 Welcome back. So, we will continue with the mean field theory of the Ising model and as I said repeatedly in the mean field theory you have the dimension enters to the coordination number gamma and which is by and large generally is a good, good, good approximation there is no, no, no, no problem with that. So, the thing that we have already done is that the decomposition which is this construction of this table that this construction of this thing that you already read the up spins is a plus and whenever I draw a line from one up spins only from the up spins to all the nearest neighbors. So, in a two up spins next to each other then there are two lines and that becomes two n plus plus and those kind of things and for example, this is the, this is the condition that we derived that gamma n plus equal to 2 n plus plus plus n plus minus and these are the kind of relations that we just did at length. So, these are the things again. So, this is the book from Carson Huang at which is the best this from Carson Huang and this is the best description of Ising model of the Klein mean field theory Bragg Williams approximation that I am talking here is done this in the, in this, in this book of Systeria Mechanics. So, then one important thing in this process going through that. So, this was the kind of things that we have been discussing all this time that the writing down the Hamiltonian in terms of this what exactly I was telling before I do the mean field approximation. And this is the partition function this is fairly universal notation we find every word in the same every book we have this notation. So, that is a equivalence that means this we feel the Ising model and mean field theories are fairly well documented and well articulated because this is the kind of so important that we have to be more or less in unison. Now I just told you that this before I go to do the next step there is to show the importance of Ising model that this is applied to so many different things one of them is the lattice gas. In the lattice gas you say okay I have a lattice where I have a if there is a gas molecule then there is not if there is. So, I want in a gas where a lot of empty spaces and I want to model that by saying okay if there is a particle then there is a black and there is no particle that is empty. So, that simple thing essentially is again n plus when there is a n plus goes over to your occupied n minus is gas particles but that is nothing there and the two next to each other is n plus minus if two occupied next to each other n plus plus. So, you can now map the Ising model into lattice gas and you can do wonderful stuff. So, then again that as I said here total number of lattice sites total number of lattice sites then these total number of atoms are occupied. So, in this lattice when they are occupied they are occupied that is n plus plus when they are empty that is n minus and total number of mirrors never pairs is n plus plus. So, exactly that thing when you map that you can now get the and so if you say when they are next to each other there is an attraction and that attraction is epsilon then that gives you that. So, it is like two parallel spins of ferromagnetic interaction and then I can write the partition function I can write the partition function like that and then I can now get the exactly decompose into this only difference is that the beta H term here is given by this term. But again I have to evaluate the same thing the n plus n plus plus and e to the power beta H n plus n plus plus that is this thing. So, I am done. So, one then number of ways of distributing atoms n plus atoms n plus atoms into the n n number of lattice sites now something extremely important comes out of that then if my number of atoms occupied atoms my black dots or my here white dots if that is equal to n plus then I know n plus and minus n minus gives me the magnetization and where does n plus gives me here n plus gives me density. So, then I immediately have a wonderful isomorphism which is goes a long way that magnetization is equivalent to density that exactly turned out to be that is the way we can transform one equation state of magnetic system into equation state for the gas system this is a far reaching consequence. So, now we now just discuss that how lattice gas model can be mapped into Ising model we will next discuss one more interesting thing and that is these are all from Carson Huang now the binary alloy, binary alloy lead brass beta brass and that is zinc and copper and then I am now occupied my lattice sites are occupied either by zinc atom I can now consider order disorder transition that takes place at 742 Kelvin in this case that you can immediately see that my up spin could be copper my down spins are zinc or other way around and I can again write down a Hamiltonian. So, sure enough I write down a Hamiltonian exactly same way I now say okay my copper copper is energy interaction energy epsilon 1 sorry epsilon 1 my copper and zinc zinc this is thing and if I want copper zinc then like that. So, this kind of binary, binary mixture we are talking binary alloy that like copper and copper like each other and zinc and zinc like each other low temperature of course here there is a more long interactions because of the metallic city and all those things but at a level high temperature when you are talking order disorder transition those things but you have to say copper and copper like each other and zinc and zinc like each other very much like a binary mixture that we do and we will talk little bit about that later. So, now I exactly have the same conservation rules that means I start with the 5 here these 3 numbers N11 N22 and N12 and then I have the conditions just I have done before exactly same that up spin up spin now copper and copper down spin down spin zinc and zinc and copper and zinc N12 and copper plus zinc is total number of lattice sites and then I eliminate and I now get my energy or Hamiltonian I get energy Hamiltonian in terms of there N plus plus N plus minus I get N1 N1 that means N1 is number of lattice sites occupied by copper or zinc whatever and N11 when copper and copper and zinc are the same as neighbors. So, these things these things are exactly same what I have reduced to exactly same that my isent model. So, now I showed two cases gas liquid transition and binary alloy which are completely isomorphic and that in one shot explained beautiful many things for example if I do binary mixture I give you just one example before I pass on against density then this is what gas liquid transition and these the critical temperature you see. Now, if I now do that with a mole fraction x of one species is exactly same graph these exactly same when I plot magnetization against temperature because I already told you density is like magnetization. So, in one shot in one formulation you are getting three very fastly different phenomena which is the magnetization gas liquid transition and this order transition and phase separation in binary alloys this is just amazing that is why one takes the isent model so seriously both in statics equilibrium properties and our language of this transition is completely dependent on isent model this is the most important system of statistical in the statistical mechanics ok we can let us continue and these I was saying this is the magnetic transition magnetic transition happening magnetization against temperature this is this is same as this is same as I can instead of that I can plot it rho versus t and exactly same thing. So, this is the classic critical point when a magnetic system. So, order this order transition mole fraction all these things are the same this is one of the most huge huge success of of the isent model. So, now we go ahead and we complete our task of the of the mean field theory and that will be now little bit in. So, the way we do now is this is order this order transition that done because I do not need to do that, but what instead I will do what I was continuing with ok. So, now the mean field theory is same the way I am the the level I am doing is is bag Williams approximation and and this is thing and I need to make it little bit better. Now, so these are introduced long range order parameter and short range order parameter l sigma n plus 1 and these are the same thing and now I make the approximation that short range order parameter is is is given by long range order parameter. So, the approximation is contained in the statement that there is no long short range order apart from the long range order that means approximation states that statement that no short range order other than long range order and once I do that I do this then I get n when I make n plus by half gamma n equal to n plus squares that translates into sigma because this is nothing but the l plus you know n plus by n equal to half l plus 1 square remember that half l plus 1 sorry and when I use that into here I get sigma this condition that sigma when I use this into I in this into that then I get this condition sigma l plus 1. Now, that actually essentially means that I have the Hamiltonian now completely in Karshan Wang h is my b. So, my Hamiltonian is then 1 over n what is energy 1 over h is half minus half gamma j l square minus b l. So, just make a note that his h is my b and his gamma my gamma but his epsilon is my j this notation I am using a little bit more modern like we use j for rising model not epsilon as Karshan Wang has used and we do we do like to use Hamiltonian h but not e as I use Hamiltonian for h that deprives me of h for magnetic field. So, I use b but b equal to h mu. So, this there is no room for any confusion here once I do that then I write down now I am in a completely free domain now I know how to go I know the how many ways I can get n plus up spins in n and that of course is n factorial by n plus factorial n minus n factorial. So, that is I have written here and n plus is half n 1 plus l the other is n all minus l. So, I can write n factorial by n plus factorial n minus factorial then that is the way or factorial 2 n plus. So, that is the way I have written here that is I have written here this is the number of ways to distribute and for a given configuration with l because I started with n plus and n plus plus that was the exact then I I made the approximation in field approximation and as a result of mean field approximation I have on the n plus then I go back I already have l and sigma and when I eliminate n plus plus I eliminate sigma I have only l but it is a consistent way in a consistent way. So, I have the Hamiltonian and then I know how to so I put the Hamiltonian here and lo and behold I have fully now I can evaluate that. So, the in the large particle limit n going to infinity I take the logarithm to get the free energy and then I make the approximation I make the starlings approximation thus to you what we done in Mayer's theory we have done in the canonical partition function again and again we have picked up the maximum term. So, of the sum I pick up the maximum term. So, I this remove that one I take the log whatever even and then I maximize with respect to l and maximizing with respect to l is same as maximizing with respect to n plus. So, remember if you know already you have not noticed note that l is nothing but the essentially the magnetization and so things are well and good. So, then we can go and once we do that then the thing that happens is that I get an equation which is the following that I get an equation by doing that which is let me know because it is not very visible here l equal to tan hyperbolic. So, that is the so this is called a implicit solution or a transcendental equation where I have a l on the left hand side I have a l on the right hand side and given a b I have to solve that numerically. Now if b is 0 the external magnetic field is 0 I have this beautiful solution which is this one that means l equal to tan hyperbolic in many places dimension k is introduced which is j by k B T then I l equal to tan hyperbolic a gamma l low and behold this has a beautiful phase transition scenario that when we so and this is given here this phase transition scenario is described beautifully here that when this has only solution of this equation is only solution of this equation is only solution is that is at a temperature is high. So, gamma j by k B T is less than one then all even less than one only solution is l average which is the maximum term actually I prefer it l star but that ok l average that only solution is the magnetization 0. So, this is a random system number of plus spins up spins is same as number of down spins. However, a solution appears when you get a solution appears when you go to lower temperature. So, when you go to lower temperature that means gamma j by k B T is greater than 1 temperature is small. So, this becomes greater than 1 then a solution appears again 2 solution appears one of them up for up spins and that is for down spins. And you get the one can show that this root that 0 must be it corresponds to a minimum not to a maximum. So, you can neglect reject that term. So, then the solution reduces to the following things that l these 2 solutions to T greater than T C you have no magnetization disorder system below T C it can be either plus or minus it has to be 2 you cannot in the absence of magnetic field they are both are equally likely there is no way to choose one of the other the way the solution is made by doing a graphical method which is made which is shown here you the graphical solution is done in the following way that you plot l and tan hyperbolic l and then you when they become equal to 1 then that is the solution and the graph goes like this it crosses here at 1 and crosses here at 1. So, these are 2 solutions that you get and it is a let me see if I can get it a little bit up there should be something here a cursor. So, this is the this line is the going selecting you when that become equal to tan hyperbolic and l. So, this is the l versus l this line l line this the straight line and this is the tan hyperbolic you know the tan hyperbolic very well because tan hyperbolic are like this and these are the solutions when they meet that is the solution of that and that is l naught plus and minus solutions. So, this is part of these things works out beautifully. So, next go to the next page. So, one beautiful thing of that is the following of all these calculations is the following that in general can be it can be obtained numerically and one finds numerically some very interesting thing comes out that near the critical temperature when you are close to Tc close to Tc is a subject of great great interest because what is called the critical phenomena and we have done Landau theory of the critical phenomena somewhat. Then you find that numerically that these approximation then l naught behaves is like that when you are away from the critical point but when you are very in a region close to critical point then the order parameter or magnetization behaves as magnetization behaves as Tc minus T to the power half this is that means the magnetization varies sharply as a fractional exponent and this exponent is called in a critical phenomena magnetic is called exponent beta one is sorry to use the same beta many different ways this is the equation of state also this is the same way rho varies as again Tc minus T also the notation beta in mean field theory Landau theory Bragg Williams theory all these cases this exponent beta is half which experiment will be found to be wrong experiment you want found yes there is a fractional exponent and the basic thing is correct that this is very singular behavior it shows near the critical phenomena but this exponent is more like one third. So, this is called equation of state exponent and as I told you magnetization same as density in order disorder transition again the fact probability of one being occupied by copper orientation or you can say composition you know one minus other density of a minus density of the copper minus density of the zinc is exactly all now it need not be a strictly one third it varies from 0.36 to 0.31 and all these things but essentially very very similar. So, this is the essence of the critical phenomena and the critical exponents and the critical experiments in many different forms but this magnetization and density is specifically it also has a critical behavior that we will be discussing it will be here. So, now here the thermodynamic functions are summarized for this. So, this is the magnetization and so this is the free energy and then you get the as I said magnetization this is the internal energy and this is the specific heat that goes like that and the df's in the following fashion it in this case it does not. So, specific in the Bragg Williams approximation that does not diverge but it increases like that. So, this gamma j by increase the I increase the this is my gamma j as I increase the as I increase the temperature KBT and it goes and saturates to a value like that it increases like that. And then similarly one can go to that is gas this property comes from the this solution comes from this solution here this thing you can figure it out. So, that we just described in previous page similar behavior you get in the lattice gas very similar equations that table of correspondence is given and this is just exactly the same one model maps into the other model predictions are exactly the same will be binary alloy. So, now this is what it is this beautiful thing given here where I have been referring to all this time that this is the gas liquid all of you know van der Waals that pressure versus volume pressure versus volume all you know this isotherm and this is the critical point and this is the inverse inverse parabola that I was drawing as density versus density versus temperature in 5 plot against temperature I get that the top is the critical point and the lattice gas equation of state in lattice gas and same in the this is the mean field theory or simplest type of main field theory that we have approximation not too simple as we have seen. But it captures some aspects of the critical phenomenon but does not aspect the full aspect of critical phenomenon which requires far more work and the same thing you get from van der Waals the same thing you get from Landau theory and you need to do lot more work to go beyond this level of approximation. So, next one level one does better with considerable more work is the is the Bethe approximation but we are not going to do that. So, we will we will stop here as per mean field theory goes. So, what did we achieve we got a beautiful equation in average or in the in the we also got a equation similar equation in the presence of the magnetic field it is a little bit more complex not too much and both has to be solved numerically but in some cases as you showed that one can be after doing the numerical work there are some radical work one can do and one get the mitigation as a function of temperature in a critical exponent beta equal to half which is we call mean field exponent. And and that does not do a good job but what is the beauty of the whole thing is that it does describe the phase transition it does capture many many aspects of phase transition of three systems which is magnetic gas liquid then binary alloy order disorder transition in binary alloy all the three. So, it shows that these three systems to certain level is isomorphic and this was further in a beautiful paper that I recommend people to study by young and the 1952 in physical review he further post this analogy between icing and Latin gas and binary alloy actually some of these things follows what we have done follows from this classic paper young and the then you know these two gentlemen got the Nobel Prize for work their work on parity but in 1952 they wrote two physical review papers one after another where they mapped the Mayer's theories. So, they connected the icing model with Mayer's theory and they showed that the cluster integrals that we do in Mayer's theory are can be expressed in terms of zeros of the again canonical partition function and huge number of results were done together and all these views of icing model binary alloy all these things were put together in an epop making paper of in these two papers and you know what level mean field theory working what level not working. So, that along with Mayer's theory and icing model youngly to those two papers played essentially the starting point or say launching point at at sometimes obtain told to our study of the phase transitions and critical point. So, we might come return later to the critical phenomena and a little bit more we already have done the Landau theory and we have done the concept of order parameter and you see here already that in the free energy then ultimately described in terms of a L square term and L is the magnetization and Landau theory order parameter is the magnetization. So, Landau theories expansion of free energy in terms of L square L4 is fully consistent with the Bragg Williams or other way around though Bragg Williams has a microscopic basis it starts with a Hamiltonian but Landau's just writes down the free energy expansion. So, it is quite satisfying to see that the Landau theory is recovered in a more consistent theoretical framework and so the concept of order parameter that we introduced in Landau theory which is same as a long range order parameter L which is same as the magnetization which is same as the density which is same as the mole fraction. So, Bragg Williams approximation gives us a beautiful understanding of the free energy that is happening the flattening of the free energy surface all these things comes out of the Bragg Williams. What is phenomenologically assumed largely on symmetry arguments and the basic physics by Landau. So, we will stop here now we will probably get to use Leon Young again and we get to use Ising model again in future course. We stop here now thank you.