 so now let us do the maximum term method as i saying here that this is one can show that the maximum term method is bounded and a very rigorous bound the bound is that means zero there is a quantity pn which is very small one can show that maximum term method and the remainder you know new and that goes to zero so now i want to take and take the replace the partition function by logarithmic term so what i am going to do now i am going to take the that ml star so i am not going to do this sum now i am going to replace that by one ml star that ml star that particular distribution which maximizes this thing so i don't have this sum anymore so this is the quantity i have to maximize right clear so now we do that so this is the maximum term method so i take this is the mistake i said i don't know how these mistakes come this should not be there anymore because we are just maximizing ml star so now i take the log of that when i take the log of that now i have to have the with the constraint that lml equal to n so log of that you can have all these uh term ml star minus lml ml ml ml sum over ml is n so this quantity is here and this is then vbl to the power m all these things you can do check it and you'll find that apply now maximum term method with the constraint that's why the next time n will become and then you get this quantity it used to be z with the back edge lml minus n n does go out lml remain and then i put it as a ln z the my my the grandian multiplier uh take the rotation ln z knowing that what will come later and then i take with respect to ml and that would this ml lml remember that is a constraint that you have to be n that's why it is there and then you get this quantity from here ln ml ln z then you as i said i put it by hand as a lagrangian multiplier ln z ml star vbl to the power z these come from there everywhere it should have been ml star actually like it should be star here it should be star here uh it should be star here i can try to do that with certain luck and kbt should be lower case these all should be star ml star so this is right here but everywhere it will be ml star the ml star is that so now you understand the details and you also understand the grand picture so n so once i get that i multiply it i know that mass satisfies n i do that hit by l and do it then i get n i get sum over v lbl z to the power l then i get density equal to lbl z to the power l this is a very important very important equation correct right so this is the this is this all right okay now i a partition function like that i take log of that then i can now calculate the free energy free energy in terms of my ml star i know my ml star my ml star is this quantity here right i know my ml star now partition function does not have the sum this is correct then i can calculate the free energy by ml star coming out the ln of that vbl and this term believe me this is all this term is here and then i can get the pressure by taking derivative of that are you following the thread or you are losing okay then right now if you go into the very detail of the equations you will get lost this you can check it up and it will take me one probably half an hour more to do that so i don't want to do that and it's probably since these are details of a keeping at that level of probably these are not details you can do it because it will take you 10 minutes to do it but you can do it so that goes inside and i when i do that there are these two terms here lnz nlnz you will go to 0 ddv but i have this term left vbl term left so i take ddv there vbl so then i get pressure doing that is kbtbl0 so pressure case we will work is p by kbt is the second quantity so these two together are called the cluster integrals the cluster expansion sorry so when you call a mere cluster expansion this is the thing is really a matter of a choice and these are exact one can do probably somewhat simpler way through the done canonical ensemble but there are some results what you won't get from there so these now as i was saying the game plan now is to eliminate if it's cause a fugacity from these expansions both are exact believe me that elimination of fugacity from here gives you another infinite series that infinite series is the following and this is a virial series that is one of the things you taught in your on high school also the pressure is expanded the density first term is a second virial coefficient which will show is exactly given by f2 term of mere cluster expansion f2 term is an integral over interaction in term of potential intermolecular potential so now from the hysterically from b2 the interaction potential the first force field was derived from b2 because it was done after mere did one finds that second virial i can get the second virial coefficient how i plot pressure versus density and this is the ideal gas but if as interactions become important i see this starts deviating so i know experimentally this quantity and i in the first part of deviating is very temperature dependent like high temperature it's like this low temperature is like that so i from there i fitting i get the temperature dependence of second virial coefficient and i'll now show you the second virial coefficient is the integral over the first irreducible cluster integral or b2 second virial okay so now so i'm going a little bit back and forth to give you the understanding of what is going on okay this will not need immediately but very important so what may i did then these bs that we have done are called reducible cluster integrals these are with the definition given here these are integrals however as we know that this cannot be reduced any further but this can be reduced so now we think is it possible for me to express bl following the same logic that i can now say okay i have a cluster we wait i have given as bl that wait now that particles now i can now define this quantity as one it is constructed of two bonds if of them are same bond but this is a different thing so i can i further decompose these group of clusters into these quantity and these quantity given that this is the irreducible and that was the next thing that may have did reduction in terms of irreducible cluster integral now is it okay let me define these are the fundamental quantities these as these i cannot reduce any further this and these are the quantities this actually should have one more because it should have the one without a bond okay that somehow it's not given here so this one now i define this is not kappa this is k k where beta k this volume this quantity exactly define the similar way as we did before we can do the exactly same thing that i now want to distribute in my ml or nl how many of them are these kind and how many are of that kind so we do the exactly same analysis we did before and you get slightly different expression but you can show that it goes over the further decomposition and this however has an approximation in the process of going this approximation that you have to assume that these are volume independent only then these reduction can be done and that turned out to be very important limitation in this theory which made the theory more questionable at high density okay so so i will not spend too much time on this if i want to do it i will do it later but right now i am going to do something else that so this is the irreducible cluster integral so now my aim is to show these quantities the second virial coefficient these quantities the third virial coefficient and some of all these things are the fourth virial coefficient and as i told you these are important historically and also for force field development even today so with time priorities have changed there is that huge amount of effort at one time went to get these integrals really huge amount of work went and that succeeded to some extent but not to a great extent so basic idea is that this is the thing and then there is a reduction you can there is a distinction you should know beta one is two particle so beta one is first of the irreducible integral that is good gives on to b2 so b3 is the three three particle clusters so this has two of these but there is one stands out which is beta 2 so when i think of b3 i b3 is the that's how it was done the way i always do or you can always do you can always i never do it this way i always write down two particle then i do the three particle then i go to four particle then i see the pattern that's how may i did it bs that doesn't come from there that come from these definitions so there is the three part of a will come three of them i think three factorial three will they are all the same value three three factorial cancels the three i think you are left with two that that's work it out but as far i remember one or three factorial into three and then you left at this two and here however nothing is there so it the three comes in because the three three of them are not there that's why you get a three five three here but let me say something very interesting so these part was done by mayor now mayor had a very very smart wife you might be knowing called gopat mayor maria gopat mayor so they were in gottingale and maria gopat mayor was daughter of a professor she was very beautiful and very smart so they are all those people are there george gammo and all the all the people mayor they are they are going to gottingale to study so finally joseph mayor managed to convince her and maria and they brought her to him they are in first in job he called me those days women's faculty position was very difficult so maria gopat mayor didn't get a faculty position she was just i don't think she was probably not even paid but she is but she was incredibly smart so she is so while mayor did the first part this part was done by maria gopat mayor and that is acknowledged in so there was a book of stratmac which is mayor and mayor very famous statistical mechanics books much of the things are taken from that book also the patria r k patria has a good job and my book also i hope the equations are all correct that was done by maria gopat mayor then another interesting thing happened sometime later there's a psst student of mayor called stop mayor who was very famous stop mayor did the theory of soljel transition when the polymers get together and from a gel there is a difficult combinatorics of this kind and that was also done by gopat mayor and uh stock mayor wrote the paper and acknowledged the contribution of gopat mayor in that theory and uh in chicago then they came even then uh gopat mayor didn't have a didn't have any uh office so in the in the uh first floor or ground floor american first floor or ground floor in james frank institute i think the she produced had in the endico farming path which is demolished all these things now except the why the farmer did the nuclear chain reaction that part is only kept and the capsule is there there's a long hallway and in the hallway one side they put a desk and a chair and maria gopat mayor used to sit on the hallway and when it would be so for me so people know his mayor's wife so once for me was going there and maria gopat mayor was doing some calculations so for me stopped and asked what is she doing so i'm trying to develop a theory of nuclear shell theory for me said that's very interesting i was thinking about it in you know it should be something like both theory you know the same kind of thing and that gave maria gopat mayor presumably the basic idea which went on to do in the the famous theory of nuclear shell which they got the Nobel prize uh ultimately so there are all these stories which are uh hangs around in in famous places like this james frank institute and the co-farming university of chicago everywhere else for example like you when you go to harvard you hear the chemistry department lot of lot of beautiful stories about bob woodward who is one of the father of organic chemistry that whatever so now uh let me go back and forth little bit now uh so these are the integrals the same as with the factor of two same as the second of reducible claustrical say is the first reducible particle there is this difference of one between them but this is the integral you have to remember so is f12 and remember f12 is mayor a function so f12 is this quantity e to the power minus beta u12 r minus 1 so then beta 1 dr right and then that is dr 4 pi r square beta ur minus 1 okay so there is a factor of volume in front so this is a strongly temperature dependence that comes from beta 4 pi comes out 0 to infinity 1 pi 4 pi v right yeah 4 pi v so this is the expression of my first reducible claustrical which is also v2 by 2 now i'll show you how that happens and we'll stop after that uh before that one tiny little bit that one can go and calculate the average number of these clusters by using this there is a what turned out it's very powerful that this exact partition function is nothing but a class of polynomials called bell polynomials a very powerful polynomials used the famous mathematician bell that has been used in many ways particularly in study of zeros or polynomials that polynomial allows you to write a recursionization it's just a bit of good luck now if i can write this then if you can give me the z1 and then give me next one i can get the higher level partition function so i can construct this called recursion relation the input of course are the bl plus one so if i have that i can construct this partition function if i can construct the partition function i can calculate what is the average number of clusters in the polynomial by using this as the weight uh this as the weight so this is the weight of your one uh of your one particular distribution right if i have the partition function they can i calculate that weight i'm not going to get the details of that but then you can now say okay lml is the number of clusters lml is the number of particles in size of cluster lml is the number of clusters of size l each of them have l so lml is the number of particles in cluster of size l now you plot it against l by n n is the number of particle l is the size of the cluster then you find when you have low high temperature then this goes to be zero but when you lower the temperature you find a second peak appears that second peaks is the is the is the liquid so a second big cluster appears with l by n close to one rightly less than one may be at point eight so as you hit the coexistence these peak signals the appearance of a macroscopic cluster in the system so liquid is envisaged in mayath's theory as a macroscopic cluster okay so when you cool down like nitrogen we have i cool it down and make the density large then the particles come close to each other they all start interacting i call it system a liquid when the all the particles are interacting each other or one of them that many others but if a one is interacting with say 10 which is a good number actually this correct almost correct number then the neighbors of that my name 10 neighbors they are interacting with another 10 then another 10 so liquid means when the particles are all talking with each other or molecules are talking that means in this description they are part of the same cluster they are talking with each other means they are connected by a bond this is not a chemical bond the bonds are breaking and forming but they are interacting so liquid is that not each particle does need not interact with everybody okay so this is the again i repeat ml is number of particles in cluster of size l ml is the then ml is number of clusters of size l ml is number of particles in that cluster okay and then this is l equal to small one or so one by n equal to very small that are the gas the small clusters when they are large this goes close to a when l is close to n that means a gigantic macroscopic clusters this close to one maybe 0.8 or so so low temperature this peak appears high temperature there is no peak this is a low temperature high temperature i should have just like this so that should have been there and it is not there so this is a nice physical picture that comes as i told you ideal gas doesn't form liquid liquid forms because of interactions and that interaction lowest level is the second or third visual coefficients that is not perfect theory by any means this theory to take you all essentially a picture and how interactions are taken into account but that is the universal scheme all other theories flew from that so if i take now l l ml and then l by n then as she asked in the high temperature gas phase this is just like this i lower the temperature increase the density it becomes like that then finally there appears this is one okay so this is this is the pure gas no liquid large clusters are appearing then at the coexistence you have lots of gas and then also one gigantic cluster so if i plot this thing as pressure versus density then this is like that which is here then it becomes more like that which is this one and then it becomes so this one is this one gas and liquid are together this is the coexistence so you have one gigantic cluster which is the liquid is amazing when your liquid is in a equilibrium with a vapor phase just like in glass of water then in one with the liquid you have this avocado number of particles interact with each other but your liquid your gas is just cluster of two or three this is a huge separation and one is stabilized by entropy which is your gas one is largely stabilized by your enthalpy which is this phase and this really took huge amount of time for people to describe this phase transition huge amount of money i always consider phase transition is where almost all our concepts came including high energy physics because that's where came the order parameter that's where came the interactions around atoms and molecules so phase transition is the one everybody should study the free energy diagram and all these things even our chemical kinetics is deeply more deeply even though independently it was developed in the beginning now one little bit because we i don't know when you will meet first one little bit the virial expansion mayors theory so then i go to the partition function and i say okay i expand this this expansion now i start putting my v is together i take only the this term then we know n into n minus one two of them i will do rest of the integrals i give v to the power n minus two correct there are only two i cannot do which two are kept here they are vector integrals then i change the center of mass to what one volume comes out this is my may or a function actually it should be u however now i take this v out it become v n minus one large n limit i can neglect this i have n square by two and then i bring it here take v n out one by v comes here it becomes one n square beta one by v beta one as a volume term okay and in general so this you can this is the partition function now i take this tiny little bit of partition function only the this part tiny little bit i just looking at a correction that correction so i am i am interested in this part i am interested in this part or this part i am i am not doing this one but i am doing a tiny little part to make it simple a full derivation can be done that will be little complex so i am not doing it so now i am going to do that put that back in my partition function v z n i put it back in my free energy this definition then i log this is a large volume and n v and then i take this derivative and then this becomes one over v that becomes one here and that derivative gives it do this this is suddenly developed but at the end of the day this is n by v by n so p v by n k b t you get exactly from here you get exactly this term so this is the first of an expansion so p v by n k b t is this the virial series so now i identify v two second virial coefficient is minus beta one by two this quantity from minus beta one by two and this is then my microscopic expression of the second virial coefficient except this is in not v and u writing in from of his room streets notes and screwed up everything very nicely but this is what it stands for this is a beautiful expression second virial coefficient which describes this bending here is in terms of mayor cluster integral and this is a general expression one can derive and well little bit i can go to van der Waals equation i can find out my virial coefficients this is a homework problem we always do you expand it pressure in terms of density and equate this thing you get the second virial coefficient third virial coefficient go back find out betas then find out betas this is betas find out your mayor cluster integral which can be done by use a very similar recursion relation so you can construct whole statistical mechanics that lies under van der Waals and that that has been done so we stop here today there are some limitations one of the major limitation is that when one goes to that level this is only temperature dependent you in doing these integrals the way you find the only when volume dependence of those integrals are neglected and that is the so this is the it it does not describe because it diverges there and it does not describe the liquid state so mayor see it does not work in the liquids but it describes how liquid forms it describes how singularity develops and it describes how which singularities are the of it so what we will do now after this we will do land out theory is the theory of phase tensor i am hurrying a little bit any questions you quickly ask yes that's a very serious quantitative change this part is quantitatively exact correct no say for example when you go to liquids then something fundamentally different appears here for that you need infinite number of them and that part where it breaks down but it as i said here it reproduces the complete flattening it reproduces this part but then it does not because for that you know people have done little better and corrected this theory but people are quite happy with this part that it starting from an interaction potential to see how a temperature dependent behavior emerging was really quite satisfying and the concept of clusters we don't want to put percolation this is the way after that percolation theory was done soljel transition done so this played as a catalyst of development of a lot of things that's why historically in statistical mechanics these things should not be ignored mayors theory people don't teach it because it's too complex as you have seen and it so next we'll do don't press transition i'll explain little bit okay i think we'll stop here today thank you