 Right, so we had started thinking about simple component substances for which I drew the phase diagram and then I started mentioning something about a magnetic model to correspond with it. Now just to recall to you what the problem is and this is the primary problem we are going to focus on. If you look at the phase diagram in the P T plane for a simple substance and you look in particular at the liquid gas coexistence curve it ends in a critical point here, okay. Now across this line you have a phase transition which is said to be discontinuous in the sense that the quantity that characterises the difference between a liquid and a gas would be both as homogenous of material media both are isotropic. So what characterises is a huge difference in the density. So if you take that to be the case this difference slowly vanishes till you get to the critical point but across this gap here the difference is discontinuous. There is a discontinuous change, finite but discontinuous change in this so called order parameter, okay or whatever quantity characterises the phase, okay. Now the reason you have problems with this point here is the point of singularity that is something I would like to explain little bit. That is the place where thermodynamics fails and it fails for several very interesting reasons very deep reasons, took a long time for people to appreciate this and then once they as I said earlier the problem was really solved in the 1970s starting with the advent of what is called the renormalisation group, okay, pioneered by Wilson above all. Now what is the difficulty? The difficulty itself was recognised long before this before 1970 and the difficulty is as follows. This is the point where fluctuations become significant so much so that thermodynamics itself fails. As you know thermodynamics is something which deals with average quantities assuming that the fluctuations about this average are very very small. Typically as we saw in a very trivial model last time one over the square root of the number of particles that fails once you get to the critical point because if that were so that for instance in the internal energy for instance, if it was so that delta E over E which is normally proportional to 1 over square root of N tends to 0 as N tends to infinity, this tends to 0 and you are alright as far as thermodynamics is concerned. But we will see now how this breaks down at the critical point. Another way of looking at it is the way Ehrenfest looked at it first, the so called Ehrenfest classification and his argument was as follows, his argument was you start with thermodynamic quantities of 2 kinds, there are the so called state variables and the so called field variables. So there are intensive variables, the field variables like temperature, pressure, chemical potential and there are the state variables which are the response variables such as volume, number of particles, the entropy of the system, etc. So once you start with that description of thermodynamics then the derivatives of thermodynamic potentials with respect to thermodynamic variables are other thermodynamic variables. Just to give you an example, I called E the internal energy D E, this quantity is T D S minus P D V plus mu D N. So if you for instance took the derivative of this with respect to S keeping V N and constant you get the temperature. So this implies for instance that T equal to delta E over delta S keeping V N and N etc, okay. Or take another example if you took the Hemmholtz free energy D F, this is minus S D T minus P D V plus mu D N and this will imply for instance that S equal to minus delta F over delta T at constant V and N and similarly P is equal to minus delta F over delta V at constant T and N etc in this fashion. So in this case what is happening is that you are getting some thermodynamic variables from the derivative, every thermodynamic variable from the derivative of a suitable thermodynamic potential with respect to the conjugate to the variable you want, okay. What happens to second derivatives of these quantities? For instance if you took this equation here and I differentiated it a second time, so this says delta S over delta T keeping V N constant is minus D to F over D T 2 keeping V N constant and similarly delta P over delta V equal to minus D to F over delta V 2 keeping T N constant. But what is T delta S over delta T? If I multiply this by T on both sides, so this is equal to minus T on this side, this is T D S differentiated with respect to T keeping V N and constant and what is that equal to? T D S as you know is D Q, this is the specific heat at constant volume, so this is equal to C V, it is therefore a response function and that is typically the second derivative of a free energy of this kind. Similarly I could differentiate this, you will get this quantity here delta P over delta V and that is related of course to the bulk modulus of the system, right. So the bulk modulus of the system, if I write it up here, this implies that N the isothermal bulk modulus it is equal to minus V delta P over delta V at T N that is equal to V delta 2 F over delta V 2 that is the volume at T N. So we see that response susceptibilities are related, these are sort, these are responses or susceptibilities they measure, they are related to the second derivatives of thermodynamic potentials, okay. First derivatives give other thermodynamic variables and this is typical, absolutely typical of the structure of thermodynamics here but we also saw something else, we also saw that, now let us be careful about my notation, so we saw that the internal energy E, this is the internal energy which is by the way by definition equal to the expectation value of the Hamiltonian in the canonical ensemble for instance, right. This quantity E was equal to minus delta over delta beta log the canonical partition function and delta E over delta beta was equal to the variance of H which by definition is denoted as delta E square, this is equal to minus delta E over delta beta but this is the average energy, right. Now if I were to write this in terms of temperature I could write this as equal to minus delta E over delta T, delta T over delta beta equal to K Boltzmann T square delta E over delta T but what is delta E over delta T? You kept all other thermodynamic variables fixed in this, so this fellow was a function of S, V and N kept everything fixed including V and then you differentiate it, right. So this is the specific heat at constant volume, the derivative of the internal energy with respect to the temperature by definition is the specific heat at constant volume, so this is equal to K B T square C V, variance cannot be negative, it has got to be positive, right. So it immediately says this is greater than 0. We therefore have a rigorous inequality which says that the specific heat cannot be at constant volume cannot be negative. It follows from the fact that the variance of a random variable cannot be negative but this poses a convexity condition on the corresponding free energy because it says now that this quantity which was C V must be greater than 0 which implies D 2 F over D T 2 V, N is less than 0. So it puts a condition on the kind of curvature that this function F can have. It has got to be a convex function in the sense that it has got to be concave down, looking from above it has got to be concave, right. The curvature has got to be positive. This is of course a function of many variables V, T and N but as far as dependence on T is concerned there is a strict inequality here which is dictated ultimately by thermodynamic stability which tells you that this quantity is now this thing here the variance being positive is in fact an expression of thermodynamic stability as you can see. In exactly the same way the compressive the bulk modulus can be negative so this quantity has to be positive, this second derivative. So it tells you that when you fix one variable and differentiate with respect to the other what is the curvature going to look like of the surface representing the free energy or some suitable thermodynamic potential. Your immediate interest was in this statement here because now we know, we know that E is proportional to the number of particles N, okay. By extensivity we know this thing here. Therefore we already argued this is an extensive quantity, that is how we got the extensivity relation and we said that E is a function of S, V and N, the homogenous function of degree 1 which immediately implies that this thing is equal to which immediately implies that E of lambda S, lambda V, lambda N is equal to lambda times E of S, V and N homogenous function of degree 1 which implies this. Well choose lambda to be 1 over N for instance, lambda is completely arbitrary any real positive constant will do, choose it to be 1 over N so this says E of S over N V over N 1 equal to 1 over N E of S V and N or you can rewrite this as E equal to N times this quantity here, something being a function of 1 is irrelevant, you can get rid of it, right. This is equal to some epsilon of S over N entropy per particle that is the specific entropy, right let us use small s for it, V over N the volume per particle specific volume so let us use V for it. These are densities, these are concentrations if you like, so they are all intensive quantities, the extensive part is sitting right here so that is why I said E is proportional to N times some epsilon so let us put epsilon of S V. Then let us use the other result that we had, delta E whole squared is equal to minus delta E over delta beta, this is equal to K T squared C V just showed that therefore delta E divided by E the standard deviation divided by the average energy is equal to T times square root of K C V divided by N times epsilon of S comma V but this is the specific heat or heat capacity of N particles which is N times the capacity per particle so this is equal to T square root of K C V, little C V divided by square root of N times epsilon of S comma V and indeed this goes to 0 as we have been saying all along as capital N goes to infinity because all these are finite quantities, they have nothing to do with N any longer except for one possibility. When is it that this relative fluctuation will not go to 0 when N becomes large at a critical point, we have already seen that at a critical point I said that the central limit theorem fails that thermodynamics fails because it is predicated on the assumption of extensivity which in turn requires that the fluctuations be neglected but that is failing here and when is that going, what is that going to show up here, what is the only possibility? C V must diverge, must become infinite. If this quantity diverges at some point in the phase plane then this is singularity in C V and then you can have the possibility that this quantity does not go to 0 as N tends to infinity, okay. So now we see the failure of Ehrenfest classification theorem, his classification said that you have an Nth order phase transition when some Nth derivative of some thermodynamic potential becomes discontinuous, okay. The previous N minus 1 derivative starting with the potential itself are all continuous but the Nth derivative becomes discontinuous, yeah, oh yeah, yeah, yeah we are going to see, we are going to see what is going to happen, no, no, no, no, no, it is not that, no, no, it is not that at all. It is saying that you may have hit when you compute C V with a quantity somehow for some special value of the control parameters whatever they are, in this case this will be a function of SN, whatever this is a function of temperature and something else in particular the temperature, there is a singularity. At that point thermodynamics fails, this whole calculation fails, right and all these things become additivity, extensivity is gone completely. So something very serious is happening, the whole formalism breaks down. If you do not have extensivity then even the equivalence between different ensembles in equilibrium statistical mechanics fails. It is completely predicated on the, I mean how did we start? Let me go back, let us go back a few steps and look at equilibrium statistical mechanics. What did we start with? We started by saying here is a huge system in equilibrium, isolated system in thermal equilibrium, by that we meant that if you take long time averages of physical macroscopic quantities they will all be time independent completely and then you made a postulate which said that every accessible microstate of the system is equally probable, that was it, that was a fundamental postulate. From that you derived all the consequences including from the fact that you can define the temperature as delta log omega over delta E and so on and so forth. And then you said all right, let us consider the small subsystem here which is so small that the fluctuations driven into it by the bath are very significant but it does not react very much on the bath and the assumptions made were very obvious. When you constructed the micro canonical ensemble to show that field variables were all equal on this side, the temperatures had to be equal if I do an imaginary partition, the temperature here had to be that here, the pressure here had to be the pressure here, the chemical potentials had to be equal and so on. What was the essential assumption? That the number of microstates of the full system is a product of the microstates here and here which means the number of degrees of freedom on the boundary is negligible compared to both these guys and the energy is in some sense additive. The entropy is additive right because we definitely said that the total number of microstates of this full system is the product of omega 1 and omega 2 here. Therefore log omega 1 plus the log of the total omega is equal to log omega 1 plus log omega 2. In other words the entropy was additive and then everything else followed. That was the extensivity, let to additivity of these entropies, volume, etc, etc. That is breaking down, that is completely breaking down here. So somewhere there is a singularity and we need to know where. Now this is the reason why the Ehrenfest theorem classification also breaks down because it only says nothing is singular, some derivative, a certain order derivative of some free energy or some thermodynamic potential becomes discontinuous at some point. Very very roughly the picture was as follows. If you had two phases, again some configuration variables here you computed some free energy as a function of some thermodynamic variables here right. And let us say in one phase it was like this and in the other phase it was like this right. Then since the potential has to be at a minimum in thermal equilibrium, this phase thermodynamic potential is larger than this so the system moves along here but then when it crosses this point it jumps to this phase and therefore at this point the slope of this free energy is discontinuous and you have what is called a first order phase transition okay. But the same thing could happen not to the free energy itself but a derivative of the free energy then you got a second order phase transition and so on. This was his original classification but we see that at a critical point that is not what is happening, something far more serious is happening. The whole mechanism, the whole formalism is completely wrong, it is breaking down okay. The fluctuations have become so large that you cannot ignore it and this classification is useless in some sense. So today we do not use this as a classification. What we say instead is there are two kinds of phase transitions, there are continuous and discontinuous phase transitions. Then there is a concept of an order parameter introduced and you say when it changes discontinuously you have a discontinuous phase transition also called a first order and when it changes continuously you have a continuous phase transition which is called the second order phase transition but it is loose terminology. We should really say discontinuous and continuous right and what is happening with going back to our original example, going back to our liquid gas example which was this in the PT plane ending in a critical point here. Definitely as you cross this line along anywhere along the body of the line except here there is a change in the density discontinuously and yesterday I drew the diagram and I was getting confused between the volume and the density, let us draw it reasonably in a way which is transparent. The way to draw this would be to put temperature here and the density here, let us put the density right. Then in the vicinity of this point close to this, as you know every point on this line coexistence line corresponds in the rho t plane to a region because it is the full coexistence region corresponding to every point here. Just to recall to you what is going on, if you plot the isotherms V versus P, these isotherms were like this. So a point here would correspond to this region here. A point here would correspond to this region here. This point would correspond to the place where you actually have just a single point, an inflection point where the slope is flat. This is liquid, this is gas and this is the coexistence region with the tie line construction put in place. What does it look like in this figure? Well, as you can see this is going to look something like this. This is rho C, this is T C, the higher density region. Now which one is liquid, which one is gas? What does it look like? T versus rho, T versus rho. That is the whole point of it. It is easier to see in this picture. So rho C, T C here. The higher density is a liquid. This is a gas. This width here is reflected here in this picture and at T C, the density between the liquid and the gas vanishes. And close to it here, one would guess, one would guess that in normal circumstances given no other information, the naive guess would be that it looks like a parabola out there. You can always fit a simple minimum to a parabola if you close enough to it. And since you have shifted T by T C, this is really saying that T minus T C, it is the vertical coordinate measured with respect to this origin is going like, since it is an inverted parabola, it is going to go like rho minus rho C whole square with a minus sign. Instead of this, it is more convenient to take, instead of rho minus rho C, it is more convenient to take this minus that. It would not make a difference to this. But let us retain this. Incidentally, this is also telling us now that modulus, rho minus rho C modulus is proportional to T minus T C to the power of half. That is the square roots or the square behaviour near a parabola. This is what we would expect. We have just said it in words, we have not proved it in any sense. So the fact that you have a singularity here is a serious matter, as I said. And we have to take note of this. We have to redo the whole calculation. Now it turns out that doing this is not so trivial at all. The whole subject of critical phenomena depends on this. But we need to keep, I need to introduce a little more standard material before we can get to this very sensible form. Let me do that. So let us make this, let us make this fluid magnet analogy. What this says is that you have a complete analogue of this phase transition. This is continuous phase transition at this point with a magnetic system. And for everything that you say here, you can make a corresponding statement there. This system is characterised by a pressure, a volume and a temperature. The magnetic system is similarly characterised by not a pressure but a magnetic field, which I call H, should be a vector but let us avoid inessential complications here. The response, if you increase the pressure, the volume changes. Similarly if you apply a magnetic field, you have a magnetisation and T. So this is the one to one correspondence between these variables. Now here you can have a P versus V diagram, you have an equation of state connecting these three variables. You can have a P versus V diagram which is an isotherm, a line which is an isotherm. Similarly you would have there an H versus M figure. Then you have a V versus T diagram as in this case or a density versus T. You can have an M versus T diagram there. And finally you have a P versus T which is a H with T in this case. Let us see now, let us draw all these six pictures and see what happens. But I need a model of a magnet to write the equation of state. I need a model for the liquid gas system for which I can write an equation of state. In either case. Let us do this since you are more familiar with this. Let us do this first. The model as we can see is the so-called Van der Waals equation of state. And we are not going to write down the algebra here but basically it is something like P plus A over V squared times V minus B equal to RT. This is the Van der Waals. Now you are familiar from elementary physics that this B measures the so-called excluded volume. This A measures the force of attraction. I need an intensive variable here because there is a V squared here. So little bit, okay. I am not too happy with this notation but okay. He has got a point. He says after all this is an intensive variable and that should be 1, 2. You cannot add arbitrarily. And similarly for B. But let us do the following. I am going to assume A and B are suitable dimensions, suitable, I mean I am interested right now in the dependence on B. P plus A over V squared. Yeah, I think that is the least harmful. Yeah, I know. It is a big mess. Okay. Yeah, in fact it is convenient to do it this way. This is the volume. Yeah, this is very convenient because this is really telling you in the naive picture it is telling you this B is the excluded volume due to the finite size of each molecule, right? And little V is the volume per molecule. So it is only reasonable that this B has this interpretation. Otherwise I got to fool around with the number of particles everywhere. Good. Now this is a cubic equation for V. That is the reason you get those 3 roots and so on. And let us see what the interesting part of it is. So if I plot P versus V, P V diagram, we have already seen that you get below a certain critical temperature, you end up with 3 roots of which there is an unstable root in the middle. There are 2 stable roots on either side corresponding to liquid and gas respectively. And you should draw a tie line etc. So let us go through this thing again, something like this etc. Very schematically. Now an aside, how the hell do you get this equation of state? Now throughout the 19th century people had the feeling that there was probably some universal equation of state for gases, for real gases, okay. And they were trying to find it very hard. And one idea was that maybe there is some scaling here and then with respect to some standard depending on each gas there is a characteristic pressure, characteristic volume, characteristic temperature such that if you scale with respect to those variables, you would get an equation of state which was universal, okay. This program ended in failure. There is no such equation because these quantities A and B in a sense are not geometrical hard sphere kind of quantities. They are quantum mechanical objects. You have to compute what this is. It is due to the repulsive part of the intermolecular potential. This is due to the long range attractive part. And it will depend on the nature of the molecules. The force between 2 water molecules for instance is very different from the force between 2 argon atoms and so on. So there is really no universal equation of this kind. But this equation is that in some very very profound sense and it is as follows. How do you get such an equation? Well getting it from an actual intermolecular potential is a very non-trivial task. It requires many body theory and getting an equation in close form is next to impossible, okay. Instead what it does is the following. We know that physically if I plot r versus the potential between 2 atoms at distance r apart, right, then due to Van der Waals interaction, even if they are spherically symmetric molecules, there is a long range attraction which goes whose potential goes like 1 over r to the 6 and whose force therefore goes like 1 over r to the 7. This is a long range attraction. At very short ranges the problem becomes much harder because quantum mechanics really kicks in here and finally it is the Pauli exclusion principle which makes sure that the 2 electron clouds do not sit in the same state at any instant of time. So there is a long short range repulsion here, hardcore repulsion. Effectively the potential looks like this. Now doing any serious calculation with a thing like this especially because you do not know what is the exact power here, if at all it is a power is very very non-trivial but you can make an approximation to this, a very crude approximation and that is to say that this is approximated by something that is essentially infinite below a certain distance and B plays a role of that distance and beyond it there is a 1 over r to the 6 attraction like this, long range attraction. So you approximate this potential by that and then you make a whole lot of other approximations and then finally you end up with an equation like this, okay. Now the fact that this is 1 over V squared is not very hard to understand because it is really saying that each molecule is seeing, you are asking what is the decrease in the pressure due to the fact or the force that the molecules exert on the walls due to the fact that they attract each other. That is why this goes to the right hand side and gives you a diminution of the pressure, right because you can write this as P equal to RT over B minus B minus A over V squared. This part comes from the short range repulsion of the potential, this part comes from the internal energy of this gas because this follows, these particles are not like an ideal gas they are attracting each other and therefore there is a decrease in the force exerted per unit time per unit area on the walls of the container due to this attraction which is this. Now why 1 over V squared? Well that is not very hard to understand because this is like the concentration, you put the ends in here. There is an N squared over capital V squared that is how you got the V squared here, right. So each particle if you assume is interacting with all the particles around it then the total interaction strength is proportional to the concentration N. But in unit volume there are N such particles, little N such particles. Therefore the total interaction energy is proportional to 1 over N square, 1 over is N square, okay which is the same as 1 over V square, okay. So this is called a mean field theory where you say that and we will see a better example of it or a more transparent example of it in a few minutes, okay. Whatever it is, this part takes care of long range attraction, this part takes care of short range repulsion here. So it looks like a very typical equation of state and sure enough it produces for you an inflection point here because these points whose coordinates are P C, V C and T C are determined completely as you know very well from elementary physics by 1 the equation of state plus 2 more conditions, one of them is says that the slope here is flat. So at that point you have this and then you have Delta P over Delta V equal to 0 on the isotherm and it is an inflection point. And as you know it is a very simple exercise in high school physics to compute what V C, P C and P C are in terms of these 3 equations. From these 3 equations you have 3 unknowns you can solve uniquely for it and you get this. Now we are interested in the critical region. So we are really interested in what is going on here in this region, we are not interested elsewhere, this whole thing. So it is very convenient to shift variables to this point and to define P equal to, I mean little P equal to P minus P C and you make it dimensionless by dividing by P C. Similarly V equal to, I am sorry for this bad notation, it is V minus V C over V C. I use this for the specific volume but now I am using it for the reduced, dimensionless reduced volume. One of the characteristics of theoretical physics is that they assume the notation is usually terrible because they, the philosophy is those who understood it, understand it, those who do not worry about them by context you know what I mean. And you use, I am sorry again T which you normally use for time, this is used for T minus T C over T C, the reduced temperature, okay. I leave it to you as a simple exercise, could have done in school to rewrite the Van der Waals equation of state in terms of these variables. So first you have to solve these three equations, find out what P C, V C, T C are, by the way V C is trivial immediately and then you have to work a little harder to find the other two, T C is also fairly simple but P C you have to do a little bit of algebra, okay. And then you write, you find that in this neighbourhood, in here you find not unsurprisingly the curve goes like this, it is an inflection point, it is not linear, it is an odd function so the next one is a cubic dependence. Now you discover that in this region, P goes like minus V cube, so there is a power 3 that emerges and I want you to show this, okay. This is crucial, it is called a critical exponent. Well you can also do many other things, you can ask what happens if I take the same equation and draw not the P versus V diagram but I do the other variables, right. The P versus V has these little horizontal pieces out here, the P versus T we already know, it is a curve in the PT plane which ends in the critical point and then finally there is the T versus V, the density thing, again I leave you to work that out because you get that parabolic curve there. But now let us go to our magnet analogy, I am in a hurry to do that. What is going to be the simplest magnetic equation of state? Well we want to look at a substance which undergoes a phase transition from a non-magnetized or paramagnetic state to a magnetized or ferromagnetic state. There are lots and lots of models of magnetism and it is intrinsically a quantum mechanical phenomenon. There are two ways in which you can get a phase transition, non-trivial behaviour. One of them is to say that there is an interaction between the different constituents which lead to the, leads to phase transitions. If you start with an, in this case if you start with an ideal gas, P V is equal to R T and there is no phase transition at all. You need the short range repulsion and the long range attraction to produce a liquefaction. You need the attraction definitely to produce liquefaction. So some interaction has been taken into account. In exactly the same way if I just start with a lot of independent atomic magnetic moments, I do not get any phase transition at all. It remains a paramagnet which is magnetized when you apply a field and demagnetized when you remove the field. To get a phase transition to a state of permanent magnetization, I must include an interaction between the different atomic moments which turns out to be fairly complicated because you need something called the exchange interaction. There is another way to do this which is to say that, which is to do what you did here. In this term, we did not write any explicit interatomic potential. We just said phenomenologically the effect of this attraction is mimicked by this reduction in the pressure out here and we gave a hand waving argument for why it should be 1 over V squared and not 1 over V cubed or V 4 or anything like that. In the same spirit, one can introduce an external field, an effective field, an internal field in the system which says the effect of the interaction between spins is mimicked by an effective magnetic field in the medium itself and that plays the role of the interaction. This gives you what is called the Weiss molecular field theory. Let us see how this works. Now we know that if you have a substance with a lot of independent atomic magnetic moments pointing every which way, etc. We ignore everything except the magnetic property. We do not worry about the kinetic energy. We do not worry about any other degree of freedom. We are looking only at the magnetic properties. Then I apply a magnetic field to this. If each magnetic moment is mu in the presence of an external field H, a magnetic field in some direction say fixed direction, the potential energy for each magnetic moment is mu dot H in the presence of a field with a minus sign. Now I make a further simplification. This of course says that depending on the direction of the field and the direction of the dipole, you get any energy you like from a maximum which is equal to mu H to a minimum which is minus mu H. Now I make a glorious simplification and say, look, I allow only two possibilities either the two are parallel to each other or anti-parallel to each other. When they are parallel to each other, cos theta is 1 and the energy is minus mu H. So this configuration, this is the direction of H and this is the direction of mu implies the energy is minus mu H and this other possibility, this is H, this is minus mu, this is mu implies epsilon equal to mu H. There are actually substances which would behave in exactly this fashion and that is now due to the quantum mechanical nature of this magnetic moment which comes in turn from the spin of the particle and the spin sometimes it can be half in certain cases such as the electron and then it can have only two possible projections along any direction, okay. Whatever, these are the two possible energy states of any of these are elementary magnetic moments, right. What is the magnetization then if this is the case and they are not interacting with each other. So what is the magnetization? You can write this down trivially. The magnetization is since they are all independent of each other and I have N of them, it is N and we want the thermal average. If you did not have a temperature, if the whole thing was absolute 0 then the whole system would go into its ground state, state of least energy which means that all of them will point along the field and the total energy is minus N times mu H, that is it. But now thermal agitation is causing them to flip back and forth and at any given temperature you can ask what is going to be the magnetization, right. We want to know the average magnetization. So this is equal to N times the average value of mu with respect to E to the minus beta mu H, etc. But this is now trivial because this is equal to N times, there are only 2 possibilities for each one of them. If it is up then the magnetic moment is plus mu and the probability is E to the power minus beta epsilon which is beta mu H, relative probability plus the magnetic moment is minus mu, so we put a minus mu E to the minus beta mu H, minus here divided by the partition function which in this case is beta mu H because this factor divided by this is the absolute probability that the moment is plus mu. This is the absolute probability that it is minus mu. So this is equal to N mu tan hyperbolic beta mu H, so let us write it as mu H over K Boltzmann T. This is my magnetic equation of state if you like for this trivial problem. It is expressed the volume analog in terms of the system size, the number of particles, the temperature and the pressure analog. So it is the equivalent of P plus N squared V is A over V squared times whatever it is V minus N B equal to N R T. But this is very trivial. We can now plot what this is. We want things in terms of intensive quantities, specific quantities, so let us define M equal to M divided by N mu. There is a dual advantage. It is an magnetization per particle and it is also dimensionless because you divide it by the magnetic dipole moment. So we have M equal to tan hyperbolic mu H over K T and we are all set now. The equivalent of the P V diagram, the isotherms is you fix the temperature and you plot H versus M or M versus H. The reason we chose P in the vertical axis, V on the horizontal axis is because you had in mind the historical reason, historical idea that you had a cylinder with a piston which changed the volume and the pressure got adjusted. Whereas here you normally apply the field and you measure the magnetization. So let us plot M here and H here and the isotherms are very trivial to write down in this case. They are just these lines. That is the tan hyperbolic function and this tan hyperbolic can never exceed minus 1 or plus 1 on either side and that is it. Notice it saturates. When all of them have moved in the direction of the field, that is it. There is nothing more possible as far as M is concerned and M cannot be less than minus N mu. In this case there is an extra symmetry present in the problem which is not present in the P V diagram as you can see, at least not overtly. And that is if you change the direction of H, M changes. So there is a symmetry present here. Now that is going to have implications. Now what is the slope at the origin? Well the isothermal susceptibility, the magnetic susceptibility is defined as the rate of change of the magnetization with respect to the field at a given temperature. So it is equal to Delta M over Delta H at constant temperature. So this formula chi is M over H is not correct because this is a nonlinear relation between M and H. It is only true near the origin where you have a linear region. So you have to take that into account by writing here H tends to 0. So the susceptibility is defined as a slope at the linear region of this isotherm. Now that is trivial to find because what does tan hyperbolic X do as X goes to 0? Goes like X. So this immediately says this quantity is equal to mu over k Boltzmann. This is Curie's law of paramagnetism which says the paramagnetic susceptibility is proportional to 1 over the temperature. Nothing much is happening in this model because there is no interaction. So there is nothing interesting to do. Now the question is we know that in real substances this is not what happens that the phase transition actually happens and if I plotted for instance as a function of temperature T if I plotted M in the absence of a field well let us go on with this a little bit. What happens in a real substance as I lower the temperature? As I lower the temperature the slope becomes higher and higher right. It cannot go beyond saturation so it does this. Now ideally if this is all this there was to it then it would go on like this and the slope would become infinite only at absolute 0 of temperature but there is always an interaction between these magnetic moments and if you are at sufficiently low temperatures then the thermal agitation cannot overwrite those and therefore there will be some kind of magnetic ordering at sufficiently low temperatures no matter how weak the interaction. In the case of iron it already happens when you hit the order of 10 to the 3 Kelvin because that is the curie temperature. Anything below that you have permanent magnetization but this is entirely a matter of what system you are looking at to see if you get magnetic ordering or not. So what looks like high temperature or low temperature depends on the characteristic interaction energy in the problem between any two magnetic moments as compared to KT which is a thermal interaction energy right. If you look at substance like helium solid helium 3 for instance that becomes an antifurro magnet a nuclear antifurro magnet at some crazy temperature which is of the order of milli Kelvin's. So first that material looks completely non-magnetic it is an inert gas for that material 1 degree Kelvin is already very high temperature. If you look at very very high purity copper which is a non-magnetic material at 50 nano Kelvin it becomes a nuclear antifurro magnet. So for that material a micro Kelvin is very high temperature as far as the magnetic properties are concerned. So this we as physicists we have to understand when you say low and high it is a question of what scale we are talking about. Now for most substances what happens is that as you lower the temperature there comes a finite temperature at which the slope becomes infinite and in fact this becomes a curve very reminiscent of the critical isotherm except it is tilted by 90 degrees because I did not plot P in the vertical axis I did not plot H in the vertical axis. What happens if you lower it even further? Well in principle it would do this if you just believe in the continuity of all these curves you would believe that once it is finished that then it is going to do this. It just keeps bending backwards so let us draw that and see what it does. So it should really do this right. I will just draw 3 curves. One is when it does that with the finite slope the second is when the slope has become infinite and the third is when it does this. This reminds us of that Van der Waals equation when you went down and up again and we said that is not true because it is violating stability somewhere. Indeed that is true because in this region from here to here dm over dh which is negative and except this is not a diamagnetic substance so this certainly cannot be true. It violates again a convexity property of one of the free energies just as in the other case the compressibility turned out to be negative and that violated the convexity property which in turn came from extensivity that similar sort of thing happens here. So if you excise this portion then what the system does is come here go down jump to this and go off and on the other way back it comes all the way up to here jumps here and goes back. What does that curve look like? Looks like a hysteresis loop and indeed that happens except in the refrigerator it did not happen but in the magnetic system hysteresis does happen all the time. However if you have hysteresis happens for other technical reasons. It happens because all the domains in the medium do not reorient at the same time and the same field and that happens sequentially but if you look at a single domain ideal substance there will be no hysteresis and the curve becomes discontinued. At high temperatures it does this, at some crazy critical temperature it does this and below that it does this. As you lower the temperature it keeps doing this cutting it higher and higher. Does not this remind you of what happened in the tie line construction? The only difference is there for different isotherms you ended up with the different tie lines but here there is only one. All of them have become degenerate here. But you can see that is because of this symmetry which is not overtly present in the fluid case. There is a symmetry but it is not immediately obvious. So we can see that we are leading up to this thing here. The next thing to do is to do a cheap thing trick which is to put an effective field there and show you that you get a cubic curve here. The 3 the exponent is going to show up. Then the half exponent which we got for the density difference is going to show up here in the magnetization and so on. So we will take it from there and then go beyond that.