 now what we are going to do again i don't need it but i will just start with this okay so this flow chart kind of thing that i am going to follow so Gibbs gave us this ensemble and Gibbs gave us the ql probability ergodic hypothesis ergodic hypothesis to an extend was already in bulge man but this was all in this ensemble that means every every system has a constant number of particles in constant volume v and all of them have the same energy so you can immediately realize that this constant energy constraint is not practical okay because your system is an interaction with surrounding media the kind of a i gave that if this my huge my number of with water glass they are in i cannot keep them in nv i can keep them in nbt but even nbt is difficult i can keep them in npt much more easily that's what i came is to box number of particles known constant pressure and temperature so this ensemble is called micro canonical ensemble this is then graduated to or generalized to canonical ensemble and then goes to other things that will not right now don't need them so this is nve and this is nbt for the time being too enough because we now want to develop try to develop the conceptual of these things why we need them and how do you work with them so when the micro canonical ensemble remember ensemble is a huge number of mental replica and the whole idea of ensemble was that i can talk of a probability that was uh uh introduced by maxillel bolchmann now probability of and i already have my um equivalent a priori probability so i need to have now how do i go ahead how do i now calculate or construct a quantity which can be now given will lead me to thermodynamics the first attempt statistical mechanics did to you know it was immediately realized it was immediately realized but all these great thinkers that the first demand on statistical mechanics is to derive uh thermodynamics because that is the one that is those days or even now explains every all natural phenomena in terms of free energy entropy enthalpy all these things so statistical mechanics was geared from the beginning to describe thermodynamics so now the idea is now i have the nve i have a huge number of systems all have constant energy all have volume v and all have total number of particles so what is different from one system to another system in the ensemble the difference from one system to another system is in the microscopic state each system is in the different microscopic state that's the most critical realization then you immediately realize that what could be a very important quantity some realize by bolchmann very critical quantity is the total number of microscopic states now what is the total number of microscopic states how do i calculate total number of microscopic states and this is a pivotal quantity from which what i am going to tell to you from which whole statistical mechanics came out that the the total number of microscopic states how we calculate it will come a little later so if i know that a system now let us think this is there is a one single in in real world there is one only one system ensemble is my mental replica now my this system particles in the systems are moving they are and randomly moving they are interacting with each other they are changing positions they are rotating and each of this tiny little movement is taking it from one microscopic state to another microscopic state now it does not take too long to realize that if a system has much more microscopic state than the system resides in that state longer time let me tell you something very important again i repeat this if a system in a given n v t condition has larger number of microscopic states then the system they are this all same energy so the system will spends maximum amount of time there now what is then the measure that that state that particular microscopic state that microscopic state which has maximum number maximum weight or will dominate so it is almost like theory of evolution in certain sense we will so that is where this came from bolchman and used by gibbs is that this omega omega has to be a property it has to be a property i will give a derivation but the derivation is again heuristic derivation it has to be a property which scales with the size of the system in a certain specific way we all know that thermodynamic properties like entropy enthalpy are extensive that means they are proportional to the number of particles in the system okay now this thing is enormously large quantity so the bolchman introduced and this will become more this is the bolchman formula i will motivate it little bit more but let me tell you also just like the postulates there is no absolutely convincing derivation of this and probably there is no need many of these things have been verified post facto but i before i motivate further let me tell you little bit more about how this omega comes about and as i tell you that this as i talk let me repeat that this is that one formula is the most important formula of the statistical mechanics whole statistical mechanics came from that okay so this is a very simple my picture of getting the giving you an idea of the in a micro canonical ensemble the number of microscopic states that you have so i have this four energy levels and four energy levels and total energy is fixed eight how many way i can distribute that so these are the here you can distribute your system into so arrows of the systems i am talking of ensemble and these are the microscopic these are all distinct microscopic states okay and i have taken care of distinct usability or indistinct visibility so each of them is one state now what happens if i increase the number of number of particles in the system if i increase the number of particles in this number of systems with the energy fixing eight then there are many more ways to arrange it similarly if i can increase the number of energy levels then again many more ways would be it okay very soon in the real world in when the atoms and molecules are moving there is a huge number of energy levels huge number of energy levels available to the system a one cc of a gas at normal temperature you can calculate at macquarie's book has given a way of that we discussed it has 10 to the power 33 microscopic states that's the kind of huge number you are talking okay so that is also the difficulty of doing computer simulation because there is a huge number of microscopic states one has to sample so the microscopic states increase what do you call exponentially as the as the size of the system size of the system by meaning that the energy levels so two things are coming energy levels of the system are determined by the system itself by the interactions for example by sardinger equation so i am having a self-consistency here because i am going taking a system then calculating it's all the energy levels then i am doing a ensemble construction is a very pretty very beautiful then i am going to ensemble construction i am saying okay i have the energy levels now and let me see my i have infinite number of billions and billions of my system which i am going to distribute with the constant that energy is concerned this is nve okay i realize very quickly that number of states that are number of arrangements that my omega is is is growing exponentially you can do a simple calculation to show that that it can a simple combinatorics that it grows exponentially as soon as that grows exponentially and i know my thermodynamic properties are determined dictated and determined by omega so in that case i am now having a beginning to have certain insight that i have i want to describe thermodynamics and i want to have an extensivity then i want to preserve the extensive property and i want to present up what is the scaling so this is now why it is entropy and that came from a much much complicated and detailed derivation of Boltzmann that why ln omega was given to at constant energy and volume is entropy that is the way it turns out that this is the you can regard that this quantity ln omega omega is more fundamental and ln omega is the quantity that we call entropy so this is essentially a definition of entropy do not consider it as though it went like that entropy came first and then came omega but for statistical mechanics it's the other way around omega is the primary quantity and that determines entropy so it's just exactly the other way around now there is a derivation that is given in many many places and i'll just case the derivation though as i told you this you better you can regard as well as a postulate and sometimes when i teach in class i say statistical mechanics should actually be considered consist of three postulates and one hypothesis and one of them this is the postulate because in in many sense there is no realistic sense derivation of this but whatever so this is the a of omega the derivation is sketched here and this is the derivation given in every textbook that you define an entropy function so this is the derivation is given in my book and the standard derivation in every textbook is to define your function entropy function in terms of probability the energy levels that i discussed that and your point systems zero one two three i think i have i remember that i have two here four and two here one just there are many like that one of them now so define this function this what actually also came from which man and i have some interesting to tell you that i'll so basic idea was that i define a function which is proportional to the probability it is occupied its probability of the system system in a jet energy level because these are systems arrows are systems they are systems of my ensemble again my energy levels are obtained by solving Schrodinger equation or even classical mechanics by all the positions and moment available to the system which is called the phase space we describe little bit of phase space in the last class and trajectory so they are obtained from a given system now i construct my mental replica and put my systems my arrows into different energy levels okay so once i do that since they are all equally probable i can find out what is the probability my system is in a given energy level j then i construct this quantity so now then the derivation says okay they are all equally probable so pj is so it is minus kb all all the sums are the same probability is the same so if they are n number of total number of arrangements and probability is ln omega and by p is 1 over ns because each of the equally probable so this cancels and then it becomes pg ln pg is 1 over omega pg is 1 over omega sorry so this cancels and then that becomes the 1 goes to 0 and minus ln minus ln 1 over omega minus so plus kb ln omega so this is the standard derivation that comes to textbook that s equal to kb you know but even then even in this derivation which is heuristic but it does bring in little bit more more physics in the whole thing that you derive a function so this is the entropic function these function defines entropy is the function that it defines micro canonical ensemble this is the this is the beginning of thermodynamics a beginning of statistical mechanics actually or a relation between statistical mechanics or be a relation of be you know perhaps the most important relation where everything flew from or these one relation that s equal to kb ln omega because because every state is equally probable see what what the reason i am little bit fumbling is because when when statistical mechanics was formulated boljman and then gives you have to understand that the theory of probability was almost entirely developed in the mathematics they were developed in a great extent by 19th century already so the people and then they had both boljman and gives had the understanding that they are going to look at a extensive property okay and then new probability is going to scale probably to 1 over omega but omega is going to scale as the something to the power n okay so that brings the lockdown so the whole motivation of introducing lockdown there and i am again and again saying this is a definition of entropy function it remains to be shown this s is indeed thermodynamic entropy and that was done later so this was introduced as a function boljman definition this was h it was not s it was shown by Gibbs that this function is the s and we are using it as s okay and i told you that i will say you something really interesting the interesting thing is that s is proportional to ln omega a proportional constant kb is boljman constant nobody yet knows why it is why kb is 1.38 into 10 to the power minus i think 2017 no no he caught in minus k but he has this was the his era of time let not get into that so that is why it did not want you here okay a is equal to ln omega this is introduced i would like to tell as as the primary postulate of statistical mechanics but you can say no this is an a function defined by ln omega and it has to be ln omega because omega is an exponential function of the total number of particle size of the system so it must scale as it must be ln that brings in ua to the power n n comes out and entropy is an extensive property okay and this as i said that the people there very quickly realize that thing but again i am not trying to justify that the justification of that s kb ln omega s is indeed the describes the fundamental equation of state of thermodynamics and entropy that is the a detailed derivation of that exist in a transition from statistical mechanics to thermodynamics that thing we will do in the next class where we will we will we will see how s equal to kb ln omega can can indeed describe thermodynamics that's that's a you know still unfinished agenda that is the thing we need to do next to convince you that s is indeed the entropy once that is done and as i told you this only equation that you need no other equation you need from there you go to my canonical ensemble you go to grand canonical ensemble your isothermizable ensemble everything omega because you realize immediately that the properties that will come that macrosport properties that particular state of the system which will have maximum omega maximum omega means maximum entropy stable state free energies minimum that also follows from this so whole of thermodynamics follows from that one equation entire entirety which is amazing no these are all the interactive these are not what i said what i said is it is the when maxwell did it the ideal gas that see whole kind of theory of gases that you read in your undergraduate is for ideal gas so in ideal gas means particles pass through each other but that was not allowed so he derived pb equal to nrt then he derived pressure equal to one third mnc square they are all ideal gas but at the same time he allows the exchange of momentum with the wall which is the billy at ball so there is a mix of interaction and non interaction which is fuzzy very fuzzy that's not the way but that's why maxwell did and got the right results boljman that's what led boljman to remove the inconsistency maxwell and try to introduce the interaction if i ever teach time difference stat mac then i always start with boljman kinetic equation that means the f2 frt that that's where you take a two particle collision and f2 which is two particle joint power distribution of r1 r2 p1 p2 and by interaction that is changing but that then you explicitly have the collision and there's a term that comes in called collisional cross-section which has molecular diameter in it see in kinetic theory of gases you don't need pressure pv nrt pv nrt it doesn't have a molecular diameter okay then all this you don't have except when you go to calculate the viscosity then you need the molecular diameter maxwell kinetic theory of gases is many many places internally inconsistent that was boljman tried to do and and he could do only partly but these are very good issues and very fundamental issues that we bypass in our undergraduate physical chemistry but i always believe the interface between equilibrium and time dependence statistical mechanics is one of the most fundamental and most intellectually exciting and pleasing to think about these things okay okay we stop here now