 Now, these are the two things, so partition function now is 1 plus Fij, but is a product, so now I can do the product, the first product all of them are 1, so I write 1 plus C, this is a binary, so 1 plus F12, 1 plus F13, 1 plus F14, da-da-da, then multiplying 1 plus F23 plus, 1 plus F24, da-da-da, then 1 plus F34, 1 plus F45, da-da-da, like that, clear no, it is not in chemistry we will do not come across this kind of somewhat intellectually demanding things where you are doing formalism, but this is the, at the one of the simplest at this level, if I do that, then every product has a 1 in front of it, so one of them will become 1, then all of them, so I multiply, then next one will be binary term, let us do the simple one to see that it is the way it is working, so let us see I have three particles dr3, then I have, is that okay, now if I do that first term is 1, next term is F12 plus F13 plus F23, okay, then next term is F, binary F12, F13 plus F12, F23 plus F13, F23, then I have one term F12, F23, F13, so I have this term, then bunch of one particle term, then there are three particle terms, but the three particle terms of two kinds, one of them is okay, so now, now remember this is an integral, there are too many things to say, so I will go back and forth, so this is the kind of the zero particle term, this is the one which is the ideal gas, so if I do not have any interaction, then may have already told us the F terms will be zero, that is why he removed it, so then if I do then only keep this term, then ideal gas, so the partition function is just the ideal gas term, next term then in the, is this binary interaction term, two particle term and this becomes your second ideal coefficient, in certain ways, we will see that and this is the three particle term is of two kinds, one we call chain diagram because this is 1, 2, so this is this kind of a term, F12, okay this is F12, F23, the one I have done here is this thing, this is F12, F23, right, so all the combinations of three particle but chain diagrams, we are not, but this one is everything is doubly connected, so this one if I draw the graph, I will find three, because 1, 2, 2, 3, 1, 3 or 3, 1, this is a different beast, these are called ring diagrams in the language of physics and chemistry, now these are the chain, so let us see what it may do with all these beautiful stuff, so this is called decomposition of a partition function, this is the universal, this is the one which is used in everywhere in your physics and chemistry, in many body, when you talk of many body physics, this is what is done, if you take Feter Wallach or any many body physics books, this is the first thing that anybody do and there it is called may have our cell cluster expansion, okay, so now there is a graph theoretical representation, the first term is unity ideal gas term 1 dot no line and ideal gas, second term two dots and a line and the black dots also has a meaning and I will come to that, black and white is man, there is a graph theoretical language that is used and there are such figures or graphs, then third term is these three dots, two lines and three lines, so one is a chain diagram which has two lines and three lines, so then the line is a bond, line can be considered a bond because this is the interaction between them and the bond is a kind of mathematical bond because the F is, has the characteristics, you have to remove the heart sphere part to make it into chemical bond, if you do that it is called a physical cluster which has also been developed but we are not going to go into that, now as I was saying the dots means in statistical mechanics and that it has been to be integrated over, however if I have an external field which I can add an external potential like in a homogeneous system like you want to take an external electric field which is position dependent or you want to take a surface effect as a wall which has an interaction potential at certain specific location then that has to be included in your description, then you need to have an extra term plus you are to be there and those particles which are facing that will be you cannot integrate them anymore, so then those will become we call them white circle, these are the things done, so this more sophisticated version of that there is a beautiful review article by George Stale 1963 and then one little before that Morita, T Morita 1961 but the George Stale essentially did the work of Morita and this is in a book Physics of Simple Fluids edited by H. L. Frisch and Libo-H and there in half of the book is George Stale's article in graph theoretical presentation, now what I have to do now, in order to evaluate the partition function I have to sum over all these diagrams, all these diagrams need to be summed over, how do I do that, it is a formidable, so we have made some progress, we have now beginning to see that how I am getting ideal gas law, I am beginning to a decomposition, things are simpler but I have not solved the problem, all the difficulties are now hidden in these terms, I have a graph theoretical representation of interaction between particles which is neat and clean but I have to make further progress, next thing what Mayor did was he said he realized that this is a dot is a kind of a zero particle, these are two particles, these are three particles, then I can go on have that the next one if I have four, of course I have four particles in a many body system, then I will have diagrams which are connecting four particles, now then what he said all these which are three particles he called bunch them together, then there are four he bunch them together knowing very well that there is a difference between them, one realizes that this one is essentially product of this one and I would be able to evaluate them if I know how to integrate this quantity but that at this point not necessary because I need to bunch them together again in number of particles in a cluster, so let me call them a cluster, this is where cluster expansion comes in and I bunch these together knowing well there is a difference but I call them together, so this is what now called the cluster integrals, so all the L particles are brought together, they might be doubly they are all connected, they might be doubly connected that is okay, it includes all of them, they can be even more than doubly connected because if I have a four particles then not only it can be ring, it can be line in between, so I can have, so four particle cluster consist of all versions of that, then these, then these, then these okay, it turns out that if we do not have any of them then it is same as it can be replaced in terms of the integral of f12, ring one also can be done with certain difficulty but these one becomes very very difficult, you have to remember these particular thing is an integration on four particles, so there are four particles into three, each particle has three coordinates, so this is a 12 dimensional integral, I can select origin as one but still I have 9 dimensional integral to pick, this was the difficulty that one could do but you some people were extremely brave, they did evaluate upto 5, 6, for hard sphere one can do upto 9 or 10 and that gave rise to beautiful theories, Karnan, Stalin and all these things which you might not have time to go through but this amazing exciting things happened in 60s and 70s because it had to wait little bit until that because of computers to come into existence and these are played a extremely important role when it was put on the lattice in critical phenomena that was called the high temperature expansion or series expansion done by dormant green and many other people, okay, so the reducible cluster integral of size L is this best, I put all the graphs together, okay, I have put a normalization here, you will see that is very useful, I have put a V here and this is, this particular thing has a volume V to the power L, okay, now when I do that then following V1 then I get 1 and I do V2 then it is this quantity dr1, d2, I put, this is a definition right now and it will be handy definition, so I have put certain normalization here knowing what will happen, so this is a definition which will be absorbed, so nothing to worry about that, so I now calculate V2, V2 now this diagram, so that diagram as I look at a diagram I immediately know I have to integrate over this, I have to integrate over that and I have F1 to there and that is the advantage of a graph or that is the whole thing of Feynman's graph theoretical technique was that they could write down the graphs and then from the graph you do not go do the algebra, instead you write down the graphs and then transfer the graph into an integration, okay, similarly we are doing so this becomes that, so now I see dr1, d2 why it has put in you can now understand, I can change my coordinate to 1, so then it becomes 1, 2 and this is 1, 2, so I can integrate over my origin, you know there is nothing so long there is an external field, so then that volume cancels this volume and then I get and dr12 I call that back r, so that is 4 pi r2 and dr4 pi r2 r1 and this is the b2, we are beginning to see certain things that you know radial distribution function kind of nature character is coming out, b3 which is a lot of fun, you have now this three three-dimensional integral F31, F21, F12, F3 all these things together, I do not know where F12 has disappeared here but it has to have F12, F23 everything, I think my student whom I told to cut and paste, I think he did not take from the book he is very very very ambitious guy, he decided that he will learn it again and in the process everybody is making mistake, okay, so it has to have the he has put it here not the right way, so F12 should come before and then 2131, so it is correct but not the way I would have written it so and this is the three-particle term, I would have write in a 12233 but that is the same thing here, okay, now comes lot of fun this is really beautiful term, now we have to say that if I go back to partition function then I have to make two things how can I group them together into one term and then I can calculate how many of them are there so two steps and this is a little difficult but please try to think so some contribution of the confession integral all the clusters of size L look at that, that sum over all the connected things so now this is the quantity I want this is still very formal I am grouping them together, I am not evaluating them please don't get it wrong, I am just grouping them together I am developing a language a semantics to be used that's how a big problem is done a big problem is divided into bits and pieces divide and conquer, I am dividing it now the conquer part will come later so that division I have now introduced cluster integral it is called reducible cluster integral and that now so what I need in the partition function is this part that part is then BLL factorial V and all the particles there that I know how to do so then the sum contribution to configuration integral the thing that I have written here all three particles are here that would be then all the L factorial to the power ML because ML is the number of clusters of size L and all of them come with the same weight VBL by definition so VBL to the ML this you can just to work out one of them you will find this is exactly includes everything now comes the important part that so this is the total contribution so if I have ML number and I will have constraint on ML if I have ML number of cluster of size L it is then I am now is going to do a combinatorics I am going to say how many ways I can distribute or I can put essentially marbles in the boxes and the boxes are my cluster of size L so I have n number of particles I have to put them in boxes so it is a very famous combinatorics I think in one of the very popular problem in IITJE in a called multinomial distributions actually very typical those who have done IITJE this is the one popular every 2-3 years this comes even then is very difficult to do so basic idea is that I have a constraint what is the constraint constraint is that total number is L playmen so then total number of molecules in ML cluster of size L so total number of molecules N and in LML is the number of particles in cluster of size L like if M20 L is 20 then there are 20 particles so in order to get the normalization I have to do 20 into number L so again you have the boxes if the boxes name 20 you put 20 balls there and then you find how many ways I can distribute that that quantity is this quantity multinomial alright you have to do little bit yourself but it is not difficult just do it with a 3 particle you will be able to do if you like there is a lot of this particular part is a lot of fun then I want to get the partition function so this is the weight of total weight come to the partition function contribution to the partition function all L number of clusters those all L like here all 3 all 2 and how many of them will be given by this the total configuration of my system so what I have done now let us see that I have instant configuration and instant configuration I have M1 equal to 20 M2 15 and M3 equal to 5 then it will change because my cluster distribution will change I have to include all the cluster distributions but each distribution this 2015 or so 5 they will come with a weight that weight is given here the total number of ways I can get that and this is the weight to the partition function so that particular cluster size distribution contributes at least much in terms of integrals to the partition function and these the number of ways I can form the cluster size distribution so I have to now multiply these 2 and when I multiply these 2 then this L factorial L factorial get cancelled and I get this beautiful expression called me as this is called me as partition function ZN by N factorial EBBL ML by ML factorial so let me write this now write even now we have just on the way to do the things we have not solved the problem but we are on our way to solve it so mere partition function is the configurational part this is exact why it is exact because this is nothing but bunch of definitions but now from there one can start playing some very interesting games ignore that last part here with not important at this point that means I am saying this part is not important at this point so this is the one that we need to evaluate so once we know ZN remember if I know ZN then I know the canonical partition function and canonical partition function Un is this 1 over H to the power 3N 2 pi MKBT to the power 3N by 2 into this is the canonical full canonical partition function and free energy equal to minus KBT QN so this is the equation number 1 this could be equation number 2 and this is the equation number 3 and this is called this one that you are saying is a fast expression of the cluster expansion and or the cluster decomposition and then one goes on doing the so I will stop here because it is fairly formidable and there is no point of going but I will really I can forward to Rajurshi this slide but take a look into this it needs certain familiarization it is done in a better in more detail in my book more explanations and everything is there but this is as I said right at this point this is an exact expression this cluster integrals at temperature and volume dependent so in this decomposition of the partition function into smaller or simpler terms BL is very strongly dependent on temperature because BL is the integration overbolchment factor it is also dependent on volume because you know your total range of you are integrating of the total volume so B clusters appear they fill the surface or they fill the volume and this can be done many different ways one way may have did is we will do other way exact recursion relations that one can solve this problem again bit by pieces so next part of the puzzle is to get the cluster integrals so in cluster integrals we know B1 equal to 1 now I want to do B2 and we so I can calculate B2 now and B2 is the FR beta is 1 over KBT so see that is what I see in the two particle that we have to have a quadrature but I can trivially put and for heart sphere I can evaluate it exactly this one that becomes your second virial coefficient exactly that is one of the great achievement then when you go to B3 B3 has all these three particles so this is a two particle cluster that means this B2 is this quantity now B3 will be this bunch of bunch of this plus now if you choose to neglect this ring diagrams your problem solved all you need is F1 2 and that already shows gas liquid transition the condensation appearance of infinite cluster so the way theory always works you have to work very hard up to certain amount of time then there are certain beginning to get your rewards and you begin to see how things happens how interactions make things different from ideal gas how interactions even two particle interactions formation of trees because F1 2 like that kind of open diagrams are the trees how the trees come and they influence the thermodynamic properties of the system yes what I am saying what you can do this is this is follow 1 2 2 3 that decomposes when you do the integration you connect you choose here then it will be this integral and this integral no no the product of the no no the product of the up to all the chain diagrams that you can do just trivial is like a convolution there is no problem and that is why ring diagram suddenly becomes so much more even at that level it becomes so much more complicated up to chain diagram things are easy this is a very common thing in all over main body physics ring diagrams we can still do there are certain ways to make certain progress it is when you have lines inside the ring so this one now had become 9 dimensional trickle I can reduce it to 3 6 dimension but I have to do it numerical again but we can still make lot of progress that we will continue to do hmm but I will change you give you the slides and you please familiarize yourself little bit because what we will do from now from we will show something very important class of relations class to just doesn't mean this it also means expansion of density in terms of fugacity which is very important part of physical chemistry you can have an expansion of density in terms of rho equal to sum over L B L Z to the over L that is exact that you can also do from grand practical partition function and that expression of the pressure in terms of B L Z L Z L Z fugacity and elimination of these two class to expansion gives you virial series so now for the first time you really begin to see and first time you see that why virial coefficients are used to extract the force field remember the first force field in this world was extracted from virial coefficient there was linear density first force field basic ingredient is the molecular size then is the depth epsilon so the remember van der Waals made a mistake he got a size 8 times in the molecular size okay but what van der Waals did was actually building kind of things attraction by hand and a kind of a priory which didn't work out well so mayor for the first time gave expression exact expression of virial coefficient and we studied the temperature dependent on virial coefficient which is very easy from the equation of state experimentally fitted to that gave you the linear zones potential 1940s that was the beginning of the our force field culture okay even now these essentially one of the part of the things that one uses okay anything else will stop here now