 We have tried to do, trying to do interacting systems where molecules interact with each other and we have done Meier's theory and Meier's theory which is we have rise to many other theories but that is essentially theory of real gases and compensation. One thing I forgot to tell is that this is also called, we did the reason of that that it breaks down at high density but it is more famous in physics, this subject is approached by famous surfaces in a very different way, this is called high temperature expansion. The reason is high temperature expansion if you look at it that the Meier function F which is e to the power minus beta u r minus 1. So when more temperature becomes large beta becomes small so F becomes small so these kind of grams that becomes less important than these and then these become less important than these. So at low density of gases anyway this will be favored over these because they are not many molecules volume is volume and these will be more favored than that, draw a little bit larger which will be picked up through the when you calculate the partition function the cluster integrals and irreducible cluster integrals but equally rigorous way of saying that the Meier's theory essentially and high temperature expansion and Virial series is a series in density but it can also be considered as high temperature because the higher Virial coefficients rapidly decrease with temperature. So that is a different view which comes from e to the power beta u and this high temperature expansion perfects all through statistical mechanics simply because of the Benjamin factor e to the power minus beta. So the difficulty comes in statistical mechanics because of the interaction between particles and we cannot evaluate the partition function we cannot evaluate the total number of microscopic states and that is why this few cases we can do rigorously plays a very very important role because they guide us and they provide us insight. Just like Meier's theory went on to tell us what is a Virial series and went on to develop many other things like I told you Soljale transition, percolation, the cluster in phenomena, the cooperativity which is really I explained is a beautiful phenomena that you can have 10 intermediate size clusters say in a particular you can have a particle of 100 you can have 50 monomers and dimers but other 50 particles can be distributed among the clusters which are 5 of say 10 particles. Now when you increase one or two more particles these large particles can get connected because another particle can come in between. You can imagine it easily in a lattice in a cell that there are these clusters which are connected lattice size another big lattice size and then there is an empty between them but as we increase the probability or the density probability of finding a particle which is same as density then that can get connected. So what happened in a very infinitesimal change this 5, 10 m can become 150 m and suddenly this intermediate size clusters disappear from the system and that is what happens when the range see after range the clouds all the things comes down and sky become clear and you do not see any any any any a humidity beyond the point goes down and everything at least upper atmosphere. So this is the same thing the cooperative so that in that sense Bayer's theory is a great theory and give lot of things but it also has its limitations as I said that it cannot be pushed beyond the point where everything diverges and I cannot extend to low temperature and high density but with that cabinet we will go ahead and do the next thing. So second interacting system which of great use in statistical mechanics and find great use in phase transition and because it is solvable in certain exact in certain limits it can be solved exactly it plays perhaps the most important system in the entire physics of interactive physics and chemistry of interacting systems and this is the ising model. Ising model is a simple one dimensional ising model is also same thing is available for or three dimensional but they are equal to dimensional 2D ising 3D ising but the original ising is 1D and then this is essentially you have a condensational chain or rod and you have the spins we just spin so you can take them dipoles whatever and they are interactively with each other so they can be parallel anti-parallel all possible so the spins can be so definition of ising model is that it consists of spins spins can take only two directions. So let me write down the assumptions of the model spins can be can be only up and down and they are denoted by plus 1 and minus 1 as if one is the plus 1 and minus comes to a great extent from the logic that you have an external magnetic field these are only for magnetic systems you have an external also it goes over for electrical systems we use that in real dipoles. So now the spins can be up and down and they are done denoted by plus and 1 and the variable that comes is sigma i and sigma i can be plus 1 and minus 1 if it is plus 1 is up and minus 1 is down and what a stop telling in a minute ago that why this is chosen essentially because you have a field in mind and cos theta and so up is the field is aligned on the top that is just a moment lecture you can do it any direction you want but up and down cannot change in ising model so it is cos theta and cos pi so plus 1 and minus 1 that is the logic of these two things then number 3 an important very very important thing that I have only nearest neighbor interaction only nearest neighbor interaction that means these spin here can interact only with this one and that one. So in a finite spin when we want to finite change we do not want to do in order to reduce the effect of the boundary we impose what is called a periodic boundary condition that means we form a ring so after that it becomes this one this periodic boundary condition which is used in many many areas of the schematic computer simulations that not a part of the assumption the part of the assumption of these three the spins can be only up and down and so you do by plus and minus 1 and the only nearest neighbor interaction is taking it out so given that you would imagine that is a trivial thing to solve but it turned out no it is highly non-trivial nobody has been able to solve three-dimensionalizing model they are at one time they are a sporadic claim so three-dimensionalizing model but has not been able to do nobody the previous people have tried nobody has been able to do and the two-dimensionalizing is extremely difficult and still called the most difficult calculation in the history of physics and that was done by Lars von Sager so the physical chemist very interesting now so we go now write down an Hamiltonian what called Ising Hamiltonian that we wrote down the other the H equal to okay let me write down the A1 then I general one so the I know spend some time on this so this is is a generalized version of Ising model where the interaction depends on the location that means this interaction can be different from this interaction so if I call that one two three four five six then one two can be different from two three and this plays a very important role later but not we don't need it now so we consider them to be constant and take it out so J okay so this done such that when these are both up or both down then I get plus for both of them and then it becomes minus and that is favored and that is called ferromagnetic interaction when parallel spins are favored similarly negative one but however is a plus and minus like in this case or in this case you have one plus one minus so that becomes minus one and this becomes plus so that is not done and now this is the magnetic field so magnetic field now if I again a matter of convention these are all matter of convention but they are perfectly generally you can change convention if you don't like this convention this also will not change so this is the magnetic field that is acting on it which absorbs in it to combine the dipole moment or magnetic moment of this beam and the external electric field that means a magnetic field is H and then magnetic moment is mu then B is H mu so that is combined in that the effect of the external field is important of course you can have statistical mechanics in the absence of the external field also which will do but this is the major thing to know and this is the ising Hamiltonian this is a general Hamiltonian valid for any dimension you can have two dimension you can again have spin up and down exactly same thing n square lattice the number of nearest neighbor here is 2 it becomes 12 if you go to triangular lattice then number of nearest neighbor becomes 6 then you can go to FCC or BCC lattices and which are just a historical comment that when you go to three dimension you cannot solve it radically and you do fall back on a high temperature expansion which is very similar to me yes it is really interesting and huge number of people building 60s and 70s used to do these high temperature expansions of the three dimensional spin very very famous scientist don't worry okay so we have now defined the ising Hamiltonian and now I spent a little time telling you why such a simple Hamiltonian it might not look so simple to you because if you are not used to kind of abstract things this is sum over i and this is sum over given one configuration this is so I have 1 2 1 3 2 3 2 just like we did okay so this ising Hamiltonian I will now spend a little time before I go to calculations to tell you what are the basic merits and certain things of this you know what I have in notes is much better than written but it is better than that but let's see this will do little bit of the work that I intend to do there too much too much needed here okay so these are lattice models so some of them are radical solution as possible okay so this is the spin that I am talking h n of n spins and this is the Hamiltonian the ising Hamiltonian and this is the you can understand now what is important thing to realize that very important to realize there are n spins now at any given instant or any given configuration these spins are all having configuration which could be which is different from another configuration but each configuration comes with an energy so a given configuration this Hamiltonian is for a given configuration okay for example let me consider this one I will pick up a minus from here let's see my integral is 0 I don't need it now I would not need it for some time now now I will pick up a positive contribution from here another positive another for some reason everything positive here the person would do it is all anti-parametric kind of interaction okay but if it were up then as I said so a given configuration a given configuration comes with a given energy which is exactly same we have in gas phase that where at any given configuration a number of particles are interacting with each other and that configuration has an energy and Boltzmann told us taught us that the weight of that configuration will be given by the energy of that configuration and this is very essence of the statistical mechanics how we built the thinking in statistical mechanics okay so again the Hamiltonian given here one says what is ferromagnetic or the anti-ferromagnetic all these things okay now how do you do the partition function now in partition function these Hamiltonian has to go in the numerator of the exponential right minus this there in front of minus beta H that minus takes care of the minus and minus here I get positive now the difference between the my earlier thing is that I have to add over all the configurations so before so the sum over spins goes here in the exponential but this is weight of a given configuration but I have to sum over all possible configurations so this is the same as in my classical or continuum statistical mechanics the integrations over positions is the same as the sum over spins there I take a configuration particles are in the position R 1, R 2, R 3, R 4, R n and then I get an Hamiltonian for that then what I do I integrate over all possible positions of the first particle second particle third particle d R 1, d R 2, d R n which is exactly same thing we are doing here we are taking over all sum over all possible spins but now you understand that it creates enormous number of configurations because these spins though only can be up and down it is a if n number of spins are there they are 2 to the power n and if n is large number like abogato number then you have a huge number of configurations there so the entropic part enters through all possible combinations and they are into a okay so now the partition function is this thing this we need not worry right now okay so okay now I want to there are going back so the isic models are explains for what is of is particularly for isic models solids polymers micelles reverse micelles this simple thing is used rather extensively and also the isic model that we do is used for gas liquid transition even condensation you can have a nearest theory version of the isic model or isic model version of the isic theory and this is the one was part of the thesis okay now what I am going to do I am not going to do this difficult thing I will do something little simple and because it will become it could become very difficult and I think I have explained that this particular spin comes with an energy that energy goes to Boltzmann factor in exponential and that gives the contribution to partition function of this particular configuration this is not good I think that is the one we discarded right yeah now I am going to consider only this part I am neglecting for simplicity now because I will be able to do something now delete it so I now have a partition function n is a function of p n number of spins at temperature 3 sigma 1 sigma 2 sigma 3 sigma n for the spins then 2 to the power minus beta j sigma 1 sigma 2 so this is the partition function I need to evaluate with the condition that sigma 1 sigma 2 are they can take up and down should be plus where plus beta so now you notice the one thing that all these are coming as product so I can introduce a composite variable sigma j c equal to sigma j minus 1 sigma now let me see what are these composite variables will take now composite variables will be when this is plus this is plus then minus minus again plus it is a plus minus minus so composite variable while each of them have two variables composite variable has four variables okay so these quantity now I can now write in terms of composite variables which is now I can now I can now calculate these contribution of these of course I have to do when j greater than equal to 2 because I have 1 2 2 3 like that so j goes from 2 so in this case j has to go from 2 to particle okay now I have to do this composite variable so now I write this as bring I bring the product outside and I take the sum inside this is my composite variable now sigma j c okay then e to the power beta j sigma j c all right now I can take each 1 j and let me calculate the composite variable so now that will be e to the power beta j sigma composite variable c and sum over say j j speed then sigma j c has four values so that will be e to the power plus beta j plus minus minus e to the power minus beta j I have plus minus e to the power minus minus plus plus both minus then I have plus minus and minus plus so that would now be 2 into e to the power beta j beginning to take the character of insulator ising model so this is a what I have done actually is equivalent to a factorization I have factorized I could factorize because of this this form of what I deleted the form of that thing so I have done a factorization just the factorization I do but that factorization suddenly gave me a wonderful form which now does not depend on j just the beauty then what I can do now I can make this product here so now I can replace now these by so many times the product n minus 1 but n is very large so I do not really care of that so I have evaluated the partition function in the absence of a field of the ising model let me do some work with that the way I always did was like this this way I always decompose factorize that is the way I learned but in the book that we have written is not like that and that is the say like now we have to really correct because this is the way one should go but most of the books jump into probably Carson Wang has this way of doing things because I remember that this is in many places the composite variable but I am unfortunately most of the places I looked into the start with the very complicated transfer matrix all these things which is at least for it can ask for this concept and now people is not a waste way to start these things okay now there is some extremely important thing coming out from there now that is then I am just neglecting n is very large so I am putting that equal to n I hope I am doing all this right now what is this quantity e to the power beta j plus e to the minus beta j so now again 2 to the power n so then KBG are area so this term I can want to write take it out so plus this second so these two cancels right these two cancels so I am left with partition function now so in the absence of a magnetic field I have obtained the free energy I am obtained the partition function of the free energy now there is something very important here you know this continues to be in the principle magnetic field the important part here is that these there are certain functions in mathematics which are called entire functions exponential is an entire function cause is an entire function and what is an entire function that has infinite number of derivatives and is continuous all through that function is called an entire function so these cause is an entire function so it cannot show any discontinuity if it does not show any discontinuity it cannot show any phase transition and these one of the primary result of the theory of phase transition is that you cannot have a phase transition in one dimension so this is also called a beautiful theorem Ascot-Merbin theorem but we no need to go into Ascot-Merbin theorem at all at least not in this kind of rather preliminary things that we are doing but it is important to know that these cannot show any singularity so it cannot describe phase transition I can calculate entropy I can get a DA dt of that and get the entropy right and get the specific heat and then I can calculate the fluctuation in energy specific heat and then I can find those two are the same all these things one can do and people have done because this is the one model which has been bitten to death but important result here is that because of these kind of beautiful expression nice neat analytical expression that we do have a completely analytical function so there was this this gave rise to a lot of logic a lot of logic and a lot of controversy that a little bit of history that why the how statistical mechanics can at all describe phase transition that means since Boltzmann is exponential and exponential is an entire function differentiable completely to all order how come then a statistical mechanics which is in the form of the sum of the exponentials as we have seen here sum of the exponentials can give rise to a singularity that was a huge debate in you know for more than 2-3 decades which was ultimately solved in a specific case by Onsager but in a very general case a beautiful by CN Young at DD Lee called Young Lee theory Lee theory two very famous physicists who also did quark and parity violation and they went on to get the Nobel Prize for that but there another beautiful work was showing how and we use the Ising model and use the mayors theory actually more mayors theory than Ising model to show that how singularity can evolve by doing a beautiful piece of complex analysis that we can do sometime okay so this is with the in the absence of a magnetic field in the presence of a magnetic field is a little bit difficult so I will not do it in the class but I will go through that little bit and then I will give you the final result is not too important that we do it for the major way course of the thing those of you are interested you can do it and if you get stuck you can come to me and I will do it.