 Welcome back and good morning or good afternoon depending where you are at what time you are listening to these lectures. We are doing continuing from the last lecture on the theory of liquids which is a big subject. There are books written on this chapter and it has been an intense area of research for the last more than half a century, aided very much by computer simulations. I will give a name of the few books along the way where you can take a look into it for more details. In addition to the book that we have here the statistical mechanics of the liquid that we have here there are several other books. In this book what we have been trying to do and I am telling you repeatedly the whole idea is to get kind of a statistical mechanical starting point or statistical mechanical view of physical chemistry and also materials and materials chemistry. So as I repeatedly tell that in undergraduate physical chemistry you have if you pick up a book say Castellan mood or Atkins you will see huge number of chapters, you will have liquids, you will have phase rule, you will have phase equilibrium, you will have chemical potential, you will have kinetic theory of gases, you will have binary mixtures, you will have polymer, huge electrolytes chapter after chapter and much of them the chapters are given you graphs, certain results and certain equations like Debye-Hooker-Lonserger equations, derivation is done in a rudimentary way. Things like solidation are described in terms of pictures, liquids are told you that is the density between the gas and the solid but they do not give good job even it do not describe in the properties of liquids which are so important for us for our chemical industry for biology. So the theory of liquids that have been developed in last 50 years with this theory it is based on statistical mechanical theories and aided heavily by computer simulation nothing would have been possible if computer simulation well not much would have been possible if computer simulation did not play the important role that it played which we will talk about more in detail. As I said in a last lecture that all it started with J. D. Bernal at Cambridge the famous extra crystallographer and J. D. Bernal got into his whims to really find out the local structure short end structure of liquids and this is what we already talked so what do you mean by short end structure? We will go back to Ising model again the famous our starting point Ising model we introduced in the spins up and down we introduced the quantities two important quantities n plus which we call long range order but there is one short range order n plus plus what n plus plus does if now if a spin is up here and the spin up there n plus plus tells me the number of such pairs now when the spins up and two up spins the next to each other that gives you a short range correlation that is why it is called short range order we use short range correlation in many places the short range or local correlations are very very important so short range and local correlations. So more quantitative way of telling of the liquid in addition to saying that the density is between gas and the solid and I already described everything should be in terms of rho sigma q sigma is molecular diameter then rho liquid is typically rho liquid is typically is star liquid is typically 0.8 is a wonderful number semi quantity which is not fully quantity but it is far better than saying density of liquid is between gas and solid density of gas is about 0.1 in this unit dimensionless in rho star density of solid 1 liquid is sitting next much of a liquid is sitting next to the solid. So as a result of the close proximity and the hard sphere kind of repulsive interaction you already talked about the potential that we repeatedly using that because of this part that determines structure of the liquids in the interaction potential as a part of this we have molecules touching each other and as a molecule touching each other they have a lot of short range order that is exactly like the n plus plus in Isaac model the short range local correlation. So my new definition of liquid would be liquid is has considerable short range order but no long range order but at solid has both short range order and long range order a ferromagnetic system lattice spins are aligned most of the spins are aligned and there is a very large degree of short range order of course there is a long range order non zero magnetization. So you need to now think of liquid as something which is ever unique we do not face this thing in the solid state phases the presence and the importance and overriding importance of the short range order that is not in crystal that is not in solids. Solids are characterized by long range order and that is enough the gas on the other hand neither short range order nor long range order. So liquid in then a new definition of liquid should be that it has a short range order okay and that is like in Isaac model n plus plus now we can go back doing next of our things. So how do we talk of this short range order how do we talk of the liquids I impress upon you or try to tell you that liquid is very unique is unique because of its this short range correlations that are in the liquid state that is what defines a liquid but then we do not have the average density does not tell me of that then we need a new language to develop this short range order and that language comes in terms of distribution function which has been described in detail in our book whole chapter is on distribution function. So what we actually want that at the simplest level that one if one by molecule here in the liquid then what is the probability of finding another molecule at a distance r that turns out would be primary importance that tells you locally at the nearest level level how molecules are packed that is the quantity that is picked up by neutral scattering in static structure factor and how the local caging that means the central molecule because of the high density as a very important quantity of liquid the density is large like the Rho star even water which we call low density liquid because of the open framework structure even that has Rho star is 0.72 and without all or a Rho star is about 0.88 or something like that go close to 0.9 as tonight tile is close to again 0.89. So you see molecules are densely packed or every liquid at a main conditions of temperature pressure are densely packed so this is very important to realize that as it is densely packed then what is happening we call a each molecule is each molecule is caged by its nearest neighbor and the strength of this caging determines to a great extent dynamics and static properties of the liquid like specific heat all these things are determined partly by this property okay then how do we describe it we start in a grand way we said okay I want to have not just two particles how the distance I want to have many particles what are this so this particle is at position R1 then say R2 R3 R4 I want to have n particle distribution so that is now defined by by very very simple by one is the partition function configuration integral of mere number of ways of picking it up then I integrate out all the ones which are not in my n so this is a n particle distribution function small n n particle distribution in a in an n particle system so capital is the total number of particles in the system and small n is the number of particles among whom we are looking at the correlations okay so now so basic this is the picture then that we are very ambitious we are trying to look into many body correlations and these terms out to be extremely important in disorder any these languages just this standard language of all many body systems whether it is glasses liquids liquid crystals everywhere this is the language that is so there is a flow flow chart given in the book and to how to go about it how to go about getting this rho n this quantity that we just discussed the n particle n particle correlation I want to know that how n selected particles are correlated that means if they are not correlated of course this probability of n particle will become rho n become will it out to the power n as we discussed in the last lecture the independent so this is a joint ball distribution this rho n that I am talking here this thing is a joint ball distribution and if they are not correlated at all that is the gas phase then I will say okay what is the probability of one particle in one position and the second particle in a position that we just wrote very weak correlation but in crystal not at all if I know one particle position I know the other particle position uniquely in the liquid that is in between but there are certain correlation have you go on particle here the probability of having another particle in a position can be more it can be less how do I talk about that that is what distribution function comes in and we do that but let us go back in the very beginning how it all started it started with the grand equation of statistical mechanics almost the equation of statistical mechanics that is level equation we will talk a little bit about it but we will not do a full job on the level equation that is that is that could be a lecture by itself and we are working on a series of time dependent statistical mechanics a book and probably a course and there we will work little bit more on the equation this equation is in chapter the derivation is given in chapter 4 chapter 4 of my stat mac book so you can look it up there is a not a bad job done there but usually as one of my student was telling today that usually it is in most of the stat mac books it is not a good job has been done but that the it actually had good job was done in the beautiful book by tall man Richard tall man in the statistical mechanics that is that is an ancient book and that is a beautiful book okay so we start in the floture everything starts the level equation then you go to a bbg equation equation born born green car could you have on five big names born max born boglio bog very famous scientist and very big famous stat mac guy and then he is involved in many many things including superconductivity he wrote a beautiful book on statistical mechanics which I used to have hard bond copy then green one of the father figure and stat mac then car could and you have one these are time dependent equations we will just discuss slide them and they can be then made into a time independent equilibrium for equilibrium that at equilibrium that become you have one born and then certain approximation is done to get the born green equations that we discussed little bit so so the level equation as I said the grand equation it work or full n body distribution n body full every particle in this system not it is not n body distribution it is both position and time dependent this is dependent it is same as the Hamilton's equations of classical mechanics essentially same at Newton's equation but this you know the difficulty of Newton's equation in theoretical work that it is in terms of momentum and momentum and positions of the particles that is great to compute a simulation many other things but when one tried to do analytical work in the kind of distribution function you have to put in the terms of delta functions and you have to go to a probability distribution that means equation Newton's equation of motion as such is not useful it is transformed into a probabilistic equation we will do that thing we do that thing very routine in time difference statistical mechanics we write an equation which we stochastic equation and then we transform into probabilistic equation that is a very common game we play that this game went back to a long long ago and the person who did first time is Leoville and Leoville derived this beautiful equation so fn here that I have defined that fn depends on is it depends actually on f as position of n particles momentum of n particles at a time t so fnt related in a shortcut version actually depends on positions and momentum position and momentum all the n particles very very important all the n particles and this is a commutator which comes from Hamilton's equations you know one can give a very simple derivation we will sketch the derivation in the next page but this is a commutator or Poisson bracket in quantum mechanics that become commutator because double equation is a quantum exactly quantum version you can write but in classical mechanics this is the Poisson bracket Poisson bracket with Hamiltonian as this and you can easily see that d a these are just the squares are just dot that d h d p i and d f d q this is this we know in Hamilton's equation but d h d p i on the other hand we know that this is nothing but this kinetic energy term of the Hamiltonian momentum gets a momentum here and this is the potential energy part of the Hamiltonian that gives to the force term so this should be the force acting on a particle i and that would be then come from interaction delta v term all right and this part remains the same these parts remain the same and one can again go back and say okay let me find out the one line derivation of Louisville equation is that let us see that in a here in a in a cylinder what is the change total change of the density this you now remember this phase space density the very first or second class that we talked about phase space density means in my 6 n dimensional phase space a point I can now talk in the 6 n dimensional phase space what are the number of particles in a small volume element in a phase space and now that allowed me divide that number of particles number of points in a phase space point that is number of microscopic states of the system in this small volume element that gives the density of the phase space and that time derivative now I am total time derivative I am trying to get however in that small volume element in phase space particles cannot be created or destroyed so they can just move through like I have shown this picture here they can moving through so there is no sink or there is no source term then when I write that that the molecules moving through then density into velocity then then the way they get is the density into velocity streaming out velocity of the momentum will be the derivative of will be the first term and that is now now this is the phase space density has to be conserved these goes to 0 and what is left here is stated is nothing but and these can be shown that these terms are nothing from Hamiltonian equations comes to these things so this is a one line derivation one can be a little bit better job on this but we are not going to spend more time but this is the level equation so level equation is fine we are we love level equation but level equation was then not it was used in many purposes but it was brought to theory of liquids in by this five distinguished scientists and they showed that the force that is there which is this term now I can try to get that force in decompose the force such that I have n particle distribution function so I have say a four particle distribution function then I have the other particles which are here I want to have these four particle distribution function then I have I can separate the part the acting from the force among themselves internally and between themselves and the one that without sight and that is what is done here that one is within that up to some is up to n other one is from n to small n to capital N all other particles when you decompose that then you can say okay I know I in order to define f n I have to integrate over d r n plus one d r n into the with a f f a full r to the power n I have to integrate that to integrate over n plus one to n those extra coordinates in order to get my reduced description that is done here exactly that thing has been done here so BBG guy if so from level equation BBG guy equation is you know one step process but the important thing to note in BBG guy that in order to get n I am connected to other distribution that means I cannot eliminate this I cannot eliminate that so my n particle distribution gets connected to higher order distribution function this is called higher entry so BBG guy equation is an exact relation with n particle and the n plus particle reduce for each so they you get when you integrate that these other ones then you can here you can show by separating out it goes to n plus one so we are landed up in a trouble that in order to find in order to find n in order to find in order to find n particle distribution we need n plus one this you can understand easily if you want to find in a say you know system where there is an external field to creating in homogeneity and I want to find what is the probability that one at R1 then in order to know that I then see I need to know the force coming from all other particles and I can integrate out and I can show this is connected with two particle distribution function because the force are coming from all these particles so when I integrate out in order to do one particle distribution I need to know two particle distribution then in order to know two particle distribution I need three particle distribution like that so this is called a hierarchy is that we are doing a slightly going at a first phase and slightly going but but you basic idea is to give you the flavor of the subject and the books are there particularly my own book and you can look into that okay so how what do we do now what is the next step dynamical approach of dynamics through BVGK hierarchy is exceedingly difficult and has not been much done very early attempt to do these kinds of things was the Boltzmann famous kind of equation approach Boltzmann tried to do this actually f1 rp r1 p1 t and you found that just like we found here through binary collision it is connected to f2 r1 p1 r2 p2 t so the first BVGK hierarchy was attempted in a little bit limited form by great Boltzmann himself and he landed up that he could do it and so he made the approximation of what is molecular chaos so the that he was heavily criticized for that this is very sad story now however we are now doing here equilibrium statistical mechanics we are at the structure of liquids not the dynamics so we now at equilibrium the time dependent of the distribution so the solution we are looking at is the time invariant that means there is a class of solution which does not depend on time we are not interested how distribution of functions are changing with time and that is done by put in BF equal to 0 when you do that then one can go to make certain change of variables and one is done here equilibrium fn is now talked in terms of you this equilibrium classical mechanics classical mechanics is very important to note here classical mechanics that position and momentum separates out so momentum goes out and so we have left with the equilibrium of the the end particle distribution so this is nothing but a definition or a nomenclature this is the condition of equilibrium when you combine these two go back to BVGK hierarchy we get an equation another formidable equation that now derivative with respect to first particle GN that gives you now separated again just like in BVGK hierarchy we separated out one particle with the others so if derivative with respect to one is now with respect to all other particles so this is the force coming from this is the force which is coming on one particle one from others now I just like in BVGK hierarchy I separate it out I separate it out that one particle thing from the higher particles so when I do that I find that nth one gets connected to n plus one it really makes sense to write down on this one that means G1 or row one then what you will find that KBT let me try to do that that KBT delta one row one R1 plus this term J equal to 2 to n so but I have one so I have now plus plus I have a delta one of G1 U1 is nothing but row one so this is row one then what I find that on the right hand side I have now on the right hand side I have del 1 U R1 U is the intermolecular interaction the total potential energy of the first particle so then I get terms which is dr2 drn and then I have G2 here and then del potential acting on the one particle this comes from all other particles so I have one tax particle here and I am looking at that distribution of probability of that particle here and but I need in order to get that I will see this earlier equation showed that I will be have G2 so this is in a nutshell a this equation for G1 and G1 is row one so row one depend on G2 G2 is the radial distribution function so this is the equation I have left out certain details but you understand that this is the way this is going to go so we are in a we are in a soup we are in a soup because so in order to find row one I need G2 you have to find G2 I need G3 very similar to what Benjamin found many years ago is better to get that perspective that is not even told much of the time that this is this is something exactly what Benjamin paid and he made an assumption which is very similar to what we are going to make now.