 We derived Pb equal to RT, we derived the Cb equal to 3 by 2 R all these things. Then we also did the diatomic. In monatomic we did the entropy which is the circuit tectonic equation which is an extremely useful as I told you of when we did it we thought it is useful at all. But I told you that we are using it every day like in paper we are writing now on entropy of water we use this entropy of circuit tectonic equation that expression we use. Amazing that how that seemingly in of course simple expression can be used in a very complex situations that is not. Then we did harmonic oscillator and we got an expression for entropy of oscillation and that again goes over to the entropy of the specific heat of solids what I am saying and divide it. Same expression you know I am just trying to tell you the reach that you do want tiny little thing through statistical mechanics. There is no other way you can do that this Einstein or Debye Einstein theory or specific heat comes through this partition function that is what I instant it. He just calculated the specific heat through V to the power minus half h mu by kBT by 1 minus e to the power minus h mu by kT square right that is and you have a density of states d omega then you integrate that okay. Then we did entropy and specific heat and specific heat of that is what we Einstein entropy of the vibrational modes also very important. So when I calculate entropy of solid or free energy of solid how do I do it theoretically. I calculate the enthalpy which I can do through computer simulations or I can some kind of Madeline constant remember the sum coming over all the interactions if they are charged and computer simulation we can do that I take an FCC lattice and I calculate the interaction energy by adding up all the interactions. But then I need the entropy you would think that entropy of solid is very small not quite entropy of solid may be significant because of the low frequency of oscillations and vibrations in many solids. Then you again use your entropy from the harmonic oscillators your normal modes are your harmonic oscillators okay. That is the first thing you would solid state physics you know like normal modes in a linear approximation. These things go very well in linear response theory that is very very important to understand. So free energy of solid if you want to calculate we just published a paper very long paper 10 years worth of work with my Japanese collaborators Shinji Saito and Yawominem and where we tried to get the super cool liquid and glass the free energy of the thing and specific heat and entropy and we use this exactly the expression should be right here the entropy we remove the kinetic energy then we diagonalize if the disorder thing we get what is called quench normal modes and we then use those normal modes to calculate the entropy and the specific heat and that was came out I think earlier this year or end of last year probably earlier this year long paper. Now many many people are doing that so the specific heat and entropy of harmonic oscillator that you will learn in the here is used in practical applications in many many cases then comes the rigid rotator with the rigid rotator and we assume a rigid rotator because we assume the rotation and vibration are uncoupled which we know is not correct because we in undergraduate physical chemistry when you do spectroscopy infrared spectroscopy you get PQ at those branches which are because vibration and rotation talk with each other corollaries coupling or centrifugal thing that it rotates very fast then the bond gets stretched or compressed this the rotation vibration rotation coupling is a wonderful stuff tells you lot of thing most importantly it tells you about the anonymity of vibration and that is a very very important quantity for our understanding a bond breaking and the activation barrier all these are very neatly and very succinctly coupled and is a unified concept that we use one side spectroscopy and other side statistical mechanics and other side quantum chemistry and quantum mechanics rigid rotator entropy and specific heat we do not have a simple analytical expression unless we make low and high temperature assumptions when you do that we get you know expressions and then we when you do the calculations you find if a molecule like water the entropy from rotation about 30% I show you the table that of the okay but all these things that we did I am just summarizing and reminding you because they are very important things where non-interacting limit I have a rigid rotator I have a diatomic molecule I have a monatomic molecule but they are interacting within themselves there are intramolecular forces but there are no intermolecular force these molecules did not interact however you want a water water or your human body unless molecules interact with each other you would not have a glass you would not have a crystal everywhere molecules interact that is where I quoted Paul you was fond of saying that I created contest matter in the means you know the repulsion were not there it would not have formed crystal the solid state the hard sphere crystal the simple potential is the role our model of studying solid state or liquid solid transition so now how do I go about doing interactions I know the partition function I know the partition function is sum over e to the power e i by k b t I will today I will give a flow chart that is what I am talking so much and then we will do the calculations which are a little formidable so one of the will be in a in rigid rotator we found the degeneracy factor remember 2j plus 1 that was in rigid rotator you have done in quantum mechanics harmonic oscillator the atomic this gi equal to 1 particle in a box gi equal to 1 right other way to do is classical this indistinguishability Boltzmann put by hand and this is the one that we know after doing quantum the best way to think about it to get this factor is to do the quantum and then come to classical from quantum which is an exact way of doing and you find that you have much of that this is other way you can have a conceptual saying okay this is normalization of the volume of a cell but that kind of thing I do not particularly encourage I would rather say do the quantum and then come to classical so and the Hamiltonian this is the total Hamiltonian kinetic energy and potential energy okay and this quantity u is the many times we do not write all these things we just I get here and we put a prime here that is the that make people standard notation we just do that then you do not have to write all this it is understood so we are further assuming that interaction there is a two-body interaction means we are saying two part that if I have a three particles 1 2 3 then this interaction is some of PROS interaction that could be a three particle interaction also which are important in some cases but understanding much of condensed matter physics we do not need this three-body interaction that is a fine tuning we need sometimes but like doing a liquid silicon we need the three-body interaction but you know much of the time that because when you are kind of near metal or metal white kind of things then three-body and many four-body interactions play important role. So now this is popular saying that the bull by the horn that means you have to do these complex things these integration where in a classical thing we can do the momentum integration because uncoupled is a bunch of Gaussians I do not need to write that I will be cavalier about it because I know how to do it so basically this is the thing I need to do so again if I need I will put them back I do not care about them so I will write Zn by n factorial I will not write the other thing 1 over n factorial so this is the configuration integral I have taken out the momentum path you understand that I have integration of beta H beta H Hamiltonian has kinetic energy and potential energy they are uncoupled the potential energy part kinetic energy part pi square by 2m they are nothing but Gaussian and not only that i and j are uncoupled x y z are also uncoupled so I can do the 3n integrals and then I get lambda to the lambda is the de Broglie wave then 3n by 2 that is the partition of that I am not just not writing because that is the part which we have to circuit to the equation and pb equal to nk but ideal gas but right now I am not interesting that many times we when you do the free energy we write free energy a equal to a ideal plus a excess and a ideal because this is free is a ideal part comes as a product here correct you understand that because it is a minus beta H and is the exponential so that become product then I take the log free energy log of partition function and then the product comes out separately that I call ideal a ideal is the ideal gas free energy which we did and this is the excess part and this is the interesting part this is the part we are now trying to do so this part comes from the interactions of particles and which is highly non trivial I will go back and forth I will use this little bit then I will go back because I want to give a flow chart I want to tell you how the thing is done so that you get the big picture and then doing the equations is filling this big picture but going directly to the details does not help if you do not get the big picture because this is the big picture that will stay with you the understanding the details will be remain in books which are always look up okay that is the way actually to do any interesting stuff okay so we did this here this this and this interaction potential we discussed at hand and this is the pair wise interaction potential is the total potential energy and as I just discussed here this is the quantity and I said I do not care about this part because this part is the ideal gas part so when I take the log of that free energy I get and then I get everything from the energy the reason you work in canonical ensemble is because I get entropy and pressure by taking derivatives okay it is far more complicated in getting a function and doing of course a very difficult because you know way to go this particular box but then in other systems with continuous potential you have no way to go when you go to micro canonical ensemble okay so this is the thing then so as I said before that this comes in a exponent u and u is a sum so I put that sum here then when I put this sum here this is like that then I realize that this quantity is nothing but product the sum here the sum comes out as a product okay and then I looked at this integral and I had to do this integral in a very very difficult way because two things this integral the first this potential is fairly complex and to make it worse this potential goes to zero when separation between two particles i and j becomes larger which indeed it should be it should go to zero but that creates the problem is that I have to do this integral and if I do that integral then that integral goes to one you do not want that you do not want the asymptotic part goes to one you know you want that asymptotic part goes to zero okay and that is what may I did then the it saturates the unity and that becomes the hard so partition function fails to convert also difficult to do then introduce may I function so may I function is the following function if this is the so the quantity that comes is e to the power minus beta you are this quantity this quantity that is the one we are dealing with that the beast and why it is a beast because this if the simplest interaction potential is that so there are two part of the interaction potential one is this part which excludes particles volumes excludes certain regions of the configuration space to my particle because they are interacting and they are harshly policy but short distance then however other than that it has a entire volume to itself that part comes from here okay so he then consider this he said okay let me introduce this function a and that then is like that his own sole idea was to make it go to zero so it now it goes to zero all right and this is my it will be sometime we call it exponential bond because to a chemist these are essentially bonds they are interacting they are not chemical bonds but we will call some them call them physical bonds but this has become minus one now now the advantage is that if I put this into there I will get this this kind of terms now it has becomes I can decompose it into smaller parts that is important thing before I could not do anything like I can do something okay so then as we discussed the other day this is the decomposition I did with three particles then the first one is I put it here sum over one term is the first term is the one second term is a two particle term next term is a three particle term so one part is just a dot go line then the second is the bond FIH is a bond that connects these two particles then this this has the three kind of then forth like that total number of particles in the system is so ML now we say okay ML is the number total number of molecules in a cluster of size well so cluster of size two how many total number of molecules two here three so now however so instantaneously at a given position like in this in this room I have oxygen and nitrogen and say I have n number of them and let us consider all the nitrogen so I have n number of nitrogen and now I any times they they are mostly dispersed in the room but now and then they come together and they form clusters they are clusters of diatoms they are clusters of triatomic there may be little bit more of that okay much of much of the time I can describe this room by the function F12 at most they are diatomics you know but you understand that whenever they are one is coming under this sphere of interaction of the other through interaction potential I draw a bond but these bonds are flickering bonds they are breaking and forming because as molecules coming together and going away okay and but the instantaneous state of nitrogen in this room is given by the ML you know and as as we are talking and if the n number of nitrogen molecule then M1 is 90% of that okay and then 10 8 of the 10% another 90% are this one binary interaction and this small number will be turner interaction so you immediately see if I want to describe the nitrogen in this room I only need the turner term that much I can I can see beginning to see an approximation at the low density that I can do with ideal gas plus a correction and that correction is the F12 now I go to denser gas I go to nitrogen little bit denser and then to liquid then I need all the terms but I have a systematic way to add them this scheme will not work all the way but it will take us far and then a different theory takes over okay but it is it is the beginning now we define the cluster integrals called mirror cluster integral which are in a given class a class is characterized by L we are not making a distinction between these and these now we can then consider all these clusters now what are the clusters just in Feynman path integral anywhere these these are nothing but this general language of physics we started 1937 Mayer's paper these are nothing but integrals so when you say graphs in physics they are integrals okay and then these most powerful language in many body physics in chemistry we use it in theory of liquids even the these these people the quantum chemists they do this t operator expansion all these the different things essentially e to the way e to the word t and very similar things goes on there so there are certain universality and understanding all these things you understand one you basically know why one is doing this so this is then we can now calculate the okay but the dot first one is this quantity v1 v2 is this quantity and v3 I sum over everything remember that these are the my these these three these three and last one the formidable one is this one I can go on writing like that so these are the Mayer introduced and this is a cluster integrals it is called reducible cluster integrals because this can be further reduced into this part and this part okay this is called irreducible we will handle we will deal with there and this is reducible okay but this can be explained in terms of this one okay now next what may I did okay as I told you that we are going to divide and conquer we had full partition function we are reducing this total partition function into the cluster integrals then we reduce the cluster integrals even further called reducible cluster integrals then you will see under certain approximation the reducible cluster integrals are nothing but virial coefficients suddenly you see oh I can now have other sum limit this beautiful thing is rather complicated theory of though beautiful we can connect it to experiments and you will see that that that that's really very very nice okay now what we did we have to now find out the what is the decomposition decomposition I want to express the partition function in terms of this reducible cluster integrals how do I do I know ml is the number of clusters of size l and then I say okay and this is my definition remember that this is kind of a normalization factor put in here l factorial because the number of ways I can arrange two factorials will come one is l factorial and this ml factorial and as I was saying this is the kind of things looked into from the multinomial theories or the way or you can arrange the particles in a box and you name the box one two three four like that and this is my l the number written on my box is l so when I have two particle cluster I put it l equal to two I have a three particle cluster I put it l equal to three now each of these l equal to three I have the three particle clusters and I number of three particle cluster in my box level three is m3 okay and this is the way one does exactly this problem is done in combinatorics okay when you do that this is what is the configuration integral of that this is just this thing because ml of them okay and this is the weight so this what is trivial right okay little less trivial is total number of ways we can combine them I have n of them into groups of l and this is the distributed in unnumbered piles of l object is this is the quantity so the total partition function is the product of the two when you do the products there are certain cancellation l factorial get cancelled ml factorial survive and VBL to the power ml and this is the beautiful expression is the called it is exact used also in quantum it is called mere r cell cluster expansion so this is an expression of the partition function it has certain advantage certain disadvantages but whatever it is it is exact so this is called decomposition is so beautiful let me write it down okay so this is the quantity I wrote down and the way to work with this exact partition function is to consider what you call thermodynamic limit we have to take n going to infinity V going to infinity such that n by V rho equal to constant all right that the limit we will take we will ignore this for the time being we will come back okay now something very interesting how did mere go further and derive two very very important thing which we call cluster expansions one is the density in terms of LBLZL I will come back but I just want to mention this two we can derive it from non canonical partition function also another thing is the pressure in terms of BLZL I am giving you and then come back and so these things that will derive now so look at the beauty density is LBLZ where D is the fugacity which is done just like beta in canonical ensemble beta become 1 over KBT how how did beta enter into the into the description in canonical partition function yeah absolutely Lagrangian multiplier here also what is the constraint this is the constraint and that constraint enters through Lagrangian multiplier now beta enters through which constraint energy exactly so and temporary energy are conjugate quantities right so here what is the conjugate of them absolutely and then this is e to the volume by KBT so fugacity is just beautiful these things are so I have an expansion I will derive in a minute density in terms of cluster integrals and now I have little bit more work I get now pressure another equation I will derive is this I told you I will give you a flow chart so I have an expansion these are called cluster expansions they can be derived also through a in canonical partition function but that that does not have the physical picture it has ok mass theory now I give you two expansions one is density in terms of fugacity then I give you another pressure equation of state now if I give you two series and I tell you to eliminate this you have done in your class 11 eliminate fugacity give me an expansion of pressure in terms of density ok I give you everything I give you all the coefficients I have not told you how you are going to calculate them I will tell you that but I eliminate I can eliminate like I have given two series one series is density in terms of fugacity other series pressure in terms of fugacity now I tell you eliminate fugacity give me a series of pressure in terms of density this is possible so long the both these series are convergent right when you do that you get the variable series that is the thing I kind of jumping and telling you the what is the flow chart now we have we start with the partition function we are getting a decomposition which is the cluster decomposition and exact partition function in terms so we form earlier on you go to cluster integrals then I get a partition function then I get two expansions which I eliminate and get the fugacity that allows me now to get the coefficients of cluster integrals and we will do that the way one works out with this equation then is that we will do again the same game as we discussed we will take a maximum term method there is a principle of statistical mechanics it turns out that when you have the see ml is a distribution of distributions please try to understand the distribution of distribution why as I said my nitrogen example there is any time ml the number of particles of number of number of cluster of size l that is a distribution however as this continuously changing there is a distribution of distribution okay now however there is one distribution which maximizes the partition function and that is called maximum term method is a very already very well established method improbability maximum term method so we have to find the maximum that particular set of ml I call it ml star which maximizes the partition function and one can now show by mathematical analysis in the limit n going to infinity the maximum term method dominates so there is a distribution that is the reason a system is stable that is the reason we free engine is minimum it is all tied together so now you can easily do a maximum term method there is a huge amount in wikipedia and all these things in probability theory because it is used in probability theory yes yeah exactly very good question excellent question we implement maximum term method to lagrangian multiplier whenever we are doing maximizing or minimizing something subject to a constraint so the maximum term method is general method which you used even before but lagrangian multiplier is the way to implement that whenever you are finding a extremum of something with a constraint extremum means you d d alpha that quantity is zero but you want to do it with a constraint if you do not have the constraint it will give you everything the constraint is down through lagrangian multiplier and that introduces one undetermined coefficient which you appeal to physics or experimental or our physical a and get what is the undetermined multiplier like we found out two thermodynamics beta is 1 over kbd okay so this is what i have been talking of the flow charge that there is a grand scheme that is going on and we need to appreciate the grand scheme that how the whole thing is unfolding