 Welcome back to the final class on statistical mechanics. This will be a summary lecture where you would like to take the student to the subjects covered in this rather long course on statistical mechanics. So in this last summary lecture, I would like to remind the students basic objectives of the course and certain important points. If you look back, we started by pointing out the scope of statistical mechanics. There is a lecture called why study statistical mechanics, where I tried to point out the huge scope of the subject in physics, chemistry, biology and also material science, particularly in chemistry and biology is extremely important. It is becoming important material science also, the synthesis of new materials. Within this course, after the preliminaries, we tried to get going and to formulate the subject and I pointed out that there are actually two theoretical subject, two major theoretical discipline in our entire arsenal. One is quantum mechanics and there is statistical mechanics. Both use substantial amount of mathematics and based on physics of like Newton's laws of motion and sorting a equation, many things. So quantum mechanics provides you about the microscopic world, about the energy levels, about your function and very intimately connected to spectroscopy. On the other hand, statistical mechanics deals with many body. The phenomena, the properties that appear because of interactions of many molecules like a glass of water. The properties of glass of water is not property of one individual water molecules. It is a sum total of properties. One water molecule doesn't have the solution properties, doesn't have its flow or many, many other things that we know and we associate with water. So the key word is that interaction, interaction among these water molecules. It's not just water, we can have ethanol, we can dimethyl sulfoxide, can have argon, many, many of metals, liquid metals. So there's a huge number of these systems all around us, including our body, where the uniqueness of each system and the properties comes not from the constituents, but also from the interaction amongst interactions in term determined by the nature of course of the molecules and atoms. Given the interaction potential, the duty or responsibility of statistical mechanics is to explain the properties. How in water acquires these wonderful properties? Why it boils at 100 degree centigrade? Why it freezes into I say 0 degree centigrade? Why water and ethanol form such a wonderful mixture? Many, many such things, just innumerable innumerable things that are around us, which statistical mechanics set out to explain. In this huge work, one was to develop a substantial formalism which we discussed. We showed that how, said with this huge task, they are forefathers like Morichman and Gibbs. They formulated two postulates, postulate one and postulate two. One is equal to priori probability and the other is imansum leverage. Before that came the introduction of the concept of insumers that the huge ideas discussed at great length, that is billions and billions of mental replica of your original system so that we can talk of a priori distribution. But that works only when you assume that the each system is in a different microscopic state, which is equally likely. So it's the equally priori probability and however the averages that we in the real world, the time average is much of the time. So time average equal to ensemble average and that postulate is the second postulate. I also discussed how these two postulates need to be connected by hypothesis, that's what we are going to do. So armed with these two postulates and one hypothesis, statistical mechanics sets out to explain the first number of its observed phenomena. It's amazing that how such simple postulates and fairly innocuous postulates can lead you to explain why water freezes at zero degrees centigrade at ambient pressure. So in doing that, that is one first start with the ensemble to discuss the concept of ensembles and that everything starts with the micro-granite ensemble which is called NVE. Then there are four ensembles we discussed from micro-canonical to canonical, canonical to grand canonical and then isothermal isobaric ensemble. Micro-canonical is NVE, canonical, NVT, grand canonical is the fugacity volume, temperature and isothermal isobaric ensemble is NPT. So these as you clearly see these different ensembles are different purposes because often our experimental systems are different conditions. It is almost never in NVE because we cannot control the energy because our systems are always interacting with surrounding. We do not cannot fully control the volume though it's little more than energy and number also. Many times molecules are getting exchanged with surrounding media like a glass of water exchanging. So we need the different ensembles to treat different experimental conditions. Also there is a very fundamental theoretical reason that doing calculations or theory in micro-canonical ensemble is rather difficult. So then we set out with these three four different ensembles we set out to calculate thermodynamic properties like entropy, specific heat, free energy, average internal energy, radial distribution function, many many other properties. But all of them go through one route that is the partition function. The partition function is essentially weight of that particular state of the system like particular state is given by say NVE. Then partition function give me what is the weight of that and that weight has to be extremized or maximized much of the time. Logarithmic of the partition function defines what is called the thermodynamic potential discussed in great detail like logarithmic of the canonical partition function is the free energy logarithm of the micro-granite partition function is the entropy. And this is then is the first thing what's describing thermodynamics from statistical mechanics which is one of the major goal. We then went after formulation of the expression for the partition function. We did several wonderful calculations. We calculated entropy, circuit-tector equation for entropy. We did the diatomic gas and vibrational rotational entropy. These are for ideal gas where molecules don't interact with each other. But just they are many of them together. Even in non-interactingly we did some wonderful properties like the circuit-tector equation, vibrational rotational term which can actually be experimentally verified in many cases not in a completely but you know 90 percent or so. In dilute gas for example is a good model of the ideal gas you know certain certain approximations. Then we did interacting systems. We did Ising model which is a system of spins and we did the Mayer's theory of condensation and from Mayer's theory it's wonderful and fairly sophisticated theory to tell you how in the presence of intermolecular interaction you go around calculating the partition function, calculating the pressure and the equation of state. And ultimately we show how from Mayer's theory you get the well-known virial series which was derived long ago. The virial theory was in 19th century, mid 19th century. Second virial coefficient, third virial coefficient all those things were done and measured long time ago. But now Mayer's theory when it was done I discussed that the expression of virial, second virial coefficient and third virial coefficient in terms of intermolecular potential allows us now to use the experimental result to construct the intermolecular potential. And this is actually some kind of a fitting to experimental data but you at the end you find a very important quantity which is is not just which was introduced from imagination that two molecules interact but we had no way to know what is the interaction potential between them. But this Mayer's theory and the expressions for virial coefficients allowed us to map and get the intermolecular potential. This is a big game now which is called the force field. So this virial coefficient first time for example that two two argon atoms interact through something like linear joules potential that came out and the parameters of the linear joules potential came out from this theoretical expression derived by Mayer for the virial coefficients. So wonderful really wonderful. That we spend a lot of time on phase transition and two different solid phases form how condensation takes place. So Mayer's theory told us how to how intermolecular interactions to be included. But that that's a very difficult theory which was further developed later through classics for other things. But there is a simpler and more direct approach to understand many aspects of phase transition, universality of phase transition without going through intermolecular potential that's called Landau's theory of phase transition. So we discussed quite a bit of time of Landau's theory which is the free energy expansion. Free energy expansion in terms of the outer parameter we introduced the concept of outer parameter, free energy as a function of outer parameter different phases of different free energy and what is the basic structure. So then we after we studied the phase transition or thermodynamics of phase transition like the discontinuity at the phase transition. We discussed first order phase transition, we discussed second order phase transition that is called air interest classification and how Landau's theory kind of unifies these different things into a beautiful beautiful scheme based on free energy. We then subsequently studied kinetics of phase transition, we studied nucleation, snow decomposition. These are wonderful subjects and they are very very useful because nucleation is something which you see in end drops falling, ice forming all around us or now that the ice melting. So nucleation is very important things, normal decomposition on the other end is a theory of pattern formation, how beautiful patterns form in nature and this is also a part of kinetics of phase transition. Then we went on and we studied binary mixtures. Binary mixtures is a wonderful subject where we studied in undergraduate physical chemistry in great detail and also biophysical chemistry that how binary mixtures, two mixtures, two liquids and we mixed they acquire new properties and that property is extremely useful in the chemical industry and also in biology. So one needed to understand in terms of intermolecular interactions how these unique properties and the composition dependent of the properties or binary mixtures how much air, how much V we mix to get this property. That also is a prescription given by statistical mechanics and a lot of work has been done along that line and we discussed that. Then we went into to describe the three lectures we found on theory of liquids, radial distribution functions and the extremely important function called GR which describes many many properties of the liquid. Then we covered in detail again three lectures on polymer solutions, polymer thermodynamics, Flory's theory, Sol's yield transition. We discussed the Flory Huggins theory. We discussed how polymer collapse takes place. We discussed how why polymers in solvage would solvent such as solene and so the interaction effective interaction between monomers connected through the solvent molecules. We discussed polymer in great detail. Finally in two lectures we discussed computer simulations which is extremely important because you could not do computer many of the complex interactions. We cannot do statistical mechanics and these are very important subjects that we dealt with with the so many so many different we went through huge number of subjects and we did quite a bit of theory, quite a bit of mathematics and however as I was saying when ideal molecules they are as I mentioned in the lecture interactions are quite complex so we cannot do them through analytical work. Even Meyer's theory you found how difficult it is pushed through Meyer's theory. So the field was suffering but at the same time in 1960s computers came in and the statistical mechanics again could go ahead because they could now treat those complex interactions using the principles of statistical mechanics by computer simulations, molecular dynamics and Monte Carlo simulation. They are all based on principles of statistical mechanics that one should not forget that but so computer simulation doing the people doing those kind of computer simulations are actually doing statistical mechanics but instead of doing an analytically pened paper and pened pencil they are doing it.