 So, the assumption that was made by Boltzmann, Boltzmann many years ago was F2 in full position and momentum I am not writing the F1 that is what he was criticized and the story is that nobody knows that he committed suicide because of the huge criticism he faced because of that because he introduced the concept of probability at any time when everybody was very termistic, very mechanical. So long, long later, 50 years later, more than 50 years later probably almost 70 years later John Kirkwood, John Gamble Kirkwood, Caltech made this similar approach. He said okay, I want to get rho I1 D2, well inhomogeneous liquid rho1 is not important. Important D2, radial 2 particle distribution function that is what we have discussed yesterday. We get the static structure factor that is of experimental. So, G2 is the essential quantity of liquid state, but I cannot get that because G2 is connected to G3. So, we made this superposition approach, Kirkwood superposition approach probably should be called Kirkwood Boltzmann superposition approach solution that G3 is now G2 G2 G2. That means if we consider 3 particle in a of course there are many others around here but we even here but we want these 3, we want these 3, these 3 R1 R2 R3. What is now assumed is that these probability distribution of 3 particle correlation is a product of the 2. So, G3 is now given by product of 1, 2, 1, 2, 2, 3 and 3, 1 is called Kirkwood superposition approach solution. When you make that and put it back into the, we get, put it back into this equation, this equation. So, now if I am doing an equation for G2 n equal to 2, then n equal to 2, this is n 3, n 3 now I make G2 G2 G2. And of that now I can integrate over 2 and 3, I want these G2 here I make as 1, 2, then I have 1, 2, 2, 3, 3, 1, then I can take the 1, 2 which is out of this integral then I am left with the integration over 3, 4, 5 and all the n particles and G2, 3 and G31 that means you get a simplification of the equations, we are not going to go into details of that but with that one can approximation the equation to a simple Born-Gren equation which is something which we, so what is the result that we get when you do that, the I do not have the Born-Gren equation here, you have to move to, so the from Born-Gren equation which is the much, much simpler equation just as I described then you get the radial distribution function of the equation, Born-Gren equation. And now you plot that against R, so if the real equation is something like that then Born-Gren equation falters on 2 cows, Born-Gren equation gets the peak but does not get the peak full and Born-Gren equation does not get this maximum in a right place. So, Born-Gren equation falters, so Born-Gren equation falters 1 does not get the right, does not get the height right, height of the first peak out of phase and Born-Gren equation did not do terribly a bad job but it was not good enough. Then there are lot of attempts made to extend it, to lot of attempts made to extend the YV equation. So, all the attempts get some results, some good results and till 1990s people are working and but then what happened the increasingly more and much better correction to Kirkwood supervision approximation brought in and improvements were shown but it did not work out that well that means it has become numerically very intensive and somehow or other a closure has to be introduced because you know as I said G2 in terms of G3, G3 in terms of G4. So, whatever approximation you do at the level of G3 turns out to be critical in determining G2 and in addition to being the numerical it did not quite it was the progress was not satisfactory and this was still those certain people are working this line of research that means from your Born-Gren equation is exact going to radial distribution function kind of as a road block this is something one should know that it did not work out that well then what happened then something very interesting happened because science always has the way to find it finds its path and so that what happened is a different approach was taken that approach goes back to 1920 when Onstein-Zernicki introduced a beautiful equation called Onstein-Zernicki equation to describe the critical phenomenon 1920 to describe critical phenomena light scattering at the critical phenomena these it becomes opaque you know the critical opalescence that this is the huge fluctuations that they they could even that time it was very clear that there is because by that time Onstein's theory of diffusion and the fluctuation theory of Onstein was there I said already so delta is square equal to Cv. So, people understood that there are huge fluctuations are taking place and by that time why this equation was not even dreamt of people had level equation but people did not have MVG KYC equation Onstein-Zernicki took it to themselves we realize something that is though it is a large scale fluctuations taking place that give into critical opalescence there is also short range correlations. So, radial distribution function or the two particle correlations diverging but short range correlations need not diverge I can build a long range correlation just like we do in Ising model out long so long range correlation can come out of short range correlation then how do I do that what the Onstein-Zernicki equation does they separate out they said okay two particle correlation which is called P R correlation H R H R is nothing but G R minus 1 and one thing that I should tell that when I the way we have defined here to radial distribution function or two particle correlations that but then it goes to 1 it goes to 1 because particles become uncorrelated and this is normalized G R is normalized P 2 R 1 R 2 by rho square. So, when they become independent of each other P 2 become rho square so it goes to 1 which is going to 1. So, it makes sense then to take out it they make sense to take out that one so then radial distribution function one take one out and get the P R correlation function H R this is a non-trivial quantity because one is the trivial thing. So, this is a measures the correlation in addition to the product thing and what they said okay introduced a function called direct correlation function. So, two particle correlation in front of all in present of all other liquid particles is there is a one thing which is direct between them the direct correlation function and then there is something which is propagating through others. So, these correlation between these two is propagating between that because these molecules are interacting in addition to direct there would be something which is propagating through the other particle like in hydrodynamics there is a term called objective term which is very similar to that or many other terms that we have that there is a direct interaction between them and there is interaction propagated through the medium. Then they said okay if it structure is very interesting if H R equal to direct correlation plus other particle is at R prime then I have a pair correlation up to R prime then these guy and these guy have a direct interaction. And similarly there is a pair correlation then direct interaction that is the one summed up here. So, Rho H R then gives the probability that you have a particle at that position alright or Rho C R you can also write but that no position dependent. So, that comes out now these a beautiful structure which can be Fourier transform and then it is a convolution and Fourier transform over convolution is a product of the Fourier transform of the two functions that is why the H K becomes C K plus Rho C K H K this is very very interesting because now I can get H K from there in terms of C K. So, H K let me solve now H K in terms of C K is I bring this H K here so I get 1 minus Rho C K so H K is now C K by 1 minus Rho C K. So, my Fourier transform pair correlation function is given in terms of direct correlation function that is very interesting that because I know the structure factor is given by S K is given essentially by G K or by H K. So, my structure factor is uniquely related to now structure factor H K is 1 plus so if I neglect the forward scattering this term is neglected then I have I have S K is 1 plus Rho H K and I have H K is C K is 1 plus Rho C K is 1 plus Rho C K is 1 plus by 1 minus Rho C K and then I get S K 1 minus 1 over C K. So, beauty is then that the H K is experimentally accessible this is experimentally accessible this is experimentally accessible. So, I get directly very important quantity is the direct correlation function. So, this makes direct correlation function of this approach so useful and then there are this another beautiful one GR now if I now take the Fourier transform of this quantity then S K almost mirrors the almost mirror GR and except this maximum is near to pi by sigma nearest neighbor molecular distance and structure factor this is what experimentally we get these probes these beautiful thing probes this nearest neighbor correlations and that is committed to C K and C K plays a very important role C K or C R plays a very important role in the statistical mechanics of liquids. But note the beautiful structure of S K and that this is very similar to what you get by GR they are not exactly the same because contribution here comes from other peaks also or second peak and third peak. But this predominate this order at the K equal to 2 pi by sigma measured in neutral scattering is a manifestation of the short range order. So, we started by saying liquid is unique because of the short range order that those short range order are you are seeing here reflected in radial distribution function and in the static structure factor. These are extremely important thing to the extent I believe that GR if not S K should be taught even in the high school level because that what determines liquid not just density you know I can have pretty high density liquid gas phase do not have at the critical do not forget above the critical temperature you know I can have I can have this is my pressure temperature plane. So, I am at the temperature here I can go to very high pressure I can go to very high pressure I can go to very high density yes but there it is called supercritical this region is called supercritical liquid supercritical water supercritical carbon dioxide is a lot of work going on there but here you see the nearest never correlations are there but much less they are much sharper and much more structured here actually if GR does not have that structure that liquid has here if you do not have this structure you would not have the liquid to solid transition. So, liquid to solid transition of freezing is critically dependent on the local structure that is what I said that the local structure trans transdents into the a structure transdent into long range order this is a beautiful beautiful thing. So, build up of short range coordination is essential for the establishment of long range coordination and the liquid is going to be restoring phase. So, then there have been several approximations that have been made in order to so now if I go back to Marker-Zewick equation then there is an equation is a beautiful equation as I said it is a lot of beautiful things many many implications but one important thing is that it has one equation and two unknowns one of the unknown is the two unknowns are H and CR now I already told you that I can get CR from HK but that already uses the percussive the on-steins on equation. So, the question of two if I really need to develop a theory of liquid state then I have to get one more equation but we have an advantage now which we did not have in YBG that we know CR is a direct correlation. So, we can make intelligent cases at what should be the form of CR and that has been done to a great extent in and in the equations like percussive equation, hypernetic integration means spherical approximations all these things came in and they are the one we will describe a little of that that like one. So, I now want one more relation between CR and GR and one popular one is given here by a definition called potential of mean force that GR is it is one minus WR potential of mean force because if I this definition if I take log of l n GR then I get WR and GR is kind of a potential because these two particles are separated by R. So, it makes sense to like GR equal WR and WR is a potential of mean force it makes sense. Then one uses several different combinations of it makes sense because we have a total you take the total for the indirect so total take and then from the total I take out all that are propagating through here that is the indirect then that of course leads me the diet correlation that makes sense this and then one uses several uses this thing to define an indirect R then you have an equation you have an equation between CR and GR the second equation that you are looking for and that then gives you and that is worked out here approximations here is given by in terms of these potential of mean force and you are and essentially CR is GR minus WR is a very famous function by function and very much like one does by the a in a theory and the this WR is nothing but GR it will be her and that has been comes from these these kind of a definitions and GR minus WR is this WR is a very important functions then you get one more equations one more equation and this equation is when is when this equation is called percussive equation and this we are not going through that the derivations which are given in my book and we might we might back come back to you and with a with a derivation of this is function a little bit more detail and please make a note that this item we should have done that and and and then percussive equation so this extra equation in addition to odd sense and equation completes our allows us to solve this for GR and CR together and these approaches very successful is the percussive equation and then here is the comparison of the identical equations with simulation results and here is far better than what born green equation did look at that that is how does it do its simulation and these are for Lena Jones all are for Lena Jones systems of argon and so this is for argon near the triple point so if we write pressure temperature then argon has much of the studies have been done near the triple point and here is near the triple point which is T star is 0.72 and row star is 0.85 as I was saying this is a typical density of the liquids and T star 1 is a good number to remember as and T star is KBT by Epsilon so if I Lena Jones potential is Epsilon so T star is a dimensionless quantity which is KBT by Epsilon then thermal energy divided by so T star is actually gives a measure of how high you are with respect to Lena Jones well depth Epsilon and these are the now you see percussive equation picks up, picks up quite a bit of the structure and it does very well here in getting the second peak that is what born green approximation with Carpenter's superposition approach then fail to do and then here one more thing hypernetician approximation which sometimes does better sometimes does worse but they are almost very close to each other for this course we do not have to go into the details of the difference but they both do so basic then take home message is that here of liquids have reached a quite a satisfactory stage when it comes to this kind of spherical molecules like argon you know also not too bad in molecules when they are dumbbell like nitrogen or oxygen this kind of molecules though force field sometimes poses a problem but reasonably okay we can have a theory of liquid also of course we can get the simulations so all these beautiful work and that percussive and hypernetic chain in spherical approximations turns to the odd sense of the equation have led us to a satisfactory understanding of the structure of liquids and the short range order that is a liquid then what happens that from 1970 or 1980 essentially huge number of simulations started coming the first argon simulation in a great way was done by Anisuraman in 1964 1972 Stringer and Rabant and Rabant did all the calculations of the water that was a pioneering simulations in a series of paper I think almost 8 papers they simulated and reported the water with a Stringer potential st2 potential which worked out quite well then much better what have been done so now simulations have become very powerful so complex liquids like you know say for example I want to do methanol I want to do ethanol I want to do dimethyl sulfoxide as we talked in the last 2 lectures then the radial since we need the radial distribution function for the further theoretical understanding of dynamical properties they are now obtained directly from simulations so the emphasis of theoretical research and theoretical effort has drifted and has moved away from these kind of analytical work analytical followed by numerical work that is percussive equation and hypernetic chain it has moved from there it has moved to the domain of simulations and then one gets the radial distribution function of 3 particle correlation functions directly from simulations the issue then become the issue of the force field in order to get simulation to agree with experiment we need very good force field how the molecules interact with each other that explains the present emphasis of all grand proposals theory research work is development of force field because computers have reached a stage where if you give me the force field I can calculate the radial distribution function I can calculate the 3 particle correlation I can calculate the diffusion equation velocity correlation function the problem remains you know when I want to do long time behavior then simulations can or cannot give you but you basic many very interesting features come with simulation cannot explain just numbers numbers do not suffice you need to have a theory you need to have equations ok so that is what the theory of liquids it stops here the one thing that I have not given the derivation is the derivation of the percussive equation that is given in my book and we might take a class want to come to that just a 15-20 minutes class to make the derivation which I do not have with me right now and well we I would need to talk a little bit about the wire function and other things and with that we will stop the discussion on theory of liquids and and we leave you with this beautiful picture of gr and r and that this defines actually quit in our mind this these two pictures that are giving you slides define the picture ok then bye for now.