 Then Mayor went on to show Mayor went on to show that this beautiful analysis that he has done he went on to show that how this analysis that he again did the maximum term method he picked up that particular cluster ML star that particular cluster which and the analysis fully given here that maximize the partition function is the same as the grandian multiplier but before that he showed that my fsm term maximum term method is the one that works that is the maximum term method that that is sufficient in the in this in the in the limit of infinite size of the system n going to infinity total number of particles going to infinity volume going to infinity such that n by v the ratio of number of particles volume density is fixed in that limit that maximum term that distribution ML star this distribution that maximizes that is enough everything else is redundant that beautiful analysis maximum term method combined with Lagrangian multiplayer and he could show that ML and Z here is the max the Z here which tells the weight of a cluster that is the undetermined in the Lagrangian multiplayer undetermined coefficient Z is the undetermined coefficient we show later that Z is nothing but the fugacity but if it is that then LML star equal to n and then we get n over there v and we get the beautiful thing which we call me a cluster expansion that density L, B, L, Z, L so now we have a beautiful relation in terms of the cluster integrals B, L and cluster integrals and nothing but those integrals those integrals that contain interval interaction so because of the cluster integrals we have now of the connection we have built the connection we made the connection with the intermolecular interactions and the way so this is a beautiful stuff then you went on and go back so this now write the partition function and about one particular distribution so we are talking of distribution of distribution because ML gives the number of clusters of 5 L and but we have picked up one particular such distribution we will call it ML star for example that can be in a 100 particle system M1 can be 58, M2 can be 32, M3 can be 12 something like that and they must adapt to A so M1 is 58 that means there are 51 monomers then I say okay they are 12 dimers that means 2 into 12, 24 particles in that 58, 24 did now 82 then I have 6 trimers so that is 18 so I will have now 100 I say I have that for distribution I have that maximizes now when N is not 100 but billions of billions then a distribution I pick up like that is one that is representative of the system then I can go and calculate the partition function once I know the partition function I get the free energy and then from this portion I get the free energy and from the free energy I can get the pressure which is the adbt here and then I can show by using this relation ML star VBL0L I get this thing okay so now this is another such cluster expansion and then further developments is done in terms of reducible cluster integrals because those are VL contents both chain and ring so you have to break down into rings and chain separately and evaluate them separately that has been done for fairly complex number of systems very large number of things and so this is called deduction in terms of reducible cluster integrals and the definitions one can do like a beta 1 is this quantity is beta 1 then this is called beta 2 now you can see if I have only chains and rings that beta 1 beta 2 is enough and if I have only chains in my system no rings like in dilute light gases then beta 1 is what turned out now beta 1 is the second barrier coefficient these become the third barrier question that is the beauty of mayors theory now we play the same game how many of these beta 1 in the system I call that NL and then how many ways I can distribute into the NL the same omega and then I make the product of the two and I get a just like me are partition function another expression which is the story goes that well when the Joseph mayor could do the earlier these derivation he could do this thing he was he was he could do the he was stuck he could not go to reducible cluster integral and the further development was done by his wife Maria Gopatmeyer who entered to give a get a Nobel Prize for his work on nuclear cell theory. So, Maria Gopatmeyer whom may have made made in Göttingen and married and brought her to Columbia University was instrumental in many many such such calculations of combinatorics and difficult things. So, when does that then one goes on and one can go on to describe the theory of condensation into in a beautiful way that means he could he could one could use these things and one could now show to get the maximum term method and one can show from the maximum term method that this is two things that one does that maximum term method give this ML start what we already showed is VBLZL and Z is the fugacity and BL one can show is this quantity it has it exponential scaling then one goes to ML and since ML goes as BL to the power ZL and since BL BL sorry ML is V BL ZL and V goes as V naught to the power L ZL. So, you combine and get V naught to the power ZL and when fugacity changes such that Z becomes greater than V naught inverse then you suddenly have very large clusters appearing in the system which is shown here as a distribution of L ML versus L ML is the number of clusters of size L and L is number of particles in that class that ML is number of clusters of number of particles in this size of cluster L and that then has this beautiful mayors picture is that a big large cluster appears in the system and that is the gas liquid transition. So, this is the co-existence this would become the liquid. So, mayor did not quite get there but he got the picture right is formulation was rather extended to in many many great directions and so there is a beautiful scaling also that is a plot depicting the distribution of cluster sizes then what one does the next thing was a wonderful wonderful thing that is a virial expansion of the mayors theory. So, mayor then went on to show that in a low density or large volume he can work this theory around and show that the free energy and the partition function and the free energy and the pressure has an expansion in density which essentially comes from these two things one is pressure is D L z to the power L and density is L D L z L. So, these are two series this is series one and this is series two now I can eliminate the fugacity from series one and two and I can get a series of pressure in terms of density and that exactly is the real series that was what was done by mayor and it is described here. So, we go now write down the partition function just like this is all the binary this is the tiner so these are all the binary chain these are these things two a bonds and just keep these two terms nothing else because this low density or large volume limit then I can do so this one is just ideal gas V to the power N then this is the way there are how many ways I can have the N number of particles get into thus the dimer that is N into N minus 2 by this and V to the power N minus 2 come out because the two integrals are here so it is V to the power N minus 2 and then I have other one will be three particles N into N minus 1 N minus 2 by 6 and V to the power N minus 3 I am not taking that into account so now I can if I take that this quantity then I can change again origin to one and this becomes one to this one to then I this becomes that thing and one I integrate I get a V so then I get this kind of thing then N is very large so this can be neglected I get N square by 2 V N minus 1 then V N minus N square by 2 and this integral now I define this integral as beta 1 and then I get N square beta 1 by V as this this term then I go and calculate the free energy from in this lower case Boltzmann constant then I work it out then this is ideal gas term then this is the term that coming from retaining only the chain diagram beta 1 dimer sorry only the dimer and then when I do that I found that this is the sorry this is all the work is this is the correction that comes to pressure beta 1 by 2 V and this small V is V by N and then you get this is the correction that comes out that means pressure P V by N K B T becomes 1 minus rho beta 1 by V and this we know now compare this with the virial series you get 1 plus beta you will find all these calculations are done here and this will be passed on to you so one gets the first term of the virial series and that is a beautiful things that you get now molecular expression the important thing is a molecular expression of the virial series and that is in terms of the series is to be u r so I now have second virial coefficient in terms of d r r square f r so this is the second virial coefficient and f r is e to the power minus beta u r minus 1 so this now tells a way to do the second virial coefficient now this is played a very very important role this particular derivation of second virial coefficient very exceedingly important role and this is the following so now I have a second virial coefficient which is given in terms of every factor I am not interested in that d r r square a square and then I have okay of course 4 pi and other things are there and density now you are so before this we did not have we had forms like linear zones interaction potential but I did not know how to get them quantum calculations they are far off and very difficult but now suddenly I have a beautiful expression of the second virial coefficient so I go equation I can I get the this part which will be quite nicely given by the second virial coefficient now I do the temperature dependence of this guy so I get now second virial coefficient temperature dependent so now I fit it so by knowing the temperature dependence of second virial coefficient experimentary and knowing their expression of the second virial coefficient in terms of interaction potential I could now get the my so this was the first time so these values of epsilon and values of sigma in the interaction potential they came out first time were calculated by using mayors expression of the second virial coefficient so mayors expression of second and third virial coefficient and fourth virial coefficient they are very very important though people started doing a molecular microscopic theory of condensation they get the virial coefficient now they see the virial coefficient which were introduced phenomenologically the virial coefficient because this was written down P P P by rho k B T P by rho k B T it was written at 1 the ideal gas the ideal gas B 2 rho plus B 3 rho rho q so these are just virial coefficients second virial coefficient third virial coefficient fourth they are phenomenological this was virial did it phenomenologically we from the experimentally determined equation of state but now we are told as what is B 2 in terms of interaction potential we are told as what is B 3 in terms of interaction potential now I can go back use mayors expression and the experimental values of second virial coefficient temperature dependence I can get interaction potential so this was the huge significance of this approach the cluster expansion the mayor a function the cluster integral V L and irreducible contrasting beta 1 which you can work out they are all done here was to get the interaction potential so though they we talk all these days the force fields all the all the all the glory of things to computer simulations we derive very complex system but the fast things came out so the virial series more than 100 years old even at the time of mayor it was 40 50 years old but one did not knew that it is coming from interaction coefficient one can expand when the was equation of state and get the virial series but one found out that does not work that means that is why actually A and B was determined but that did not work well because they are not robust but mayors theory gave us a robust way to look into the interaction potential and evaluate and that played extremely important role in our development so even today the values of the force field that we call for argon the sigma and epsilon come from these kind of things that epsilon is 119 K that comes from these analysis and so so there are certain limitations of mayors theory that this is the ideal gas and this is the mayors are unfortunately if you do mayors theory you know it it it kind of gives the condensation but then it it remains flat and there are many many reasons for that and we are not going to go into that in in this detail you can get that in the book then there are many studies beautiful beautiful studies that have been done so we are talking of this finite finite through the bell polynomials the bell polynomials are a class of generalized polynomial very powerful polynomials which exactly has the same formula with nobody mayor did not we did not know for many years but the advantage is that if that is so then they have a beautiful recursion relations which we can use now and this is the recursion relation bell polynomial recursion relation which now can be used to evaluate mayors function function and that is a beautiful stuff and you get this beautiful relation of mayors body so now if we can have some reasonable cluster integral then we can do a good job but the difference is that this problem remain we made considerable progress but we did not solve the problem because the reducible cluster integrals are not available even today they are not available a useful cluster integrals have been evaluated by great pain and it did not it works out somewhat better for many techniques are there for hard sphere kind of a system when there is no attraction it works out is in the well there is a very large number of real reducible cluster integral virial coefficients has been evaluated up to 11 and that means beta 11 beta irreducible cluster integrals of the beta 10 has been done by graph theoretical methods method may are initiated but but for linear zones it has been done only up to some 6 or 7 reducible cluster integrals that is not enough because when it goes to liquid phase then all the particles become connected so there is kind of jail that forms and the mayors theory breaks down but you know but you should be happy what you have got and some things that so we will stop here today and there are some old articles that we wrote long long time ago and this described in the book that the applications of mayors partition function to the real systems and so we stop here now.