 Till now we have studied quite a bit of statistical mechanics. We started from the need to understand statistical mechanics and I tried to impress upon you that the great multitude of natural phenomena that we experience in our everyday life and also in laboratory experiments can be explained only by the use of statistical mechanics. I emphasize that we essentially have two different branches of theoretical knowledge. One is quantum mechanics, the other is statistical mechanics and they are complemented to each other. In fact, the information from quantum mechanics is translated to statistical mechanics and that is done routinely in a everyday theoretical research. Remember that quantum mechanics came in order to explain spectroscopy. Similarly, statistical mechanics came to explain thermodynamic properties. So, I will briefly recapitulate some of the things and then go over the derivations of monatomic gas once again and with certain more insight and certain repetition. But I think they are important because the working of statistical mechanics is based understood in case of ideal gas, in monatomic ideal gas. So, it all began with Maxwell in mid 19th century when Maxwell wrote down the famous velocity distribution law. So, the very first point that we need to think about is the Maxwell velocity distribution. So, this was quite a unique departure because Maxwell introduced the concept of distribution and a probability distribution, probability that the velocity of a particle can be within volume V and V plus dV and that we all know is the famous distribution where A is the normalization constant. Now, as I told you and the story goes that Boltzmann was completely taken you know rolled over by this paper of Maxwell and he carried that one paper which has the velocity distribution rest of his life. Then there are certain problems in Maxwell velocity distribution there is the molecules do not interact with each other or interaction between molecules giving rise to that distribution was not very clear. So, Boltzmann took up on himself to develop a theory where these intermolecular interactions collisions between them can lead to a time evolution of the system and where the equilibrium distribution will be Maxwell distribution. So, Boltzmann then tried to write down if he tried to write down a distribution function that a particle at position R will have a volume V and R and V at three dimensional vectors in three dimensional space. So, then he tried to derive an equation for this F or VT and then he found that if he wants to describe it between F or VT that means if a one particle this is my intact particle if you want to do that then an undergoes a collision with another particle 2. So, in order to describe the time evolution of singlet distribution the single particles probability of having a velocity V at position R at time t gets coupled to that of the second particle. So, then that in order to describe this process one gets into then the second distribution. Now, the F R VT itself the first one already very complicated and this one is even more complicated. So, Boltzmann tried to make certain approximations and that led to some advances but he was not fully successful to accomplish what you wanted to accomplish that means develop a theory of the kinetic properties like kinetic theory of gas of illiquid and ultimately also get some idea of the thermodynamics. Remember that Maxwell's distribution gives rise to the equipartition theorem that half m v square 2 k v t and so Boltzmann attempt didn't work. So, let me put out that Boltzmann failed or Boltzmann's attempt failed he derived many beautiful things one of them is entropy and but he did not he was not successful in accomplishing basically what he wanted to do. So, he wanted to describe he wanted to have a develop a theory of interacting systems that molecules moving they are interacting they are colliding with their among themselves and the wall with the wall they are giving rise to a pressure the effective interactions that would give us their y carbon monoxide is different from water or methanol those kind of really goal was not accomplished. We don't know to what extent Boltzmann really had this goal in mind but when one looks back then we understand if Boltzmann would have been successful then that would have been a complete theory and in fact that is still a very active area of research that people are still trying and working along that line to do a many body distribution function. Now, when Boltzmann was trying all these things and trying to develop and didn't work out then on the other across this other side of the as I also mentioned before the other side of the Atlantic one very very smart person was watching the efforts of Maxwell and Boltzmann and his name is Willard Gipps and Willard Gipps was interested to develop a theory of his equilibrium and phase transition and he was interested not in dynamics but in equilibrium properties. Willard Gipps knew very well the the beautiful work of Wendell Wall equation of state but one thing that Maxwell Boltzmann and Willard Gipps was quite aware the thermodynamics however beautiful and the most successful theory as Einstein pointed out this is a self contained in the sense that you can predict the changes of entropy enthalpy free energy or you can if you choose to be integrated from zero Kelvin to get certain absolute values but they don't have any expression that means other than g equal to h minus t s or a equal to a hemorrhage free energy equal to e minus t s e is the internal energy h enthalpy I do not have any expression I do not have a method starting with intermolecular interactions to calculate the properties to calculate the thermodynamic properties to calculate the free energy this is a serious limitation of thermodynamics however beautiful and self contained and self contained thermodynamics is it does not provide us any microscopic picture or microscopic insight. So Willard Gipps was interested also in the same thing that Maxwell and Boltzmann was interested to develop a theory of matter starting from kinetic theory of gases but where molecules interact with each other not like in ideal gas where molecules don't interact with each other. So then Willard Gipps realized one thing that the reason that Boltzmann failed or Boltzmann's approach failed was that it was just wanted to achieve too much that means it wanted to describe the time evolution of its large number of particles. Now when the large number of particles interact with each other like in real gas like in this room the nitrogen and oxygen and they are interacting with each other and we are breathing our life going on so this when the particles interact with each other with the potential then we call that interacting many body system it is a jargon that we use that interacting many body system. So it just means that there are very large number of particles typically we are fond of saying that there are number of particles N is typically Avogadro number. So now such a huge number of particles and if we try to Boltzmann approach then what we it means means we would like to have a single particle motion couple to two particle motion then two particle motion couple to three particle motion four or five like that there is a hierarchy and it is never ending and we don't know how to truncate it we don't know how to stop it without making a serious approximation and not only serious approximation uncontrolled approximation but Boltzmann of MMA Willard Gipps realized one thing that he was interested in equilibrium property he was interested in the relation between pressure and density and at between between the you wanted to understand van der Waal's equation of state how could he do that. So but Willard Gipps realized one thing that if there is a glass of I have a glass of water that the thermodynamic properties of the state of that properties of the water at a given thermodynamic state does not really need we don't really it doesn't seem that we need to know the details motion of the each of the water molecule then he realized something or in this as the following thing what happens if I have a very large number of my glass of water identical identically free now all these water molecules have the same thermodynamic property they have the same density they have the same specific heat but at a given time the individual water molecules and there is this large number of water molecules in one glass and in in different classes the water molecules are in different position they are not in the same position and if I tag a water molecule and then I said okay I tag water molecules and each of them I give number one two three four like that and even they start the same time then after some time these velocities and positions of the water molecules will be different but that doesn't matter the fact that the dynamical evolution or dynamical motion of water molecules different from one glass to another glass does not affect its thermodynamic property that made Willard Gipps realized one very important point and it is a brilliant observation of the most brilliant observation that a theoretical scientist ever made that's why Einstein went over to tell about Willard Gipps the most brilliant mind of America now Willard Gipps realized that then if I could now think in a of my system my body system my original glass of water and I can make a mental replica of many many many of such water molecules many many of such mental replica of the glass of full of water molecules then what happens my in my mental replica in each system the water molecules are differently moving now that could be now then a way to describe what Boltzmann tried to do Boltzmann tried to do take one system and try to map out the evolution of the distribution of velocity and position of water of the water molecules instead now I constructed a ensemble where each microscopic state of each system is different from the other and microscopic state is defined by the positions and velocities of the water molecules of the abogato number of water molecules now what Boltzmann tried to do Boltzmann tried to the goal was to derive a distribution function then properties can be obtained by averaging over the distribution function now at equilibrium my system this ensemble or my mental replica billions of billions of glasses of full of water molecules billions of billions of mental replica of my original system or each in a different microscopic state so instead of doing Boltzmann's approach of following the motion which is called the trajectory of each water molecule and averaging over the trajectory of the system I can now talk of averaging over the instantaneous state of my mental replica of by ensemble of system these are these are little tricky and these are the translation of picture from very straightforward Boltzmannian view where you want to follow the time evolution of water molecules each of the water molecules to a view where you now don't have time evolution instead you have this billions and billions my ensemble of the original system each in a different microscopic state and now I am going to take average over the properties of the system of my ensemble so I can go on talking about it but you need to think about it that is this is my system original system let me call it zero then I constructed many of the mental replica all in the same same thermodynamic state same number of molecules each of them total volume V and maybe total energy E all of them are the same but they are in different they are same macroscopic state same thermodynamic state but each of them the molecules are position different the microscopic states are different now this is wonderful but what did Gibbs achieve by introducing what did Willard Gibbs achieved by doing this doing such a kind of a very abstract logic what he achieved is something just very very far he now said okay I will now follow more or less what Boltzmann did I will say now I have all these systems in different each of my system in different microscopic state but now I can start talking of a distribution of distribution of microscopic states so from a trajectory dynamical evolution I go to a distribution for example I can now talk even in NVE I can say what is the for example temperature of a system or I can talk of what is the pressure of a of one of my system so I can now go from I can start talking of a distribution of different properties because because microscopic state of each of the system is different so the basic idea then that we go over to a distribution we go over from a very and this distribution that you are talking of now because every system every system is in different microscopic state but they are at equilibrium that is the beauty of it they are at equilibrium I do not have to talk of dynamics I do not have to talk of time evolution I do not have to talk of trajectory I do not have to solve mutual situations or I do not have to solve these very difficult equations that Boltzmann tried to develop so the distribution so the what I achieve that is by constructing of the ensemble is that I can now talk of equilibrium distribution okay that is a wonderful thing because suddenly again this concept of probability that is what Boltzmann was criticized so much for but it was also introduced by Maxwell so concept of probability distribution comes back and it now Gibbs could take it much further so Gibbs introduced the concept of ensemble which is the mental replica of mental replica of original system and now one comes to this beautiful okay now how do I construct the distribution and remember what experimentally one does is an averaging but this is that is the we get the property of a system like the pressure by or temperature by at equilibrium by measuring it over certain time so the experimental things are at the time average but here we do not have time anymore we got rid of time we said time is not important for equilibrium probability we constructed an ensemble and we are going to average over all these members of the ensemble which are identical thermodynamic systems so now given that condition how do you go about first thing that if that will that gives rents these are now we have discussed to postulate we will not spend too much time on that anymore and one hypothesis the algorithmic hypothesis and these two postulate we have discussed and is also given in the book in a you know fairly detailed form is that one is the one is the ensemble average that the average you know the ensemble is ensemble average equal to time average and that is the first postulate second postulate which is kind of interesting that ensemble average allows you to go through a large number of microscopic states of the system because each system of my ensemble is in a different microscopic state now this is something you should think a little bit about why why should you should not believe what I am saying why should it be that the different are these billions of trillions and trillions of mental replica my systems which constitute the ensemble why they should be in a different microscopic state the reason is that in a large system the number of microscopic states accessible to the system is huge they are not just trillions they are trillions of trillions like a once it is a water once it is your water molecules that means it is something like say 10 to the power 22 water molecules their microscopic states is huge 10 to the power 100 or so so there will be there are much more many more microscopic states than the number of systems that you need in ensemble so naturally all these systems from the microscopic states from the system are distributed into different microscopic states so that part is okay that part is actually these are the things we know much later we know through computer simulations and many other information that filled in from the last more than 100 years so ensemble average go time average is the first postulate but then what what is the probability now I need to talk of probability because I am trying to talk of averaging so I need a pro distribution the what is the probability that a system is in a given microscopic state that is where now Willard Gibbs made another brilliant observation he said okay let me construct the following every system he has a constant number of molecules in volume V and energy and then all the systems of my ensemble and the same energy same number of particles and same volume V now since they have the same energy he could now postulate he had no other option but to postulate that turned out to be correct is equal a priori probability all the all these states are every microscopic state is equally problem they are all have the same energy so this was wonderful so now I have equal measure of all the microscopic state that now allows me to calculate ensemble average because I have huge number microscopic states all are equally probable if I can get the thermodynamic property in each of the system in a different microscopic states I add them up and divide by the total number of systems in my ensemble I get the ensemble average that we discuss before if I not discuss before I am discussing it now when and good but now things got a little much because ensemble average is a time independent but I am telling it is equal to time average so if I have to make it an experimental time then I have to make sure that in my time averaging of my original system that one system which I started with in my original system my I must wait long enough but the all the microscopic states are a good number of their fraction of the microstate which is explode so not only that what happens if I want to go from one microscopic state to another microscopic state in my original system there is a large barrier and there are some states I cannot access then of course if I do ensemble averaging my system is in that state but in my time averaging that state might not be that easily accessible so then they are so I in order to implement one in order to ensemble average time average in order to implement ensemble average I needed equally probability but now my time average is in danger so then it was it was started bulge man and I think it was gives that the hypothesis was introduced this color body hypothesis so so the statistical mechanics is the two postulates and one on top ensemble average and time average and equally probability and then your the analytic hypothesis okay so that is essentially that after doing that what did you do after doing that we went on to develop to be postulate with the fluctuation went to develop the different ensembles micro canonical ensemble canonical ensemble and canonical ensemble and some others will do later as I said micro canonical ensemble that we had omega is the number of microscopic states in the system and remember that this is that is the reason this KV is the bulge man constant okay and omega is the total number of microscopic states and it is a beautiful relation or the most fundamental relation in the in of statistical mechanics and a one you can derive it by some simple derivations that we have done that the is writing is as in probability distribution but they are identical because if all the states are equally probable then PI is 1 over omega and then you from here you get to that one you can as well take these in the equilibrium statistical mechanics you can take it as a postulate also or starting assumption now and this of course of origin I was derived by bulge man in in in its kind of kinetic theory approach then we going to do this omega can be calculated in some cases that we did not do too much we will do some later then we went to canonical ensemble where we separated the now we allowed the systems to exchange energy will form there is a masterful construction again done by Gibbs to put this different canonical ensemble energy is exchanged with with in touch with each other allowed energy to exchange and then put it in then insulated it and put it in a bath so that the one gets a temperature is a it reaches equilibrium and with a temperature tip the temperature bath then you put an insulation around it so this each member of the system interacting energy with the other but the temperature is kept constant equilibrium and then one goes on to develop a theory of this partition function and other things so the partition function that we derived is the following this and in if I write in terms of one term system energy levels then so I sum over all the energy levels so this is the classical and this is the quantum they are identical as you can see that these can be written in terms of the e to the minus beta e putting away function here and putting away function bra and ket on the two sides and then integration over this these things and then this becomes e to the minus e i by k B T and when you do quantum a bit of quantum stat mac we get back to that little bit more seriously but right now I think we do not need to worry about it so we can take these as the partition function of the classical partition function of the system now we what we will do now what we have done before but I want to revise little bit after this preliminary division and starting with this equation this classical expression of the partition function and we have already done that the partition how the partition function is related to free energy and all those stuff but I want to go through the two important things because that's where working of statistical mechanics becomes very clear and I want to go through the derivation of the ideal monotomy gas where we can get the partition function exactly done and then we can get from there the entropy we can get the free energy we can get the entropy we can get the equation of state so many other things and some of these things are extremely useful new things and in the process we get an expression for entropy the microscopic expression for if you call ideal gas law which which thermodynamics couldn't give us thermodynamics has no way to give the absolute value of entropy so what we got here in the process is enormously important and it is very important that even though it is the ideal gas diatomic monotomic and diatomic and polyethylene that you understand them well because from there starts the understanding of the rest of the system.