So, let us take a look at another aspect of the Langevin model. We have seen that there is a connection between the general diffusion type of stochastic equation for a diffusion process and the Fokker-Planck equation corresponding Fokker-Planck equation. Before I go on, let us take care of one of the questions that was raised. The question is as follows. In a multi-dimensional process, if you have an equation of the form x dot equal to some vector valued function of all the x's plus a multiplicative noise, so this was g of x times as zeta of t where this is set of noises, independent noises and the dimensionality of this was nu. So remember that this was nu times 1 column vector whereas this was an n times nu matrix, rectangular matrix and the idea was that if x is this quantity is an n dimensional noise represented by an n dimensional column vector then the question being asked is these components are independent of each other, they are different noise components, 1 to nu but because of the presence of this, does it not mix up the various noise components? Yes, in general that is the most general possibility because even in the most trivial of instances when nu was 1 and n was equal to 2, remember that you had an equation of the form x dot equal to v and v dot equal to minus gamma v, the drift term on this side plus a noise which was square root of gamma over m times the white noise, Gaussian white noise. Now the question is is it not true that this noise affects the x itself? Sure it does. So there is already a coupling, it is a couple set of equations in any case. So what this is saying is that the most general possibility is that the strength of each of these noises is dependent on the current values of all the random variables which are anyway coupled to each other dynamically. So there is nothing inconsistent about it, it is perfectly reasonable. What is true is that these noise components are specified independently. So the statistics of this noise is taken to be quite independent of what is happening to the output variables, the driven variable x. You have to prescribe for me all the statistical properties of this multi-dimensional noise before I can tell you what the dynamics of the driven variables is. That is certainly true. And the whole assumption in this kind of modelling is that the driven variables do not affect that noise. So there is no feedback on to the noise from the driven variables. We take this in the simplest lines of our cases. We took the noise to come from the heat bath and I said that the effect of the motion of a single Brownian particle on the heat path is negligible. It does not throw it out of equilibrium or anything like that. So it is in that philosophy. That is the very general philosophy that you prescribe for me an external noise of some kind, statistics of this noise is prescribed independent of what is happening and then the statistics of the driven variable x is dependent on solving this set of equations, coupled equations. So that is the philosophy behind this stochastic differential equation approach with the driving noise, okay. The different components of this are uncorrelated with each other. In the simplest case, we take them to be Cartesian components for instance. They are uncorrelated with each other. But there is no reason why the coupling should not be such that this instantaneous strength of this noise as it acts on any one of the variables depends on all the other variables. And that is where this g comes in here. So is this clear? Okay. Now let us go back and let me point out what is wrong with the Langevin model. We will see where some serious problems crop. And we will do this in the simplest context which is that of a tag particle, a Brownian particle moving in a fluid. And again the essential physics is already contained in the one-dimensional case. So let us go back to that. Let me call this inconsistencies of the Langevin model. So first I will point out the inconsistency and later on we will fix it on physical grounds. We will so to speak derive or deduce an improved Langevin model which will take care of this immediate problem of what main problems flaws are in this model. So we call once again that for the velocity process, velocity of the Brownian particle, we had an equation of the form v dot is equal to minus gamma v plus square root of gamma over m zeta of t where this was Gaussian white noise, zero mean and a delta correlation. And this was taken to be Gaussian, stationary, Markov and delta correlated. And what was the output process? It was the Anstein-Rohlenbeck process. It was also Markov. It was also Gaussian. It was also stationary but it did not have a delta correlation of some kind. But we certainly proved that it was stationary. In particular we proved this relation. We proved the fact that v of t not, v of t not plus t in equilibrium, this quantity, we calculated what it was from this using the consistency condition that asymptotically this correlation will factor the product of averages and that the mean square value will remain in thermal equilibrium at the Maxwellian value. So this was found to be independent of t not and it was k t over m e to the minus gamma modulus t. It is a stationary correlation. Now if you go back to the general linear response theory that whose formalism we worked out, this is actually a response function. And in the language of the response function, what we have is phi AB in this problem, A was equal to x and B was equal to v because we perturbed the system by applying a mechanical force minus x times this f of x, f of t and then we measured the velocity, the average velocity. So the observed variable is the velocity and the variable which is cos coupling to the force is x, the dynamical variable. And this came about as a response function. So in a sense this quantity is in fact phi x v. Recall that in classical apart from some factor of k t or something like that. Recall that in classical physics, in the classical case we showed that this response function is beta times the equilibrium average value of A dot at 0 with B at time t. And A dot at 0 is v of t not. So this is what we derived here. And this is independent of t not by stationarity. But look at its consequence. If you took that seriously, if you took this thing seriously, it immediately follows that if I differentiate this function, this quantity with respect to t not, the answer should be 0 because it is independent of t not, right. But what happens if you differentiate it? You end up with v dot of t not, v of t not plus t average plus v of t not, v dot of t not plus t equilibrium should be equal to 0 because there is no t not dependence on the right hand side by stationarity. Now set t equal to 0 from above or t equal to going to 0, approaching 0 from above, right. Then it immediately says that at any instant of time v of t not, v dot of t not equilibrium must be 0, must vanish by stationarity. So what we have to do is to let t tends to 0, let t tend to 0 and immediately you get this. And they are classical variables, so it does not matter in which order they appear. So what it says is that the velocity at any instant of time is not correlated with the acceleration at the same instant of time. That is consistent, certainly with our assumption that the force drives the acceleration and not the velocity. So it is sort of reasonable that the velocity is not correlated with the acceleration at the same instant of time, okay. And in any case it is demanded by stationarity. But if I do that here, if I differentiate with respect to t and then set t equal to 0, it is like differentiating this guy here. So let us take t to be positive. If t is positive this goes away, I differentiate both sides with respect to t and then set t equal to 0. And that is going to give you v of t not, v dot of t not in equilibrium equal to minus gamma k Boltzmann t over m which is not equal to 0. All I do is to differentiate this with respect to t and put t equal to 0 from above. And you get a non-zero answer. So it says the velocity at any instant of time t not in equilibrium is correlated to the acceleration by this number here. The equal time correlation is not 0 identically but some finite number and yet if we stationary which we showed explicitly, this must vanish by stationarity. So there is obviously an inconsistency somewhere in this model, okay. So one or the other must be wrong. Either this or that must be wrong, right. Now the stationarity, this thing by the way, we derived this by making an assumption when we computed this correlation function. We assume that the velocity is uncorrelated with the force at later instance of time including that particular instant of time itself. So we made that assumption and that now leads to a result which is inconsistent with that assumption. So there is something wrong somewhere. Stationarity on the other hand we proved under very general considerations for from linear response theory we showed the stationarity of this correlation of the response function and yet this is happening. Let us see if there are any other problems. So you agree that there is a serious problem here. The equal time correlation between variable and its derivative should be 0 if the variable is a stationary random variable but according to this model it is not, alright. Any other problem? Well let us again look at this response function. It is phi x v so let me drop the subscripts for a moment and argue in the following way. I would like to find out what is the mean square velocity is k T over m in thermal equilibrium. What is the mean square acceleration? That should be some finite number. It is a physical quantity. It should be a finite number. So if I compute v dot on both sides I should get a finite answer. Now let us call this correlation phi of t minus t prime equal to v of t prime v of t in equilibrium. That was our call our response function which is e to the minus gamma mod t minus t prime in this model, right. Now the Fourier transform of this is a spectral function and the moments of the spectral function can now be computed because we have the spectral function explicitly. So what is that equal to? What is the spectral function equal to? Phi tilde of omega equal to integral 0 to infinity d t t to the i omega t times the phi of t. That is the definition of the spectral function, right. In this model, in the Langevin model we know what it is. We know explicitly what this thing is. So this is equal to, well let us compute, let us compute the power spectral density of this process, the v process which means we take its autocorrelation and we find its Fourier transform, okay. So what we need is the Fourier transform which is minus infinity to infinity. This is the power spectral density and what is this equal to? Well, we need a slightly different symbol for this whole thing. I call this the spectral function. It was, how did we define the spectral function? It was the Fourier transform of the response function, right. So it was this. I am just trying to get my factors right. This is in fact equal to phi tilde of omega. The susceptibility was the one sided Fourier transform. The spectral function is just this guy here. Now in engineering, the power spectral density is defined in various ways, apart from some factor of 2 or something like that, sometimes 2 pi, sometimes 2 times the Fourier transform of the autocorrelation function. So modulo of that, let us just define it to be the Fourier transform itself and let us compute what this number is in the case of the velocity process, yeah. So this is equal to k Boltzmann t over m integral minus infinity to infinity d t e to the i omega t e to the minus gamma modulus t because recall that for t negative, this is e to the minus modulus t. It is a symmetric function of t, e to the minus gamma modulus t and why should it be symmetric? Why should this quantity be symmetric? Because under time reversal, v goes to minus v. Both these fellows change signs, so this fellow does not change sign. Or yet another way of looking at it is, it is the time parity of a dot times the time parity of b and a dot is x dot, which is also a velocity. So it is completely consistent. But this is a trivial integral to do. This is twice the integral from 0 to infinity because it is a symmetric function and the sign part of it vanishes. So you have got 0 to infinity with a 2 factor k Boltzmann t over m, 0 to infinity e to the minus gamma t cos omega t, which is of course gamma over gamma square plus omega square. It is a Lorentzian, okay. The Fourier transform of the e to the minus x is a Lorentzian in k. So that is all that we are saying here. But this has a physical meaning, this quantity here because you can differentiate this on this side. Each time you differentiate with respect to t, you pull out an i omega from this, okay. So let us do that. Let us write this. So you have v of t prime, v of t, this is the correlation function. This is equal to 1 over 2 pi integral minus infinity to infinity d omega e to the minus i omega t minus t prime phi tilde of omega by definition, that is the inverse Fourier transform, right. And we know what phi tilde of omega is. So now what I do is v dot of t prime, v of t in equilibrium, again in equilibrium is with the i omega pull down and if I differentiate with respect to t as well, it is with the minus i omega pull down, right. So immediately you get this, 1 over 2 pi integral minus infinity to infinity d omega e to the minus i omega t minus t prime omega square times phi tilde of omega. The second moment apart from this time dependent factor. Now I set t prime equal to t. So that tells me v dot of t square in equilibrium. It is stationary, it is got to therefore be independent of time. The time dependence disappears from this thing and this is equal to 1 over 2 pi integral minus infinity to infinity d omega omega square phi tilde of but that is equal to this fellow here. So if I put it in, it is omega square over omega square minus square apart from constants. So it is equal to 1 over 2 pi, 2 k Boltzmann t over m times this fellow. But this is a finite physical quantity, it says mean square value of the acceleration but that is infinite. That has turned out to be infinite because this is not dying down fast enough at infinity. It goes like 1 over omega square and therefore goes to a constant as omega tends to infinity on either side. So this is tends to infinity. It is clearly wrong, which is clearly wrong. So again we are faced with a serious problem and of course once the second moment goes to infinity, the higher moments will all go to infinity, omega 4, omega 6, etc. will all blow up. So the velocity process therefore will not have, its derivatives will not have finite mean square values, which is clearly unphysical. Now what is the origin of this disease? Where could we have gone wrong in this business? It is after all a very very simple model. So where could this have been wrong? There was one assumption made in deriving this model and the assumption was that the model is valid as long as you are looking at time scales, which is not at the level of the molecular time scale, the collision time between molecules but may be 5 orders of magnitude higher. So time scales of the order of gamma inverse and greater. If it becomes very much greater than gamma inverse it goes to the diffusion regime but certainly it is not valid on time scales going towards 0 on the other side. So right there we made an assumption which is not valid and now notice that all these diseases arose when you took the time difference to go to 0 and in this case when you took the frequency to go to infinity, so infinite frequency is like 0 time, very small time in the Fourier language, high frequency is like small time and vice versa. So that is where the problem was, how do we fix it? In the context of this model itself how would we fix it? We would still like to have retarded response, we would still like to have causal response and we would like to stay within the regime of linear response. So clearly the one place where we made an assumption was that the acceleration of this particle we wrote mv dot is equal to the force on the particle, so there could be possibly some external force on the particle which could be time dependent, we do not care and then there was a random force on the particle but this random force, so let us write it as f random of t and we said this random force has two components to it, two parts to it, one part was a systematic part and the other part was the truly random part coming from molecular collisions.  Now this friction arose saying that the friction is proportional to the the frictional force is proportional to the instantaneous velocity assumes that this frictional force acts instantaneously, the reaction of the medium acts instantaneously on the particle that is not true, that is not tenable not mathematically instantaneously because the medium itself has sometimes scale in it, the molecules they cannot act instantaneously on this. So this model does not go all the way to t equal to z to instantaneous response, okay. So this is the point that had to be fixed, this model is untenable at very very short time intervals, how will we fix it? Staying within causal, retarded and linear response, how can we fix it? Well we will we will do the following, we will say look it must depend on the velocity but not on the instantaneous velocity but on the previous, the history of the velocity earlier velocity, right. So the way to fix that model is to replace this by minus m gamma and integral from wherever, from the infinite past, okay, minus infinity up to t because we certainly would not want this part to go beyond t, we are looking at the acceleration at time t that is not going to depend on what happened to the velocity at later times, right, times an integral dt prime and it must be a linear functional so v of t prime must be there, it must be causal, it must be retarded that means it must only depend on the time elapsed since t prime happened, so this must be some function gamma of t minus t prime, so this thing here is a function, it is a kind of memory kernel, oh there is no gamma here, so I make this gamma a function of time, defined only for positive values of its argument, non-negative values of its argument and what other property would you like of this gamma? Well, we would certainly expect the following, we would like gamma, let us call it gamma of tau to be neutral, gamma of tau memory kernel, we just want to write some general properties of it, we do not want to make specific modeling of gamma but some general properties, gamma of tau, we should take it to be identically 0 for tau less than 0, does not make sense to go, that fix is fixed by causality in any case but by the fact that I cut this off here but you could formally say this gamma is defined only for positive values of its argument, okay. Then I expect the effect to keep going down as the elapsed time increases, so I would certainly expect that gamma of tau decreasing function, so as tau increases I want this effect to go down, okay. Anything else you want of gamma? Well, we would like it to go to 0 at infinity sufficiently fast, if it does not wear in trouble then we have to fix it, we will do that, okay. And right now I am not saying how fast it should go to 0, as tau tends to infinity but what else do you require? There is one more very important criterion we need, one more condition, it represents frictional force, right, so you want its sign to be positive, you do not want it to be negative, right. So non-negative, non-negative decreasing function, let us put that in and see what happens, so my Langema equation now reads m v dot of t plus m, I put this m here purely for dimensional reasons so that this gamma whatever it is still has the dimensionality of 1 over time, so minus infinity to infinity d t prime, oh sorry, no, no, no, there is an extra time factor, we will fix that in a minute. I want to put this m here so that this m cancels out on this side completely, this we will see what dimensionality this gamma has but from this equation itself t minus t prime, v of t prime equal to any force you exert on the system, some applied force, deterministic force plus the random part which is the eta of the force we denoted by eta of t earlier, so let us leave that as it is but important not necessarily white noise, in fact I am going to show it cannot be white noise, it is noise, okay, it is random but we cannot assume, we cannot, we have to prove that it is consistent to assume that it is delta correlated, we will have to see, we do not know at the moment. So pardon me, oh I am sorry because I said already 0 so it cuts off but okay 0 to t, by the way you can recover the original version by saying if gamma of tau equal to gamma delta of tau where this is a constant implies implies the Langevin equation, Langevin model. So the physical dimensionality of this kernel is gamma divided by time, there is an extra 1 over t so it is 1 over t squared is the dimensionality because there is an extra t here and when you hit this you get a 1 over time and then it has the same dimensions as this, okay. So this eta of t at the moment is just noise but not the systematic part of it, the noise from the medium, okay and what are the assumptions we have permitted to make of this eta of t, well exactly the same physical assumptions namely that on the average it is 0, with or without the external force because again the assumption is you are in a heat bath and the fact that you applied an external force does not change anything, does not change thermal equilibrium, remains as it is and it certainly does not get affected by what is happening here. So we think, we do not know for sure as yet but this is a harmless assumption here. So we have no clue at the moment what sort of noise is consistent, the whole idea is to make this model consistent with causality because it looked like we are violating causality and to make it consistent with stationarity, it looked like that too was getting violated, okay. So this is this thing here is called the generalized Langevin equation and we need to fix this model in such a way that it will give us the results that we know are already true namely this velocity should finally the probability density should turn to the Maxwellian, should remain in equilibrium, that it should have a stationary autocorrelation function and that the mobility of the system is governed by the velocity correlation, that was a very general theorem that we deduced and that cannot be violated. So that result has to be checked to do this. So the whole idea will be to fix this model in such a way, precise way that is the only way we can fix it such that it will be consistent with stationarity, causality and lead to the correct Kubo formulas for the mobility and so on. In the process we will discover the fluctuation dissipation theorems. Now remember always keep this at the back of your mind, when this eta of t was Gaussian white noise with a strength square root of gamma that capital gamma and this little gamma here were related to each other by this relation. So we had this consistency condition gamma was equal to 2 m gamma k Boltzmann t. We call this the fluctuation dissipation theorem because this measured the strength of the fluctuations and that measured the dissipation in this system. We need to find out what is the new fluctuation dissipation theorem. So that will that also tell us something about eta? Yes, yes. So that is going to tell us something about the correlation of eta here because ultimately if this fact that the system is in equilibrium is happening because thermal fluctuations throw you out of equilibrium, the dissipation brings you back. The noise itself has to play a role to bring you back to equilibrium. So there has got to be a consistency condition between these two. Precisely what it is we do not know at the moment. But what we do know but what I can assert and we will have to prove this is the following that it will be inconsistent to assume this to be white noise. It will show that it cannot be white noise. It has got to have a finite correlation time. Not surprisingly that correlation time will be related to this memory function. They have to be related to each other. So there will be a relation between them. That will be the replacement of this theorem. That will change this. So let us see if this is going to work. By the way, this kind of business is used in understanding the movement of particles in liquids but it is a much more intricate formalism than this simple stochastic equation formalism. But this is a standard model. It is next step to improving the Langevin model before we go to the full glory of something like the BBJQI hierarchy or something like that. But this thing here will already tell you a great deal of, give you a great deal of information, okay. So now let us do the usual game. We are going to find average quantities. So let us find the mobility for instance. And we do this by the standard trick of saying well the average value is 0 and what we need for the mobility is for unit applied force, what is the average velocity. And so the average velocity per unit force, applied force is called the mobility of the system. And we also define the dynamic mobility mu of omega by saying suppose I apply a sinusoidal force with frequency omega, what is the response look like, average look like. So a short way of doing that, I am not going to start from the beginning like we did for the Langevin equation, is to simply say alright, let us take this thing, apply a portfolio transform on both sides and see what happens. So we are going to define V of T equal to as usual, our convention was 1 over 2 pi minus infinity to infinity d omega e to the minus i omega t V tilde of omega and the same for eta, the same for f, etc, etc. So any function of T is written in Fourier language in this fashion here.  And we will also take averages so that the average value of this goes to 0 anyway and where does that get us? Well if I substitute that, let us write down the equation for the average value of V of T directly or rather the average value of the Fourier transform of V of T directly here. So I apply this, I put this in on both sides and see what happens. So the first term gives me m times minus i omega because there is a dot, right? And I am going to equate Fourier coefficients on both sides. So minus i omega, the average value of V of omega, this term here plus something or the other plus and let us do this properly, m divided by 2, but I am going to read out Fourier coefficients, okay. Let us do this slowly, pardon me, it is a convolution here, I am not okay since it is forced on me, this is a convolution, it stops at T and we can write down what this is going to be. It is going to be the one sided Fourier transform of this V of this gamma because the integral gets cut off and then you equate Fourier coefficients. So I hope you see that directly and the 2 pi is cancelled out everywhere here, here, here, etc. And you end up with this V tilde of omega times minus i omega plus I need a symbol for this quantity integral 0 to infinity d tau e to the i omega tau times gamma of tau. That is going to be the integral over dT prime. When I change variables from T minus T prime to tau, this integral will run from 0 up to infinity, it will get inverted and the minus sign will go away, right? I replace T prime by tau which is T minus T prime. Then this becomes gamma of tau and the e to the i omega minus i omega T which came in in the Fourier transform is going to give you e to the minus i omega T times e to the i omega tau. So let us give this a name, it is not gamma tilde because this is a one sided Fourier transform. It is if you define it to be... But I do not, I want to keep the notation straight so that I do not want to be confused that sometimes I define it that way and sometimes I do not. We have said gamma of tau is defined for positive values of its non-negative values of its argument. So only this thing, this definition, let us call it gamma bar of omega. For the response function I use chi, I just said that is the susceptibility because that is the standard symbol. For this, as far as I know there is no standard symbol, let us just call it gamma bar. I have used star for complex conjugation, not a bar. So it is gamma bar of omega minus i omega in this fashion here is equal to f tilde external of omega, m times. There is nothing to average in the deterministic applied force. So that is just f tilde. And the eta tilde goes away because the average is 0. So if I divide this by v tilde of omega, so therefore, pardon me, yeah, I have done that. I have done that, sorry. Thank you. Yes, of course, of course. It is the average. It is the average, why do not I write equilibrium outside this average? Why, why have not I put a subscript equilibrium? I am not in equilibrium. I have applied an external force. So this is the perturbed system. And that is why the system, the average velocity is not 0 because it is moving. In the absence of this external force, the average velocity is 0. It does not go anywhere. But I will still retain this notation. I will still retain the notation, this bracket here because I am doing a full average. There is an average over the initial condition v naught and then there is an average over all possible v naught. So I do not care. I am not even going to specify what that ensemble is. This is and the average force is 0 and that immediately gives me this. But we know that this fellow divided by this f tilde external of omega, we know this is by definition the dynamic mobility which in this model is 1 over m gamma bar of omega minus i omega. This is the dynamic mobility. It is a susceptibility. And what kind of generalized susceptibility is? Is it? It is chi x v of omega. Because I am applying a force which is purely mechanical, minus x times f external is the change in the Hamiltonian of the system. And therefore you get this. But there is no reference to Hamiltonians or anything like that here. Once I put in friction, it is not a Hamiltonian system anyway. But I do not care. The model still works. This still works. So in the sense of generalized susceptibilities, this is what the mobility is. But in simple and physical terms, it is the average velocity per unit applied force. And when you apply a force which is sinusoidal, you do this division for each frequency component and you call it the dynamic susceptibility or mobility here. We know there is a general property of this guy which says that the singularities in the omega plane cannot be in the upper half plane, not with this Fourier transform convention. So I am going to leave it as an exercise to you to show that this, the singularities of this function, by the way, this is not a constant. If it were a constant, the matter is trivial because then you know that the pole is in minus i gamma. But now you have to show that the singularities of this are in the lower half plane or better still, there is no singularity in the upper half plane. So any root of this omega equal to i times gamma bar omega must be in the lower half plane. That follows from this representation. So that is the susceptibility, the general in this form, in this model. We still have to see what it does in terms of the velocity autocorrelation function. We still have to compute that explicitly. And we have to show that this quantity is, in fact, the velocity autocorrelation function apart from some KT factor and then e to the i omega t integrated over t. We still have to show this. So show that u of omega has no singularities in the upper half plane. And of course you know that in the case when this fellow is constant, when the memory kernel is just gamma times delta of tau, this thing becomes gamma, comes out and then this is the gamma minus i omega. It was already the dynamics susceptibility in the large of my equation. We had already found that. So it goes to that correctly. But now we have to know, we have to see what is going to happen when you put in this memory kernel here. So now we set this aside and let us go back and compute the velocity autocorrelation in this equation in the absence of the external force because the whole point of linear response theory is that the response in the presence of the perturbation to first order in the perturbation is given by some physical quantity which is a response function dependent entirely on equilibrium fluctuations or correlations in equilibrium with no reference to the external force at all. So that is the crux of linear response theory and we now need to show that. So now in the absence of this quantity of this external force, this is the quantity that we have. So let us put this as 1 over m and remove this m. I want to find the autocorrelation function. So instead of trying to solve this integral differential equation, which is by the way a mess because one way to do this is to do what we did, namely do Fourier transforms and compute formally a solution. That is not going to help us find autocorrelation functions very easily. So we will cut a long story short and anticipate the fact that V is a stationary process and directly calculate the autocorrelation function in equilibrium. In other words, in the absence of the external force and we will check if this assumption of stationarity was right or not post-facto. So let us write this equation down with an arbitrary time. One point I wanted to make was yes, as an equation, as an equation, mathematical equation, it is an integral differential equation and as you know when you have an integral differential equation of this kind, in principle you can convert it to a differential equation but it will be an infinite order differential equation in general. So as it stands, there is no guarantee that, there is absolutely no guarantee that you have a Markov process as the output because for a Markov process I would expect a Fokker-Planck equation which has only the second derivative in the velocity etc. But this looks like there is infinite memory in the problem. So yes, it is indeed true that this does not lead to a Markov process. This output process V is not Markov even if this were Markovian which we do not know right now. So it is not, it goes beyond the usual Markov process. The next question, if it is not a Markov process, will its conditional density satisfy some kind of Fokker-Planck equation? In general, you do not expect it at all but in this case it turns out that it does. It satisfies a Fokker-Planck like equation very similar to it okay. And what is needed for that? I am just trying to motivate my sort of, heuristically what the answers would be, eta should do something, yeah. Now if you are familiar with Markov processes, you know that when you have a driving force of this kind in general, if this were a white noise then you have that Langevin type equation and sure enough you have a Fokker-Planck equation. That is a rigorous mathematical equivalence. Otherwise, the partial derivative order on the right hand side in the variable goes to infinite order. It is called the Kramers-Moyal expansion in general. It gets cut down to second order very crudely and roughly when higher order cumulants vanish. Now what sort of random variable has all cumulants beyond the second equal to 0? Gaussian random variable. So if this is a Gaussian noise, then it turns out that this output process in spite of this infinite memory here still satisfies a Fokker-Planck like equation, okay. It is not Markov but it still satisfies such an equation, a great simplification. We are not going to do that. We want the more physical aspect of this namely we want to fix the problem at stationarity, causality and to find the mobility. We would like to find that here. So let us do that. So let us write this equation at T naught plus. But T naught is an arbitrary number and T is any positive of this plus integral from minus infinity to T naught plus T dt prime gamma t naught plus. T naught is an arbitrary point and I am writing this equation at T naught plus T where T is any positive number. And now what would be the way to find the correlation function in equilibrium? It would be to multiply this by T naught on this side and then do an integral from 0 to infinity after multiplying by e to the i omega t. That will give me what the left hand side of the cobalt formula has, right? So I do that. I am going to do this here. But in the normal derivation for the ordinary Langevin equation, we could have done that and we did do that. We found the correlation function directly by multiplying by V of T naught, taking averages and arguing that V of T naught is uncorrelated to eta of T naught plus T, put that side equal to 0 on the average by saying that the force at a later time cannot govern the velocity at the present time. And then we ran into this problem with stationarity Gaussian, with causality and so on. We certainly want this property. We certainly want V of T naught, V dot of T naught in equilibrium to be equal to 0 because we would like this thing to be stationary, certainly. It turns out that the only way to do this with this equation here with this model is to argue that V of T naught, the correlation between V of T naught and the random force is not V of T naught with eta of T naught plus T correlation equal to 0 but rather a portion of this memory should be taken out on the right hand side and called that is the effective force on the particle. Now the question is what portion? What portion? That is left arbitrary at the moment but we know the result that we are aiming for and we want consistency here. So the argument is the following. Look at what happens if I break this integral up V dot of T naught plus T plus integral T naught to T naught plus T, dt prime gamma of T naught plus T minus T prime V of T prime is equal to 1 over m eta of T naught plus T minus this fellow here from minus infinity to T T naught dt prime V of T naught plus minus T prime sorry gamma of T naught plus T minus T prime. There are many ways of motivating this but the simplest is going to be the following. I am going to multiply both sides of this equation with V of T naught e to the i omega t and do a different on both sides of the equation after I average. So if I average this guy here that is going to be the average that exists in the cobalt formula for the dynamic mobility. If this is stationary if I put T equal to 0 this part of it here should become independent right? It will be equal time and should vanish this fellow should vanish. When will that happen? If I put T equal to 0 this integral vanishes and I am going to get this correlation equal to 0 provided the V of T naught is uncorrelated with this guy. Not with eta of T naught plus T but this portion of the velocity history subtracted out and that is the only way you can split this to achieve this point. We will see as we go along that how it becomes consistent right? And then we will justify it. So let us call this something else. Let us call this noise some h of it is a function of T naught plus T but it is also a function of T naught because there is a T naught sitting here and there is a T naught plus T sitting here. So it is dependent on both the starting point T naught and the current time T naught plus T and it is not stationary. This fellow is not stationary by any means. It is a function of T naught plus T and T naught separately so therefore it is a function of T and T naught separately. Now we will assume and we have to see if this is consistent or not that V of T naught is not correlated with h of. This is for any positive people okay. So that will be the next step. I have to stop here today since we have run out of time but we will take it from this point next night. We would like this to be true. We would like this to be true so that we do not have the problem which we had in the Langevin model where on the one hand stationary said it should be true but the exact computation of the velocity correlation said it is not true. We definitely like it to be 0 and that is achieved by saying look let this thing trivially vanishes if T is equal to 0. So I split that portion and the rest of it but it has got a physical meaning. There is on the time axis there is some arbitrary instant T naught and this is T naught plus T and we are writing the velocity or acceleration at this instant of time by saying that this is affected by a friction which operated all the way from the infinite past up to here but that is broken up into two pieces part of the velocity history which operated till here and the rest of it. And this portion of it I am subtracting from the noise at this instant of time and arguing that the effective force, random force with which the velocity here is uncorrelated is in fact this portion, this subtracted portion. That is the only way in which I can ensure that if T goes to 0 this correlation here is going to vanish identically so it preserves stationarity but now that is fine as a kind of fix but we need to see if it gives you the correct Kubo formula and it gives you the correct fluctuation dissipation theorem. By the way there are two fluctuation dissipation theorems here. One of them is the statement that when you apply an external force the response to first order depends on equilibrium fluctuations. So it is a kind of Kubo formula for the susceptibility. The formula for the susceptibility itself in terms of the response function is called the second fluctuation dissipation theorem because historically the fluctuation dissipation theorem was written in the context of the Langevin model where you have a specific stochastic model Langevin equation or model and then there was a connection between the dissipation constant and the strength of this random noise which you put in by hand and that was called the first fluctuation dissipation theorem. So the second one is in fact the consequence of linear response theory directly the first one is specific to a stochastic model but the second one must be valid as well clearly otherwise linear response theory does not make sense okay. So since our Langevin model is a linear model we expect that the linear response theory formulas will also be true it is linear causal retarded and therefore it satisfies the conditions required for the linear response theory formulas. So I hope this is clear that there are two of these theorems and we will write them both down explicitly.


what's up guys ruxen 34 back with another opening and this opening is pretty crazy we're doing 100 packs a search for multiple cards that i've always wanted to pull and we have yet to pulse but before we get started i want to remind you guys to like the video for the giveaway i'll be giving away an entire ultra rare exodia set near mint condition from yugi's legendary decks every piece including both arms both legs and the head summon exodia if you want to win the game yeah so um that probably won't work but this set is pretty awesome so if you guys want to win it all you have to do is like the video be subscribed and let me know in the comments which of the chase cards from these packs do you want to pull and i'm about to talk about what we're trying to pull but i also want to remind you guys about the 20k special we're opening five of these packs and more to be announced that is going to also be equally as epic in my opinion maybe even more epic so we have these five first edition blister packs that we're going to be opening so look forward to that guys make sure you guys are subscribed we're getting close to 14 000 at the time of recording this maybe we're there by the time you guys see it and yeah let's get into it so so what you guys see here 48 tune chaos first edition packs yeah 48 packs we got two battles of legend heroes what's called revenge i think whatever the one with the black blaster soldier is there we go yeah heroes revenge we got some more dimension of chaos we have turbo pack three and judgment of the light the cards are trying to pull out of all these sets obviously we want to get a collector's rare and we want to get the black blaster soldier or the starters dragon or even the chaos emperor again but the first two are what i'm really looking for out of here we need to pull the last ultimate rare from the set that we haven't pulled the odd eyes vortex dragon out of the turbo pack threes we want to pull kaius the mega monarch the ultimate rare and then the star eater ghost rare so we could pull some insane stuff in this video i think it's gonna be pretty awesome so make sure you guys like the video for the giveaway and just to support the channel it's gonna be awesome so let's get started i will continue to talk while we open because we have a hundred packs this is gonna be a little bit of a longer video hope you guys do enjoy it fright for meister i know some of you guys like the longer one some you don't so i'm trying to mix it up you know do some regular old 10 10 to 15 minute videos and then some videos i'm gonna do longer like this so i'm guessing this would be about maybe 25 30 minute video skilled red magician fluff of wings there we go you guys know about that one yeah so make sure you guys uh check out my tcg player and ebay affiliate links below i sometimes do not remind you guys but i won't remind you guys if you want to support the channel all you have to do is click those links buy anything on those sites and it supports me so that's pretty awesome okay let's see we get kaiku xx saber regura all right let's keep going all right judgment of light let's see what we can do yeah 100 packs this is gonna be pretty fun i just felt like it i have these two i still had a few tune chaos boxes sitting around figured i'd open them i could have sold them i was like you know what it'd be way more fun to open them so that's what we're gonna do all right tune chaos i said yeah a few judgment of light boxes as well so hopefully we could pull a star eater ghost we did pull the secret rare make sure you guys check out that video up in the eye barfler dragon fright for sabertooth super rare and tune world yeah i really want to pull a kaius that thing looks really awesome sweet really a sweet card the only ultimate in the set so hopefully we can do it didi crow that's a nice common and fault troll all right yeah let me know let me know in the comments guys which of the cards that i mentioned that i want to pull which one would you most want me to see pulled would you want to collect your rare bls would you want to see me pull a ghost rare would you want me to see pull kaius or probably not the odd eyes vortex dragon but i do really want to pull that to kind of have all the ultimates because i pulled everything else there's only a few dimensions of chaos i had more but i i wanted to keep it even at 100 packs so let's see if we can pull something insane out of a few packs to brush headlong didi necro slime our favorite card yeah so so far we've only pulled comments we haven't pulled any foils or anything hopefully we can at least get like a pot of extravagance to pursue chaos that would be pretty awesome let's do this fry for meister there's the chaos in bro dragon judgment of the light star eater even the ultimate rare is really awesome so that'd be a cool pull and look what we got guys you guys know what it's time for medocie chicken lattes chicken lattes chicken lattes yep there it is our chicken lattes has appeared we're gonna see a few of those intercept tomato really struggle with that pack okay tune chaos we have the chaos emperor dragon in the background to help us pull chaos emperor dragon here we go fluffle angel so far we're slow but we do have cards we do have good packs in the future i know for fact they're out of booster boxes a lot of them okay here we go i'm bro horror mecha phantom beast something yeah it's confusing when you open tune chaos because it has all rares and i'm trying to have a pile with the rares in it and then i put the rares in there because they're basically commons and you know gamma oh immortal phoenix gear freed there we go ultimate not ultimate ultra rare three nights it was an ultimate there's a pot of desires there we go mortal phoenix gear freed all right we're back i've got more sleeves ready to go sleeve up our epic cards we hopefully will pull i hope walker soldier sublimation night rares aka commons and that's it all right let's go for a chaos hopefully we can at least pull something in terms of hollows out of here they're not very good ratios treacherous trap hole there we go as soon as i say it treacherous trap hole super rare let me know if any of those was it cloak and dagger somebody was telling me it was decent nope i don't know let me know i'm gonna put it aside if i pull any decent commons and rares treacherous trap hole super rare especially from the turbo pack i think that's a decent card i don't know if it's worth anything in common but i know that some of the cards are worth a little bit here we go let's get something good here we go come on oh what is that oh a secret rare brotherhood of the fire fist rooster there is a secret rare from judgment of the light check that card out that is pretty awesome looks like a little ninja i mean fire fist obviously but pretty sweet so we've got a secret rare we're doing all right tune chaos gave us the immortal phoenix gear free which is about a 20 card i believe all right come on mass hero acid sanga stardust dragon dimension of chaos we have four packs left let's see what we can pull can't we pull the odd eyes vortex dragon ultimate rare come on let's do this db from it cradle impact and dr franken derp i keep pulling two out of the judgment of the lights they really want to be open maybe there's a star reader come on fencing ferret phantom beast halyard talia princess of cherry blossoms i think i remember that being like a dollar 50 which is not bad sleeve it up here we go next pack let's do judgment of the light again just because we pulled two out by accident maybe we can pull back to back foils let's see mech phantom beast halyard chigalates chigalates there we go and there's the decent rare intercept tomato rare tune chaos it is let's get a collector's rare i've not pulled one in a while because i haven't opened much tune chaos since the two cases we open fright for sabertooth chaos zone i haven't even pulled a chaos space left let's go back to back tune chaos proglia blackbuster soldier number 75 it's really hard not to spoil with these tune chaos packs because they're super slick they just kind of slide over here we go poke and dagger again super ancient deep sea i think this was a decent common too and then this was a decent rare so i'm just gonna set all those aside from what i remember opening a few packs here we go keeper of dragon magic chaos face i asked and i received pot of desires that seems to have happened a lot recently i just mentioned it and then boom it happens i'm thinking about uploading this video on the day that i my 100th straight day of videos which would be like next week i think or something um i might i'm not sure though so you may be seeing this on my 100th straight day you may or may not may see it a little earlier i can't remember what the day was exactly it was like the 20 something oh the july not sure okay here we go oh not this vein betrayer all right let's go with more tune chaos because we do have two booster boxes so the chances we have about a 67 chance to pull a collector rare it has been a while since i pulled one so it would be great to pull one of the good ones especially it'd be pretty crazy all right let's see syph gamma sigran pot of desires the there has been one pulled out of this case i can't remember which one it was vintage yugioh pulled one on his video i think it was a mortal phoenix gear free to actually and that's the one that's been pulled out of this case because we split the case so we don't really know i could have zero honestly you could have all of them which would be really terrible or um i could have all not well he has one so i got three so i probably don't have three if he's already pulled one here we go i think he opened two boxes and magical android there we go tune chaos get the collector's rare come on gamma eternal chaos stardust dragon let's go another turbo pack let's get the kaius come on let's see that epic ultimate rare saber slash polonosis okay come on judgmental light can you give us a star eater this isn't really a battle but in a way it is you know to see which one we can pull the big card number 104 number whore vain betrayer let's go tune chaos again let's get a collector's rare black luster soldier gamma and chaos zone i need i haven't really checked the gamma and the pot of average prices i think they did go down quite a bit because there are just a lot of them available being only commons basically they're technically rares but not really sine kodak all right let me know if you guys do enjoy this 100 pack opening and if you'd want to see other variety 100 pack like this whether it's not just the same cards there's another gladiator beast it's not just the same pack like a hundred times it's different packs to make up the 100 packs oh this pack's a little crazy all right we have a super defense robot leo jingy turtle number 39 utopia ray victory i think this is like eight or nine dollar super rare so that's super good and i mean it is the cover card isn't it yep so pretty awesome card this is a nice ultimate rare as well so that would be a good card to pull so definitely a good super rare happy about that i think i have pulled an ultimate once so pretty good it was pretty good chaos valkyria i still have my collectors rare of that one i sold my pot of desires my pot of extravagance but i kept my end of my chaos space but i kept my chaos valkyria and the other one chaos space there we go the uh tune black luster soldier i actually sent that one into ps a maybe in two five years we'll have it back we can see what we got all right more tune chaos can we hit something epic hit the like button for luck guys here we go oh tune page flip we've got another tune card i don't know if the values held on this used to be like 15 20 bucks but it's probably still decent i think that's like a better it's a better what's it called tune table of contents it's but it's like a once per turn or something like that yeah i'm not even reading the effect i'm just what i remember okay i could just read the effect but you know that'd be too easy that'd be too easy guys here we go diamond dude dd crow nice and a dark corruption let's go with a dimension of chaos i haven't seen this in all battles of legend i forgot about that let's try that soon let's get the odd eyes vortex come on great illegal great illegal twice dd necro slime so that's interesting two great illegals we're gonna have to try and get that soldier of chaos here we go supervised tune rb late it's a nice super rare play the second best behind the chaos space judgment of the light oh man give us a star eater please please yellow dustin the creepy toast all right nothing in that pack let's go with a battles of legend heroes revenge and pull the blackluster soldier soldier of chaos oh there it is i forgot cyber stein okay that's a cool secret i don't think it's too valuable but it is a really awesome one because there are only three prints of this card the sjc uh shown in jump version the dark beginning two rare version and then this so pretty difficult to get that card even though i mean it's really cheap but it's still not easy because there's only three versions oh tune rb late see it skipped right to okay another tune rb lady that's actually pretty good i always had trouble pulling those when we were opening those 24 packs all right triple pack three we haven't hit anything too great yet we've hit one secret rare uh kai ku saber slash um the secret rare i don't know what it's worth i don't think it's too great we did hit a nine nine dollar super that's pretty good but we haven't hit anything too crazy in terms of rarity other than the secret here we go come on yellow dustin back to back brotherhoods and nothing all right we we've hit two ultras but we have not hit a collector's rare out of here let's see what we can do side frame fluffle angel blackluster soldier let's go another dimension chaos we have one more after this so we're getting we're getting pretty close oh here's a short print pot of the forbidden that's a nice card okay right to the rare by accident didn't across line i got a couple of decent hits in there but nothing too crazy back to tune chaos the most popular yugioh set in a while for sure definitely pretty cool sublimation night and speaking of new sets i will be opening the new battle's legend armageddon or whatever it's called i think that's what it's called and with like the 10 000 dragon or whatever it's called and then the utopia ray japanese so i'm gonna try and pull those guys sigran and i think it's gonna be a lot of fun so we're gonna open a few few packs worth of that and by a few i mean a lot here's the last image of chaos and then we're definitely gonna get some rise of the duelists as well so i hope you guys are excited for that stuff hopefully we'll get something cool in terms of the new stuff tune barrel dragon it's nice de necro slime we have not hit a foil out of there in a while okay actually at all in this video we didn't hit any all right judgment of the light here we go give us something epic howlyard trap tricks trap hold nightmare all right that's a pretty cool card he's got those spider on it pretty sweet super rare let's go back to tune chaos and let's pull something really awesome like a collector's eternal chaos and sign that code let's go turbo pack three can we get the kaius come on let's get something we want cloak and dagger again that thing's everywhere i don't know if it's worth anything hope it is tune chaos so far we've gotten two ultras so we got we've got some stuff to come i think oh the chaos creator as i say it the chaos creator ultra rare that's a pretty good one pretty valuable ultra rare there we go now maybe we will hit a kaius uh i think there should be ultras in here as well right ultimate and ultimate ultras and supers magical android all i've hit so far is one super from the turbo pack threes judgment of the light we have hit a secret we haven't hit an ultimate from here yet so maybe we'll get one cheap cheap cheap mistbird colossus or colossus seems like colossus i think it might be let me know in the comments how you say that all right back to tune chaos i'm i'm eagerly awaiting a collector rare please be in here number 75 pot of desires pull that out let's go another tune chaos two booster boxes by the way tune chaos booster boxes like 135 bucks so this opening is over 300 which is i know not that crazy compared to some stuff i do but still pretty expensive at this point i got it for a lot cheaper fortunately because i bought those cases early on but uh yeah it's not going to last oh dark greffer from turbo pack three ultra rare there's the ultra rare we're talking about this is a really sweet card check that guy out i believe originally a secret in fan of darkness i think that's right so that is really cool ultra rare so finally we got something out of those turbo packs we did just super earlier i don't remember what it was what was the super it was a trap card i think i can't remember junk blader okay come on now let's keep up the fire there's a cold wing our first one it's a nice common now we need our tune cast has done well it's done pretty well we've had three decent ultras we just need to collect a rare tune page flip again okay let's make sure it's not a collector rare it's not it's regular okay ultra rare we got two of these pretty decent card though okay let's read the effect control tune world reveal three different tune monsters different names your opponent randomly picks one for you to special summon and shelf is rest your deck okay i don't know if that's that isn't better than i heard that was better than table of contents but table of contents you get to pick exactly what you want so i don't i don't know if that random thing is that's good i don't know let me know there's probably some reason why it might be better in certain decks and stuff but i don't see it oh there it is collector rare the chaos creator oh that is an epic one i don't remember exactly the price on it but that is a sweet card because it is a new card it is a pack um art wrapper whatever you want to call it that is awesome oh my goodness we got a chaos creator collector rare so this is a successful opening we've we've done pretty well so let's let's set that back here if we can pull like the kaius too that'd be crazy that was a that was a great have it kind of snuck up on me it came right after that page flip so that was really cool all right so we got a collector's rare we've gotten a dark grapher that was pretty cool all right glad to be rich arie we pull that dude constantly collectors rare let's go pretty good one def i'd breath i'd definitely rather have a monster than like a like a spell like even if it even though the prices on the pots are really high it's like i just want to sell those for the price i don't really care to keep them all right let's see what we can get now micro coder yeah so that was pretty awesome this this opening is done well now i think last time i saw he was around like 120 or something so pretty good value dark eruption it's not as high obviously some of the other ones like the black oyster soldier which is 300 400 but still pretty good i think it's pretty low though for being what it is a new card and everything all right here we go cockadoodle do we seem to get this guy every time it's crazy all right we'll sleeve that up ultra rare cockadoodle dude i think he's actually worth a little bit i was like surprised i was like this card's actually worth something okay now our goal out of here is we're probably not getting another collector we might that'd be pretty crazy um but a desires would be crazy or not desires sorry well there's the desires i accidentally asked for it and we got it i meant to say extravagance that would be a great pull okay we got a two turbo packs by accident all right so far we pulled a super and an ultra let's try and get the ultimate xx sabre we need to look at the ratios let's see the ratios for this stuff oh wait there it is no ratios it just tells you what it has it has one ultimate in the set so it might be like one per box hundred packs that would not surprise me at all okay black oyster soldier judgmentalite here we go we still need an ultimate out of judgmentalite i think you're basically guaranteed one per box so hopefully we'll get one xz's agent all right turbo pack let's do this can we make this an extremely epic it's already pretty epic let's make it extremely epic by pulling an ultimate rare kias come on oh wow this didn't cloak and daggers than like literally every pack so i don't think that's worth too much but it's also turbo pack people aren't really opening that constantly so maybe maybe it's just gotten popular who knows eternal chaos kias simple dragon judgmentalite here we go fencing fire ferret umber horror the calibrator okay come on we're waiting for that next fire i can feel it i can feel it that's the first side frame in forever okay i was in the middle of my song and i got distracted why now that i haven't seen one of those in a while i probably skipped over it but it's you know maybe you just have seen one it's possible too tune arpey lady haven't seen that tune in a while but i also was pulling like none per box so and the previous stuff i've opened okay tune chaos let's see what we can do let's pull something epic let's go fright for saber tooth judgmentalite we're getting down to the very end of this 100 pack opening yeah what do we want to pull star eater trap tricks that's a nice one whoo i'm just waiting i feel i feel i feel the big pull here guys like the like the button for luck that's what i was gonna say like the like button enjoy the like but okay that's yeah never mind pretend like i didn't say that bro let's keep going oh i forgot to have battles the legend if that has the blacklist your soldier i'm gonna lose my mind here we go ooh reynaud this is actually almost a 30 dollar card now this is i think the second best card it passed um the immortal phoenix gear freed i think it got more expensive than that one and now it's right behind the extravagance not right behind but it's behind all right that was the legend here is revenge you gotta be careful with these because you gotta do the pack trick because if for some reason they put the secret at the front so here we go miracle stone guardian of order artifact lance yeah that's a pretty good one actually that's a nice secret rare that's a pretty good pull i don't know what the value is but i know it is a decent card so probably not too bad so that's a decent card let's finish off strong finish strong with those let's finish off strong with the other packs mastic acid i think we should have at least one more ultra in these two chaos if we get lucky we'll have a collector's rare judgment of light i'm hoping we have an ultimate i don't know if we will or not i don't think you're necessarily guaranteed oh oh super rare first it looks like the borders but then i was like never mind we got we get this dude almost every judgment of light opening it's crazy super rare all right what do we have left we've got a few we've got four three a few more turbo or a tune chaos so let's see if we can get something good oh straight to it chaos valkyria pot of desires my stacks getting crazier here because there's so many good commons and rares all right turbo pack let's get the kaius here we go we'll have to continue searching if we don't get him here there's another deep sea king all right more tune chaos i think we're down to what six packs left to tune chaos six packs to pull an epic collector rare another epic collector rare i should say because we already pulled one check it out chaos creator or a pot of extravagance let's do either one of those side frame gear gamma that's good chaos space good pack definitely a good pack boom chaos space gamma two chaos okay now we'll be down to three of every other pack after this one which is black forest magician of hope not magical hope all right turbo pack i really want the guy let's do this a rescue cat we haven't pulled yet that would be good too chariot and ritari again oops okay come on judgment of light give us either utopia ray victory ultimate star yurt ultimate or star yurt or ghost that's what we want cheap cheap cheap i have pretty bad luck when it comes to ultimates from the set there we go cult wing i always get that uh susan owo guy literally every time it's crazy all right because they're soldier another the chaos creator again we got wait how many did i get two of these and the collector rare that's crazy we got this dude so much but that's okay i mean that's a pretty good one could be a lot worse all right so we got two more packs let's go turbo pack and save one of them for the end come on let's get some good cloak and dagger polynosis and dark eruption all right let's get one of these judgment of lights and let's pull an ultimate rare come on let's do it ultimate rare here we go jinky war coach captain bare man ultra rare another guy i pull every time i feel like i pulled the exact same cards out of judgment of light every single time it's crazy okay probably nothing in these tune chaos unless there's a collector rare so that's why you can't really know if you got everything or not master acid pot of desires you of course can't miss those one pack left of each set or three of the sets of our 100 pack opening hope you guys have enjoyed it make sure you hit the like button if you did and leave a comment on which you wanted to see me pull or that you enjoyed in the video for the exodia giveaway oh don't want to miss that there we go all right nothing okay so i didn't get an ultimate no ultimate that's kind of sad all right let's go to chaos and we'll finish with turbo pack chance for the kaius that would be pretty awesome here we go chaos space a good way to end it and pot of desires can't complain about that another pot of desires another chaos space turns out our boxes were pretty decent and let's go into the turbo pack three our chance at a kaius can we pull it here we go glad i'd be sports chariot kaius the mega monarch yes boom we did it unbelievable unbelievable the last pack the last pack not gonna lie guys i saw it back and i was like i think that this is an ultimate i'm not sure unreal we did a 100 pack opening and we pulled the ultimate rare what an ending what an awesome awesome opening okay let's go over what we got i'm not put it all together i can feel it coming in the air tonight all right guys we got a bunch of decent commons we've got some good rares in here i mean we got a whole i got a huge stack of all this stuff decent stuff bedulce chicolates don't forget that the gammas the desires the necro slides dd crows pot of the forbidden we got a few of those racharis however you say that card the trap tricks mermelio colt wings you know all that good stuff i got a bunch of those then for our actual foil pulls we've got this dude colossus the cherry blossom sublimation night trip some trap tricks tune harby ladies we got one two three four chaos space artifact lancia secret rare brotherhood fire fist rooster cyberstein coach captain bare man cockled doodle do to tune page flip yeah it's hard to say to the chaos creators one immortal fiends gear freed one ray not our ultra rares were actually really good pulls dark refer ultra rare from turbo pack that was amazing a collectors rare chaos creators we pulled three chaos creators in two boxes that was nuts and a kaius the mega monarch so that was pretty awesome and incredible 100 packs ended on last pack magic again that was crazy make sure you guys like the video if you enjoyed it and yeah i'll see you guys later thanks for watching peace elements are here dark bright