's it's not a spin one half model spin one half model with a usual q device term here and plus i add in one dimension this is a constant here forget about the constant with a coupling k and there is a coupling a nearest neighbor coupling the solution of the model in the canonical ensemble was done by negal and kardar a simple you just as we did for the other model you just express the q device term using the Hubbard Sartonovich and then what you get you get you get this is what you you should do as an exercise okay so what you get you get an effective free an effective partition summa which has this form is the integral over the auxiliary field of the Hubbard Sartonovich as usual so there is an x beta si here and then there is a minus k with a j j to n and others okay there are constants okay and then you have a k si si plus one and this is the ising model in the in an external field okay in the external field contains the the auxiliary field so you solve the ising model in the external field using transfer matrix take the the logarithm of the largest eigenvalue in the transfer matrix and so you express these the the eigenvalues in term of x and then you maximizing x and then you max sorry this there is an exponential here of course and then you maximizing x okay and you from maximizing x you get the the partition summa and and okay so maybe the counting is for the next I don't know are you are you tired we can go on or or not it's okay we can go on because we had a quite a long break so all you want five minutes of rest five minutes okay okay okay five minutes of rest