 and the very first lecture i talked about why we need to do statistical mechanics is a difficult subject very difficult subject and in the in the evolution of statistical mechanics physicist and chemists almost you know they were hand in hand many many things were done together when Willard Gibbs many times is one of the father was considered to be chemist when i was doing phd he was referred to as a american chemist and now i think he is claiming as much as now so very fast lecture that we did we did the preliminaries these the we talked about how if you take a undergraduate physical chemistry or even ms level physical chemistry book what you will find is that the chapters except the the web mechanics one or two chapters and spectroscopy if there are 32 chapters you will find 20 25 chapters are things like kinetic theory of gases thermodynamics entire four three four chapters of thermodynamics then you have a phase equilibrium then you have a phase transition then you have binary mixture your solution electrochemistry all these chapters that you study they are all and you know remember that when you study of your conductivity of the ions lithium sodium potassium then you plot it against the size they are supposed to go as one over size of the ion crystallographic radius but it just goes instead of going like that straight line it just goes the maximum and falls back that if you remember that was called conductivity times viscosity is called faulted product and that non-monotering behavior was called breakdown of the faulted product the reason i am talking of the breakdown of the faulted product is that because i want to make a point and a very important point which is the following so i this is limiting ionic conductivity in electrochemistry you have right d by hooker remember and viscose at one over r sorry one over r ion since i am this is nothing but the what is called the diffusion and diffusion is inversely proportional to viscosity so this product should vary as one over r ion which is like this instead of that what happens that it just comes down like that and this lithium then sodium then potassium it goes like that and this is here where cesium rubidium and all these guys are maybe potassium will be closer to that now so in the undergraduate textbook of castellan or wood or glass tone all other atkins they call this breakdown of halden's product and give to some pictorial description like iceberg formation around the small ions so basic idea of my telling this is that in march of physical chemistry in undergraduate whenever we have anything interesting going on we have a picture that came but that picture was mostly very approximate however behind all these pictures there is a quantitative theory that was largely developed in the post 1950 and or maybe 1960s 70s and aided enormously by the computer simulations so potassium mechanics has come of age now and like in quantum chemistry you have all the packages so a student working in a quantum chemistry laboratory they can always have these packages and they use these packages for example doing organic chemistry calculations and other things statistical mechanics those kind of packages are just coming now in terms of grommocks or amber or the different force fields that have come so it has become much more institutionalized now so there are a lot more calculations going on and it has played a very important interesting thing that it has gone it has gone again okay it's a rapidly developing field huge amount of it's not that much visible still in India though there are now significant number of people doing statmaking chemistry physics always the huge huge in physics so this was the preliminaries that I explained that why that you need to invoke statistical mechanics to understand the what you call the large-scale phenomena large-scale phenomena many particles like phase transitions like this conductivity or understanding thermodynamics or so phase equilibrium phase transition all these things we discussed that in the first lecture second lecture we had little bit of mathematics where we did probability and statistics and because this name is statistical mechanics which combines mechanics mechanics is very deterministic it starts with neutron situation and you know everything is initial conditions given you can predict the the future but as I told you that the unfortunately that the we cannot even solve what is called a three body problem even if you have three particles we cannot solve that analytically even two particle having a little in a complicated potential not in a complete radial potential say lennard jones potential even that has to be done through a quadrature yeah so the i the whole a of so this is a interaction between a and b r a b this is the lennard jones potential we call it 6 12 potential the form is this is a universal notation sigma r by 12 minus sigma r by 6 yeah i let me say a and b are the same two particles both are a then i don't need the b i can do without this index then is a separation between them that is r and this is the form it very important to understand these things interaction potential between two particles when they come from a distance they attract each other but when they come too close electrons overlap and there's a huge repulsion this simple potential apparently simple potential plays a very important role in the understanding of many many phenomena a solvation phase transition many things however coming back to the point even these potential in a two body we have to do by quadrature and by the time i go to three body potential if i bring one more i cannot solve the newton's equation anymore okay but however when you think of the properties of water then you think of properties of water in a glass which has avocado number or molecules and they are strongly interacting and we cannot use newton's equation anymore to solve them if we want some really very complicated and sophisticated effect like polarization that one molecule is polarizing another molecule then we have to do quantum mechanics that is even more difficult okay so we are faced with then a situation where we have to explain natural phenomena like why ice males and then why steam 100 degree centigrade becomes steam why i put a solute there then depression of freezing point and elevation of boiling point that you have learned in school and these kind of things if you want to understand you cannot on from first principles and what do i mean by first principle these are term we use again and again statistical mechanics you also you know quantum mechanics quantum mechanics when you see first principles we think that we are starting with third anger equation and interaction potential and going about it of course they are looking at electronic properties and there are there are there are approximations and there are answers in statistical mechanics why one can do quantum statistical mechanics right now let me focus on classical statistical mechanics that because the large amplitude large scale phenomena like phase transition or as i am saying why steam bites you more than water at the same temperature depression of freezing point like you add salt that is what the principle of ice cream that you get you can go to minus 20 degree centigrade that is where ice cream forms you can do that experiment doing that now you can ice cream ice cream maker and there is a liquid there which is can goes very low temperature and you put water and you can make the ice cream and not water milk and water now so these the many many large amplitude phenomena that you want to understand we cannot do by following neutron situation anymore so then we need but how do you go so neutrons mechanics classical mechanics means the neutron situations or hamptons equation or whatever let's continue with the neutron situation which we know neutron's equation then is thus now is not going to help us because we cannot even solve a three body problem that is where the statistical mechanics come seen it was formulated starting with Maxwell Boltzmann then we did Gibbs now so what is then i do i cannot do mechanics as i know it so i have to get there this was a huge huge conceptual and philosophical problem in the 19th end of 19th century when people started introducing the concepts of statistics and the what it all started with the work of Maxwell you know Maxwell all of you know the distribution Maxwell is the first guy who told okay if i have in a glass jar a bunch of atoms and molecules at a temperature T i do not need to follow at each of their each properties of each atom or molecule instead i can talk in terms of distribution so Maxwell said okay what are the Boltzmann also what are the properties we want to know i want to know the viscosity of the gas and later viscosity of the liquid i need to know the pressure equation of state you know how is the pv equal to nrt is the ideal gas but when you go to little high density ideal gas now breaks down and then comes the virial equation okay how do i get then virial coefficients so and so then Maxwell said i don't need to know individual atoms and molecules in a departure striking departure from classical mechanics instead i'll talk of a probability distribution as soon as the concept of probability distribution came in then came the question of statistics how do i define probability i need statistics to define probability okay so that is what it was the second lecture and which we discussed probability and statistics then the last lecture third lecture we started talking the postulates so fundamental concepts and postulates i'll revise a little bit of that but then i'll not do liable theorem in this lecture i'll do it later i'll go to directly to something which is little bit more application ensembles and partition functions and what these are from my book and what i written there that actually earlier i have this from postulates to formulation so we'll go to the formulation of statistical mechanics today so it's a very important class in that sense these are next class and what i had before the title is that from promises to realization because there was this promise that was made by bolchmann wheeler gifts or wheeler gifts mostly realized it so the postulates both bolchmann and wheeler gifts played a role in giving us the main postulates of statistical mechanics from there we'll directly go now because this is more to do with dynamics and we'll do that at some stage but not now so the postulates there are two postulates what are connected by the one one hypothesis and based on that statistical mechanics promises to explain the natural phenomena which is a very high a so first postulate so there are postulates that i did last but this is so important it's no harm done doing it once more so first postulate is called time average and equal to ensemble average and second postulate is called equally probability and what is not told in a not any book is that why we need the hypothesis and these two are connected by our godly hypothesis i i'll spend five next five minutes talking about it then we'll go to the starting the partition function and ensembles how did that come it came because bolchmann tried very very hard to develop a kinetic theory of gases so you when you read the i mentioned this last time also you know like in sardingar equation in in quantum mechanics if you know noticed i didn't notice but i realized later partly because of a book i picked up in one and a half rupee in presidency college the old books next to presidency college a that fans that when you study quantum mechanics is all the way from hydrogen molecule you don't have any name you know there is no name because particle in a box rigid rotator harmonic oscillator hydrogen atom hydrogen molecule that because everything was done by sardingar alone you know when you first hit upon the idea you went to a resort area and stayed three months there and solved everything so the whole quantum mechanics as you read in msc you know is that done by one man similarly kinetic theory of gases again there is no name almost entirely of the kinetic theory of gases was done by Maxwell and a bit here and there by bolchmann later so when this beautiful paper of Maxwell appeared there are other papers around that but not as clear as Maxwell and also not meant to English speaking world then bolchmann fell completely in love with that paper until the end of his life he died early 1906 he carried that paper and then he tried to extend the Maxwell's Maxwell had this funny mix of assumptions that particles are like billiard ball but then on the other end he doesn't he is also talking of an ideal gas ideal gas don't interact they pass through each other so there was the discontinuctions in all the ideal gas in our study of kinetic theory of gases so bolchmann set out to extend it to real gases and separate and there's a famous equation on bolchmann kinetic equation but bolchmann did not fully succeed he tried very hard he could go only to very dilute gas and he also made some assumptions which were heavy criticized to the extent that probably cost his death now when bolchmann tried very hard and could not take into account even with the he is the first one Maxwell had it the probability distribution concept of probability but he did not explicitly stated that in his formulation and that was done by bolchmann he explicitly added probabilistic concepts he said okay if i have a probability of a particle at a position r with momentum p i call it f r p t another particle at position r1 i say this r1 p1 this r2 p2 then i have then the together i have a two particle distribution like i call up f1 f2 subscript to r1 p1 r2 p2 t that a given time a particle has r1 and p1 another particles r2 p2 that is a two particle distribution but it is so difficult because it has r1 and r2 all are three dimension three three six and p1 and p2 another three three six what a beast it is so he made an approximation that f2 r1 p1 t2 this was called is random random chaos approximation and that he was immediately and hugely criticized for making this approximation but here are no other option once subsequent whole century people have tried to 100 years people have tried to extend that and have done to some extent so when bolchmann tried this he could extend only a little on the other side of atlantic one person who looked with concern at the difficulty faced by bolchmann who was equal fan of maxwell and maxwell distribution his name is wheeler gibbs wheeler gibbs then thought okay what bolchmann did is not tenable anymore we cannot go that way because we cannot do two particle three particle is out of question and there are many other complexity so then wheeler gibbs make the important observation that as i told last class also then let me have 10 glasses 10 glass of water half full now now if i look into by that time the microscopic motions and all these things were somewhat guided theory of gases was already understood so if this 10 glasses all of the glasses if i think of the microscopic state and a microscopic state now is defined by if giving the position and momentum of each particle so i have n particles then i give you r1 r2 rn i give you p1 p2 pn that together determines my microscopic state of the system now my 10 glasses always have water now all of these microscopic all of them have the same properties they have the they are in same temperature they have the let me copy same volume they have the same pressure they have the same specific heat they have the same conductivity same entropy every property is the same but certainly the microscopic state of all of them would be different because there's such a huge number of microscopic states atoms and molecules are moving around so then wheeler gibbs realized that if i can now have a mental replica i have just one this is my system in question this is my real system now i mentally construct billions and billions of my mental optical such that they are thermodynamically same but their microscopic states are different so they are mental construction these mental construction is called ensemble then he said okay my over if i wait for a long time then my this system my system in question going to go through all the microscopic states who are essentially same microscopic states which are these particles are these systems are so now if i now in my mental replica i create a mental billions and billions of copies of these things which i call ensemble the collection is called ensemble if i can take average of that then that would be same if i now study what boljman tried to do study the detailed trajectories trajectory means the path that the particle takes all the particles take together that you will be the same that means if i can do a time averaging over infinite time with boljman tried and if i can replace by the ensemble averaging averaging over all these my mental i'll be little bit more exact because sometimes words don't language don't carry the difficult but so now they so that was the first assumption first postulate of statistical mechanics that time average equal to ensemble average now as soon as gives did that there is a problem came problem came that when i'm doing time averaging yeah i have one system which i'm studying for a very long time and i'm studying the motion of atoms and molecules their positions and momenta and then i'm averaging a property for example pressure i'm averaging a property for example the internal energy their enthalpy entropy but now i have replaced by the my mental construction but what is the guarantee that my mental construct my my what is the guarantee that the system i'm following for infinite time will go through all the microscopic states and second even if they go through all the microscopic states what is the what how do i how do i give a probability to it so that is the time the he introduced that i am going to talk of the systems with a constant number n constant volume v and constant energy so all my mental replica have a constant energy so the all the microscopic states have the same energy if all the microscopic states have the same energy then i can now assume and i was no other option but to assume under these ensembles that there is equally probability all the microscopic states are equally probable there's no other option than to do that it turned out it's connected worked out but that means all the subsequent works said that is okay all the microscopic states with equal energy are equal people still work on this so so these are the two first postulates first postulate was introduced the ensemble the brain what are the most brilliant idea that mankind has ever come up with it probably doesn't get the sufficient credit that how brilliant this construction is the ensemble and then time average equal to ensemble average but as soon as this was made here to average over all the microscopic states and he needed a probability of being in a microscopic state but since they are all the same energy the natural postulate was that they are all equal problem that's equal right they are now no other option but it turned out to be okay now came what us in in between started talking about that now came the problem following or then gives face the following problem okay i have done time average ensemble average and i have said equal probability but what is the so in ensemble average i am going to billions and billions of microscopic states but what is the guarantee that when i do a time averaging the system goes through all the microscopic states in a given time capital t say what is that guarantee then comes introduce the constrain of the hypothesis which is called algorithmic hypothesis which means the system indeed given sufficient amount of time it visits every microscopic state equal like this so first postulate required the second postulate because to average over ensemble and once this is set in but then i have to make sure that my system visits every state so this is then the made the ergodic hypothesis which connects these two hypothesis the these two postulates so this is the very important point of the statistical mechanics and why it is important because armed with these two postulates and one hypothesis that is where everything whole of statistical mechanics is built on is amazing you know this is what you can say equivalent to there is no equation equations will come later it is equivalent to you are saying that okay this is a wave function and the wave function has to have these properties satisfies Schrodinger equation psi square has to be positive and square integrable the similar kind of things that comes here in this in this statistical mechanical formulation this very very important to understand this part is more important now to understand than before because when we do now as i was talking you a while ago about the different packages time average is the what we call now the more molecular dynamic simulation and this is ensemble average what is called Monte Carlo so these are the two major branches of computer simulations that we do and the same problem in research by Gibbs so many years ago more than 100 years ago is exactly the problem we face now so when i do equilibrium statistical mechanics equilibrium properties like phase transition i land up the problem by one phase to other phase that cannot be done in molecular dynamic simulation because it cannot explode all these things it gets stuck in the minima we'll talk about it now that's where you go to Monte Carlo which is much better to do time independent equilibrium properties now so what she was asking me today why we do mix these two the reason we mix the two is that many times molecular dynamics get stuck so we follow a hybrid where you start with a multiple not too many maybe 10 or 20 initial condition then do molecular dynamics on that and then probability of initial state given by the energy so the hybrid simulation is a very popular thing now to study really really really complex problems like in biology or like in like in water in a supercooled in a very low temperature there are lot of interest now in water in low temperature because there is some presumably liquid transition and people have gone banana over that okay so but now i'll slightly talk if i have okay now this is the kind of thing that i want to point out say i have a particle which is undergoing some kind of a so this is a particle but i have a a a a kind of a landscape energy like this in that case the particle can get stuck here this is a very simple thing a one-dimensional walk random walk but in ordinary random walk drunkards random walk you don't have the you have a flat energy but you have a call equal to rugged energy landscape such a simple problem one-dimensional with the energy distribution it just hell to get a molecular dynamic simulation going into these kind of a system because it becomes non-argodic very quickly and so if you want to calculate equilibrium properties you do a Monte Carlo you just sample that's very easy to do on the other hand if you want to dynamical properties it becomes very very difficult so this is an example what we call compromised ergodicity so the ergodic hypothesis was particularly made to make sure these kind of situations are not done now why it does not is less serious many of the cases in nve because all of them are same energy like this will be ruled out by if i can do the simulation nve because these kind of things are not allowed however in real system we do not do nve we do the other ensemble which i am now going to going to discuss