 Welcome back, we shall continue with your study of Isaac model as I stated several times in the last lecture on Isaac model that this is the perhaps the most important model of statistical mechanics. This at this its parts it serves several purposes on one side this is the simplest model of many body interacting system and at the same time it captures most of the essential features like phase transitions, critical phenomena and many many other aspects of many body problem. One thing one should know remember while doing this that this Isaac model is not just a model by itself it is it goes on to explain lattice gas models or gas liquid transition it goes on to explain many aspects of order disorder transition binary in binary alloys it goes on to be used in polymers. So Isaac model has this multifarious things at the core however the Isaac model is a very simple thing it just has nothing but spins and spins are in a in a and so I can consider that this is a spin up and down and there is so the spins essential element is that spins can be either up or down second is that is the nearest neighbor interaction very important that is nearest neighbor interaction. So these two are essential parts that means this guy here can interact only with this one and this one. So this nearest neighbor interaction makes it really simple and one may one may wonder how such a nearest neighbor interaction can explain so many different things the reason is that this is interacting with these but these also interacting with these so the effect of the interaction propagates. Now and the spins can be up and down they can also flip between the two states so this is called two standardizing model and as I told you before that this is the model which was introduced as a by a PhD student Isaac in 1925 and he solved it and then it people immediately realize the beauty of this. However one dimensional Isaac model does not have a phase transition as we discussed in the last thing there are two cases that we solved for one dimensional Isaac model and one of them is that in the absence one deizing model one is in the absence of the field in the absence of an external field the other one we have a field on. Now the solution in the absence of an external field is very simple we describe it is just if n speed is cos hyperbolic 2j by kBT to the power n that is a simple solution in the presence of field the solution is far more difficult and but still analytically doable and the solution briefly to go to that the solution of this one dimensional Isaac model so this is the Hamiltonian of the one dimensionalizing model that this j is the you know the coupling coupling term and so when a sigma I can take so the AI we take the notation we take that when it is up spin is up we get sigma a value plus 1 when it is down we get the value minus 1. So when it is and these are nearest neighbor so one should realistically have a thing here that is saying that these interactions of course i and i plus 1 does that also the same purpose it does usually one write something like ij here and then put a because some is over both i and j that is the other notation that one use otherwise one can also use this notation. Now if you look at the character of that so if when the one these i spin and i plus 1 since say this is the ith spin and this is ith spin and this is the i plus 1 when both of them or rather take well this is not the best example but let us say when both are up then both of them brings plus 1 plus 1 then these become minus j and when they are down then again minus 1 minus 1 then also you get minus j however when one is plus and is minus like here then this will be plus and this is the minus and this will be plus and this will be minus and this whole thing then contributes this particular pair contributes a minus term and then becomes plus. So when they are parallel whether they are both up or both down they contribute negatively to the total Hamiltonian and that is favourable so when two parallel spins are favourable up or down that is called ferromagnetic interaction however when they are opposite spins so in that ferromagnetic interaction opposite spins are not favour because this comes with a plus j it contributes positively increases the total energy of the system and this one now is the external field and b is the external field but b actually has any two terms one is the external magnetic field h and the magnetic moment mu so bh mu b is h mu and this is the way the Hamiltonian reduces this is a very simple Hamiltonian it just takes into account certain favourable interactions and certain unfavourable interactions and then one attempts to solve this thing and see what kind of result that comes out. So one dimension is a model as I said repeatedly can be solved exactly and we discussed it next time so this is again the Hamiltonian written all the nearest neighbours and everything now so partition function partition function of these things will be written as the sum over all the configuration so when you take the sum or all the so sigma 1 is plus minus sigma 2 is plus minus and sigma n in plus minus so you have all in all 2 to the power in configurations so the way it is good to draw an analogy with gas that we have done in a gas if I take a one snapshot at a given time then I have the molecules at different locations so the molecules at different locations give at a given time a given configuration of the system so now the molecules interact with each other and when the molecules interact with each other they you know they you get an interest in energy and you of course in this case you have the kinetic energy in our case in IC model we do not have the kinetic energy but we have the potential energy and we have discussed at length the potential energy is sum over the pair wise additive term here you have many nearest neighbours in a 3 dimensional or 2 dimensional systems and you sum up the potential energy of interactions between them so a given configuration a given configuration of a gas or liquid is equivalent to specifying positions of atoms and molecules at given positions and that in a in an instant of time that position is changed and the new configuration is created and your Hamiltonian reflects or captures this change in energy with respect to the change in configuration so when the system goes to a configuration which is low energy configuration or favorable configuration then in the Hamiltonian that is reflected with a low energy or negative energy which then in the partition function in e to the power minus beta e to the power minus beta h term which is here e to the power minus beta h term it has more weight so all favorable configurations come with a larger weight now let us now analyze how that is reflected in IC model in IC model with a ferromagnetic interaction where parallel spins are favored whenever the spins are a domain in spins are parallel then that comes with a lower energy so those configurations are picked up so this is the reason why at a low temperature you get a transition to a ferromagnetic phase now one dimension as I discussed in the last class does not have a phase transition it does not have a phase transition put you bluntly and very straightforwardly simply because it does not have sufficient number of nearest neighbors it is it has just two nearest neighbors in one dimension and that is not enough to give rise to one there is a beautiful theorem in this thing that goes that is I discuss in my book I will just refer to it little bit to the colastropter and Benjamin theorem which formalizes what I said that why 1D is cannot have a stable ground state or a ferromagnetic state except at temperature T equal to 0 at temperature T equal to 0 1 over beta is 1 over k B T that diverges so only this one particular configuration is picked up so now so the idea is then is to calculate in this partition function but I would like to tell you that the purpose of this class is different from the last class I am just preparing little bit spending some time to talk about it and then go what to what I am going to do in this class so here I write now the canonical partition function which is the function of the temperature total number of spins and the magnetic will be and that is written in state forwardly from the from the Hamiltonian which is given here that Hamiltonian is written and gets into this partition function it is rewritten again essentially for certain periodic boundary condition which is we do not need to discuss now but it essentially says that you can make it into a ring and that is possible only in 1D this beautiful ring thing so that the these spins so the one next to that is the same configuration and so make it into a ring that there is an advantage of mathematical representation which allows us to solve this problem. Now I discussed little bit that this is solved by making a method called transfer matrices and I do not want to this essentially discussed here I do not want to go into that detail that is right but I solved at length in the last class the case when there is no magnetic v equal to 0 when the v equal to 0 then this Hamiltonian if this part is not there is very simple then I can show that this will be e to the power minus beta j plus e to the power minus beta j and that that goes over to the cost and then you get cost to the power 2j by kb e to the power n that partition function and then one can show that partition function does not have a phase transition which is similarly I just tell you give the results of the very well known results of the in these I discussed little bit in the last class how you go through a transfer matrix then how you use a unitary this aim is a symmetric matrix which is nothing but this x and x and y are beta j and beta b and the diagonal of diagonal terms but you know I do not want to go into too much detail into that today but I just want to show you the final results of the partition function which is for the time being for us is okay so this is the final expression of the partition function of the ends in the presence of a magnetic field and if the magnetic field is goes to 0 then one can show that then this term goes to 0 and you can combine the two things this goes to 1 and you will get e to the power beta j plus e to the power minus beta j to the power n that is in the apps when b equal to 0 but important thing is that this is already you can see this is a considerably difficult and complicated partition function however this even this complicated partition function which was derived by icing many years ago does not show a phase transition you can now go and get the standard this ln you get the free energy from the partition function and kbtln qi and that quantity becomes just this quantity this probably is not needed at this point so you end kbtln cost and these term now when you plot in the magnetization magnetization is nothing but total number of so up streams so my integration is number of spins that are up so that can be obtained by amount of spins up minus my number of spins down so it is just some over the state of the spins this gives you the magnetization when you plot the magnetization against beta beta is 1 over t 1 over kbt then you see that when beta is 0 large temperature large temperature up spins and down spins are equally favoured so you have then up spins and down spins equally favoured means you have m equal to 0 which is starting here now on the other limit when t goes to 0 absolute temperature then beta goes to infinity and then as I told that then the parallel spins are picked up because they have the lower energy but see the most important thing the most important thing is that that continuously goes to infinity so this is a very important result that shows that one dimensional iso model does not have a phase transition again the details are done here as I said that this magnetic field is sum over sigma i which is given here sum over sigma i which can be obtained by taking derivative of and this we all know that magnetization is very very free energy is magnetic field and these are things we have done many many times then like that then you can do this little algebra but even in this case the little algebra does not too little because of the sinche and cost terms that are involved but this is the final result that one obtained after doing this calculations here so this is the in the presence of the magnetic field in the absence of the magnetic field is much simpler and you essentially have the same behavior that means you do not have a phase transition this situation is completely different in so this is completely different so the 1D 1D iso model does not have a phase transition this is a very significant result does not exhibit any phase transition but even then the one dimensional iso model plays an extremely important role so even then I will just spend 1 minute on the importance this is the model which is used to do polymer dynamics it has been described my cells and reverse my cells many many models many many cases this even one dimensional iso model has been used and one of the most successful theory of dynamics interact dynamics so interacting system is a based on one dimensional iso model okay so now that one dimensional iso model we did and we know how to solve it I worked out in the last class the zero magnetic field and in the presence of the complete magnetic field is a fairly demanding calculation and even though it is just one dimension so I just refer to the book and the transfer matrix method which you can you can you can do so now we have to go to the next next case which is so okay one dimensional iso model gave us some understanding of the interacting systems and the temperature dependence so it does not so fast phase transition but still it is a pretty nice features in many different ways the course from the solution in terms of caution switch now we want to do okay what happens and that was the question we really asked after icing that what is happening in high dimensions and that turned out to be a really really highly enjoyable and but very difficult difficult journey for persistent and chemists so the higher dimensions people though it was 25 then it took almost 20 years to get the solution in the 2D so two dimensional case the two dimensional iso model now we have a say square lattice a three dimensional you can have a center or boy center or simple cubic lattice also here you can have all different varieties so in this case what is what you have you have spins in again up and down but you have spin at every lattice side so these arrows are in a lattice side so you have now a spin on each lattice side and the spin can again to the same thing spin can again spin and again point up and down so that part remains the same so in the Hamiltonian we write the Hamiltonian this remains the same so Hamiltonian remains the same but we are working on two dimension so ij again and here is neighbor so now in the case of higher dimension I can say number of neighbors is not denoted by gamma in this in this class so here then in my in my 2D square lattice I have gamma 4 if I go hexagonal lattice then I will have a gamma 6 if I will simple cubic three dimensional lattice then again my gamma number nearest meter is 6 but if I now go to face center cubic lattice I will have number of neighbors 12 and body center cubic lattice I will have 8 so okay so then in this treatment in this Hamiltonian the information about the lattice is coming through the nearest neighbor number of nearest neighbors so not only that we have spins up and down we also have the nearest neighbor interaction so those two basic features of one-dimensional IE models carry on to two-dimensional and three-dimensional and it is ready now but the two-dimensional things are exceedingly difficult and it was solved it took you know a large number of people tried to solve the two-dimensional problem it was solved by Lars von Söder in 1944 by a solution which is considered the perhaps the most difficult calculation or ever done in the history of physics or science the Lars von Söder solution then it was what Lars von Söder solved it in the absence of the so what did von Söder solution the very difficult solution but I will show the basic features it exhibits a phase transition it exhibits a critical point Tc below with the system of spins is ferromagnetic and above which it is not so there is a beautiful result and a huge huge the difficult calculation the Unziger solution was in the absence of magnetic field this was with finite magnetic field was done another took the force by calculation by CNEL and he found then the other beautiful thing that in the presence of a magnetic field the phase transition becomes faster than any the exhibit see stress so the basic things of magnetic phenomena was captured by the two-dimensional model three-dimensional model nobody has been able to solve three-dimensional model it remains an answer problem people have done huge amount of numerical work and people verified that much of the features for two-dimensional model remains same as three-dimensional model and people have used three-dimensional model for experimental systems and it has been found to be remarkably successful in explaining many many things but the fact remains that the three-dimensional model is has not been it has not been possible to solve that many people have tried many many people I know have tried and they could not go anywhere two-dimensional is the model itself is very difficult and it is I can talk you of the solution but and I can tell you of the basic features which is more or less here and there are some many things that are relevant to critical phenomena that I want to talk a little bit later today that how these things go on okay but before do that this also solution I want to go little different path and I want to do something which is captures the physics of the problem that you are trying to deal with without really doing this amount of work that is required in two-dimensional is a model and this is a approximate method that has been invented over the years and has been a successfully employed this is called the mean field approach and the mean field approach is also goes by the many other name it goes by the name of a van der Waals that was one can show the van der Waals situation of state that we have studied that van der Waals is essentially mean field then we have done land out theory at great detail then land out theory also is this a mean field however what we are going to do today now do the mean field theory in the context of the ising model and will get the results of the ising model of the results which are of what not 100% correct approximate but it will capture many of the physics of the two-dimensional and three-dimensional system mean field theory is really bad in two-dimension it becomes more and more accurate as you go to higher and higher dimensions and there is always a critical dimensionality where mean field theory becomes exact but in three-dimension is bad but in four-dimension it becomes pretty good so we will now go on develop the mean field approach the reason will develop the mean field approach that it is a beautiful insight and very useful insight that it provides in addition to giving results which are highly useful and in our more on place routine applications is a mean field theory a place a very important role that is like land out theory and we actually do not go back to a very formidable ising model kind of completely interacting system just like we thought that may have theory breaks down beyond the point so then what do people do people of course try to extend it like people try to solve the three-dimensional ising model exactly and couldn't do it but then the an alternative approach that appeared is to take on the education for there but then build a completely different model like the land out theory then land out theory wanted to take correlation in the account it went on to go up something called Ginsburg land out theory so basically one one takes a step backward and developed an approximate theory they since the really exact path is kind of blocked for you you can do it simulation something but you are not going to do any analytical work so in that case you take a step backward build an approximate theory like land out theory then go forward and to to make that theory far more a understandable and to far more useful so we will now go to the mean field theory and try to describe what is the mean field theory.