 Today I will discuss a topic which should have been discussed a little earlier when we did Langevin dynamics but which is somewhat formal and mathematical and not directly connected to the rest of the topics that we have done earlier but it is very crucial now becoming more and more important in the context of non-equilibrium phenomena, specific stochastic models of them and therefore I thought I should at least mention briefly what it is all about. It has to do with our old friend Brownian motion and if you recall so let me start by recalling a few facts about Brownian motion. If you recall we wrote down the Langevin equation for some Cartesian component of a particle of mass and moving in a fluid and to quickly recapitulate what we had we had a formula like V dot plus gamma V is equal to 1 over m eta of t anything like this where this was Gaussian Markov stationary white noise and we also discovered there was a relation between the correlation the strength of this force and the friction constant gamma. If you put that in we could write this then we are a capital gamma here so this would go to 2 gamma k Boltzmann t over m times zeta of t where this white noise had the following properties it had 0 mean and zeta of t zeta of t prime was equal to just a delta function delta of t minus t prime okay. I slurred over the fact that this quantity is not very well defined it stood singular in some sense in a strict mathematical sense so we will come to terms with that now and do this little better than what we did earlier. Now of course in the diffusion regime what happens is you are at long times t come compared to gamma inverse and then the effect of this inertia get term gets negligible and this term dominates here so let me go straight away to the diffusion regime and write the equation in the diffusion what we call the diffusion regime or equivalently high friction very high friction this equation gets replaced by x dot equal to square root of 2d times zeta of t because recall that d was kt over m gamma if I divide through by gamma after neglecting this you get precisely 2d here so all the factors are right now this is not a very comfortable equation to work with because it is too singular an object so mathematicians like to write this in the following form and I will define this term so this is written as dx equal to square root of 2d times dw t this is called a Wiener process the W of t is called a Wiener process let us write that down I should put up t inside the bracket really as far as this would but mathematicians like to put it as a subscript because more notationally easier to handle there and the practice standard practice is to set 2d equal to 1 so rescale matters in such a way that 2d is equal to 1 and then it is called standard Brownian motion so standard Brownian motion or equivalently Wiener process it is a Gaussian Markov process that is delta correlated that is correlated in a specific manner which I will derive in a second okay this Gaussian Markov process eminently has most notable property is that it is not stationary non-stationary it is a continuous process that is important with the following property essentially it is this x and x we recall is the position instantaneous position of a particle diffusing on the x axis with the diffusion constant d and the probability density function of this x obeys the standard diffusion equation delta p over delta t is d times delta 2 p over delta x 2 but for this process here we have set 2d equal to 1 it is clear that for this W so the pdf pdf of x satisfies of W say satisfies delta over delta t let me just call it rho equal to 1 half delta to rho over delta x 2 okay pardon me w of t is not white noise it is the integral of white noise because in a crude sense quote unquote dw t equal to z of t dt so w of t that is what I did here when I wrote this as dw t we have to be careful about what I mean by this differential and that is the whole point right. So in this heuristic way of looking at it this w of t is like the integral of white noise because that is what zeta of t is but the problem is that zeta of t is not a well-defined mathematical object it is got a singular covariance it is got the singular correlation function delta function which is not pleasant okay now think of what so w of t is essentially x the position of a diffusing particle okay that is it apart from this constant here but thing we know already what this guy does we know that it is not a stationary process and if the particle starts from 0 the origin at t equal to 0 then the position is a Gaussian and distributed by Gaussian with the variance increasingly nearly with time etc and we also know that this process has an autocorrelation function which eminently is not stationary which tells you so w of t w of t prime the average is equal to you recall what it is yeah the minimum the lesser of the 2 now we have set this 2d equal to 1 so with that normalization it is minimum of t t prime we are only considering positive values of t and t prime right. So it is clearly not stationary because autocorrelation function has this behavior here so let me let me stick to standard notation like this and this chap here has got the standard Gaussian solution if you recall this is implies fundamental solution rho equal to 1 over square root of 2 pi t e to the minus x square root of 2d I have set again 2d equal to 1 okay so this is where it came from now let us try to understand this a little better what this thing is this is the covariance of t and t prime that is the word for autocorrelation function in statistics if you divide it by the covariance of this and the standard deviation of this and that so as to normalize it so if you write it as sigma x sigma t so how should I write this sigma w t sigma w t prime and w t prime here over sigma w t sigma w t prime where this is the standard deviation of w t and that is of w t prime it should be sigma squared here by definition it is equal to this quantity then this is of course equal to min t t prime over and we know what this is the square of this is 2d t and 2d has been set equal to 1 right so this is equal to t square root of t t prime this is 2d t and this is t prime and we are taking the standard deviation so root of t t prime and it is easy to see that this is equal to min t t prime over max t t prime the whole thing square root that is a convenient way of writing easy way of remembering this now what can we say about the paths of this w t okay it is clear formally that the Brownian particle has got Brownian motion as infinite velocity because x squared scales like t not x and therefore delta x over delta t is formally infinite you need to put another delta x on top in order to bring it to finite value right okay so what are these paths look like that is worth looking at and a number of results are known in one dimension Brownian path or sample paths of Brownian motion we are talking about Brownian motion starting at the origin so here is x and sorry here is t and w t the Wiener process starting here and this process takes off and does this kind of crazy thing it goes up and down etc. And the following can be established rigorously from the properties of the Wiener process the average is 0 we already know that so over all set sample paths you take the average it remains 0 here okay but the question is is it differentiable or not is this differentiable and the answer is no I will just quote a number of results here so nowhere differentiable almost surely so non-differentiable except on sets of measure 0 non-differentiable almost surely how badly is it non-differentiable and there is an answer to that it is a continuous process that is very very important to remember in fact had we dropped this and had we required it to be stationary what would the process be pardon me on signal and back that is the only continuous Gaussian stationary Markov process and it follows that it has an exponential correlation and vice versa the converse is also true in that case but the moment you make it non-stationary but require it to be continuous then you end up with Brownian motion it is non-differentiable but it is continuous therefore there must be some kind of holder continuity for this process it turns out that the continuity looks like this W of t plus epsilon minus W t modulus is less than or equal to some constant times epsilon mod epsilon to the power beta where beta is less than half but can be arbitrarily close to half and not half except on sets of measure 0 arbitrarily close to half from below so that sort of tells you how jagged these paths are it is a measure of how jagged it is because you can see that this will imply the function is not differentiable because for differentiability yeah if you divide by epsilon so it is clear that W t plus epsilon minus W t divided by mod epsilon tends as epsilon tends to 0 it tends to infinity because of this property it is going like half so you divide here you are going to get a half little more than half in the denominator and it is going to go to infinity as epsilon goes to 0 so it is therefore not differentiable but this weirdness that it is holder continuous with an exponent which is arbitrarily close to half from below means that this increments in this process are acting somewhat like square root of time this epsilon is like the increment in time right so it is acting like the square root of dt the infinitesimal dt one can formalize that and then you have to be little careful but by a clever amendment of the rules of calculus one can actually handle this process quite rigorously here is what happens go to continue on the sample another property of the sample path you can ask when does it cross 0 in a given time interval how long does it stay above the axis and how long below on the average and so on in fact you can ask where is this particle going to be most of the time and it turns out there is a law of the iterated logarithm this is kinsen's law so you plot for Brownian motion starting at t equal to 0 you plot the following function this function is square root of 2 dt log log 1 over t for small t sufficiently small t plot this okay and you plot the negative of this function here this is minus the square root of this stuff and if this is 1 plus epsilon times that function and that is 1 minus epsilon times that function so this is 1 times square root and this fellow is 1 plus epsilon and this square root here and similarly here then the statement is almost surely the particle is either here or here in this range for arbitrarily small epsilon okay so you not only have a statement about the probability density function of this x as a function of t or w as a function of t but you also have a statement about where it actually is what this part does and yet and yet the particle crosses the axis and infinite number of times it does so in any arbitrary small interval and you can in fact ask what is the measure of the 0 set the set of points where it crosses the axis so 0 set this would correspond to the set when wt equal to 0 okay that 0 set in the limit has a fractal dimension which is a half so while it can go arbitrarily far away the 0 set is such that by the way this thing the fluctuations obviously they are going to go way up there way down here and that is why the variance itself will increase with t because the fluctuations are such that it is getting it is diffusing although the mean remains 0 always you can ask further interesting questions you can ask what is the given given some t here what is the fraction of time for some some capital t starting at 0 what is the fraction of time that it spends above the real axis above the on the positive side and what is the fraction it spends on the negative side that is a random variable it is a yes but it is a random variable one can ask what is the probability distribution of that random variable so let us suppose I am being incoherent you could ask what if t is very large you have a law of the law iterated logarithm for t large as well all you have to do is to replace that log log 1 over t by t okay but the crucial point is not that the crucial point is this process is reinventing itself at every instant of time it is what is called a martingale I am not going to talk much about that but we will see some consequences of what it does it means that if you are here at this point say you are here at that some instant of time then the Brownian motion is as if it starts from there at this instant of time and it again behaves in precisely the same fashion as it behaved earlier here so at every point there is a law of the iterated logarithm every time the process as the memory is very short it is a Markov process okay. Now one can ask the probability distribution function so let t plus be the be the amount of time such that Wt is greater than 0 for 0 less than t less than capital T okay so we look at the Brownian motion up to some capital T and ask what is the total time spent where it is on the positive side and what is the total spent in time spent where it is on the negative side of the x axis the pdf of t plus and similarly as you say t minus it is completely symmetric pdf t plus is proportional to 1 over square root of t minus t plus which implies that the cumulative distribution function cdf of t plus it looks very much like yeah the time instead of if you have a regular oscillator and you ask dt t is distributed uniformly over a period you ask what is the distribution of the angle it is precisely this 1 over square root here many such distributions okay. Now the cumulative distribution function namely the probability that t plus is less than equal to some given value tf this thing here of course now is proposed is equal to normalize it and you integrate this fellow from to 0 to t plus you get 2 over pi sin inverse t plus over t it should of course be 0 when t plus is 0 and it should be plus 1 when t is t plus equal to capital total probability must be equal to 1 right this is called the levy arc sign law okay. Now let us see what the fact that the sample parts are irregular very jagged they have very specific kind of irregularity essentially characterized by the fact that the the older continuity is half essentially half that says that the following property can be rigorously proved and this is part of what is called eto calculus and I will mention only the rudiments of it essentially one formula and it is the following what one can prove what one can prove is that for Brownian motion or a Wiener process if you have on the time axis any 4 points so let us start with t1 some point t2 and then some maybe t3 and some t4 and you ask what is this quantity W of W t2 minus W t1 so this minus that and you multiply it by W t4 minus W t3 this thing here the value of this product here is equal to the overlap between the two intervals so you have one interval from t1 to t2 another interval from t3 to t4 and the overlap between these two is this guy here so this is equal to overlap length of the overlap between t2 t1 and t4 t3 it has an immediate consequence this by the way this statement can be proved by looking at the correlation the autocorrelation and then it is a simple proof here now this has immediate consequences the first consequence is that dw t dw t dw t equal to dt because imagine a completely overlapped infinitesimal interval the length of it is dt and that is the square of dw t so this is the one that formalizes the fact that this dw the increment in the Wiener process is like the square root of the increment in time okay this immediately has the following consequence the several ways of doing this and what we are interested in is asking for the behavior of the properties of functionals of Brownian motion x is a random variable in the normal diffusion problem you ask what is f of x look like where f of x is some function etc so we are going to consider what happens if you had a functional just write it as F w t you could extend it to the case where it has a explicit t dependence as well we will do that in a second what happens to this functional and what it is what is its differential look like okay what you have to do is to do a Taylor series about any particular point and keep track of the fact that this thing here is like dt okay so we will assume that assume that f of whatever argument f of x is a continuous twice differentiable at least twice differentiable we will consider functions which are at least twice differentiable functions of the argument then we could ask what is df w t this is equal to f prime w t d w t that is the first term right but that is not enough because it is a function of a function that could also be an explicit time dependence here so let me in fact write that time dependence let us look at the more general case where you have this then is well let me use the following notation f prime and I will explain this notation in a second so df of w t t equal to this plus a portion coming from the fact that the second term in d w t will still be of order dt and that term turns out to be one half f double prime w t t d w t d w t plus delta f of w t t over delta t dt but we know that this is dt so it gets added to this term okay but this is the fundamental rule of ito calculus where df over d w t is equal to f prime d to f d w t so you differentiate with respect to w t alone those are the two derivatives you already assumed that well I should really write f of x comma t is twice differentiable in x now the addition of this extra piece here helps you that is the amendment that the rules of calculus need in order to be applicable to as singular objects as Brownian parts as a weiner process because this the intake this is the differential form the integral form of this will tell you what the correction is normal cases so remember this rule this here now let us look at what is the integral of say t 1 to t 2 t equal to t 1 to t equal to t 2 of something like f prime of w t let us look for the moment at functions which are not explicitly time dependent so you see what the correction is d w t what is this guy equal to well we are integrating both sides of this equation here and we are asking what is this fellow equal to so it is this integral by definition it is a differential therefore you can write this as clearly f w t 2 minus f w t 1 that is certainly true but then there is a correction due to this piece here so it becomes minus half integral t 1 to t 2 d t f double prime of w t d t so there is this extra piece which you require so let us see what this is doing for us what it implies let us first look at a case where this f prime of w t let us say is w t itself then it tells you that integral t 1 to t 2 in time w t d w t I should write t equal to t 1 t equal to this is equal to by this applying this theorem applying this blindly it is the integral of this fellow which is w t squared over 2 so it is half w t 2 squared minus half half comes out minus w t 1 squared minus half well f double prime is 1 in this case so just t 2 minus t 1 but this is equal to t 2 right we already had that overlap rule and so the square of w t squared is t itself so this is equal to half t 2 minus t 1 minus half minus t 1 is equal to 0 is identically equal to 0 because of the way this ito calculus works now what is that actually mean I mean this is an important result because it tells you this thing tells you that there is a specific interpretation being given to this this process here this integral here so it says that what we mean by that integral so since we can write t equal to t 1 equal to t 2 d w t d w t can be written in the following way so I start with time t time 0 time t and I break this up into n parts so this time t n and time t n plus 1 is equal to t and this is 0 so this is t 1 this is t 2 etc break it up then this is equal to limit n tends to infinity summation from i equal to 1 to n times w t i times d w but this d w is w t i plus 1 minus w t sub i here it is the forward difference I am sorry yeah I chose it to be 0 yeah I started with yeah this is a general result this follows a general result here but you can start at any point in this process I start from 0 and it is a meaning of this so it is this quantity that has been written there and this follows is this difference here in the limit as the number of intervals number of subdivisions becomes infinity now this is the forward increment and there is a current value of the variable we know that the overlap between this 2 is 0 because this goes up to t i and this goes beyond that there is no overlap between these 2 integrals in these 2 factors therefore by our basic rule for Brownian parts this they are independent this is 0 but that explains why you get 0 here now you could have interpreted it differently when you do normal integration from a summation you converted integration there is no reason so you normally write f of x dx etc now you write it as a sum this dx is say x plus delta x minus x but this quantity here f of x is x at which point it could be at x plus delta x over 2 it could be at any combination of f at x and f of x plus delta x this is a specific choice of the way this integral is interpreted the interpretation so this this the fact that these 2 factors no overlap and this guy here is the forward increment it is this that is gives you the ito calculus and all the other properties that I wrote down so clearly it says that there are other choices possible you could choose this to be t i plus 1 for example or you could choose it to be t i plus t i plus 1 divided by 2 so you have any number of choices here and each time you get a calculus stochastic calculus okay now if you choose the ito interpretation and you have this feature then and only then does it turn out that the interpretation we have for the general lanshama equation or general ito equation now so general ito equation stochastic equation I should not call it lanshama anymore this is of the form some process dx is equal to some f of x dt plus g of x remember the lanshama equation that we wrote down with a drift term and a diffusion term there was a g of x times we called it white noise and put this as x dot but the correct way of writing it is with the dt here and this is dw this is an ito equation this is to be understood in the sense of ito so the x here and if it was time dependent the t is such that this is the x at instant t and this is the forward increment in the venal process okay you need that interpretation otherwise it is ambiguous multiplicative stochastic process is definitely ambiguous so the question is you have a noise in physical terms you have a noise that is dependent on the state variable itself the question is state variable at what time okay it is the state variable when you write the increment in x it is the state variable at time t where this dwt is the forward increment between t plus delta t and t okay then this implies and is implied by the Fokker Planck equation that we wrote down for this process it implies that the probability density of this thing is delta p over delta t is equal to this fellow here is equal to minus delta over delta x f of xp plus half g squared of x d2p over dx2 so when you have a non-trivial noise a multiplicative noise you have to interpret it in the ito sense and then this is true the Statenowich interpretation where this would be t i plus 1 would give you a different Fokker Planck equation for the same stochastic differential equation interpreted in that sense the point is that the physics has to be the same in both cases so what would happen there is that you would have what is called a spurious drift term so in going from the you go from the engineering equation to two different stochastic differential equations so as to get the same master equation once again for the random variable because that is its moments are physically measured so you start with an equation between averages the so-called engineering equation you add noise to it if it happens to be stochastic multiplicative noise you have to then specify is the equation this equation in the ito sense or in the Statenowich sense or anything in between there are other interpretations and then each time the prescription to go from the stochastic differential equation to the master equation changes in such a way that the final master equation is exactly the same for a given physical process we have used all the way while I have used the ito interpretation it is all a satisfying because of this that it is not anticipatory that is the current dependence on the rate of the state variable at the current time and then the forward increment but it is not sacred in fact in physics very often one uses the Statenowich interpretation but then you have what is called the spurious drift term there is a correction to this fellow here and the prescription to go from here to the stochastic to the Fokker-Planck equation changes in the two cases we have consistently used the ito equation okay. Now of course if you if you use the calculus carefully then questions of uniqueness come for solutions of this process questions of uniqueness arise and here is an example due to ito himself by the way I should give a reference to this whole business it is the mathematical treatment can be made very rigorous and one of the best places for it is the original book by ito himself so ito and McKean it is called diffusion processes recall that we called an equation like this a diffusion process we called x a diffusion process and you have to you have to worry about the uniqueness of the solution to this sort of equation we solve this for the on serial number case it was completely trivial this was a constant that one was linear in x and then it did not matter we wrote down the solution but if you do it rigorously properly then you have to be a little careful because you could have non uniqueness and here is an example due to ito himself we will work backwards write the equation down so if you have example this is due to ito and what an army and the equation is the following x equal to Wt cubed for example ito it is a nice functional of Brownian motion right then what is the equation give you it says dx equal to f of x that is 3 Wt squared so it is 3 times x to the power two thirds dW dt plus the second derivative you have differentiate this this is 3 x squared 3 W squared and that is equal to 6 W divided by a 2 and W is x to the one third and the 2 goes away in the calculus so get plus 3 times x to the one third dWt sorry that is the coefficient of how did I write this it is the other way about this is f prime and then you had to do a dWt right so it was this is the correction part so if I ask what is the solution to this equation we know this because we work backwards to get this equation so if I ask what process is this x you would say it is Wt cubed from this guy but you could also put 0 there and it gives you a solution so you have this or that therefore the uniqueness of solutions is not guaranteed fortunately this is a kind of academic example fortunately it turns out that if you put in sufficient holder conditions on the continuity of this process then you get unique solutions for instance if you can show that mod x at some time t minus x at time t prime is less than sorry the function so if this fellow is some function x of Wt if you can show that x of in some alpha minus x of some beta modulus is less than some constant times modulus of alpha minus beta for all alpha and beta all argument then that suffices to show that the solution is unique this is not this does not satisfy that these two guys are not smooth enough at the origin they are too singular but if you have something milder like this then the solution is unique and that is what happens in most physical examples now you can ask can this whole thing be generalized but before that let me point out think I have run out of time but I should point out what the Feynman-Kartz formula is at the very least okay this requires a bit of a preamble let me let me again motivate this on physical grounds so let me give a little preamble to this recall that the original diffusion equation so let us write it in the physics notation our original diffusion equation was of the form delta p over delta t for the position of a Brownian particle d d2p over dx2 and you can ask what is the solution to this equation for some given initial distribution not necessarily a delta function then of course you take the original Gaussian solution use that as a kernel for the green function for the different operator differential operator and you integrate over the initial distribution whatever it is so you write the solution p of xt equal to 1 over square root of 4 pi dt an integral dx prime over all x prime e to the minus x minus x prime whole square over 4 dt could have started at any point t minus t0 the initial time it will replace this by t minus t0 we are talking about processes starting from the specified at x equal to 0 this multiplied by p whatever be the initial p initial of x prime this this was p of x at 0 it acts as a green function here now the Schrodinger equation for a free particle is exactly the same behavior right so there you have delta psi of xt over delta t equal to ih cross times that is minus h cross squared so like write that as minus ih cross whole squared so this is equal to ih cross by 2n d2 psi over dx this two has a similar solution except there is this i so it is as if the d were an imaginary quantity and then it is not clear if this this is an oscillatory function in that case it is not clear if this thing converges but formally if you do a weak rotation here in time and go to imaginary time i times t then the two problems are mathematically equivalent and well defined so you have exactly the same sort of solution this this would imply that psi of x, t is equal to 1 over square root of 4 pi and then instead of d you would replace it by ih cross over 2n t integral dx prime e to the minus x minus x prime whole squared divided by whatever it is 4 times d d is this so twice to ih cross t and then an m or something like that times psi initial of x prime and if you recall this is the starting point of the path integral formulation of quantum mechanics then you do time slicing and stuff like that and you get the path integral formulation of quantum mechanics right now mark cuts notice this and said now this operates in two different can give a form gave a clever formula which essentially says that you can use this fact the fundamental Gaussian solution in order to write down solutions to some deterministic equations or conversely starting with the deterministic differential equation which is essentially the diffusion equation with the potential term on it you can find expectation values of certain functionals of Brownian motion it works in both directions and in its simplest form it looks like this so this is the so called Feynman-Kartz formula and let me write it in the simplest form it is got all kinds of generalizations higher dimensions etc but here is what it looks like suppose you have a partial differential equation a parabolic partial differential equation in one space dimension in one dimension equal to let us write the standard diffusion equation down this is the pdf of a Wiener process if you like but along with that let us suppose there is an extra term v of x times u of x if this were the heat conduction equation it is like there is some external cooling which is paid state dependent x dependent here or in the context of the Schrodinger equations like a potential somewhere then the question is what is the solution you have to specify an initial condition and the initial condition is so some conditions are put on this this is a non-negative continuous function and u of x 0 is equal to some specified initial function so this is u0 of x and we will assume this to be continuous bounded so it is a continuous bounded function then the statement is that the solution of this the unique solution of this has the following form it is equal to the average value of e to the power minus integral 0 to t v of x v of Wt dt times u0 of Wt where I have to say what is Wt and what this average is over where Wt equal to standard Wiener process or Brownian motion standard Brownian motion started at x at t equal to 0 so you have Brownian motion started from wherever you want the solution whichever point you wanted and then you let it go and equal to average over all all walks all random box all sample paths subject to the above conditions namely all Brownian motion starting at x at t equal to 0 at any speed no starting at x is going forward right in time and then you do this integral so this works both ways if you want this where you have an arbitrary functional out here and here some given functional then you have this back again but the simplest case would be v equal to 0 then you see where this is coming from you put v equal to 0 this goes away you are going to take the initial point and you are going to put the Gaussian kernel and you are going to integrate that would be one way of computing the average and that is exactly the solution we wrote down to the diffusion equation but this generalizes it to the case of arbitrary functions here subject to very mild conditions okay if you were quoted back it is just the path in time Feynman path integral for the potential v of x exactly but what is interesting is this is capable of enormous generalization first of all to higher dimensional Brownian motion for which we know how to write you have a general equation of the form dx equal to f of x dt plus g of x dwt where this is an n dimensional vector so is this this is an new dimensional noise and this is an n times d matrix with suitable conditions on the G's we know how to write the Fokker Planck equation down now for the process x a general diffusion process x you have an analog of this formula it does not have to be standard Brownian motion alone that will correspond to the case where f is 0 and g is 1 but you can instead of wt you can use x and you can write a generalized formula once again for a more complicated equation here which involves the first derivative of this u with respect to x with the drift term and the second derivative with that diffusion term here okay. So instead of this operator you replace it with g square times this guy plus f times f of u f of x times the first derivative du over delta x and it is still true. So higher dimensions more complicated parabolic equations it still works this formula and people have been at it for a very long time I should finally mention that the case where this fellow is linear in x and that too is linear in x is the famous Black-Scholes model that is used in financial mathematics and it corresponds to exponential Brownian motion. So the random variable is e to the wt if you like times some constants okay and then an explicit function of t to take care of ito calculus once again. So that is what I wanted to say here in this context we have this entire course we have sort of started with very simple Langevin dynamics and come back to it in a very general form here. On the way we went through all kinds of applications of Langevin dynamics that is a paradigm that is a model and I tried to point out very briefly that the same model is generalized to the case of fields order parameter fields and then it has connections to both equilibrium and non-equilibrium phase transitions how the help of the Landau functional you got handle on non-equal critical exponents including dynamic critical exponents we did not take an excursion into critical phenomena per se that is a separate topic by itself but we did study some linear response theory on the way both classical and quantum and some topics we have run out of time and most notably chemical kinetics and the thermodynamics are very small systems all the fluctuation theorems these are two notable exceptions perhaps at some future date okay stop here.