 I start doing the nearest neighbor interactions so I consider again a lattice and I will tell you where I make the mean field approximation to a great extent you will see the things will be done exactly to a great thing it will be done exactly then at some level you will see a very intuitive approximation will be made okay so in order to get start let us see the following things since we are talking of nearest neighbor interactions I have the following variables so at a given configuration at a given configuration I have so if I want to write the Hamiltonian so let me write the Hamiltonian again h as minus j i here nearest neighbors then I have minus b sigma i now look at this look at this so what does this term depends on this is a nearest neighbors it depends on whether my spins are how many of my spins are parallel how many of my spins are anti-parallel so now I introduce the following things number of spins which are up and down is here so I have number of spins up one variable n plus I have one number of spins down one variable n minus I have number of nearest neighbors which are up because that comes naturally here in plus plus then I have number of spins which are down parallel then I have which are plus and minus one parallel one anti-parallel that means in these Hamiltonian I have five variables these can be expressed in terms of five these five numbers so however one thing we need to now know that and this is something you should ponder about that in there is a relation between n plus and n minus that means n plus n plus plus n minus equal to n right so there is one condition so I do not have all five independent variables I have right now four independent variables then there is a very interesting thing there is a very interesting thing which is that if I take all the spins up I just I just I ignore all the negative things right now I take all these spins up and then I draw a line between from the up spin I draw a line to let me draw that a green line I draw a line to the nearest neighbor so so I pick up only the up spins and from the up spins I draw a line to the nearest neighbor because I have only nearest to me now this is something very nice so if my these spins also up then I draw one from this then I draw another from this so in this case what you will have a double line between I will have a double line between two up spins but if it is a down spin then I will have one line so to begin with let me summarize I pick only the up spins and from the up spins I draw a line to the nearest neighbor how many lines I will have I have gamma nearest neighbors and from each up spins I am drawing the name F1 so then I will have gamma n plus that will total number of lines that I will draw what then will be equal to there will be n plus plus will come with two so this is the conservation condition that must be obeyed so in addition n equal to n plus plus n minus which is the conservation condition I get a second conservation condition now I say okay I want to do a if the same thing now starting only with the down spins again I get this relation so what do I do it now I get three relations so I started with five variables and now I five variables that comes into the that enters the Hamiltonian but now I find that those five variables are not independent they have three relations within them so I now have the essentially two independent variables now that now tells me okay if I have I have a choice that I can have two independent variables so what are the two independent variables I choose can you make a guess one of them I will make either n plus or n minus but only one of them is possible so I take n plus plus so these are now my independent variables everything else now has to be so I have to now express n plus and n plus and n plus plus in terms of all other things so I can eliminate all those other variables and I will get the I will I will express in plus minus in minus minus and n minus in terms of capital N my total number of spins and these two so once I can do that I can now write the Hamiltonian in terms of only these two variables everything still now everything is exact everything is exact so we will go very far by being exact then we will make a very significant contribution and that is the significant assumption which is the reason that I want to do this mean field approximation in a very robust and systematic way because this is the something with majority of the students will at some stage just I will end up using day in and day out and similarly this is a very general way of doing things which finds lot of applications and in the future in the future research okay so now we now need to be here with me for a minute now we need to we have the relations and we now need to eliminate and get a so we have done that now now I do go back to Hamiltonian this is a large number of steps involved this even the main field theory calculation is fairly non-trivial any any any a land of type theory is highly intuitive and not not technically non-trivial van der Waals is you know not non-trivial but those don't have the details of the interactions when you want to build in the details of intermolecular interactions like here we are doing skin-skin interactions and you have already have the experience of mayors theory which was of course not being things are not easy so one has to do a I remember one class I was taking at Brown and is to have a small story to tell you that and we had the great great Leo Kadanoff was teaching such mechanics I was fortunate to have his class twice now one at Brown and one in Chicago now Leo Kadanoff was really was doing gave these very difficult problems and in his in his in his books and so there is this post doc or a student I don't remember from Nobel laureate Leon Cooper and who was you know pretty good and then so one the Kadanoff asked what is your opinion and what is about your influence in general of how the course going how do you find it that is the way actually people go they always ask you at some stage that how are the things going and this student whose name I forgot then told your Kadanoff that is all great and nice but I your problems are not elegant your problems are too laborious and elaborate and boring so it is not elegant so Leo Kadanoff got very upset then Leo Kadanoff went on saying it give a long lecture and the summary of that was is the following he said that look you know in physics it is 90% of the time or 95% of the time you just do your hard work and you have no chance of being elegant you are very lucky 5% of the time you get to be elegant so better you know that doing theoretical physics or any theoretical science the huge amount is hard work but you do that hard work because you like this kind of work but don't expect that you will get the chance of being elegant so we will see now a large number of steps that we will do and we will go back and forth and so I to make it easy for these students I will give you a flowchart and go through the flowchart and then do the so this is the way I hope you can read but I will read it for you if not because it is not I don't think it is very big but this will be passed on to you there are certain problems on the right side that would be then corrected in the notes that will be handed over with the course that means what we do you will be able to read it here anyway but you will be seeing that the final thing will more complete so we consider a lattice of n spins we consider a lattice of n spins as I said only here is our interactions write down Ising's been Hamiltonian H so this H I write down in terms of sigma i sigma j minus j minus b sigma i okay now if I write down down then this is the following thing j sigma i sigma j gives me so sigma i sigma j with nearest neighbor I get the following results please check it out n plus plus plus n minus minus minus n plus minus okay now I can do sigma i this time I can do sigma i and sigma i I get n plus minus n minus right that because plus comes plus spins comes with plus one up spin stop spins come with plus one the dose pointing towards the up they get plus one those and I minus one so that is this thing this is this thing and then sigma i sigma j is then plus minus so when they both are plus I said both are minus as I said so before so then then you see this 5 variables are coming 1 2 3 4 5 so 5 variables 5 variables come that is from the Hamiltonian we are not introducing them just like that it is a necessity of the calculation every step that I am going to do is very logical and they go from one step to another step and very systematic and the beauty of this thing that I am doing why I am not doing the 2D as a model and doing this thing is because this gives you you know piece by piece minute by minute more value for your time than the other thing now then there are these 5 natural variables come note that these are they are obtained by picking so there is a relation there are 3 relations as I said between and we have done that between them they obtained by picking first only the up spins and in the up spins then you write down that what are the ups for the up spins and the condition we remember gamma n plus equal to 2n plus plus plus n plus minus then gamma n minus equal to 2n minus minus plus n plus minus and of course we have the other condition n is n plus plus ok that we have only 2 independent variables and we choose as I said n and n plus 1 ok now we continue with these things these amazing suddenly we see that things have become very simple we can express the Ising Hamiltonian in terms of these 2 variables only and they gives and so all these all these proportional to n plus and n minus up to this point everything is exact now we will go on till now for a quite some time which will be exact so now I go back to partition function some over all possible values of n plus and n double plus and then I write the Hamiltonian now in terms of n plus and n plus plus well I have not written down for you what would be that I will do in a minute what will be the real Hamiltonian in terms of n plus and n plus plus that I will do in a minute because I have to eliminate those things which are little time consuming ok we have as I said we have done everything exactly and now after that one does a shortcut with the mean field theory ok and now before I go to mean field theory I go to I just want to draw this thing that the why I need the main field theory ok so at this point with this Hamilton we turns out with this we everything is exact up to this point and here we get stuck because we cannot evaluate the partition function with n plus n plus plus this is the main problem of two-dimensional model three-dimensional everywhere and but this is where we will be able to make an approximation very enlightened approximation so now I am going to go and do a little bit more calculations and go a little bit further and let you know so I have now used this I have to have these three relations we now so I can now show by using this three relation is this you can verify everything in terms of now only in terms of n plus plus n plus plus plus and sigma i a is now and sigma i ok this is alright so I have this is the one trivial this one I get by substituting into n plus plus then I have n minus minus minus n plus minus that is this thing then I eliminate in minus minus and n plus minus in terms of n plus and n double plus when I do that then I get this equation you can just trivial trivial exercise and you can all of you can do that and then this one so now and I have j in front so I now have the Hamiltonian in terms of n plus so I now go I said ok I have the Hamiltonian h as a function of n plus and n plus plus I have the Hamiltonian so now what I need to do then I need to do I have to write the partition function of ising as a function of t temperature total number of spins in and magnetic field be what do I need to do I need to do g this is if I can do that I am all set I but I I can express h in terms of n plus but I cannot evaluate this quantity this is now the final barrier that we faced till now everything done exactly but you cannot do that ok so we have done so let me write down I have already written down the Hamiltonian but by combining those things still I can combine one I can give you once more the total Hamiltonian h equal to same I am doing minus 4 j n plus plus plus 2 gamma j minus b n plus minus half gamma j minus b n so this is Hamiltonian so the e to the power minus beta h that has to go this h or goes there but in front this part is ok but I need to evaluate g if I could evaluate the number of ways I can number of ways I can find out n plus I give you n and n plus and if you can give me number of ways you can realize that configuration with the free the given n plus n plus minus 1 then if I can give you an expression then I will be able to play the trick that maximize the partition function minimize the free energy and pick up the term but I do not have any clue to that now how do you do that now I will go one step backward and I will do something really very very interesting so now I introduce a two terms one is called long range order parameter l and the short range order parameter now now I introduce remember l long range and s I will sorry I will I will change the notation little bit I will again call it sigma but without a subscript so short range order parameter sigma for s this is the universal notation so I am going in many books in order to separate they say sigma instead of sigma i sigma j for the up and plus 1 and minus 1 values of the spin up and down they use si and sj but sigma i and sigma j is the more common thing and short range order parameter sigma is also fairly common but since we do not have any subscript to subscript or superscript to sigma we will have no problem that means short range and long range order l for long range sigma for short range just easy to remember so now I define the following way definitions very nice definition the fraction of up spins and double spins so l clearly you can see l is less than 1 greater than minus 1 sigma is less than 1 greater than minus 1 so in l is 1 when l is 1 then this is 1 plus 1 2 and these 2 gets gets it 1 that means all these spins are up so in l long range order parameters 1 means all these spins are up system is fully fabricated short range on the other hand is the following way again sigma is 1 1 and this is half gamma n gamma n is the gamma is the number of nearest neighbors so you in order to avoid double counting you put the half in front of it because you are n plus plus you can count this one and this one twice so half gamma n for that now a sigma is 1 when sigma is 1 then it becomes 1 that means n plus plus becomes equal to half gamma n and then on the other hand if that means all the short range all the short range parameters are up all nearest neighbors are up and on the other hand when it is minus 1 then these 0 then you do not have any any up spins everything is surrounding your nearest neighbors all spins and down so these are the definitions of the short range and long range these long range now they look at this long range order parameter is very very intuitive and very interesting this is nothing but as I told you total number of spins total number of spins is this is nothing but when all these spins are up I know n plus I know n then I know n minus and n plus minus n minus is nothing but the magnetization m you have that is the magnetization so if I know n plus I have the magnetization the problem that I want to solve n plus plus on the other hand is a microscopic quantity and I do not have any handle on that except I know when many many spins are up then I have when there is a ferromagnetic state not also it is around you know to be a ferromagnetic state all you have to do n plus is greater than n minus and you want that ferromagnetic state to be established or a significant number of spins other than by a phase transition that means it has to jump from low magnetic state to high magnetic state that happens at certain temperature so it is the essential part of change of magnetization m in a in a singular way now I want to capture that part but I do not have as I said much of a handle of n plus plus except that you know when magnetic state then more n plus plus would be more than n plus minus and more than n minus minus because there are more spins that will be up other than that I do not have any handle that I know of course I know if the ferromagnetic interaction of low temperature there are more of that so this is the situation that we are now going to so we have done till now all the things correctly but now we are going to towards something physical inside because we have gone as far as we can exactly by decomposing into n plus n plus n and n plus plus eliminating other theory of everything is exact gamma changes into dimension triangular lattice is 6 well that is it is 4 then your three dimensional lattice cubic lattice is 6 BCC is 8 and FCC is RHCT are 12 so we know that and so the only thing that is taking care because of the nearest number interactions is gamma I am tuning gamma I am changing I am kept changing the powers of the system but we have gone as far as we can because we cannot evaluate the number of state g and n g n plus and n plus plus so now long range and short range order parameter had two physical chains can you work with that and that is what the infill theory does they work with this so now I want to express the next following thing I can now go since n plus and n plus minus are again in terms of so L is in terms of n plus n plus plus that immediately tells you now I will be able to write the Hamiltonian in terms of L and sigma so in state of a microscopic kind of the oil is not microscopic this is microscopic I am going to go into a order parameter and then I can hope to make an assumption and an approximation okay so let me write down the Hamiltonian in terms of L and it so I can write this thing like that sigma I sigma I is very easy to say it is nl right because n plus is n plus by n and now I can also write I can show that this you can do because we have done sigma is sigma j before a few slides before we have evaluated that now I just put the definition of L and L and sigma and I get the following thing I am just writing it down so I can now write the Hamiltonian so I write the Hamiltonian h in terms of L and sigma now what do I do so what they do now is make an answer so I have expressed Hamiltonian I have expressed h in terms of L and sigma so I have introduced two qualitatively nice things but I have really not made any significant process other than that so but now I look one now I am going to make an approximation now I say the following things I go back to probability and I say okay my probability of having two spins up two consecutive spins up next to each other nearest neighbor if I now assume that this probability that it is up is there independent of each other so they are multiplication of the being up then probability of the n plus plus of the fraction of n plus plus would be same as n plus minus n square so I now make the approximation the fraction of n plus plus of the possible is so this is my answer this is my mean field approximation that means there is no short range order other than than what is contained in long range order so this is the essence of mean field approximation that you ignore the short range correlations and you say short range correlation is given by the long range correlation so once I make this approximation what do I achieve after this approximation so what do I achieve I have now eliminated so once I do that I already have an in terms of l sigma or n plus n plus plus I went to l and sigma to get the physical essence long range and short range order and I say short range order is those same that close from long range order probability I can argue that n plus plus is given by n plus so I then eliminated from by one logic one more variable so I have eliminated the n plus plus by giving that condition and once I eliminated n plus plus what I do I go back I already have h in terms of n plus and n plus plus and I get that now h in terms of n plus only I have the Hamiltonian in terms of these two now I have in terms of that then what do I gain after making the approximation now if I have this Hamiltonian n plus my ising model partition function my partition function for any dimension for any dimension d sum over all n plus that can be from 0 to n number of states I can distribute n out of n spins and now how many ways I can get n plus out of n beta h n plus I know so what is the whole crux of the thing is this reduction and once I can do that I can write and this is approximate of course this is approximate is an answer but I can now write this but low and behold I know that that is nothing but n factorial by n and I know this quantity so now I can go from the sum under the summation I can find out the Maxwell term so at the end of this long road we are in a position now to exactly evaluate the partition function exactly where the free energy and we will take a short break now and we will come and do that by following the most elegant description that I know of this problem that is the current walk but basically what I said I will probably in a faster phase evaluate that and take you through the results.