 So, now we go to the Landau theory which I will spend about I think some 15, 20 minutes on that. So, Landau theory is saying that although it is largely phrenological, but it is the beginning of the conceptual framework of understanding phase transition, ok. So, read this, but also let me in the meantime also let me briefly tell you. In 1937 when Landau did the theory of phase transition, it is a general theory of phase transition and it paves the way for everything that followed, a very simple theory, incredibly simple theory and but however this generality that it allowed and the new concepts that brought in which was not there before and one of the new concept that Landau brings in, he just did, he asked the questions that I was asking here, how do I take the characteristics here in the old phase which I know contain the key to for the phase transition. So, how do I do that? What are the way I go about incorporating this? So, he rightly now guessed that the way to do that that I need to have the, so I have the minima, I have the minima in the liquid and I have all the correlations here. So, now if I can expand the free energy in of these phase in terms of free energy of all the characteristics of these phase, then I can look for a certain discontinuity. So, that is of course ambitious and it is the reason it does not fully work, but that is the one it is called the, so this is again my favorite line, some of the predictions are not quantitatively correct, but the main merit and the strength of Landau theory is its tremendous generality that comes with a simplicity that is unmatched, unmatched in theoretical discussions and you will see that, but the reason I am preparing you for that though it is a very simple theory, it opened the door for many theories and it is the generality that is absolutely mind boggling. So, what it is? After saying all these things, what Landau did, let me write down the free energy of the new phase, free energy, F is the new free energy of the new phase in terms of a simply called Landau expansion, expansion in a Taylor series, in a Taylor series of, see people are all very perplexed for many years, decades, nobody wrote down such a simple expansion, but Landau wrote down, that is what actually probably the simplicity or whatever is the genius, so he expands. So, this is the, when you expand like that, then eta is the order parameter, so I am expanding free energy in the order parameter, in the old phase order parameter is 0, so f equal to epsilon, now my aim is to find a solution of the free energy in terms when the eta is normally 0, so this is the search of a solution, what are the conditions then, where then the answer lies, the answer lies as I said answer lies in this branch, answer lies in this branch and where are these branch are going to, all these quantities alpha, A, B, C, all of them are temperature dependent, this is explicitly put out for reasons I will describe a little later, all of them are temperature dependent, now what are these quantities, how do I get alpha, A, B, C, exactly you take the derivatives, so that is the characteristic I was saying, we need the derivatives and once you try to find the derivatives, you go a little deeper, you find the derivatives are in fact not only just specific heat or compressibility, but they are also given little higher level theory intermolecular correlations in the old phase, which we are not going to go, so this is the famous Landau expansion, there are many other terms there, these when you put the gradients here, delta eta from that becomes a Ginsburg Landau and that is one of theory of superconductivity people use or super fluidity, the Ginsburg got the Nobel Prize later for this and many other things, so now we start playing some games, so these are first important things they are determined by the derivatives of the free energy in the old phase, so these are determined by the old phase and we are trying to buy Taylor expansion just like we do, how do we do a Taylor expansion, we always do Taylor expansion by going from known to unknown that is the Taylor expansion, interpolation, extrapolation, so Taylor expansion always like we do f x plus a, we do a Taylor expansion f x plus then first derivative, but evaluated at x a, so whole idea is that of going from known to unknown and known we know, we are entitled to know everything about the old phase and then we are trying to do new, now first thing is that because it is a free energy minima, because it is a minima here in the liquid phase this is 0, so now it becomes simpler that the free energy is one term less, next very important one quantity is done, now I make a distinction between a first and second order phase transition and this is a very important, so I will now describe something else that is coming here but coming at a later stage, so I better do it here, this is extremely important thing and essence important part of the Landau theory, so in a first order phase transition free energy surface behaves like that, this is against order parameter free energy and I am lowering temperature towards the transition, so this is the old my parent phase this is the daughter phase parent daughter, now parent and daughter becomes stable, now what is in my pressure density what is this point? Exactly this is the co-existent point, so this is the metastable, characteristics of first order phase transition that there is always a metastable phase, one of the major characteristics and what is the manifestation of the metastability, we see it all the time what is the most important manifestation or a signature of metastability hysteresis, you see in the presence of magnetic field huge hysteresis that is because system even strapped even when this is stable system strapped here that is one of the reason, another reason even at co-existence there is a large barrier in first order phase transition, what is the consequence of this large barrier in a first order phase transition gas to liquid to solid nanomaterials forever, nucleation you cannot spontaneously go from one phase to other phase because this is a macroscopic barrier which is proportional to number of particles in the system, so it must be new phase must be nucleated, so two important characteristics of first order phase transition is A, metastability hysteresis be it must be nucleation because there is a macroscopic barrier which so system must be is like a winning a war you must get into the fort, these are first order phase transition what is happens, so this is the land of free energy surface for first order phase transition, what happened to land of free energy surface for second order phase transition like the order disorder transition like super fluidity, now this is very nice and very nice this is what is the land of theory is gives you a nice physical picture, this is low temperature now I am increasing temperature or you can say decrease whatever you will see so uniquely different, this is time I really miss a big board, this t equal to tc, this t less than tc, t near tc and then t greater than tc, let me see t greater than tc, t greater than equal to tc and t less than tc, so there are the four diagrams, so there is no metastability in a second order phase transition there is no nucleation, so as I go to the transition point the systems remember I told in the response function second derivative are the springs that hold the system together, here this spring is compressibility or susceptibility or the specific heat and that spring slowly goes away that is why you are infinitely compressible at critical point and the free energy surface becomes really broad and shallow in the minima and this is where things are diverging and just below little below that you get this spontaneous symmetry breaking of the system, so this is land of provides this free energy landscape, the free energy provided this beautiful picture how a phase first order phase transition happens and every second order phase transition happens, this is how first order what is shown here is first order what I have drawn there is first order and the so there is a huge change in going from melting is a case where there is no second order phase transition and there is a beautiful logic again land of logic that in a gas liquid however there is a second order phase transition this becomes narrow remember the isotherm then this is the critical point inflection point this is Tc and so as you approach that then free energy surface start become even in the minima start become flatter and then the whole second minima completely disappears and you get the so this is the one that is shown there is the first order phase transition is metastability and then I will do the land of theory in a minute and this is the flat and then this is the spontaneous symmetry breaking in a second order phase transition. So let us do the land of theory quickly and so knowing that free energy surface are very different land of trivially now he said okay okay first thing I again mentioned that is an order parameter that is why order parameter is introduced so that I can expand the free energy order parameter is the change and you know the free energy expansion is an expansion is done in Taylor expansion have to change so in fx plus a our Taylor expansion a is the order parameter you are expanding around that and so order parameter is the smallest parameter okay that is number one second one now we going to do the second order phase if we do first order phase transition then this quantity b is non-zero and if b is non-zero I can make b negative and by making b negative I describe this transition and that is exactly the way land of Dijon theory of liquid crystal was done the Dijon was given a workplace for 2 watt one is liquid crystal another is polymer or 3 or semi-contact also in these two liquid crystal and that it is he did the land of theory so is the beautifully one can describe and it turns out I can do the calculations of b and in first order phase transition they are negative what is an example let us do the van der Waals gas and we take again and again so this is the van der Waals gas or famous van der Waals and then this is the free energy for integrating this one you get the free energy and I can plot the landscape free energy landscape from van der Waals I can also do expand free energy in a Taylor expansion van der Waals gives you exactly this what we discussed the other deferrals these are my material series beauty is that the second coefficient is this thing now free energy is very very go over you get a key term cubic term you integrate that get the free energy these term becomes negative at a low temperature simulating the phase transition first order phase transition and these goes to the cubic term goes to become small near the gas liquid critical point that does not fully describe this theory but land of theory is essentially van der Waals theory these are called mean field theory in the sense that I am a neglecting fluctuations I am using a simply Taylor expansion to describe the if I integrate that whole square row cube that comes in so van der Waals free energy is exactly land of free energy except I now have some little bit of ideas where are it b b and b 2 b 3 are there ok it already has now the strength of land of theory is in providing a description of the second order in a trivial way now second order phase transition here there is a symmetry the symmetry is that free energy is invariant if you 0 this plus this minus is the same this is called the degeneracy so if it is so then my this quant this free energy the condition often called the parity a symmetric condition and then this becomes simple like that this is very famous you know particle phase 5 4 theory many many theories you know cage will say himself did it it has a huge impact on the enormous field of field theory this is this land of thing that is it went in so many different ways and whole polymer is I told you liquid crystal everything is done in this language this is more I like to tell that this actually gives you the language of the phase transition that teaches you how to think about phase transition how to think of free energy change ok and order parameter so this is now simplifies and now land of made this brilliant thing is that ok a is a strongly temperature dependence I assume a t this is the not the hemorrhage free energy this is constant a and this is a universal language of a land of theory this kind of expansion is the absolutely standard now you get so I make this why do I do that I make this why do I do that I did it because when t goes to t c so this is this term then this term is the one that is the spring that is the frequency right and told you that is the response function this we know will be specific actually one over specific it but that so this thing is going to be flat so that is why he assumes that it goes to 0 as a t minus t c and then I solve for eta with that condition put this here this is here now I go and find out this take a derivative when I take the derivative I get eta equal to 0 always remain a trivial solution because this phase always remains a trivial solution this is minimum here but in maximum the same condition also holds and then I solve that I get this is the so the order parameter is a remarkable prediction that order parameter has this non classical dependence that means even when look at this beautiful thing if t c minus t c is 0.1 I go below 0.1 I am in this regime I am going below the critical temperature what is square root of 0.1 0.33 right I go to 0.01 what is square root of 0.01 0.1 what does it mean this kind of fractional exponent means he is a huge dependence and this is the real hallmark of critical phenomena this fractional exponents these exponents is called alpha is the specific heat no sorry this is called beta because it is in terms of order parameter and beta is the universally 0.3 or 1 by half close to this is amazing you do so many different systems define the order parameter and consider the temperature different order parameter and you find the fractional exponent all almost always 0.3 0.3 that is why it captured the absolute the a of imagination of scientist across that is this universality is a characteristics of critical phenomena and these are the called critical exponents yes no what do you have actually you have absolute and there are lot of these exponent beta above and lower called beta and beta prime and there have been lot of work on that and beta and beta prime are almost always the same they might be slightly difficult but very difficult do the experiments exactly very close to that but experiments were known for a very long time this is exactly exponent you find from van der Waals theory also these are called the landau theory does not get correctly exponent 1.3 but it captures one of the essence that the so eta is the change so eta will be if I go below this I come from here to here then this will be the order parameter. So what I am saying is that by giving change a small amount the eta is going to change by big amount that is a singularity that we talk about in order parameter. So there are all these different kind of order disorder transition isotropic nematic is very closely second order and then gas liquid is a thing so going back to first order phase transition now like liquid crystals or van der Waals discuss the first order fairly well that then B is non-zero anymore it is the B that changes the sign. Now what Dijan did in liquid crystal transition in isotropic nematic he noticed that isotropic nematic is nearly second order though first order so he kept landau theory he kept landau this path and then he also kept the B term and it developed a theory again same spirit same phenology nothing great but he also again captured the describe the liquid isotropic nematic phase transition in liquid crystals which is given here. I think this is what I wanted to tell the landau theory and the free and the free angel landscape of landau theory then the critical exponents so there are some other critical exponents which one should not I do not like to add that but the other critical exponents are the following specific heat alpha eta just even I might be making something wrong this is beta and specific heat alpha I should have had that then order parameter is a beta and there are some other critical exponents that is gamma delta a series of critical exponents which compressibility compressibility goes as gamma space with alpha alpha is typically 0.1 beta is typically 0.3 2 gamma is typically 1.8 landau theory predict this is this is half landau theory finite discontinuity here and this one landau theory predict 1 so landau theory gets is values wrong but it does get the basic physics correct. So, should I write it separately these Cp Cp or you can see these things Cp eta and chi and these are the exponents critical exponents alpha beta gamma and as I said they are universal that means they are nearly the same across all the systems this is a good question and all is a mean field Ramakrishna Nimswap is a mean field landau is a mean field all are the same at the end of the day the reason is that when I expand that these are the ones with guys working on phase transition right the day in and day out ok. Now in all these things now let us consider gas liquid simple thing in a gas liquid near the critical point what happens there are regions which are liquid like there are regions next to it which are gas like large scale density fluctuations you look at the density and now I look at the density as a fluctuations around the average density in my critical rho star language critical density is again universal in near point 3 it is very interesting near point 3 I take that out and look at fluctuations then what do I see I see there is a region like here where is my origin I draw like that there is a large value then after that it becomes negative then become large then becomes negative I find out that so delta rho this is liquid this is gas this is liquid like that need not be periodic there is not periodic but there are so what are these now these are interfaces what does interface bring in bring in surface tension at an extra energy where is that in Landau theory it is not. So Landau theory has to be supplemented by and lo and behold you have one of the most powerful theory of the world these Ginsberg Landau. So Landau Landau walls does not have this extra term which is we call effect of fluctuations so mean field theory does not have the effect of fluctuations you have to go to Ginsberg Landau to do that that we all do when you do surface tension in my book we have dealt with it at length surface phenomena nucleation these are one of the main at the topics in my book which are not dealt in either in physics or in chemistry books no you do not have the co-existence you have no you do not have co-existence you have a point see co-existence a line. So as I drew there that line coalesces it becomes a point point of singularity here this is where a large fluctuations are going on gas like and liquid like in a magnetic phase transitions you have the up spin and down spins and you know or ordered and disordered that means one place is ordered followed by a disordered region. So these are the then the order goes away and becomes fully disordered so the co-existence so this is a flat free energy interface there is no they are they are are inside it gas like and liquid like but those are microscopic or mesoscopic domains they are essential and ingredient for the fluctuation large scale fluctuations we call the exact language is large scale fluctuations but those large scale fluctuations are affecting the thermodynamic properties but there is no macroscopic interface. This is a very important question that you asked that the and the very essence of a critical phenomena and so one I go over take care of those fluctuations in this term and then like a huge amount of difficulty that comes in cannot be solved and that is where comes an important way how to go beyond next step and that was done by first thing was done by Kadanov by he said how the introduced how the lens scale diverges that is called Kadanov transformation. There is a beautiful book by Shankang Ma Dynamical Innovation Dynamical Innovation Dynamics of Critical Phenomena he described the beautifully Kadanov and there is a wonderful review article for high school students essentially but first year is students called Mattis and Kadanov physics today 1979 or 78 wonderful that is the best of critical phenomena that I know of Kadanov and Mattis or Mattis or Kadanov both were at Brown and they wrote together and we got the we got the print everything was done in one dimension which is done that particular article almost in Toto in my book the critical phenomena. So doing that was done step by step Kadanov did the Kadanov transformation says how to treat the emerging correlations long length correlations that happens things become correlated over long say because the fluctuations become correlated that because here if I have liquid like regime then I have a gas like regime this is called fluctuations getting correlated the one of the most powerful theory of modern times K. G. Wilson in 1970 or 71 did the transformation which is lot of fun lot of fun to that I did that once I was happy to I was very lucky in Chicago to work with one guy called Yoshio Ono we did the transformation group calculations of polymers Mattis amazing theory okay.