 So, those are the relation between thermodynamics and microkernical ensemble that I have given to you. I have given in the book certain problems and I should mention to you that the entire solution book, all the solution of the problems are I think one of you probably should work out in the class the problems. Another problem said that is now available online free if you have bought a book or anybody in the bought book then you will be able to download the solution book. So, at that solution book itself is about 200 pages because this is a huge number of problems in this book ok, but you need to do the solution because there are examples there are there are problems which let you calculate omega and they calculate the properties. So, that is very essential part of the problem. Now going over to canonical ensemble, canonical ensemble now is the ensemble where the system is characterized now by nvt. So, you cannot change number that is kept fixed, you cannot change volume kept fixed, so your volume is fixed. Now so these are well the volumes are different nvt, nvt all are the same and the I am thinking of systems of ensemble, but the energy is changing now. So the systems are in a connected with the bar. Now we will do a beautiful mental construction that gives it which is how to construct so basic strategy that gives it is the following. So here we have number fixed, volume fixed, but energy exchange is allowed, energy exchange allowed such that temperature is, temperature is constant, so you put it in a bar and we in chemistry always do these experiments you put it in a bar and to preserve the constant temperature. Now then how did it gives, did the calculation okay now let me first ask you one thing that when I let my energy constant go which is you know we grew up with the conservation of energy and constant energy we let it go, we let it go and replace it by a constant temperature. What is the change in this microscopic state of the system? Yes absolutely, so it can have many different energies. So conservation energy is gone, conservation energy is a huge constraint, all of them they are constraints, but we work in a natural system or everywhere in the world with constraints, but here we let go the energy constraint, we were having a temperature. So we kind of let, we not kind of we let a macro variable go or extensive property go and replace it by an intensive property okay. So what will happen now to the microscope, number of microscopic states, it will increase, it will increase enormously right okay. Now this is an example, I believe is a very nice example okay, so basic idea then is the following, these are not the best pictures, but we worked around with the pictures the way it is a canonical ensemble is NVE, so I have a huge number of systems, I can put them together and I can, so I can, this is a construction gives it, so we will come to that construction in a minute, this is the I think one of the best thing in these whole of statistical mechanics here. Now I have the same example, my 4 systems, but now I make the energy levels change okay, so now heat energy is 6, there 8 is 10, I have same energy levels, but now before I had only 4 right, I had only 4, but now I have this additional microscopic states, you are keeping the energy temperature fixed from outside, you are putting it in a bath, you know the temperature, just like earlier I knew NVE that was my choice of my system, here my choice of variables is NVT, so I have the systems which are same as number of each of okay, in my system NV and T that means, you know this is very standard experiments we do that we have this number of volume and density fixed, total number of in a beaker, total number of then in a volume in a cell, volume is constant then you put it in a temperature, outside temperature, so as an experimentalist or in my third experiment I can always keep these 3 fixed from outside, so they are fixed from outside means what? Connection to both of you asked the questions that means I am freedom to vary these things, I do not have freedom to vary other things, I have freedom to vary these things like I had freedom to vary NVE and as a varying NVE pressure is varying, here also I have these are my 3 independent variables okay, now I do not have energy conservation energy constant, however I know if I keep the temperature fixed then energy is getting exchanged, the system is exchanging energy debath, if the system is exchanging energy debath then I have different energies, so as the different energies are there in the same number of microscopic states because the system is the same, I will have the, sorry the microscopic energy levels, I will have many more there I have 4, I have many more microscopic states, but of course let me ask you a question, of course constancy of temperature and variation of energy allows very different states, but there is a very powerful, very very powerful equation, actually one of the most important equation that comes out of statistical mechanics that came out of Boltzmann's work that allows you to say which of these elements is more probable, which is that, so when I allow energy to change, recently carefully I think I am not formulating it well, I am telling you I am allowing energy to change, but systems always have a spring in them, systems just like volume there is a spring and in energy there is a spring and I tell you the spring, the spring in energy is the specific heat, spring in volume is the isohermal compressibility, okay, these are all the springs that means which hold system together, it does not allow it to change, actually you can write down the free energy or any as a function expansion and you can see that they are first order is zero because of the minimum energy free and the minimum condition, second term of this we call the response function which we deal with at quite extensive related, that is one of the very important outcome of this, as in my book when I wrote this chapter I called realization of from ensemble realize and the fluctuation of chapter I said realization of the promises, because that is first time stat mag wave which was done by Einstein, the most remarkable predictions that comes, okay, so I am asking you a question now, if I give you this kind of thing in a more general sense, many more energy levels, much higher energy, but in the system that which the energy is more probable, what is that, there is a distribution which will come out, but all of you know that, you know that from high school days, both distribution, so though energy is fluctuating across because of constancy of temperature, there are in probable distribution of energy, so when we relax the constraint of constant energy and in as a result of the constraint, we get all these different arrangements or different occupation of the microscopic state and different microscopic states, there is some another there is a constraint that comes, so we relax a constraint, but you get a constraint in terms of distribution, this happens again and again the same thing happens in the case of grand canonical ensemble or other ensembles, but that is one of the beautiful of these thing, okay, so go back, let us see how a constructed Gibbs went around doing it and there as I told you it is a beautiful construction that gives it, what Gibbs said, okay, I assume that all my systems are at made at I put all my systems together, all my NVT together in contact with each other and then I billions of my microchemical ensemble systems put together and then I put them in contact, then I put them in a isolated, I put a box around them and I solute it constantly, completely, so now each of the system is an NVT system, this is NVT, this is NVT, this is NVT, but I have put them in contact, they are NVT, they can exchange energy, so instantaneously each of the state are in different energies, okay, now I from a super ensemble, I all of them I put in contact and I now added the energies of all of them and it is a very large number of ensemble of ensemble and I put, isolate them, then I can talk of my super ensemble, I insulate it, my super ensemble is a total number of N, so I have this number of Italy size N and this is my total number of ensembles, total number of microchemical ensembles in the system, so now I put, I create an ensemble out of this ensemble, that means now my NVT, each of my system here are at NVT, but instantaneously they are NVT, but they are different energies, right, so when I put them together, I have a huge number, N of them I put them together, then I isolate them, then I make ensemble out of that ensemble, my super ensemble, then I that my super ensemble, each of them is a contrast of an ensemble, they themselves are in having this number of, this number of volume and this number of particles, but energy will have a sum here, there is a constraint and this was then, because the only thing that I have, you have to understand, the only thing that I have is this formula, that why I said that everything follows from that, the only thing that I have is formula, so now what I am trying to construct, I am trying to construct when I go to canonical ensemble, I want to construct an ensemble of ensemble such that my, in my super ensemble, each ensemble is a micro conical, ensemble is a micro conical ensemble, okay, is a, I saying this is one of the brilliant construction, it is very difficult to a of that, then what one does, there is a, I have discussed it quite a bit here, but I just want to do the math. So now, if I now have in my ensemble, these the number of systems that have my energy here, then that has to be, this is here, the index is not capital N, it is total number in S and so then N i, total number, so let me denote this, N j by the number of systems in A 0, E j, so then total number with the total number of things in N n, C j are the total number in my, in my super ensemble and then the population N j is in energy will be E j, then multiply the, I got the total energy E j, so the notation as I personally when I teach I actually, this is Italy size N, but here the Italy size N has become capital N and capital T, so this is total energy, E j is energy of one of my system in my ensemble and I have made, I said microgrammical ensemble of the ensemble of ensemble and this is the condition because this is the total number, so N s is the total number of systems in the, my super ensemble and this is the total number, total energy, so each system in my, in a canonical ensemble constant temperature, but they are in a different energy state, is there any confusion here, yeah you are allowed to make the systems interact with each other and you are giving the condition that they themselves, you keep that system for a, a equilibrium with long time with a bath, so that the temperature almost the way we do the simulations, so the temperature it comes to a constant level at temperature T, then you isolate it, you insulate it completely and that is the super ensemble, that is how and then you call that, so the ensemble of number of particles of microgrammical system you put together and construct a canonical, a system in canonical ensemble N with E, then you put ensemble of those things and that has a constant total number of particles, it is not the number of particles in one system, it is the total you know and then energy also a, in fact this is the constant, is it clear, this is not an easy construction, but what do you get at the end of the day is of a huge importance, it is a very serious business and usually this thing is not taught in the classes because it is a kind of complicated thing. Now we are, as I told you again and again whole, so we have created a super ensemble which is micro canonical because I have nothing but of this equation to go, so I have to go back to that equation, that is why I have to construct the micro canonical, super micro canonical ensemble and I am working on the super micro canonical ensemble, so I have the super micro canonical ensemble wherein is the total number of systems, so it is the total number of systems in my ensemble and total number of system, each of the system has in that energy level, so in my super ensemble there is n1 number of systems, not particles, n1 number of systems in hw1, n2 number of systems in hw2 and then I can write total number like that, so I do the combinatorics. Now I can now develop a saying and this is really extremely important, one can define that probability of observing a given state ni in energy ei is this quantity, so here I have done a partitioning, I have said I have ni systems in energy level ei that has since they are in constant energy that has an omega associated with it, so that is this omega then this is normalization because I sum over all the energy levels, so total number of microscopic states is the total normalization, but then this gives me probability of observing ni particles in the ei, so I am done an averaging here, so this is a very important definition in the whole scheme, give scheme of developing the expression of micro-granite ensemble, is it clear? That means you have to think in terms of little bit because this nj are number of systems at a energy level j, since reward you are getting, since you have fixed these things as constant energy I can talk of this and it makes sense, because omega is number of microscopic states of a system with a given nv in energy, now game we are going to play now, so this is like you can guess what you are going to do, so we have defined probability of observing a system with a energy ei and we have the total number, however we know in the total number there are these two constraints, but so this is not an unconstrained, this is not an unconstrained variable anymore sorry, so this has these two constraints, so what is the way we do it, so anybody has any clue how to go about it now, just like before we are going to do a, we will do the, we will maximize ln omega and from the combinatorics I get this equation, any confusion there, now I want to maximize this quantity, any confusion there, however there is not the way you do that, there is a way to do this, there is a condition called, they call the air condition of an advanced multiplier, when you have a variation, but with subject to the constraint, so when I have these two constraints, I have to put these two constraints into mine, these two constraints I have to combine with this equation of this equation with this quantity and do you guys know the method of Lagrangian multiplier, this is known, this is the method of a constant, Lagrangian multiplier is called method of Lagrangian multiplier of variation of constraints, you have done that, no, no, okay, so basic idea is that we have to maximize this quantity, we have to find that distribution of Nj, remember Nj is number of systems at a energy level Ej, however even though I am allowing systems to vary over different energies, there are priorities, because of NV and intermolecular interactions, there are some energy levels which are more populated, there are some energy levels which are more populated than other energy levels, actually there is a Gaussian distribution and the coefficient of the Gaussian distribution is this basically, but that comes later, those very fundamental things comes later, right now we are trying to get okay, I give you the energy levels, I allow you to choose your energy, but the system has a constrained system because of the total energy is conserved in the super ensemble, total number is conserved in my ensemble, the variation, so the omega, that particular distribution, so what I am talking of this Nj is a distribution and I am after this distribution, okay, so Nj is number of systems in energy level Ej, okay, let me write down and I am allowing these energy levels to change, because I do not have a conservation of energy or constant energy anymore, so these those my arrows, so I have many different energy levels, however my systems prefer certain energy levels, all the energy levels are not, when you do the computer simulation, what do you see, you show the energy and the energy is fluctuating, but they fluctuate around the value, that value is the average and also the most probable, now you plot a probability distribution of the energy, what do you find, you find the Gaussian distribution, who determine what is the average value and who determines what is the, that the width of the distribution, width of the distribution is given by the specific heat, who determines that what average energy it will be, it is by the intermolecular interactions and density involved in N and V, intermolecular interactions N and V, that determine your, so now those things must be there put in a constraint and those are the constraint, so a distribution is there, which I have to maximize, but that any distribution will not do, those distributions obey, the total number of particles that I have is constant, I have written those equations here, total number of my energy in the grand canon, because after all I am, I will be working in the grand, going back to my principle, so the way that is done, picking up a distribution of many distributions, so I have a distribution of distributions, picking up a distribution, out of that distribution is, that those distributions that follow this constraint and this is a method of Lagrangian multiplier, please a that, I have a appendix on my, Lagrangian multiplier in the book, so then we do this, maximize with this constraint, this is the way to make a Lagrangian multiplier and then you take this derivative, you can, you get omega, you put the omega, which we did here in this omega and you do the Stirling's approximation and then you get this constraint and this is just a constant, you can see now in j e to the power alpha beta j and so in j star become this quantity, so this is the yellow equation, this is the equation that comes out from these things, exactly that is a very, very good question, actually that is a excellent question, so we will come to that, that is an excellent question, this is the Poznan distribution, but you are right, but I will come to that in a minute, so I think we will probably, with that answer of your question we will stop today, as I said that is one of the, again the questions just like the questions you guys asked is a cause of confusion, so but you are ok with this now, now I have to get this quantity here e to the power minus so I do sum over nj, if I do sum over nj and I sum over e to the power minus beta j, then I find sum over nj is the total number n and then this is the quantity, this e to the power minus beta j e to the power alpha because this n is the a, so then you get pj is nj by n from there, you get e to the power minus beta j by e to the power minus beta j, this quantity is ok, because just I sum over that, so this is the definition that we started to do, the probability of finding a system, because we work with systems, number of systems energy able ej, it is not number of particles in energy level ej, it is number of systems in energy level ej and so the probability of observing a system in a energy ej is this quantity and this quantity is the canonical partition function. And so if I write the canonical partition function here that I have written later, then this is the one of the most fundamental equation of statistical mechanics, which now tells you how we have not yet shown that beta is temperature, but we will show that that beta is temperature, for that we have to do an analysis very similar to what we did with Euler equation in micro conquerance of that we will do. Now, I will just before we stop I will answer this very good question, what you have to do now in order to get out that, so probability of pj is you know pj is e to the power minus beta h e j, now what you do, you define the average energy e and then average energy e is all right because this board is solution it is this quantity which is Gaussian. So, now you consider your average energy is e this you make a fluctuation delta e and from here now you calculate the that is exactly we will do later, we calculate the variation of energy. So, now you say since energy your average energy where the system is the one which minimizes the free energy, so first derivative of that goes to 0. So, your energy then it is like that energy e is average plus first derivative is 0, then second derivative is d 2 a d 2 delta e square. So, your variation of free energy and then because first derivative is 0, so now probability in in this e to the power minus delta a and that is e to the power minus e goes out and then e to the power minus delta e square. So, a really very good question that Boltzmann distribution has within it the Gaussian distribution, but this is Boltzmann distribution where you are asking what is the individual energy, what is the population of individual energy levels, Gaussian that you observe is the total energy. So, in simulation when you find a Gaussian distribution of energy that is the total energy not the individual energy, yeah. Now, on any observed system, when you look at the observed total energy of system is the total energy average energy or in observed energy that you are, this is not the observed energy, this is the one confusion you always have because when you do spectroscopy in quantum mechanics, we consider this is the energy that you observe in spectroscopically, but those are isolated single non-interactive picture. Then you are allowing energies to exchange like in in in a macroscopic system, so homo dynamic system, then this is the pj is the probability that your system is an energy ej, we use that same there also, but when I talk of Gaussian distribution is the total macro variable, total macroscopic energy that is the Gaussian and that comes from there by same definition, but you define the total energy by that equation average energy of the equation, then you say my free energy is the one that is minimum at the average energy e that property use and then you get the Gaussian distribution at equilibrium. At equilibrium, at equilibrium, okay look look let us look at that, these are the construct we gave we did because as I told you there is only equation that you have is entropical kb ln omega, we did the construct made of a super ensemble which is micro canonical. So we started with ensemble which is canonical, then we put many canonical ensembles together and I put a super ensemble micro canonical ensemble, in the micro canonical ensemble I now can talk of number of system in energy variable nj, then I can do and construct a omega now and that omega has the constant of total number of systems and total energy because I have a micro canonical, so micro canonical has constant energy and constant n. I put those constant by using Lagrangian multiplayer and I get that out, that is the whole game that he played, gives played, okay now. Now you are asking say the energy is Gaussian, volume is Gaussian all these things. So in a computer simulation we are starting is you are talking of a single system but you are talking of trajectory for long time or you can take a Monte Carlo simulation where you are doing, you are creating many many configurations of the system which is same as creating many systems, okay, right. Now you are talking of average energy. So when we plot, we plot the energy of the system either on a trajectory or in Monte Carlo and I have a histogram, I have a this many energy, this many time my system is in this energy but that is the macroscopic total observed energy. This is not what you observe, this is what the microscopic occupation of the system and we find out from this distribution, what is the energy, average energy that just what I wrote down, we will do it tomorrow and what is the entropy, what are the other properties, we have not done that. We have just like in micro canonical ensemble we have done, we have done God that important to it, that is a Boltzmann distribution and from there now we will construct the rest of the things. So that we will do tomorrow 10.30, right.