 So, what we, these some, I am telling that these are one of the most difficult part of statistical mechanics which are usually not taught in courses and probably some of you who are here know the difficulty, some are not there, so I will try and repeat about that. So, basic thing to remember which makes life really much, much simple in equilibrium statistical mechanics. Now, equilibrium statistical mechanics is the discipline I explained at large, you have studied that we, we, we, at length you have studied that this is the one we use to describe phase transition and all these things, so the say pressure of a system or energy, enthalpy, what we do in terms of the radial distribution function, the, the thermodynamic properties and that a very important quantity is two particle correlation function that we will do probably somewhere for 10th lecture from now will come the radial distribution function which is the quantity which is observed experimentally in a many different ways. So, at one side of this equilibrium statistical mechanics is this thermodynamic properties like entropy, other side the, is the information about microscopic arrangement, how molecules are arranged around each other and as you can understand this microscopic arrangement plays a very important role in chemical reaction. If two reactants A and B are going to react in a solvent which is the common scenario that we do then that is done through these, through the certain equation of motion but the intermolecular arrangement plays a very important role. All this information comes out of equilibrium statistical mechanics and equilibrium statistical mechanics is a very major discipline not taught here but in when I did doing PhD abroad then we had three courses in chemistry department on equilibrium stat mac and going over to time dependent in the last semester last a, 201, 202, 203 from 20 the graduate level course used to start. So physics also had three courses so this was the kind of importance given to that and everybody was made to take 201 so then quantum used to start at that level somewhere like 2667 quantum used to start however in between there is a difficulty but now again stat mac is becoming very popular because we are having all these computer packages and we can calculate many things. Another thing that is of much interest these days is the nanomaterial synthesis and the nucleation phenomena and the phase transformation which is connected to phase transition that is also comes under the essentially view of or at the borderline between equilibrium and non-equivalent statistical mechanics. So this is the preamble I probably given every class that why we need to study and what is the basic perception but every day it will be different and different things I will bring in to motivate you and keep you focused into this subject which is either formidable theoretical discipline we do lot of on quantum mechanics in chemistry particularly but the advantage of quantum is that by the time you do quantum one or by the time you do suddenly quantum two you are into Routhans equation and all the perturbation theories are done and you go to numerics so it is fairly easy from there and there are all the packages which are amazing packages so huge number of people doing quantum and it is easy to publish papers because of the interface with organic chemistry or materials and they deal with molecules and chemist love molecules in statistical mechanics we also deal with molecules but rather with molecules is taken in term of intermolecular interaction and also molecular shape and size so there is always good to keep these two in mind now coming back so what we did yesterday was that I said the most important relation is the entire statistical mechanics is this relation and as I said that these was derived in as a function which gives a measure of the state of the system by Bolchmann however these in it was brought in equilibrium statistical mechanics used in space input by Willard Gibbs GW Gibbs and there is really no derivation of this expression as such so this is to be taken as a definition that is a very important point remember this important thing from which whole of equilibrium statistical mechanics follows entire in its entirety is because of this Bolchmann formula so that is why in his grave and also in his bust in University of Vienna where he was they have just written this formula in his also graveyard just this thing is written you can do a Google you can see them I saw the bust in Vienna and of course there that's the University of Vienna those days because of Austro-Christian Empire Austro-Hungarian good Austro-Hungarian Empire the Vienna was the capital right so it was a great great people were always all the including Mozart and many many people were in Vienna it's much less crowded now actual population of Vienna is now less than what it was I think 50 years ago or 100 years ago okay so how do we go from here and start so I would prefer to consider these that we have two postulates connected by an ergodic hypothesis that we start the statistical mechanics but two postulates connected by hypothesis is great to say that we really have ensembles and everything come in which is so much in the fast multigrammical ensemble but these also should be regarded actually more of a postulate except that one can show unlike the two postulates where you do not you cannot is prove them a priory you prove them you know posterity that is similarly thing here also no priory motivation probably there is motivation there is no priory proof of that what is subsequently shown is that these function is is the same function that we call entropy in thermodynamics and entropy in thermodynamics you remember is given by the Clausius and it is dq reversible by T right so that is the way if the connection between thermodynamics and statistical mechanics is very important comes so I will go and then I think one of you had a question so please ask the question so now the basic idea then was that we start with consider a small variation in microgrammical ensemble this is the a system is characterized by total number of particles volume V and energy which are fixed but I can always consider that I can change the number by little or volume by little or I can change the number energy by little so that's so the then as a result of this variations that I do there is nothing wrong in this and nothing nothing obscure about this okay so I get the variation of entropy as ds however this already beginning to tell you something very interesting that because you have seen this kind of things dsdv dsdv because this is we routinely use it as a temperature in our derivation but even aside from that you have this oiler's equation oiler's relation you know oiler was a mathematician who started doing his study at the age of 24 and for mathematics 24 is really really really quite old and then he did some amount that means for few years not very long time but he has done huge amount of work in a relatively short I think he died young also but he is in every field you go to hydrodynamics there are lot of contribution from oiler so essentially to oiler they were certain combinations and a very interesting combination that temperature is the conjugate of entropy or entropy is the conjugate of temperature pressure is the conjugate of volume right and the chemical potential is conjugate of number you know it in many different forms if I call it free energy g then I will bring it on the side e-ts e-ts is a e plus pv e plus pv is the enthalpy and then enthalpy minus t is the free energy these things goes over both in the in the g and a where hemorrhage and gives free energy but it is the same however this one now can be used with certain variation because we know it dd equal to tds minus pdv that condition we have in these are the relations that follow from these things why you are doing that why you are doing oiler equation all these things because you want to get an handle on these things and that then gives us the one that I want ds dv from this relation I get ds dv equal to p by t and you have to understand just like in the Newton's equation or Schrodinger equation we write down things they are a very strong plausibility argument we write them down you know as you know very well there is no relation of Schrodinger equation it was written down by transformation so here also one writes down things and then see that whether they work or not like this most important equation okay then this is of course we know very well is our multi component not necessary for multi component but this is the very your familiar equation so now you can call that you know it very well that this is then indeed is the entropy the advantage of that now if I make these identification the advantage now is the following I can know from micro chronicle ensemble these are the condition that means temperature is the derivative of log omega with respect to energy which intuitively makes sense because omega is the total number of microscopic states of a system belong to a system in the micro chronicle ensemble characterized by nve then if I change a little energy then the number of microscopic states that increases by this quantity 1 over t okay now similarly you can have definition of pressure in terms of micro chronicle ensemble then you can have chemical potential in terms of the omega v now these are the very very interesting one corollary which I will discuss a little later if I come so this is what the thing that I wanted to review of the micro chronicle ensemble then you may continue to do canonical ensemble now you have one of you had a question yeah please tell me in micro chronicle in md right very good question yeah excellent question so when you do these calculations these like phase transitions and this creates an enormous problem and I give you problem we do NPT like I give an example right now we are doing and also do we gives a gives ensemble that means because number of particles exchanged and that is new PT that means that's the gives ensemble NPT is isomer isobaric so all these things all these different ensembles are brought in but at certain cost and I tell you the cost right now we have been trying very hard to study ice water interface other people have studied you can see several papers are there so two things we are trying to do one of them is that the why is very interesting question we are just writing the paper that if you take like argon which are characterized by energy potential then the FCC lattice and the liquid interface is rather broad it's about 4 to 6 monon layers or molecular layers but water interface is sharp so we try to understand why water interface is sharp and it's a huge importance ice water interface in in many many different contexts and we came to that from there is a class of proteins called antifreeze protein which is lot of theonine with hydrogen bonding hydrogen bond ray so these antifreeze proteins with the large number of theonine in this top ice growth ice remain static so you want not just to growth we also so from there we went okay let us just study the interface first and I knew this problem because it's a very difficult problem because the when the something an interface is 4 to 6 or 8 monon layers then you can cannot study them by any experimental techniques you know like I cannot do neutron scattering on that that means micrometer or millimeter length I can do imaging but the spectroscopic imaging cannot study that length scale I can do TEM but TEM means it's static it is not dynamic I wanted to do the fluctuations at the interface and then when we try to my student is trying to two students are trying to stabilize NPT it is a hell of a problem because they just either the ice melts back or the ice forms that because there is a barostat an idiotic barostat is giving forces into the system and then of course there is a thermostat so so the people who are continuously studying an NPT or gives ensemble they have this difficulty and you are right there is no way I can do NVE in that system because if I do NVE then I would not be able to remove the latent heat see latent it has to be removed you know otherwise I can stabilize and you move everything and say it is NVE then that will stay in that system unless I can have such a large system and humongous system then NVE will be able to capture because this growth of crystal or melting will be captured as a fluctuations in the system but that's not possible here so but the kind of systems we do we have maybe 5000 water molecules and people are doing those who have done successful simulation they were very long a they were crystal slab and then liquid on the two sides and different people ask different questions so we absolutely right that there is this huge problem and that's why there are two kinds of simulation people one of the people like us who are application and not respected and there are people who are developers who develop potential develop methods like this guy is in Italy what is his name Perinello those who then toady and value who did umbrella sampling Perinello group did many things have been issue plus meta dynamics so the way to come around these things in phase transition then use these specific sampling you know that means you which are called method of constraints which are like our lagrangian multiplier which are method of constraint you move things around in the direction you want it to go and you equilibrate along that that's the way you do the simulation so there is a huge problem one probably should at the I have two chapters at the end of my book on computer simulation I want certainly to do the chapter 31 which is on computer simulations where I have devised new ways of doing periodic images minimum age convention I did in terms of ising model and it's so easy to see what is the method of images not sorry minimum image convention periodic boundary condition but that's in the chapter 31 of my book it's very simple I was not happy with the conventions give the explanation given a alien tildesli or other books because it seems too complicated it's so simple a method but then you take the one-dimensional ising model the all these explanations and all these statistical mechanics becomes much simpler okay so does that explain partly your question is the difficult question and difficult to answer at the same time because that is not done okay now so we now from then on we went so what will happen I am I'll be spending more time on this because I want you to be comfortable because this is intimately connected with all our simulations and all the interpretation of our results that how we do statistical mechanics and once you become familiar with this is really very very nice things and you can you see when you for example you are doing lattice calculations very recently all the surfactant work all the lipid bilayer work were done by saying how many ways I can pack things and then how many ways you pack things from there you get omega and then you go on from there so this is the partition function so in micro chronical ensemble omega is the partition function or sometimes it is also called so in micro chronical ensemble this is the partition function and this is the thermodynamic potential thermodynamic potential is the quantity that takes an extremum value actually if you think of that you always maximize the thermodynamic potential so free energy we write as cavity but we way it should write at a minus cavity so that is the one that follows from Boltzmann's arrow of time which was Stephen Hawkins and all these people wrote so much book on that that the arrow of time is because the entropy is a system left by itself goes to a state which has maximum entropy that's the Boltzmann's principle and okay so then from but Mike as we know micro chronical answer please ask questions micro chronical ensemble is not very useful so there are several issues one issue is system at equilibrium but away from phase transition the properties of water properties of isotonic trial or diamethyl sulfoxide or properties of a block of iron or or whatever solid there is one thing like that another thing is phase transition and the many other aspects that we study so when a system is at equilibrium in a stable state away from phase transition then if I do describe that then NVE is not the greatest greatest of the system because we are too much constraint real world NVE is not there so we have to go to with now now we start relaxing we go to NVE NBTE then we go to NPT one by one we'll do that