 Welcome everybody. The basic idea is no while statistical mechanics is very big in physics department, in chemistry department sits across India hardly taught. And my motivation came from once I was visiting the central college in Bangalore University and I was going through the syllabus. I was giving some valedictory lecture. Then I went to the syllabus. And they have statistical mechanics irreversible found dynamics. So I said what? But they say they don't have anybody to teach it. And then I tried to talk with physical chemistry one quantum chemist there and they said you know that they have never been taught and it is difficult for them to pick it up. So then it occurred to me and I noticed one thing in undergraduate colleges that Indian students not that many IITs and other places but across the board in the state universities and even JNU they like to read books written by Indian authors. That is the very important observation. So I said okay we don't have a single stat-mark book in neither by physics nor by chemists really in the sense of a textbook. So I already had written two books and this was a Herculean task to get this done and involve a lot of help from the students and it took a very long time. So this is the standard book that we read all read. The first edition now it is the I think second or third edition and this is the statistical mechanics by David Chandler. So but this book is somewhat different in the certain sense that I will go through though the course will be kind of a mixture and not from any one book. And it I would like to have put an elementary here but I thought that will lessen the appeal because we will do many things which are not elementary okay. And I will go on doing it for some time and I think those who are present here will enjoy it and it will take you through many things which usually you don't think about. Now let me start by saying the following thing that when you go through the take out the textbook you know when you go back you can take the Moore, Walter Moore or Castellan or Glaston or even Atkins. Atkins of late has done some amount of more modernization but if you take the undergraduate textbook you will see there is a electrochemistry, there will be binary mixtures and the phase eutectic diagram. You have liquid molecule liquids like water ethanol your polymers and then you have the liquid solid transitions huge number of things that you study in undergraduate physical chemistry almost except spectroscopy and atomic structure all the chapters essentially comes as a phenomenology. For example take the ionic motion in water you have read and about the Solvenberg model that means the water forms an ice like structure and that reduces the mobility of lithium ion. Again and again that kind of a structure or structure breaking and structure making when you do binary mixtures we say okay this is structure breaking and this structure breaking like water ethanol or water DMSO. And they have depending on the whether they are structure breaking or structure making you have a composition dependence of viscosity which have this curve going up with composition going down. I am spending so much time on this essentially to bring home the point that what we read is essentially statement of facts they are observations and they are hardly any explanations there. But that does not help you you can memorize the fact and you will forget what stays with you is you have certain amount of understanding because then you can use it later. This is one that I will come back this particular slide that because we hope to do quite a bit of that earlier part we will do non interacting systems then we do interacting systems. Ultimately we want to touch on this very important thing for a chemist particularly in modern days of nanomaterial synthesis and understanding you know you cannot give a really meaningful lecture without knowing certain amount of see I hear lecture after lecture and in my because the solution and structure chemistry. And basically they know little bit of Ostwald step rule Ostwald ripening but they do cannot explain many faculty member a prospective faculty members do not get selected simply because they cannot answer very simple questions. So you know we look for faculty we look for students who can explain things okay. Now I will do something little bit which is very important and profoundly important see when you do a quantum mechanics. We start with black body radiation and planks introduction of quantum and quantum theory. Then you go to Einstein for electric effect then we go to Niels Bohr's theory of atomic structure electrons stationary orbits. Then comes the very important the broadly wave particle duality. Then it was Peter Debye who said there is stupid to talk of silly actually talk of wave equation wave particle duality or waves without an wave equation which presumably prompted Schrodinger to do his work. Then of course Heisenberg and all these things and Dirac. So these evolution of quantum mechanics is a folklore. Most of us knew it when you are doing BSc and by MSU is certainly new. Such does not exist in the historical evolution of statistical mechanics though it is very well known and you all know quite a bit of that how did it start. So I will do a little bit of the historical evolution of statistical mechanics which will allow you to think and put the things in the perspective okay. So it is all started around 1850 there are several people though Maxwell who gave the clearest exposition they were thinking of the distribution of velocity of a gas particles. The very idea that people were thinking of distribution was quite unusual because that time it was of deterministic which you again know from quantum mechanics. Everything was kind of deterministic perspective that people were doing. Maxwell I am not going to go into very detail because I will just want to give you the history now. Maxwell came up with his famous Maxwell velocity distribution which is the Gaussian distribution. So now suddenly people like Maxwell proposed that you may not know. You might not know the velocity of an individual particle. It does not make any sense anymore because when there are so many particles moving around instead of that we should be able to talk of a velocity distribution. And this is the normalization constant. So this distribution we all know is the Gaussian distribution. There is this path velocity if it is vx and then positive and negative and this is the pv. Now that impressed Boltzmann so much that all his life rest of his life he carried that paper of Maxwell with him. So he was obsessed and he realized that this distribution is a very fundamental way can explain many things in nature starting from very corpuscular. It was called corpuscular view of matter. And so Boltzmann then went on trying to develop theories and Boltzmann suffered enormous amount of criticism. Maxwell died early. So this distribution related criticism was he did not quite suffer as much as Boltzmann suffered. To the extent he probably committed suicide and that is the main certain mystery there in 1906. Another very important thing happened around that time which all of you know is the Van der Waals equation. Why it is so significant I am now trying to tell you. And I am again I am repeating that I am giving you right now the history how statistical mechanics was developed. And very important to establish the history and the chronology of evolution. Van der Waals came up with that equation his famous equation and which is you know p plus a by v square v minus n b equal to RT. Now what was interesting and very interesting was that Maxwell and Boltzmann they already derived the equation from a molecular view a particle view the equation pv equal to RT. The universal gas law. You know if you look at that that was a very significant we cannot over emphasize the importance of that. The way they derived that because Maxwell that a particle though Maxwell is the whole kinetic theory of gases you do not see any name. Because we are all done by Maxwell maybe little bit by Boltzmann. Similarly when you do quantum mechanics all the way up to hydrogen molecule you do not hear any name. Because you are all done by Schrodinger everything was done by Schrodinger particle in a box harmonic oscillator rigid rotator hydrogen atom hydrogen molecule. You know I had that collection of Schrodinger's 7 papers you know 1.5 rupee picked up on a college street second hand bookstore. And I was just impressed that here is a guy and he did not do the way you do quantum mechanics. Very important to know that very rise and loss in the first edition did the way Schrodinger derived Schrodinger equation. Not the operator thing that you replace del square by d2 dx. No that is not the way it is. It is very very important that you know the history and the chronology because that gives you much much deeper insight. Van der Waals had those terms pressure P but Van der Waals introduced one more thing which he did not have. He had Maxwell did not have Boltzmann had ledger and I will come to that the size of the molecule. Remember this one more everything. So if I do not have if I take V going to infinity then I get PV equal to RT. So Van der Waals introduced the interaction which is A and this size which is B. Yes it is some approximate way and we all know the criticism of Van der Waals equation. But this equation was a kind of a thermodynamic equation. But it was u it used Maxwell's essentially method that the particle is going to hit the wall and it is drawn back the way we all know the derivation of Van der Waals. So interaction when atoms and molecules were beginning to come though classical mechanics and continuum people were quite a bit against and very critical of the Maxwell Boltzmann approach. Van der Waals was a thermodynamic equation. It was an equation of state which was verified in a very spectacular way by law of corresponding states. I will talk a lot about law of corresponding states. The beauty of that and the universality of law of corresponding states the whole critical phenomena originated from law of corresponding states. But that that again can wait. Now comes this Telofeginius Williard Gibbs. He realized something very very interesting. He was absolutely again obsessed with Van der Waals equation state and he set out to develop a statistical mechanical the approach. Now again a bit of history Maxwell did with ideal gas law. Boltzmann now tried to put in interaction between atoms and molecules. That there is a size and he assumed heart sphere. He did not have an ideal gas law does not need an interaction. But here there is a ballistic collision between two particles. So Boltzmann introduced this important important the collisional event that one molecule going and hitting another molecule. And then getting deflected. This scattering event or binary collision event was taken into account by Boltzmann. But Boltzmann could not go very far. What Boltzmann tried to do the evolve the distribution function of such event of binary collision. Quality distribution that one particle at one position has at position r1 as velocity v1. And another particle at position r2 as velocity v2. This joint quality distribution was a very formidable thing that Boltzmann tried to tackle. And he had to make certain big big approximation. One is that a molecular chaos. And when Boltzmann made he was heavily criticized. And remember the whole concept of entropy in a molecular definition A, C, Q, L and W that came from Boltzmann. Wherever there is a statue of Boltzmann I saw in the University of Vienna at Austria on his mast that famous equation is written from which so much flow. So much further was developed. So it is important to realize that Boltzmann tried to develop a time dependent approach to statistical mechanics. He had a molecular definition of entropy. He tried to go as far as he could. But he could not go very far. There are things are extremely complicated. On the other side of the Atlantic he was great fan of Maxwell and Maxwell was great fan of Willard Gibbs. And Willard Gibbs looked at very carefully the approach of Boltzmann. Then he realized one thing that even if I could solve what Boltzmann was trying to do. Boltzmann couldn't go beyond two particles. But we are in a liquid or in gas phase we will have many particles. They have a number. At least we need to discuss in a locally few hundred or thousand molecules interacting. So Boltzmann approach as Boltzmann tried was too ambitious it will not work. But what is the brilliant thing Willard Gibbs realize that if I have a glass of water. Then that glass of water and I put water into another glass that is standing there at equilibrium. The property with a density one number cc specific in our old unit one. All these properties are time independent. They are time independent. They are not time dependent. But Boltzmann tried with a q from Maxwell develop a time dependent. Which ultimately after a hundred years bore enormous fruit but that's much later. Willard Gibbs realized that then I should be able to develop an equilibrium approach. That means I don't have to solve the equation of motion. Boltzmann tried to solve Newton's equation couldn't do even now nobody has been able to solve the three body problem. And Gibbs realized that then I would be able to do something if I assume I create many many many many glasses of water. Millions in my mental in my mind frame. And then at a given instant all the water molecules in one in each glass are different positions. They are moving at different velocities. They are a little bit of different positions. So he realized if I now go back kind of Maxwellian way of thinking that if I can think of a distribution. An equilibrium distribution not the time dependent distribution which Boltzmann tried and couldn't do. I can do an equilibrium distribution of my millions and millions of glasses. Each of the glass in each glass the water molecules are occupied different positions. Then I should be able to then I'll be able to if I have a distribution of the locations and positions of the water molecules. I should be able to use Boltzmann distribution. Boltzmann distribution essentially e to the power minus beta e what is your Hamiltonian. So now he combined Maxwell Boltzmann and his brilliant mental construction. So we all know Boltzmann. So now he say if I have this millions of glasses and with water molecules millions of them. Each of them my at a given time my water molecules are located differently. Each of them different energy but that essentially obeys this kind of distribution. Then I'll be able to calculate the properties. So he avoided and is extremely important that we let gives avoided the construction of Boltzmann. He avoided by this brilliant which is called ensemble picture. He avoided the whole difficulty Boltzmann faced. And we let gives is considered along with Boltzmann the father of statistical mechanics. He is the one who created this ensembles picture and he is the one who created this micro canonical. Boltzmann did micro canonical canonical grand canonical all these things are all done by we let gives. And why we let gives did that? One thing you should really know when you are doing science that we are no different from Carpenters in certain sense. Science is a very pragmatic thing very practical thing. It is nothing to do with kind of philosophy. Science is science we are great teacher Shadhan Vasu who used to come like this with the patent disorganized and he was a wonderful teacher. Which that we had way of those days could video his thing and he was talking of free will and all these things in context of Boltz theory. And he was famous line we always to say that science is science philosophies philosophy and science is not philosophy. So we let gives really did something lot of philosophical discourse went into it but he had nothing no philosophy in mind. He just wanted to calculate. He just wanted to calculate he wanted to calculate this thing. He wanted to calculate van der Waals equation states and he wanted a formulation that starting with atoms and molecules starting with intermolecular interaction. Can I how do I go there? Not the heuristic derivation of van der Waals. And on top of that you know the van der Waals had this loop which again Maxwell corrected it called Maxwell tie line. Getting the correct pressure out of van der Waals. So these 3 guys or 4 guys are just hand in gloves they together created the whole statistical mechanics. Now so now I have said that it started Maxwell but I should emphasize again Maxwell was not the person not the only person. Some other people also had the Maxwell distribution but Maxwell gave a very clear exposition. Maxwell motivated Boltzmann and Boltzmann went to develop the solved Newton's equation for many body problem. They introduced the definition of entropy and Maxwell Boltzmann couldn't do beyond binary collision. Even that he couldn't do very well. Then Gibbs trying to develop and understand van der Waals equation and phase diagram and phase transition. Van der Waals is again the guy who did the first theory of interface surface tension. All these things were done by van der Waals which will do a little bit of that. Another very important area that you have faced in your DSN MSC is surface phenomena. But again that was not done in a very systematic way in your undergraduate or MSC. But this is a beautiful subject the surface phenomena and surface tension which is again connected with nucleation as some of you know a granular theory. Now Einstein played amazingly role. In many many statistical mechanics he played an outsider role. He created the theory of Brownian motion which we used to do the diffusion. Einstein contemporary when they are doing the great work he was senior to Einstein. Around the same time Einstein came up with what is called the theory of fluctuation. Einstein realized one thing that many probably from a Boltzmann the thing we call specific heat. The thing or conductivity, viscosity these are all what today's language we call response function. Like when you get the first thing we got the rock from the moon what is the thing we calculated. What is the thing we not calculated we measured is a specific heat. First thing that was measured was density and specific heat. Any rock that's what we do. What is the specific heat it's a response function. What is the conductivity it's a response what is the isothermal compressibility that is a response function. Einstein realized that response functions are actually natural property of the system. They are mean square fluctuations. If I have energy energy in a system interacting with surrounding media is always fluctuating moving up and down. You now take the average and then you can take the delta E energy flag different at any given time and then you square it and take an average over a long time that is your specific heat. Then take the volume and take the fluctuation of the volume that gives you the isothermal compressibility. This was done by Einstein that you know that's why he really showed the path in certain way that what how to calculate the things gives was trying to develop in a certain sense. So just like as I said in quantum mechanics all of us know very well the tree of development. In statistical mechanics however I have not seen the tree of development not even the great book of Tolman. Or any of the books that we say. So this pathway that I traced here is the way statistical mechanics was developed. And that is not the way it is told in the books but that is the way it should be told. When you try to explain statistical mechanics the beauty of it is through these things.