 So, we have looked at the Ising universality class of systems for which we wrote down a Landau energy functional and from which we got the equilibrium configuration. Now, I would like to do that in a slightly more general context and show what happens when you introduce fluctuations on the one hand and the idea of inhomigenities on the other and how there will be relaxation to the equilibrium state. So, to recall to you, we begin right away with the order parameter M of R, the magnetization or whatever the order parameter be and if I if you recall the Landau free energy, it is not exactly the Hemol's or the Gibbs free energy, we saw that we constructed a functional in such a way that you got the correct equilibrium value above and below the critical temperature. This free energy L, let me just call it L, this is equal to an integral the in d dimensions of R and then the terms inside, first of all, in the absence of an external field, there is no linear term but if you put in an external field, there is of course a term proportional to that. So, let us put in a field H of R, so this is minus H of R M of R, there is such a term and the next term is a quadratic term but if you recall it had a coefficient which was proportional to T minus T c, so that it would give you the right critical behaviour. So that term let me call it A tilde M squared of R and this A tilde is a positive coefficient A times T minus T c over T c which is A times T, little T was this difference between this T minus T, reduced temperature T minus T c over T c and in the magnetic case, there was no cubic term but only the fourth order term and the temperature dependence of that term, the coefficient was irrelevant. It just had to be positive so that you would have stability about the equilibrium, stable equilibria, so this was B times M to the power 4 of R and then to include inhomogeneities cause not just by the fact that you have an H of R but in general you have a term which includes gives you the gradient energy when you have inhomogeneities in the magnetization and this was of the form one half, that half is just for convenience, some coefficient times gradient of M of R mod square, okay. So that was the Landau free energy. The equilibrium solution is found by minimizing this free energy, so equilibrium configuration given by delta L over delta M of R equal to 0 and we also have to ensure that it is a minimum rather than the maximum but the structure of this makes it clear that it will be a minimum. So what is that equal to? Well we use the rule for functional differentiation, make this R prime everywhere and use the fact that the functional derivative of M of R with respect to delta M of R prime this is equal to a delta function d dimensional of R minus R prime that is the basic rule. Then of course the only non-trivial term is this and I argued that the functional derivative of gradient M of R mod square R prime say divided by delta M of R that is what you are going to have to differentiate out here. So if you make all these guys R primes yeah I need to do the integral as well but let me look at what is going to happen here this is going to be twice so it is going to give you a twice gradient of M of R prime times derivative d d or whatever it is of R minus R prime times the functional derivative of the gradient itself but the functional derivative because it is got to be a vector right. So this is times dot product with delta over delta M of R gradient M of R prime and this you do integration by parts yeah only one oh yes yes so you are right the gradient of the delta function thank you yeah I use this gradient with respect to R prime yeah so this is R prime out here but what I really have is an integral over d d R prime so really you should insert this d R prime on both sides and when you do that and you integrate by parts this delta function is going to fire and this gradient is going to operate on this factor here so you got a del dot del and then you have del squared this already half so just gives you minus del squared okay so let us cut that short and write this whole rule as delta over so once you put that in we can write the solution down so let us just do that so this configuration is given by minus I want to write this properly so 2 a tilde M of R prime plus 4 b m cubed of R sorry the R everything is R R prime is gone minus H of R minus C del squared M of R equal to 0 that is the equilibrium configuration so it is a solution to a partial differential equation in space like the Schrodinger equation but there is a nonlinearity here there is an inhomogeneous term which is not serious but if you had just this term it would be nice be linear but unfortunately there is this nonlinearity okay which you cannot get away from so the equation the thing from to the from the start it is obviously a non-equilibrium situation a non-linear situation okay alright now let us suppose we solve this in principle and now I ask the following question if there is a local fluctuation say due to thermal noise from this equilibrium configuration how does the system come back to it so this is now in the spirit of our old friend linear response theory and we would like to find out what is the way the relaxation is going to proceed okay and now there is no rigorous way of doing this except in a specific model but you would guess the following exactly as we did in the very naive Langevin equation case you would guess the following you would say well a good assumption would be to say okay for a given configuration delta L over delta m I am not going to write all the arguments this quantity is 0 in equilibrium and away from equilibrium for a non-equilibrium configuration this measures the deviation from the equilibrium so a good way to find out what the relaxation is like a good guess is to say delta over delta t a configuration m of r relaxes to equilibrium by minus this times a constant times this deviation from the equilibrium so that is a relaxation equation typical relaxation equation exactly in the same spirit as we saw relaxation occurs in the Langevin equation or in the Boltzmann equation etc but there is no guarantee that you give a configuration it does not relax to a local minimum so in general once you give me an equilibrium configuration of this solved by this there are local maxima and minima and this is presumably a global minimum which is the thermodynamic equilibrium state but if you give me a model of this kind with some initial condition configuration there is no guarantee that it does not tend to a local minimum and stick there that does not happen in actual practice because if you plot this m as a function of configuration variable it might have this kind of behavior and you if you start here you might relax simply to this point whereas you really have a thermodynamic equilibrium state over there so it is clear there must be some fluctuations which take you out of these local wells and put you into the global minimum therefore to this you must add a noise term which mimics the effect of fluctuations but this noise must itself be inhomogeneous because the function of r because you have got an order parameter which is a function of r now so it is it is a field plus 8 of so this is thermal noise and this is exactly the structure of a nonlinear Langevin equation but nonlinear so its solution is formidable because you have a stochastic differential equation for a field and it is got nonlinearities it is space dependent so it is a partial differential equation space and time derivatives are on top of it you have nonlinearities so it is formidable and the only way you can make any headway with it is by functional integration methods but we need to specify what sort of noise so we make the simplest assumption that it is uncorrelated Gaussian noise so we will start by making this assumption that 8 of r and t over all realizations of this noise over all realizations will say that it is got assume that it is got 0 average and more over and here is a crucial assumption 8 of r t 8 of r prime t prime over all realizations this is proportional to delta functions so we assume that the noise is uncorrelated at different time points and at different times so this is equal to some constant 2d delta of r minus r prime delta of t minus t prime in d dimensions of course this is not so obvious and I am slurring over certain technicalities here for instance if you have what is called a non conserved order parameter then this is no longer true and consistency demands that you should put del square over here acting on the delta function but in the magnetization case that does not apply so this is okay as it stands then we need to specify the probability distribution of this eta so what would one do you would assume that the distribution of eta so let me call it p sub eta the probability yeah this is not a quote unquote dissipative order parameter it is magnetization unlike for example concentration that relaxes to an equilibrium configuration but there is no conservation of total magnetic moment or any such thing so there are 2 classes of problems and if time permits I will mention the other class but in this case this is a consistent thing here okay so you have p of eta of some configuration m of r of eta of r t this probability is proportional to e to the power minus some variance and because you got a 2d here you need to have 4d out here integral d d of r eta squared of r and t so that is the probability distribution of it is a Gaussian probability distribution generalized to a field this constant of proportionality whatever is out here is a normalization the it will be a functional integral over all eta's so they would this this will be some constant and k inverse goes like integral well d eta e to the minus whatever it is 1 over 4d integral etc so I am not going to get into we will not try to normalize that we just need the assumption that it is a Gaussian here okay which means you in principle know all the joint probability distributions as well once you make this assumption that this delta correlated it is like noise this is the sort of spatial space dependent extension of white noise that we did in the Langevin equation okay and we are now talking about the probability distribution of whole configurations of eta at every point are for a given for each given t then not surprisingly one can actually write down a corresponding Fokker Planck equation because we need to know what is the probability distribution of m what is the probability distribution of the configurations m of r at any time t given this so it is the old question exactly that we the same as what we solved in the case of the ordinary Langevin equation given a Langevin equation the statistics of this eta the fact that it is Gaussian etc how do you get the Fokker Planck equation from it now I just made the statement that we have in the original in the other case that we have a Fokker Planck equation we looked at its equilibrium and so on but one can make this a little more rigorous one can make this one can do that fairly easily as follows and we want to do this in this functional case so let us go about it in the following way this quantity must obviously be equal to by definition the expectation value of a delta function generalized delta function of m of r t minus m of r for a given configuration minus the m that you get from the Langevin equation so let me write it as m l e by solving the Langevin equation for each value of the noise each realization of the noise and then averaging overall noise realizations so if I call that m l e this is a function of r and p over eta in other words the averages of this delta function oh we waited with p sub eta that is by definition the the probability distribution of this configuration in m okay where m l e of r and t equal to m l e of r and 0 you give me the initial configuration and then you have to solve this equation so it is just minus gamma times integral 0 to t dt prime delta l this is what I give there is no need for le it is whatever initial configuration you give minus delta l over delta m I would not write the argument here plus integral 0 to t dt prime eta of r t prime that is the formal solution of course this quantity here involves that derivative with respect to L so that is a formidable nonlinear object but in so it is not a solution it is just a representation of this m l e and you have to put that so everywhere here in bracket the argument is m l e in a self consistent way of course so I should really mark the argument this is l of m l e and then I have to take this delta function multiplied by p of eta and integrate over all realizations sum over all realizations right so that is the way one would do this and it of course start by saying look we do not write the solution down I find the derivative of this with respect to time so I am going to head towards a Fokker Planck equation and for that I need the derivative with respect to time that will give me a derivative of this quantity so it is a theta function to start with and then there is going to be a derivative of this so all the t dependence is sitting here and for this quantity I go back and use of course the Langevin equation here so that is the way you derive the Langevin Fokker Planck equation for a given Langevin equation even in the ordinary case the finite degrees of freedom case so the sum and substance is that you end up with the following Fokker Planck equation so we have delta over delta t p p m of a configuration at any time t this quantity is equal to not surprisingly not surprisingly you are going to get gamma times a integral d d of r prime times gamma times that is the drift term there was a minus sign in the Langevin equation and remember in the drift term you get another minus sign so that gives you a plus out here plus and the way we have normalized this with a 2d here you have to do 1 over this guy so this becomes plus d times delta p m over delta m so there is a second derivative term because the noise in this case is not multiplicative it is got pure delta functions no r dependence here by assumption so it is d times that which is what you expect plus this guy here this is the drift term the only difference is it is not linear this thing is completely crazy it is not linear equation at all but we can write down what is the equilibrium value what is going to be the equilibrium distribution that is found by putting this equal to 0 which is equivalent to putting this equal to 0 so the solution the equilibrium solution it is write it down the equilibrium configuration it is not a function of time this fellow is obviously apart from normalization constant it is e to the minus gamma over d because essentially it is equating this to 0 right so it is e to the minus gamma over d times integral d d r prime of r let us say the function of r here I am a little confused here because of the notation no it will be l because I want to check if this equation is valid or not right so it is just going to be l it is just l right because if I differentiate it I am going to get delta l over delta m here but we would like it to be the Boltzmann distribution remember this is the energy density and you are going to have to integrate to get the full land of energy so it is of the form e to the minus whatever the energy but we would like it to be the Boltzmann so this goes to the Boltzmann distribution which goes like e to the minus integral d d l minus 1 over k Boltzmann t provided d equal to gamma k t that is the fluctuation dissipation theorem right that provides a consistency check okay I am sorry for using the same symbol d that we use for diffusion in the position space earlier this is the diffusion in the velocity space the analog of that because it is the Pfokker Planck equation for the analog of the velocity if you recall what that was way back when it was m v dot plus m gamma v equal to this force 8 of t and we assume the strength capital gamma for this guy here so we assume that 8 of t 8 of t prime was equal to gamma delta of t minus t prime and then we got gamma is 2 little m gamma k t but more important remember the Fokker Planck equation for it we had delta p of v t over delta t equal to gamma times delta over delta v v p plus gamma k t over m d 2 p over d v 2 so this was the diffusion constant in velocity space now the m has essentially been put equal to 1 there is no m sitting here so it is not surprising that you get capital gamma plays the role of d capital d here plays the role of gamma so that is that is the way the consistency thing works out okay so we have some idea exactly exactly so there is an specific l we model this l we took care of the m square term the m 4 term the gradient energy term etc such that you get the correct equilibrium distribution and we impose this condition here now the next question is to ask how does it relax how does it relax to equilibrium that is a harder question because you really have to go back and ask look at the Langevin equation itself but that is a harder question however one can do the following one can linearize exactly so we are going to assume so relaxation and let us do this in the simple case when t is greater than t c because then m equilibrium of r this guy is equal to 0 I use this for the average itself so I should really put brackets here because I have used this m in the Langevin equation without putting brackets but okay so we write an equation for this m to go back to the Langevin equation I have delta m over delta t this quantity a little bit away from equilibrium is equal to minus gamma delta l over delta m which is equal to minus gamma and I have to tell me what all those terms were there was minus there was a minus c del squared m so let us do it in the absence of an external field plus 2 a tilde m plus plus there was a 4 b m cube I am going to throw this out right we are taking average values not going to indicate indicated specially everywhere this average so the eta term has gone away so this is equal to gamma c del squared m plus 2 a tilde I am unhappy with the sign minus minus minus delta gamma because there is a del squared term here and this m is space dependent the obvious thing to do is to resolve it into Fourier modes take a Fourier transform let us let me put a tilde for the Fourier transform here so it says delta m tilde over delta t for a given k so this with respect to the space variable r spatial Fourier transform this fellow is minus 2 gamma a tilde m tilde and then this is going to give me a k squared del squared with a minus sign so there is going to be minus gamma c k squared and I pull out the m tilde right I do a Fourier transform here I am going to get a minus k squared because it is i k the whole squared and this is m tilde so this quantity is equal to leaving out the non-linear term so for small deviations from equilibrium the linearized equation gives me this incidentally this Langevin equation the this relaxation equation the full non-linear relaxation equation is called the time dependent Ginsburg-Landau equation in super when applied to superconductivity so just this alone with the full non-linearity is the time dependent Ginsburg-Landau equation and then you add to it the noise term it becomes a stochastic differential equation and it helps you to analyze the way fluctuations lead to state of equilibrium in principle within this model it tells you everything about the time dependent magnetite configurations if you include the external field then it tells you in principle everything okay but whether the original assumption this is valid or not is a different question it is clearly reasonable and plausible sufficiently close to equilibrium because you are saying the rate at which it relaxes to equilibrium is proportional to the deviation from the equilibrium it is in it is absolutely a linear response the statement just like fix law for refusion or heat conduction etc it is a it is a linear response statement okay linear in the field it is linear in the field not not linear in the variable nor in the order parameter no but it is linear in the field that is what linear response does right okay. So in in particular it will give you some statement about the susceptibility because remember the susceptibility is the derivative of the order parameter with respect to the field at 0 field so this thing is equal to minus m tilde over tau of k where the relaxation time is given by relaxation time tau of k is given by 1 over tau of k equal to 1 over tau of 0 that is this term plus gamma c k square right tau of 0 by definition this quantity is 2 gamma k tilde so the infinite wavelength or k equal to 0 the uniform background configuration relaxes at this rate with tau of 0 but this is equal to 2 gamma a times t little I should not use t t minus t c over t c and as t goes to t c this goes to 0 so 1 over tau 0 goes to 0 therefore tau 0 goes to infinity that is called critical slowing down so this is where critical slowing down comes let us write that down t then remember we are in t c greater than t greater than t c we started with that assumption so it diverges goes to infinity implies critical for k not equal to 0 so finite wavelength fluctuations even at the critical point will be okay they will relax with the finite time they go like k square 1 over tau k goes like k square but as long as k this thing is 0 if k is 0 then you have a divergence of the relaxation time but the finite follows do not do that which is reasonable it is only the infinite long wavelength overall background mode that makes that gets slowed down. The shorter wavelength ones will have finite relaxation time because those are controlled by this quantity as this becomes larger this is finite and therefore you have finite relaxation time okay so this helps us formulate it is a start of something called dynamic scaling the dynamic scaling hypothesis what we have therefore is the following and from here yeah the static exponents will get related now we will see when you have so let me call it dynamic scaling look at what tau k did tau of k it went like t now let me use t for t minus tc over tc for a minute there is no time appearing explicitly anyway in this formula so it should not confuse this goes like t to the minus y y equal to 1 for k equal to 0 right and it went like k to the power minus z these are standard symbols z equal to 1 2 for t equal to 0 at the critical point the other modes the 1 over tau k was exactly proportional to k square and therefore tau k is proportional to k to the minus 2 so we have introduced 2 new exponents y and z out here now both these can be subsumed in one relationship by again making a hypothesis that at the critical point this guy here is some power multiply and that close to the critical point this guy is some power law in little t multiplied by a scaling function exactly in the spirit of widown scaling right and then experiment will have to tell you that is correct or not okay so when hypothesis is that tau of k goes like t to the power minus y that is this in general even when this kind of simple mean field theory is not valid multiplied by some function of k times the correlation length the old order parameter correlation length to the which is a function of little t and now we need to get these 2 from it out here for that you require since so let me let me let me write this fellow as in the critical region t to the minus y phi of k times t to the minus nu apart from some constants because this diverges like with this exponent nu if you recall which was one half in mean field theory right so how is this going to be reconciled with that if you put k equal to 0 you should have this divergence here which is this term here already provided phi of 0 is finite so we require of this function phi phi of 0 must be finite and non-zero some finite constant but as t goes to 0 this guy goes to 0 you want this behavior that means this thing must be cancelled out with the t to the power y so we want phi of whatever it is argument phi of x let us say phi of x phi at x equal to 0 must be finite and phi at x tends to infinity must go like x to the power minus y over nu right because then this term will go like this will then go to t to the power minus y k to the power minus y over nu t to the power minus nu to the power minus y over nu which will give me a t to the y which cancels this and gives me a 1 over k to the y over nu so this goes like k to the power minus z where z equal to y over nu because remember we wanted k to the minus z so it is a simple trick it is a same trick being played all over again that you have 2 different limits then it forces the scaling function to have this behavior that is the only way in which you can be consistent here so this will immediately imply or y equal to z nu so this implies that this tau k goes like t to the power minus z nu phi of k t to the minus so we have introduced a dynamic scaling exponent z this is one more exponent here now we can relate this to the susceptibility because what you need is the formula the susceptibility chi t now we do a Fourier transform of k omega this fellow here is the derivative of m tilde in Fourier transformed in space and time and average taken with respect to delta h tilde of k so we can now write and we know that there is a gamma here in exponent gamma divergence near t equal to 0 for the static susceptibility right so once again we can make a statement about the dynamic susceptibility make a generalized scaling hypothesis here so essentially we will have to assume that this fellow here is some power of t little t multiplied by a function of k psi and omega tau naught where tau naught is this thing the k equal to 0 relaxation time right and that turns out to give you relationship between this dynamic exponent and the other exponents okay and so on now this whole business can actually be generalized to much more general class of problems including the way this is the beginning of the dynamic scaling theory which is now applied to a very large number of problems both in and out of close to equilibrium and far away from equilibrium such as the growth of surfaces spinodal decomposition etc etc so the wide variety the trick is again the scaling I have not talked this is the starting point of the renormalization group approach to critical phenomena which is where this comes into full play the power of these scaling arguments and you can rigorously show what the relations are between various scaling exponents what the upper and lower critical dimensionalities are what the nature of the critical point is in every case etc see it is we got all these exponents by looking at just trying to we got relations yes when we try to write everything in terms of the object which came from the correlation right right now what we are doing is we have got from this thing we have got something yes to correlation time exactly exactly exactly so that is that is the whole point once we have a relation like that we got rid of this intermediate thing why and we already know new and the time behavior is given this thing here will tell you what the dynamics is with suitable modifications yeah with suitable definitions of said etc yeah the power of this whole thing is not apparent here because unless we do the renormalization group which is another way of saying that you use scale invariance near the critical point the system becomes scale independent all fluctuations on all length scales and time scales become equally important and the trick was as opposed to the original ways of tackling equilibrium statistical mechanical problems where you try to solve or trace over a fine partition function you trace over all degrees of freedom simultaneously this divides and conquers so it breaks things up depending on the k value or the wavelength of the fluctuations integrates over variables which either very slowly or very very long period times along spatial extent and then or the other way about and then ask for a case and impose the condition that the system will look scale invariant on all scales so that forces you to have certain relationships between various exponents among other things right it also gives you a calculation method of computing systematically computing critical exponents outside the framework of mean field theory I use the symbol y and z here although in the case we looked at y was 1 and z was 2 but the whole idea is that you hypothesize that these exponents can have different values other than these values here and indeed they do there is a closely related related to this Langevin equation there is another one for growth of random surfaces growth by aggregation for instance called the Kpz equation the Kardar-Parisi-Zang equation which again is like the diffusion equation with a noise term added to it we saw the original diffusion equation was for a probability distribution but now we are saying this noise added to a diffusion equation itself in the same spirit as this is already partial differential equations and on that we added noise a similar kind of approach that leads to so called roughening exponents and similar results okay so I think I will stop here with this topic and refer you to some text for the rest okay. There are a couple of interest good textbooks on this many many good textbooks on critical phenomena but I will write out a list of these useful books and give it.