 Now, I come to the rational between the two postulates of statistical mechanics and the I already talked about the algorithmic hypothesis. So basic idea is that we do a huge number of interacting systems and this section dimensional phase space that Boltzmann and Gibbs developed Maxwell did not. Now, so this motion just the question you asked motion is determined by forces and the instantaneous positions and intermolecular interactions that is what Hamiltonian sequence comes in and so the path followed by the represent you are executing the natural this is very important this Kubos language we are a great stat mac guy called Rigo Kubo their natural motion so what we are following is a natural motion of the system in the phase space so trajectory is very important no external force is there okay so relation then we want to establish now the relationship between such a microscopic trajectory so a trajectory is a microscopic thing and measure properties and this is I could do that I could calculate the pressure but if I could do Newton's equation hold the system then all the if I could tell the atoms and molecules bombarding against the just the way Maxwell did in in his ideal gas you know calculating the pressure pv equal to one third mnc square I could do that but unfortunately I cannot do that because I cannot calculate all these things okay that's why we are now going to do fast we do a on some time averaging this is the time averaging that I already introduced that you follow it for a long time this is the kinetic energy mv square the average over a time trajectory which particle this is exactly what I wrote down sometime ago here I am just kind of recapitulating the whole thing yeah yeah well a little bit of a problem yeah that it should be yeah I think one has to do yeah this is always a problem okay so so time average from theoretical point of view it is impossible to generate a long-term trajectory of a microscopic system with large number of particles that I have been telling so we had no other option but to go pure probabilistic description the one that I describe fortunately the probabilistic description can be developed with the help of only two postulates and one hypothesis that's the beauty of the whole thing so then that is in order to do that Gibbs came out we let Gibbs came out with the brilliant brilliant mental construction of ensemble and that was really amazing though it was one should remember that he followed the same ensemble which Boltzmann followed which is NVE so all these postulates two postulates and they are got a hypothesis are valid in the micro canonical ensemble NVE ensemble is called the micro canonical ensemble as we will do more later so so this is so this is a a language is that the kind of uncertainty or the imprecise is the same statistical description so if these are important interesting thing again I am as I am saying recapitulating the concept of an ensemble based on the realization that the system and equilibrium must have a very large number of microscopic states this partly probably probably my query the natural motion of the system at non-zero temperature takes the system to a finite fraction of the states there is a very delicate question here which we will deal much much later in a in a time that is comparable time of measurement is very important even when you are doing experimental measurement we are going to a very tiny fraction of the total sample space very important and there is a concept of self-similarity and these things will come in to do that measurement of my body if you now consider the system difference at fixed energy at all this time then the trajectory moves on a constant energy surface that is the NVE and according to the laws of mechanics a classical quantum and that generates the trajectory but we do not need the detailed information is extremely important we do not need the information at least for the ensemble based equilibrium statistical mechanics that we let gives developed we do not need there is a again again saying but we need certain realistic understanding of this trajectory and realistic understanding of what we are talking of averages and I instead of taking the time evolution of the system or the microscopic states as required by Newtonian mechanics we constantly a huge number of states and as I am keep saying is NVE that is where hold this thing was set up but we going from NVE to temperature T is a little different thing we have no control over microscopic states of the system and neither do we want to have any control we do not want them but we want them to flow over the phase space that is why equal probability and organic hypothesis is important the flow in the microscopic state in the phase space in the trajectory should cover a significant part of the system okay so ensemble represent a collection of mental replica and is called micro canonical ensemble trajectory system travels in a constant energy surface all the points in trajectory are equally likely that is what the as I said there are no other option and good fortune of us that this works out okay actually it was still very doubtful for a long time even in 1950 there is a guy called who wanted to push Shannon entropy called Jains who wrote to Ullenbeck saying to making that okay because of your difficulties stat mac you should probably consider information entropy and this kind of as a mainstream stat mac so Ullenbeck gave the famous answer saying that since the Lianian proved that one can explain statistical phase transition within statistical mechanics we don't need your Shannon entropy and that turned out to be correct we did not need that so complete trajectory of the system can be obtained if we cancel anybody neutral situation that is the one partly we do when you do time-dependent statistical mechanics that get part of it the trajectory generate of the few tens of nanosecond and few interval and we time interval we dump it even then huge data is there which comes handy sometime so this is kind code on details of the understanding what how could be the phase space can be quantified the smallest parameter turns out to be Planck's constant and we this analogy is beautiful these but I am not going to go into two detail on that right now I just want to say the following so these are the different kind of ensembles that I will do in the next lecture so the main idea of this lecture is something which my intensely fond of is these two postulates because I figured out to an extent the interrelation between the two postulates the necessity of the postulates and the algorithm hypothesis is only when I wrote the book so all these things can be connected and all these done was done by again it is very strange the person who does does everything like Maxwell did everything of kinetic theory of gases all the equations that you do under-graduate Schrodinger did all the same thing and then Willard Gibbs did much of it almost alone there so this time average and ensemble average you all did that one thing as this question was asked that when does why the problem is that this is a question a problem we worked out that if you have a system like that let's see it's stuck here these are energy energy landscape and the particle is random executing random work here and if it gets stuck here did we simulate it and we simulate it and we simulated four years we couldn't get a converged result the guy who was doing or not stupid that is because the system as you make it is larger then you face larger and larger barriers larger larger deeper minima and you are thrown out of the track we never got converged result this is the reason this is the thing you did the ergodic hypothesis if ergodic hypothesis so these systems was ergodic later I found out that there is a famous paper when human and stein who said diffusion and these are beautiful paper by I go to Sekisan and Sholgoth Banerjee that on this this issue but we were really put on the right track by Newman and stein two mathematicians Santa Fe Institute and current institute saying that one-dimensional random is called random energy landscape diffusion in one-dimensional random energy lessons pathological in one day that exactly what turned out to be correct so then postulate the quality probability and then its state is visited equal number of times so it a very long time the probability every state is equally problem and then the of course these two are connected very very important two postulate system and up to be supplemented by ergodic hypothesis these are with assures the practical validity of the fast postulate without the ergodic hypothesis your fast postulate as you are asking is mute okay mathematicians and this is then this done by sometimes called bulgeman senai the senai is the one who first proved ergodic for a system of hard disks like me at balls but on a 2d he did that real proof is amazing is the boobinovich and senai all mathematicians you know system hard disk is a beautiful beautiful thing that was done and so the ergod states that during its trajectory in phase space a system is free to explore all the microscope it has to be free here it is not free that guy is not really getting stuck there it has to be free sufficiently long period of time spends time in a state that is proportional to the volume of the state in phase space that opens the room for NPT or NNVT ensemble where energy now plays an important role so it's like that I think I think I will stop here today the measure of ergodicity and all these things I am not going to go into but but I think today yeah all these beautiful measures are there this is essentially done by Dev Thirumalai and mountain by doing the how it is connected to diffusion is given in my book the all the details things and how you talk of diffusion is very interesting as I was telling one of my lecture here that you have to if you want to talk of many body diffusion whole system is diffusing then you have this kind of quantity that's what Thirumalai and mountain used to show the breakdown of our gaudy series glass transition is approached but that means you are not talk of one particle you are thinking of whole phase whole configuration space of the system all right I think I think I'll there are many other things but I'll stop here to recap started with we started with the probability and statistics central limit theorem and several other interesting things then you went to space space trajectory and we discussed to next what will happen we will do starting with these postulates of so in my book I talked about realization of promises we now go from micro canonical ensemble to canonical and canonical ensemble and then when you go to fluctuation we start deriving the beautiful equations like mean square fluctuation specific heat and all these things and then the other other systems will follow any any questions sure right right right right absolutely so whenever you have orientation the phase space becomes larger absolutely so for example when you are doing water molecules you have the orientation sure that you see that at a level formal that doesn't pose any problem the reason is that you just expand the you know you expand your notation x now has position and orientation and P will have now momentum all all kinds of momentum angular momentum so angular momentum so one reason is that you know it it it decays very fast usually so its conservation hardly pose a problem for that one momentum conservation is also doesn't pose too much of a problem in most of our calculations but yes but the formalism part nothing changes see this is a essentially like in sorting an equation you write down a wave function psi you don't know when you are writing down sorting an equation of a many body system you do not think of solving it at that point you are writing and so all the all the theory that you prove with a psi and everything that is just it's just formal formalism to whether it has particle in a box or whether interacting systems and electron gas nothing matters so you have to what I am trying to say you have to separate out the formal path and separate out I once in I was the good friend of a friend called Jim Lille who was developing a theory of scattering with John light reactive scattering I think called Lille light theory and he was all the time writing this very very difficult equation with momentum angular momentum position so I always told him Jim how are you going to put these things he said Bhima let me develop the formalism first he developed the formalism then he put in computer and it became very famous theory so when you are developing the formalism you have to think of applications but do not get bogged down by applications at that stage so yes to you when you are writing down phase space or writing down these equations or partition functions it does not matter whether you are x as orientation or position you have to be able to think like that you have to be able to think in a very formal way but at the same time back up your mind you know when you are going to do applications these formal things will become real things and that happens that much faith you have to have science is always for optimist people science is never for pessimists I have already told you that this is the third lecture fourth will start with the partition function and then we will what level equation is there but I might not do level equation to seriously level equation is the equation that allows you to track to generate the trajectory in phase space is a nothing but a Hampton equation I am not going to do that instead I will continue with the verma ensembles I will go to fluctuation and from fluctuation which is a wonderful chapter I tell you that from there we will do monotomic gas and diatomic gas but everywhere it will be very practical like when I do monotomic gas I will tell you want to tell you about the use of that in modern day real life problem like protein dn interaction dark dn intercalation and that has been used to do find out the binding of chemotherapy to act down on my sin to dna and then again we go to see the very interesting thing that entropy comes with the monotomic gas beautifully but the concept of rotational vibrational entropy those are very important thing rotational entropy in molecule usually 30% of the total entropy that comes to diatomic and then all your drives like the normal modes your the fact that CV is 3 by 2 R all these things comes from just a diatomic molecule is amazing that how much you can know from doing monotomic and just diatomic then we will do is model which is the first really so monotomic gas with a non-interacting diatomic we will do non-interacting first time interacting system we will do is model okay