 ఈంల్సాడిు నేన్నింహ్టాా is to discuss . . . . . . . . . . . . . . . like nennad joneson cannot even do that analytically. So, this is one reason that one has to have statistical that is a technical reason and there is a more profound conceptual reason is that even if we could solve newton's equation for all the all particles I have heard a number of particles in what do we do with that because that contains lots of irrevalent information this term in statistical mechanics what is relevant and what is irrelevant is an extremely important concept we have to extract from lot of things that are going on something which we observe in real world and also what is experimentally accessible okay so having said that so that was the difficulty which man was motivated by Maxwell's velocity distribution to introduce the probabilistic description he took it halfway but he couldn't do very far as I said other side of the Atlantic our great wheel at Gibbs realized that there could be a different way of doing the things wheel at Gibbs did completely equilibrium statistical mechanics these has to be understood clearly while Boltzmann was trying to do transport properties even today transport properties are time dependent statistical mechanics starts with Boltzmann's combining equation so this is a very important distinction what makes so wheel at Gibbs realize that you know that many different systems which observe they have same macroscopic properties they must be in different microscopic states so is a different microscopic states of the system that belong to a given macroscopic state then there must be way to look into equilibrium distribution in terms of a probability so the important breakthrough then came going from the trajectory or time dependent stat mac or time dependent distribution function which gives time averaging from there one wants to go to an equilibrium distribution so wheel at Gibbs realize that the simplicity gates in doing the equilibrium distribution that could allow him to do a probabilistic description and even by that time early 20th century his book was published I think 1903 by that time probability theory was well developed mathematics of probability theory central limit theorem and all other things were well developed so he had now the he could rely on that so he they said if I want to describe a probability distribution then what are the probability distribution I want I want a quantity which is a I could talk in terms of thermodynamics a term in terms of entropy energy fluctuation these kind of quantities he already had one result in front of me which is Boltzmann definition of a function entropy to what extent that entropy already thermodynamics was rather well developed entropy function or heat function was well known that correspondence had been had occurred to Boltzmann I am not too sure of but probably it was but suddenly wheel at Gibbs realized it so he had that formula s equal to kb ln omega kb is Boltzmann constant very appropriately put there and so now he had to find the omega and of course if it and rest of the thermodynamics how how did he go about it so he said okay let me construct a an ensemble which is a huge collection of the system each system has characterized by same number of particles n same volume v and same energy all are constant now and then in this ensemble he said okay how can I describe now probability distribution he then introduced two postulates one is that he wanted to do the average how do I do the average instead of one way to of course time average follow one system for a very long time the other one you follow many identical systems now the idea is that many identical system billions and billions of identical systems of nve they are all in different microscopic states why because my number of microscopic states is enormous it is billion to be to the power billion so all these systems will be in different microscopic states so that is a very critical understanding that all my I can construct an ensemble means a collection of identical particles all of them in different microscopic states then if time averaging that I do by following in one system a trajectory for a infinitely long time or very long time that can be replaced by averaging over the these different systems low and behold what is the advantage instead of following the trajectory which is a time dependent quantity I am going to talk of something which is time independent and that's it enormous simplification I don't have to solve new prosecution anymore which was the prescription of Boltzmann then when he did that of course he had these these ensemble picture of systems in different microscopic states and averaging over them is what is today is Monte Carlo simulation trajectory is molecular dynamic simulation however in order for the trajectory to be equivalent to ensemble ensemble I have been all different microscopic states you have to understand these are still at a conceptual level things are assumed these are postulates and the verification comes you know with the results and comparison of the experiments then he said that okay my trajectory then must go to all the microscopic states that I am representing in ensemble then comes the second postulate which is called equal a priori probability that means all microscopic states are equally probable to be visited by the system and in a long trajectory they will be visited equal number of times these are limiting things infinite number of states infinite time infinite number of systems so everything asymptotic where things are can be proven exactly now when I say things could proven exactly that of course happen only 1981 case more than 100 years later almost so these things that a system equally probable but equally probable doesn't mean that it will be visited so the two things I jumbled together to be separated that comes as a hypothesis that all these states will be visited that's the ergodic hypothesis that each state will be visited in the course of time that means its system will not get stuck in a minimum that haunted us in molecular dynamic simulation that's why you cannot do phase transition just in envy because it gets stuck in one minimum then all the new techniques like umbrella sampling meta dynamics that have been invented and created are essentially to take the system out so the reason that one has to be conceptually clear and study these things even more now than before is because of the connection with the modern stat mac and which is so powerful now it has reaching the state of quantum chemistry in this sense the packages are being available I see my students other students getting like one of my student without understanding too much but great results in rotation of carbon monoxide nitric oxide and cyanide ion in a aqueous binary mixture something which was unthinkable few years ago such complex system but that can be done now I cannot claim that the student knows very great of the concept of statistical mechanics and how to do coupling theoretical work but the results at the hand and at the end of the day results are that are important that's what everybody wants so statistical mechanics has come to that level now so we have to understand this little bit more detail to have a certain sophisticated certain sophistication your understanding sophistication in our so then from that micro canonical ensemble using s equal to k l n omega we derived the expression of pressure expression of energy ds dt is energy so everything is now number of omega number of microscopic states omega well done that part was done but then nv is a very artificial system because it is a system which has same number of constant number of particles which is not changing constant volume constant energy that's not the real system real system with a volume is fluctuating transition ice melting density goes up by 10 percent then you have the energy fluctuating because these systems are far more open in at least in chemistry and physics and in chemistry we have also systems were chemical reaction taking place so any so nv microcarbons is too restrictive so the next stage gives d construct a beautiful and brilliant mental construction that you went over to constructing and super system out of nv t where energy exchange allowed you put all we all nv t together allowed interaction between them flow of energy then put it whole thing in a bath let it be equilibrate then put an insulator around them so that the whole super system is an nv e of Italy size n is my n number of particles then Italy size n by normal n it is as n by v Italy size n by e are the number of particles volume and energy of my super system now we formed a micro chronological ensemble of the super system and then showed by doing constant method of constant variation or Lagrangian multiplayer we can get a quantity called partition function which logarithm gives free energy and whose derivatives gives all the thermodynamics after doing that he was my genius of high degree he didn't waste time in seeing movies or other things he was single and he was always going coming back and forth from starling that what he had all the time but this amazing what he did he did the grand canonical ensemble where now number of particles also exchange allowed the same thing was done and he wrote a beautiful beautiful expression of equation of state pv equal to kb t we remember all these are energy dimension pv equal to kb t nm grand canonical partition function then we also discussed npt which is used most in simulation and experiment so till now we have done all the groundwork still keeping things fairly simple we have done four ensembles and we could have spent more time on npt but okay that is i told you that it is essentially same construction but the thermodynamic potential of npt is gives free energy so thermodynamic potential of micro chronically is entropy thermodynamic potential of canonical is hemorrhage free energy thermodynamic potential of grand canonical is pv and thermodynamic potential of npt is gives energy that what exactly thermodynamics tells us that exactly comes out to be true so one of the major goal of statistical mechanics is to explain thermodynamics remember one of the things that we you read in thermodynamics i don't know we are a great teacher professor meh choudhury who again and again told us one thing that a thermodynamics cannot give you anything a priori give the results it gives certain relations so thermodynamics does not give you what is the value of entropy cannot give it unless of course you integrate of a cpt it does not give so that means there is a it is not a theoretical machinery that gives you anything from first principles that's the that the correct statement it doesn't have first principles it has certain working principles which equal laws of thermodynamics which will be fun to teach some time the thermodynamics in a in a very deep and fundamental level because that's a wonderful subject however here we don't have a for that time or course content for that but here thermodynamics then comes out of statistical mechanics so one of the first idea of statistical mechanics was to explain thermodynamics and what it does it does in a beautiful way starting from microscopic principles intermolecular interactions which gives you energy levels then you everything flows you get all the thermodynamics exactly so these are the theories we are going to do applications of statistical mechanics and we'll do the simplest thing first which is monotomic ideal gas ideal gas monotomic so these two things are both very important so so this is what we are going to do and it is an extremely illustrative and very very nice and i saying when we studied these things we thought is useless but then later much much later when i did drug the an interpolation okay so we'll do the this is the following thing what we will do today derivation of the thermodynamic equations applying principles of statistical mechanics these two derive micro-supertion model system we'll do now entire volume available to all the molecules maximum translational entropy in ideal gas this is a very important thing again because just you know in a bark i gave the lecture where i had a path which is diffusion and entropy there's a very hot topic now even in active matter everywhere diffusion and entropy what is the role entropy plays and okay so what goes in there the entropy called excess entropy and excess entropy entropy system minus the entropy of the ideal gas and diffusion is e to the power excess entropy you know this is amazing how these things haunt you come back to get you the results so you know those of us who thought that ideal gas monotomic gas is not important we are using it every day now okay so let us now do these things Hamiltonian of the system is an interacting particles is h n tv is equal to n number of particles so Hamiltonian is just kinetic energy there is non interacting so there is no potential energy if they are not then partition function is just n over integration over all the particles so so partition function of n number of particles chinese people detailed you know they don't know nothing to worry about India because most of the indians are sick which i think is by and large true i myself got sick yesterday so this is the configuration part and this is the momentum part and then these are function of all r n that will write like that and function of without writing so this is the integration that you have to do so n particle integration over n vector of one to n momentum of n number of this and this is the Hamiltonian then i hope you understand this very important because this will be repeated again and again it will be repeated today and it will be the important class that will start from Wednesday morning with the mayor's cluster expansion theory we will use this again and again and again and again okay momentum three n integral here three n integral here and this is the Hamiltonian in of my case Hamiltonian is just kinetic energy and then you can see this will come out straight with a since this is additive i can app to the power n i hope you understand additive means is just multiplication this is the thing is it clear this thing how it is happening because this will happen again and again that's why i'm spending some time here what you have done that since this is some all of them have the same mass now i can write it as a product and then the product means there is a n number of them okay this is the h to the power three n here that has been missed we have to care we have to put that back this is indistinguishability called bolshman's factor he put it by hand with then quantum mechanics came it became okay the basic idea is that you don't get a extra state but just by exchanging two particles in the eye by keeping everything identical so formulation factor is this time and we have h to the power three n and there's a great new reason for h to the power three n anybody can tell why h to the power three n should be there and must be there a very nice and elegant very simple exponential you guys did text at my course no yeah absolutely it is close to the answer here answer that it has to be dimensionless quantity see we put log in front of that you cannot have a dimensional quantity after log or in exponential there has to be dimensionless that the first thing we usually ignore that fact very much second that we actually normalize the volume integration by the volume in the cell and the volume in the cell is given by planck's constant and if you do that this quantity that q q is one of them so let me say the so let me also put it here h three n n factorial then now dr one dp one e to the power minus beta p one square by two m the whole thing in the exponent this is to the power n because the n part is there okay now this is volume this is momentum okay now if i make now this is l cube this is momentum cube if i make length in momentum then what is the dimension of length momentum okay so length and momentum so dimension of length dimension of momentum is ml by t right yes so this is the the dimension of this is also the dimension of the planck's constant so the momentum l by t ml by t energy is ml by t square ml square right that's what i was saying some something is missing i was mumbling ml square now in planck's constant remember h omega is energy that is ml square by t square okay that means we are having that's why h dimension of h bar if this one ticket goes out right you get ml square by t this is also called the dimension of the action so that is the volume in the phase space which is normalized that should be h to the power three n here okay now we most of the time don't write it i actually don't like to write that one but it's okay this is the dimensionally correct one has to be so now one does this integration of one particle this is a single particle that decomposition is possible because there is non-interacting when you do that then dpx dp z dx dy du z with one minus beta h so one of them this one gives volume so that volume comes out then this integration is same integration because next decomposition comes p p momentum square same as p x square p y square plus p z square right and p x square p y square p z square so this integration is this one what is this in of this integration value of this integration root over pi by a yeah so root over pi by a means this is one over kbt two here m here two pi m kbt right root over pi by a a both are a this also comes here yeah two pi m kbt so this is the integration that integration in its full glory comes here and these since it is root over three of them one two three of them becomes cube and since root become three by two if i have to put Planck's constant h to the by three and here that goes here at h square okay so this is the small q i have to one over n factorial in front of the big q this is the partition function of the monatomic gas