 Welcome back. In this course of statistical mechanics we are dealing with equilibrium phenomena. So just a brief reminder that statistical mechanics is broadly divided into two, one is equilibrium statistical mechanics, this non-equilibrium, non-equilibrium stat mech. This is the one that we deal with the thermodynamics and very, very important part, thermodynamic properties and also the phase transition. Here we do relaxation phenomena, chemical kinetics and then time dependent processes. Now in these, so we are dealing with in these equilibrium statistical mechanics and we already started with the basic postulates and the hypothesis and with that we went to construct the methodology which was built on the failure of well failure in a very, very essence. He succeeded a lot but he failed to develop a theory of kinetic theory of matter or at those days used to be called heat. And the reason was that Boltzmann tried to go all the way by starting a time dependent phenomena. So what Boltzmann was trying to develop was a theory of non-equilibrium or time dependent phenomena, non-equilibrium statistical mechanics. And then he landed in the difficulty that the hierarchy, I discuss the hierarchy that he started with single particle distributions and then that he tried to develop a distribution for position of a single particle in position and momentum but he found that gets connected to two particle distribution then he made it approximation, physical molecular chaos and that landed him into a lot of difficulty but anyway that this project did not work out. Then however Willard Gibbs sitting there at the Yale University in New Haven, he had brilliant idea he realized that this approach will not work. So he went on to develop what is the modern equilibrium statistical mechanics. He realized that to do an equilibrium stat map we do not need to know the trajectory, we do not know the detailed dynamics because thermodynamic properties are independent of time. But it was the concept of ensemble is one of the most brilliant idea that mankind has come up with is somewhat less given credit to. And he then said okay I can thought of doing equilibrium statistical mechanics, I need to do a probability distribution. But probability distribution mean how? I need to consider particles in the system is in different microscopic states. How we will now create this microscopic states without going Boltzmann way? Then he created a mental replica and the basic idea is that the number of microscopic states is so huge that the systems, the mental replica of my original system all reside in different microscopic states. So now he came up with these two postulates you know ensemble average equal to time average. So now the microscopic thermodynamic properties of the system are the average of the systems of the ensemble and each system is residing in a different microscopic state. So as if Boltzmann trajectory is going through all the systems but however he now had to after making the ensemble average or time average he had to make sure the system really indeed goes through all that that is what and you have to talk of distribution. So all the microscopic states are constant energy they are equally problem that is the assumption second postulate and then he had to say okay my system must go through all these things and the Boltzmann trajectory as I was saying a minute before and that is the Argonautic hypothesis. So that is started then we went to micro chronicle ensemble and micro chronicle partition function which is entropy actually partition function obey a number of microscopic states and thermodynamic potential each ensemble comes with the partition function and each partition function comes with the thermodynamic potential and from the thermodynamic potential we can calculate thermodynamic properties in micro chronicle ensemble envy ensemble we have already discussed omega total number of microscopic states is the partition function and entropy is s k b l n omega that is why k b the term Boltzmann constant comes because with the Boltzmann the Boltzmann is the one introduced it. So that is the canonical micro chronicle partition function now that of course becomes very difficult and also not practical because systems are not in constant energy you can have constant volume a but an even number but energy is always fluctuating because we cannot isolate systems. So then which gives with another brilliant thing introduced the canonical ensemble where now the particles the systems particles in the system is allowed to exchange energy with the surrounding. So the energy is not conserved but energy is replaced by conjugate variable which is the temperature. So the ensemble has now constant number constant volume constant temperature so the n v t and that gives rise to the canonical ensemble and the canonical partition function which is denoted by q and the logarithmic of the q minus k b t l n q gives you the Hamiltonian energy and as we discussed that is the advantage of this Hamiltonian energy is that I can get equation of state like pressure by a volume derivative of free energy entropy by temperature derivative of the free energy and then specific heat by the temperature derivative of entropy. So these are just very beautifully flows that it was kind of fortunate that in n v t we get that we are not that fortunate we want to n p t because n p t the thermodynamic potential and partition functions are not that helpful they are still useful quantities. Equilibrium statistical mechanics however one very important thing of in addition to this thermodynamics which we get from ensemble the important thing is the phase transition that is one of the real mandate of statistical mechanics is to describe the phase transition. Phase transition as I described is a amazingly beautiful thing and is also really very striking because you are changing a control parameter by infinitesimally small amount and you are getting a huge change and we said that that is the definition of phase transition that control parameter is changed by an infinitesimal amount but the a is changed by huge amount so it is like this that I can entropy against temperature then this is the way if this is the liquid this is the gas so look at the sharpness look at the sharpness all the derivatives diverge in a this is the first order of phase transition or first derivative of the free energy we discussed that air and phase classification the first derivative of free energy discontinuous give you the first order of phase transition but when the second derivative of the free energy like specific it acts funny then we call that a second order of phase transition this particular is a continuous type second order two kinds of second order of phase transition that we this is the gas liquid or order disorder or magnetic ferromagnetic paramagnetic transition. Superconducting transition is there in first really jump discontinuity like this kind of discontinuity in the second derivative that is resistance or resistivity okay so these are the basic element things of phase transition the definition of what is the phase transition then we want to know why and how and why does the phase transition take place and then we come up with the thermodynamic logic okay phase transition takes place because now we have a another minima appear so these in first order phase transition that another minimum appears and that minimum now becomes deeper as I look at temperature and this old phase is gets replaced by the new phase so these are free energy description on why now in this case the free energy becomes flatter and flatter and we get huge fluctuations on a flat free energy surface because fluctuation does not cost energy and that is the way this was described this description all these came by in terms of land out so land out is introduced the kind of free energy landscape picture of phase transition and these then led to the land out theory of phase transition the definition of order parameter and many many other beautiful things and we learned how to talk of free energy expansion in order parameter we learn to calculate the changes in entropy from these theory and we learn to change these how to describe these kind of divergence the land out theory is not a perfect theory it took a lot of time for people to correct it by almost 40 years but people did correct it and we have a much more complete theory of phase transition now so the land out theory which describes the free energy in terms of things as I told you was a parallel almost parallel with the other theory which was developed more quantitatively from physical chemist perspective so it was more specialized but more quantitative so that was the mayors theory so mayors theory describe that how gas can go into liquid and how you can develop the virial series so and the picture that may have developed and this beautiful that we get the definition of virial so this was almost last century the virial series was introduced or something like that in the beginning of 1900 or so however we did not know what is the second virial coefficient third virial coefficient mayors theory to give us an analytical expression in terms of intermolecular interaction and that is just beautiful because that I discussed last class we have the intermolecular interactions B2 we can measure experimentally because this is just nothing but I P versus rho and this is ideal gas this goes like that so I know this so I can now have an expression for B2 in terms of intermolecular interaction you are then I can put that and play around with that and I can get a potential and that was the first force field that was developed as I described once before so is a extremely important theory one more success of mayors theory is that mayors theory gave us a picture a physical picture how a gas can go into liquid and that was used in many other things like in lattice theories and many other cases it is very very important to appreciate the continuity and the appreciate the flow of development of a subject the historical perspective now mayors theory then told us that okay there is suddenly large clusters appear in the system and we know that this is not perfectly a physical cluster because this everything is very mathematically defined it does not there is real chemical bond lasting for a long time not there they are mathematical definition in terms may are a function F that we described but it nevertheless tells you how a big picture appears now before I go to the next thing in a mayors theory there are two more things that we must discuss and not in great detail but in a semi quantitative level one is the solgyl transition these are very common phenomena that when you have a sol and you have polymers and then you have here this you have the polymers which functionality three or four then when you increase the concentration or lower the temperature though it is not true thermodynamics then what happened that it they form a gel phase how do I know they form a gel phase the viscosity diverges so solgyl transition is a clustering transition is a clustering transition large clusters appears there is a very weak thermodynamic signature the solgyl transition is a very common phase transition usually it is a subject of polymer physics and polymer science though it is not done that well in polymer thing but this is very important in the in the industrial context in industrial chemistry the formation of a gel we routinely form gel not on just the gelatin that you eat as a desert but in many many industrial functions we form solgyl transition the rubber is can be considered as the formation by these kind of chemical process where solver acts as the vulcanization or different carbon groups okay so then what is the essence of solgyl transition how do I describe solgyl transition qualitatively I will then describe how it is done quantitatively so this is the very very interesting the theory that was developed of solgyl transition was done by two people at the same time one person is Paul Flory and who is considered father of polymer physics and it was also done at the same time by Walter Stockmayer this I think 1943 I think this is little before 1941 what both of them considered okay this is not really thermodynamic transition is a clustering transition will look at these are real clusters now chemical bonds are forming okay that you are bringing a monomer which is a functionality three or four and they are connected together to form a branch sometime there are rings also but the huge branch that is forming so suddenly so there is again a phase transition because suddenly at some point the clusters becoming infinite so how do I describe that both of them came out with the identical description and I will describe the Stockmayer's one because that follows Joseph Mayer's treatment and Stockmayer was a student of Joseph Mayer. So now the way they consider both of them let me consider a ML exactly the way it did Mayer ML is the number of clusters or polymers or polymers of size L they are now there in number of monomers so I have a condition L ML equal to N now L ML these quantity then these quantity then these quantity is number of monomers in a cluster of size L is L ML then there is one more quantity both of them considered which is a very interesting quantity which is L square ML so this is second moment so ML is a distribution this is the first moment and this is second moment and let me call that a susceptibility. Now what the way Stockmayer went about doing it is very interesting Stockmayer now comes to consider exactly with Mayer did that I give you that at a given time my system at a given extent of reaction that means certain amount of reaction has taken place then the that this ML has formed how many ways now I can distribute monomer into a polymer so then you went around doing again saying the same way Mayer did that how many ways I can form omega out of a ML and a number of clusters then what Stockmayer went around doing he maximizes these quantity these D ML with respect to ML or L N maximizes the amount of reaction and found out the most probable distribution exactly the way Mayer did in there there is a non-trivial calculation just Mayer had a non-trivial calculation of finding the which led to the partition function you are in non-trivial calculation to calculating that if each each of my monomer has a functionality F then how do I calculate omega that is the crux of a very very difficult calculation once you do that calculation and you can find out the distribution ML start then he found that if I now plot L ML start which is the same I discussed that when in a system large size the most probable distribution is same as the average L then one finds that I plot it against L then before the say I am increasing the density and putting more and more monomer in my box and they are solved but certain critical density is reached and then the huge cluster appears so now this is now will be given like this at a low density it will be mostly monomer may be few dimer then I am going on increasing then there is this kind of tail appears then these these actually comes down number of monomer comes down and then there is nothing in between the intermediate population disappears because they become a lot so this is the jail phase the analysis I am not going to going doing great detail but this is almost a because one can go into doing a very detailed job that is not necessary in this course but one need to know that the soldier transition formulated in a stock bear we is a is exactly exactly same as as a may as theory of condensation so then the one can calculate the viscosity of the system in a polymer we can when you do polymer will do that in more detail the this this kind of things but here I just want to bring it to an attention to this and one more transition which are very similar and then I will describe if you want to give aspects of that which is the percolation this you might have heard a lot this is essentially same as the soldier transition but the difference is that percolation transition is in solid state physics and we consider almost always on lattice so basically you have a lattice I will briefly describe what is the percolation transition then one or two quantitative aspects soldier transition and percolation transition are essentially same from the point of view of phase transition though they are very different systems as I told you this one of the greatness of the universality of phase transition that you do first order phase transition of one once I say it is the first order phase transition I immediately it comes to my mind okay then this transition will have a first derivative or a discontinuous jump discontinuity then I will have a latent heat I will have a volume change and this these are the properties I say second order continuous phase transition I know okay specific it will diverge so first order phase transition of many different materials and just amazing universality that we we think and I will talk a little bit of the universality little bit now before we go to the thing that all these things will be done little bit more detail later but right now percolation transition is in solid state physics it comes in the conductivity problem basically when you have two kinds of species in the system one is a conducting and there is a non-conducting and you have dispersed them in many micro crystallites in a in a in a lattice phase gold nanoparticles and that is with another carbon nanoparticle maybe you have a binary mixture of the two now one of them contact current other does not that does not so when you are very low density of the conducting one then there is no no no conduction no current goes through that this is a very important industrial or practical day however when you have the conducting one which is certain concentration then suddenly you find out the current comes in and so the way it happens is that these guys they have to form a chain so so it has to form a connected cluster so when this connected cluster is formed it forms a path and current flows in the path so this is the percolation transition percolation transition then again is the formation and then appear in software connected cluster so this the both percolation and solid transition are there is a something called the critical critical occupation probability critical what is the this the red ones they have to be certain number and that number goes by critical percolation is given as Z is the coordination number so when for example in you can imagine that in a two dimension lattice square lattice where Z is 4 there is there is 4 your percolation probability is 1 by 4 minus 1 is 1 over 3 so this space is now the these red ones now has to be 33 percent when it reaches 33 percent before 33 percent is again completely dramatic and sudden even at 30 percent you see that there are disconnected clusters intermediate size disconnected clusters but as soon as you reach the within 0.01 percent of the 1 3 this cluster suddenly appears so it has both these and solid transition has all these things these kind of expression of these kind of expression also there exist one one for a solid transition they are also you we describe in terms of this quantity L square L square M L and L square M L this guy again diverges with a at a given concentration of the monomers so in certain sense they have the characteristics of a phase transition both solid transition and percolation but both of them are clustering transition both of them has to do with appearance of large clusters both of them has the second moment source divergence this L square M L quantity these consider those source divergence and like this divergence is happens with exponent something will talk later and there I will discuss percolation little bit in more detail and the derivation of these things will be more detail but I do not want to go into the these things but to know that the this this description of solid transition and percolation transition are very similar to the mayor's theory of condensation and in certain sense all three can be considered as a is the appearance of a gigantic molecule the appearance of a gigantic molecule that happens at a certain critical concentration of the monomer and they live they do not have the very weak thermodynamic signatures thermodynamics you do not see much but you see their properties like in solid transition is the viscosity that diverges in a percolation you suddenly in a lattice you see the current current starts flowing percolation also used very much in the flow of water in the sand boxes or in the in the sand that the where there are in the conductivity is to be made for water to flow so these are very very similar things are quite universality and I cannot go into very detail of these things right now instead we will now go on to do something different but before that a little bit more of a transition that I want to just review and then go over to that so so these are the so we have done first order phase and we be and first order phase transition and second order phase transition they are universal and we have not talked of any specific systems similarly in solid transition and percolation they are clustering transition in the mayor sense and we have not described any detail any particular system and though some examples I will be given they are given in my book and we will probably do a little bit later before we go into something completely different I just want to do one thing that you know when you do first order phase transition I discuss of one thing which is called hysteresis and this I also called is the same as metastability and I also told from Landau theory Landau's free energy surface it means that we have free energy landscape if I plot against the order parameters eta then this is if this is the minimum then something else lurks there so when I go then further then this becomes this then it becomes this is so this is the metastable phase this is the stable phase when I lower the temperature then so top temperature going down and but this is T equal to transition temperature T T and these temperatures below the transition temperature so this is the old and this is the new old and new now new is more stable but the minimum at old remains minimum at old remains and that means the system can even trapped and this we see when you do hysteresis we plot H against magnetization M we have a large H so it is now when it comes back it does not go so even though I have switched the field spins remain up magnetization remain positive and these we know in pressure versus density this is Landau's loop the exactly same thing is a beautiful actually this analogy between magnetic system and the castley system is one of the most wonderful thing because we gain tremendously by comparing one with the other so here this is called hysteresis or metastability here so when I pressure against density this is the Maxwell so beyond this point it should go over to liquid but it does not experiment not just in Reynolds theory in experiments this continues to stay and then an explosive transformation to this is metastable gas this is the metastable gas that goes over to liquid it just then goes like this bang with a bang it goes transferred to that so this is the essence of first order of hysteresis is the metastability now why metastability happens the metastability happens because once you have this kind of free energy but the system which is trapped in the old phase the barrier is the thermodynamic system this is a thermodynamic barrier there is a huge barrier no way the system can cross this barrier so well there is a way but it is a in bulk transformation of the old to new is not possible the way it is done is another beautiful thing that we will do next in the next class that is nucleation so this is the end of this particular lecture and so I discussed to you just why is I discussed to sol-gel transition and percolation transition and I did not go into very detail but there are the similar thing may as theory that mean the large clusters appear in the system and the method that is done sol-gel transition was done by Walter Stockman is same exactly same as Stockman as mayor and percolation done in a more detail quantity way somewhat later in the context of more of a critical phenomena and that is still critical phenomena is best understood in terms of the percolation that we will do in a later class and I want to describe the man I said the percolation probability 1 over z minus 1 exactly similar thing is there in 1 minus alpha alpha is the extent of reaction and in sol-gel transition so the extent of reaction when certain which is given by the total number of bonds formed actually 2 n minus m by n m that is the extent of reaction and your critical thing is happens when extent of reaction reaches certain value which is matches with the data percolation the very similar thing that means the second moment diverges with the exponent minus 1 but I do not want to talk about exponent right now I will talk when I do the critical phenomena but both have the signatures of phase transition in the cluster plane both so very weak thermodynamic anomaly and almost no thermodynamic anomaly you cannot detect them that the percolation transition has taken place with thermodynamics you detect them by doing dynamics but they are something very fundamental change in the nature and organization of the system has taken place and that is this appearance of the large gigantic cluster same as gas liquid termination so we stop here now and we will start with nucleation in the next class.