 We have been discussing and deliberating on the ideal gas law where molecules do not interact. We have done monatomic gas which has been given rise to an important expression of entropy of translation, new translation degrees of freedom which is circuit-tector equation. Then we did diatomic, everywhere we had certain payoff. In the case of diatomic molecules, we did the vibrational degrees of freedom which though for molecules do not directly affect the thermodynamics but the vibration of partition function model as harmonic oscillator with the sorting solution from sorting equation gives us the expression of the entropy and free energy and specific heat which turns out to be extremely important in context of the specific heat of solids and crystalline and amorphous solids and that is the way to think. Then we did polyethylene but however all this ideal gas the molecules could be reached internal structure like in polyethylene molecules in water and methane and sulphur dioxide or methanol. However they are non-interacting that means the molecules are modeled as phantom thing they can pass through each other, they do not see each other. Now in a high temperature and in low density the majority of the contribution to phase space or to partition function comes from non-interacting part. So this high temperature or low density gas works out quite well you know though this ideal gas is not a poor approximation. Yes it breaks down but as soon as you enter dense liquids dense gases dense gases or the liquids then this non-interacting will be done and it was very very difficult. So you know people realize that very quickly and they were trying to get around it by this solids to get around by doing quite a bit by normal mode approximation like it was done by Einstein and Debye. Then in the quantum cases this both Einstein statistics and Fermi Dirac statistics that was very useful by explaining super fluidity but that was also again we did not we did not do to consider interactions there are we went to a representation and where we could do the like you know in statistics it is just non-interacting particles in both studies Fermi Dirac statistics is so important in electric describing electron gas but there is free electron gas we did not have to take interaction in the electrons at all. It was like that till 1935-36 but in 37 however a very significant development took place and that came from Joseph Meyer. Meyer they realized that that the difficulty Meyer realized one very important thing that in order to understand the real gases and in order to understand gas liquid transition he was not thinking of liquid solid transition that was far from you know he was he was really just tried to do real gases dense gases because you know from van der Waals you know from van der Waals equation of state we know from van der Waals equation of state that if we can if we plot pressure against density then this is the part very low density part 0 very low density part goes like ideal gas then it bends and then it undergoes a phase transition. Now if I think of a then ideal gas that goes over like that this is the ideal gas this is the ideal gas but this is the real gas this is how real gas real real so real gas bends bends from here then it bends like this and then it flats these equal resistance and this is liquid however van der Waals when tried to describe this behavior it is a wonderful equation van der Waals equation of state he got something like that that called van der Waals loop which Maxwell fixed by doing the Maxwell tie line and this what we call spinodal and all this kind of stuff we do not want to get into it right now. However the important thing is that that this departure so mayor was interested to describe this departure the departure from the ideal gas and mayor just like van der Waals see before that van der Waals did that van der Waals took at in attraction and the molecules so mayor already knew that how to proceed he knew that we have to take the molecules if the even if the spheres they attract each other in intermediate distance very far of course they do not interact and when they come very close to each other bang they repulsive because that is a overlap of the electron so he is a famous equation of state that p plus a by v square v minus b equal to RT the van der Waals equation of state where a talks of attraction b is the size of the molecules what is the size of the molecules that is nothing but because they repel each other the repulsion defines the size of the molecule diameter of the molecule so mayor knew that mayor knew that this is the way to go that I need to have attraction and I had at an intermediate distance I need to have repulsion is a very short distance when the molecules touch each other and they are to the interaction has to fall to zero when they were there but how do you go about it van der Waals did it phenolotically remember he said okay the total amount of volume accessible to a molecule is total volume minus the volume of individual molecules and then when a molecule going to hit the wall to define the pressure remember he was still following the Maxwell Boltzmann kind of a kinetic theory picture of pressure that pressure comes because molecules interact molecules go and hit a wall and that molecule is pulled back because there are certain other molecules there it will attract it so idea the ideal gas it would have pressure p now that pressure gets decreased and volume get decreased okay so basically what you added wanted to add it okay I want a redo a effective pressure p and effective volume v so that I get it so I still working on PV got what ideal gas law but then he said okay the V actually is little renewed and the pressure that I really get that pressure would be if I want to get the ideal gap pressure then I have to add this term to get the ideal so I want to write PV got to RT but P and V are different because of the interaction so the he got there that is the logic he gave to get the available situations but that will not work right because we now need we need a molecular description we are talking of microscopy so we are going to have a the not this kind of hand waving argument which works reasonably well but still hand waving we want to have real molecules real interaction potentials that how do you go about it there is nothing there before mayor did it that is why mayors contribution so important and that is why you are spending a time on discussing mayors theory okay we want a fully microscopic statistical mechanical analysis of interacting many body systems then what do you mean by interacting many body systems let again go down a little bit now and let us see so I have a potential these I described that interaction potential say you are are between two molecules I and A and this is Rij and this is Rij then the interaction potential I have in mind is something like that and this is called inner zones interaction potential so you are equal to 4 epsilon sigma by R to the power 12 minus sigma by R to the power 6 this is the attractive term and sorry this is the repulsive term because of the 12 and this attractive with the negative sign so this is the 1 over 12 part when they and this is the where the molecules kind of touch each other so this is the molecular diameter sigma and this is sigma then it zoom the energy goes up when they separate a little then there is an attraction is called London forces because of the induced dipole in those interactions and I cannot go into that and then it goes like that we will at one point of time talk of intermolecular forces but this is the kind of thing that we have one important thing to note that these are radial potential that it depends only on R there is no angle so these are just fears the simplest possible thing so if we have this kind of interaction potential then we want to so we have starts with intermolecular potential so given intermolecular potential so I give you the linear zones is simple to one then we want to do calculate the we attempt to calculate the partition function from past principles like sum over energy levels interaction over e to the power minus beta is Hamiltonian that is the thing so that was the thing that Joseph may have did and we are going to do that in the process he introduced what could be considered first graph theory of liquids and gases and that he did the cluster expansion and he got a huge number of results that came out of these things not only that that made the beginning of classical statistical mechanics the most significant step after which we let keep which one and that gives so it did it derived it gave a derivation of the real coefficients it explained many of the things of van der Waals language of class the size distribution of clusters which are so popular and microscopic picture of gas the grid television all these things came out and so next to this is interaction we already done that now so let us start the working now we have the n number of particles n number of particles so in a system I have box volume v and I have n number of spheres they are not too low density and the spheres are interacting with that with the Lena Jones potential Lena Jones potential and this is also diameter sigma so two spheres are teaching that is sitting and I give the distance it goes little bit faster than drawn here okay and this is the at high height of the attraction depth of the maximum attraction potential and this is epsilon so the that these this is epsilon that epsilon is epsilon okay alright so we have this of this Hamiltonian kinetic energy plus potential energy and we have to evaluate with that Hamiltonian we have to evaluate this partition function we have to evaluate this this and look at that I have these are vectors I have n vector integration over position n vector integration over momentum I have Boltzmann factor 1 over n factorial and I have h to the power 3 and the the blank constant as we told in the very beginning that these are formidable thing of statistical mechanics to evaluate this integral is just this is the holy grail of statistical mechanics to evaluate the partition function if I have partition function I have everything I have free energy I have entropy I have so if you need I have compressibility but how do you get the function that is the problem so whenever you try to do something so formidable as this one the idea is to divide and rule and we divide and conquer and we do that divide and conquer okay so then there is a partition function if I get the partition function I get the free energy and I get the pressure and the equation of state and the beta h this thing now the h can is written this is the kinetic energy and this is the potential energy potential energy is sum over so we assume the potential energy is pairwise additive that there is very good approximation but still a simplification it goes wrong now and then but it is perfectly okay to start with this pairwise additivity that means total total potential energy can be a as a sum or noted look at the notation here so that we avoid avoid double counting we have to say that the i and j runs like is greater than equal to 1 but j is greater than 1 so the terms will be written as u12 u23 plus u34 plus u13 plus u34 or 1 4 like that that means you have rather 1 3 1 4 then 2 3 2 4 like that so that I do not count it twice so with this u n now formidable because of the energy zones I set out to calculate the partition function what do I do it now so I write down the partition function again putting the whole thing here now I immediately notice in a classical partition function I can evaluate this part like I did in translational monotonic gas because they decouple from that and they decouple some of themselves moment of one particle is not coupled moment of one other particle and x coordinate is not connected to y coordinate so it is px square px square pz square for one first particle second particle third particle and I can do that integration we did that just exactly like that we did and that gives I know this part that we did in ideal gas law and we know that part is used to circuit the refrigeration you can see the decomposition of beautifully since 3 energy is the log of that term they already start the composition log either this part the ideal gas part and the interactive part and this part is called the configuration integral is now contains all the non-trivial contribution all the non-trivial contribution of the effective interaction from the system this part which gives makes gas to condense to liquid that cause liquid to go to crystal the this interaction part with let you and me to talk and walk around so that is all in this configuration integral so we are now going to find out how to evaluate that thing and we just rewrote that that there are this this is some e to the power minus I can write now that quantity I write as a product because e to the power minus a plus b e to the power minus a 40 to the power minus b so this is some the sum I e goes to product and this is just a notation no no big deal now difficulty of doing this integration was people tried that this being with the potential unfortunately goes to long distance goes to 0 and potential goes to 0 e to the power minus that potential that goes to 1 and that is bad news for us because I have to integrate and it something which is less than 1 because e to the power minus beta u can be less than 1 in short distance greater than 1 in intermediate distance but then tapers off to 1 is a very complex function e to the power minus beta u ij so may have did a brilliant thing at that point and you know he said okay these integrals that I am trying to do in configuration integral are not convergent because they are giving lot of difficulties because if they long the separation they are coming with a contribution unity contribution 1 and that is not a good thing for me because I cannot do I want to separate it out so that I do not have to worry at the long separation part so introduce this function which is called may are a function which is f so this is called may are a function you said okay let me write the following way I write f r ij f ij e to the minus with the minus 1 I take out the minus 1 part so that now r going to infinity r ij going to separation between 2 molecules become very large beta ui are going to 0 so this quantity going to 1 and then this going to 0 f r ij going to 0 so this is embodied here so it is a beta ij which is causing the problem was going and this actually it was this is 1 so it was stopping at 1 but now I have taken it out 1 so it start negative it was before that it was starting when it was infinity u at short distance infinity it was starting from 0 starting from 0 and is intermediate distance when there is attraction is becoming greater than 1 now I am taking it out so it starts from minus 1 and it goes above can go above 1 or so depending on these depth of the attraction but now in long distance is going to 0 this is 0 this is a 0 line is going to 0 so it is very good now I can do the try to do this integration then what may I did or something really very very interesting you said okay let me write the partition function now with the partition function is written as the product of these now only have positions and e to the power minus beta u r ij as I said is the product and f r i so this quantity is now replaced as 1 plus f ij this become because f ij is this quantity minus 1 so this quantity 1 plus f which is written at 1 plus f so my canonical partition function is becomes like that and then I the beauty is that now beauty now I can decompose it now I can do that look I write it out like let us say the 3 particles 2.1 if 1 2 1 plus f 2 3 3 particle system for 3 1 just let us consider that we have n equal to 3 3 particles then then I have first term 1 second term is f 1 2 f 1 2 3 f 3 1 f 1 2 plus f 1 3 and f 2 3 then there are another set of particles the term that are binary term f 1 2 f 2 3 f 2 3 f 3 1 but there is one term with the f 1 2 f 2 1 f 3 all three are present so this is now done here symbolically written you know one come then these are these isolated terms which is 1 2 plus 1 3 plus 1 4 then these are the product terms. Now may I introduce the following to beginning of the graph theory and this is as far as you know the first application of graph theory to study the mechanics is it okay first term which comes with the value 1 is just one dot second term which is this series of term are two dots and joined by line and line is the mirror function then there are three dots and their chain and there are three dots and I can form a ring total number of and then he said okay let me now consider that the total number of single particles as m 1 these guy which is a cluster of size 2 total number of particles that m 2 then total number of particles of then these things we call them m 3 which includes this one. So now m l is the number of clusters of size l that does not make any distinction where there is a singly connected doubly connected and then I can of course write in equal to element you are beginning to see something there is kind of things we played the game in particle canonical going micro chronicle to canonical that kind of game we are going to play because these are constraints that will come in my writing a partition function and I will do exactly like that I will try to write a total number of ways I can distribute particles to this cluster and then you can imagine that I can have the constant and I will get a Lagrangian undetermined multiplier that the game that is played and then we have done it before then what may have did okay he did something very smart he said well I have defined a number of m l at the number of clusters of size l now can I now give a weight to a cluster of size l okay I will do that because you know looked at the partition function he showed that they are nothing product any of Fij so now he said that I will now have all the integrals which are connected all the connected diagrams that means say three particle will be F12 F23 plus F23 F31 plus F31 F12 plus F12 F23 F31 so these are all three particle clusters they are all connected they are chain diagram like these and these are chain diagram and ring diagram so he said I put them all together and the weight of these to the partition function I call it BL the cluster integrals and then I define it as 1 over L factorial V a V is very important for reason I will say for normalization and then I have a integral over D1 and I have the product this is the product F12 F that is the product then this is the sum and this is the sum 1 2 3 4 you just write it down and you will see what are beautifully then BL at sum of all these things so for one single particle there is L factorial V and there is nothing inside DR1 I get one two particles is the ring I can do that and I say 1 over 2 V and DR1 to FR12 now I can change from my coordinate system I can go to particle 1 and say then this becomes already it is 1 2 this becomes 1 2 then I get a integral and then say R1 to I R so DR4 as VR the DR1 is not involved I can integrate out and that is what the origin I get a V that cancels this V so I get this quantity B3 the exact what I wrote down this is the B3 what is the advantage now the great advantage he achieved by decomposing it and then once you decompose like that you sum contribution to configuration integral because that is what we are trying to do the total configuration integral of clusters of size L so it is a graph theoretical decomposition that becomes this quantity so there is there are ML clusters ML clusters of size L and they come with the weight VBL and then I VBL to the power ML where ML is the number of clusters of size L and clusters of size L all of them put together and they bring together a size ML so that becomes that and this part is I have ML cluster each of them L so I can now in classical statistical mechanics I can rearrange them so L factorial is the way I can distribute and there are ML such clusters in my system at any time there are ML such clusters of size L so the L factorial to the power ML so that becomes that now comes the important thing that how do I now so but these clusters can be many different size and they are fitting they are not real clusters they are clusters which are we call mathematical clusters because they are connected by this bond F bond may be bond so everything arrangement is possible also I out of in fact a total n number of particles how many ways I can form these clusters that is n factorial by these just the combinators that you all of you have done in the cool multinomial so now this is the weight of one set of ML and that one set of ML I can do this is the number of ways omega and then so I need to make the product of get the partition function then the partition function is one over L factorial ML and this ML this L factorial this cancels and n factorial remains here and then I get VBL to the power ML and they sum over all possible combinations of this kind then I get this beautiful expression VBL to ML is exact and these call me a partition function now one can go do a lot in with these things we still have not done one thing so we that we have not calculated VL we have not evaluated these things that what may have been but in the process introduce the class tentacles there is a recursive way of if you know the VL then these actually nothing but a polynomial called bell polynomials and then that is described in my book then one can have a beautiful relation recursion relation between the end which allow between Zn plus one and lower series so we can build up the configuration integral the which is the total actually the non-trivial part of the partition function we can so if I know these cluster integrals if I know the redo with these called reducible cluster integrals then I can calculate the partition function