 Continuing with the Brownian motion theory, we showed in the previous lecture that the velocity of the particle starting with its initial velocity v naught in the presence of a fluctuation acceleration called a t prime evolves according to this formula v v t equal to v naught e to the power minus beta t plus e to the power minus beta t and integral over basically the fluctuating acceleration. So, very first principle solution to the first order differential equation it is Newton's law for v. So, then when you do ensemble averaging, ensemble average of a is 0. So, we get the mean velocity just decaying of the original velocity and we also did an exercise to find the v square average as a function of time and if you subtract this v bar from that, we basically got sigma of in v the variance that would be gamma by 2 beta into 1 minus e to the power minus 2 beta t. So, this evolves with the rate of 2 beta as opposed to v bar alone which evolves with the rate of beta. Very interestingly, we then showed that as the time goes to infinity v square average because v bar will tend to 0. So, v square average becomes gamma by 2 beta and then using thermo dynamical arguments that is the theory of equipartition of energy, we established the relationship between the fluctuation coefficient gamma and the friction coefficient beta and that relationship was quite consistent with a relationship that we derived using gas kinetic arguments for the for the coefficient gamma. So, that shows an internal consistency of the whole process that we are discussing. We continue with this how the velocity fluctuates in a Brownian theory. We have to understand the velocity fluctuations well. So, that it is from velocity fluctuation that the random walk originates. The real variation of x the randomness in the position variable comes as a kind of integral over the fluctuation of the velocity variable. So, that we need to understand the velocity autocorrelation function. So, velocity autocorrelation function v autocorrelation. Now, when we say autocorrelation it is a the correlation of the same quantity at 2 different time scales 2 different times. So, by definition it is nothing, but velocity at some time say t 1 correlated with velocity at some time t 2 and its ensemble average. We also can write it as v t 1 v t 2 and over bar we can use because it is sometimes it is more convenient to write. So, this is the definition of an autocorrelation function 2 time correlation. So, we try to now establish from all that we have done how this quantity between 2 different times evolves with respect to their difference in the times. How does it depend on t 1 itself and how does it depend on t 2? t 2 can be greater than t 1 can be less than t 1 we will establish all this. So, we first make use of the expression that we have given here for v t. So, we start from first principles. So, v t is given. So, I know v t 1 and then I can write v t 2 and putting them together I will have the expression v t 1 v t 2 I am not still ensemble averaging I am just writing the two quantities that is e to the power minus beta t 1 plus t 2 e to the power minus beta is a common in this expression. So, it can be taken out and that will be t 1 plus t 2 then there will be a term v naught plus integral 0 to t 1 e to the power beta let us say t prime is the dummy variable a t prime d t prime this is 1 with respect to t 1 and with respect to t 2 again it will have v naught plus 0 to t 2 let us say e to the power beta t double prime let us distinguish because these are dummy later when we put them together there should be no confusion. So, this time we will call it as a t double prime d t double prime and this bracket closes. So, it is a product this is common. So, this is v t 1 and v t 2 let us expand this it is because there will be now four terms. So, all the four terms we can expand. So, moving over to the next we can see that v t 1 v t 2 expanded form will be v t 1 v t 2 it will be e to the power minus beta t 1 plus t 2 will be there then v naught v naught will give you v naught square. Then there will be two terms multiplying v naught 1 will be an integral over 0 to t 1 of e to the power beta t prime a t prime d t prime and another one will be that integral over d 2 e to the power beta t double prime a t double prime d t double prime only brackets close. Then there will be the double integral which will be 0 to t 1 and 0 to t 2 which will involve e to the power beta first with respect to t prime and then with respect to t double prime and that will be a t prime a t double prime d t prime d t double prime and here we close square bracket. So, the four terms are like this it looks quite messy, but when we take the expectation or take the ensemble average many things drop out. As you can see when I take the expectation of v t 1 and v t 2 overall realizations or ensemble average the first term does not depend on any statistical variable it is original velocity. So, it will remain the same the second terms within the curly bracket. Since a t prime expectation will be 0 a t double prime also the ensemble average expectation when I say I actually mean the ensemble average that will be 0 and here of course, this is the two time correlation of the acceleration function which is a delta function. So, we input this to arrive at an expression v t 1 v t 2 we use over bar for ensemble average that will be e to the power minus beta t 1 plus t 2 will remain the same. So, v naught square will remain the same because it is not a statistical quantity thus this term within v naught will become 0. So, just to remember that we will call it as 0 the next term also will be 0 and the third term will have gamma. So, gamma we can take out. So, I will write it as gamma 0 to t 1 0 to t 2 e to the power beta t prime plus t double prime delta of t prime minus t double prime dt prime dt double prime square bracket closed. So, for future purposes we write it more neatly remove all the zeros. So, this is to create space e to the power minus beta t 1 plus t 2 into v naught square plus gamma double integral 0 to t 2 e to the power beta t prime plus t double prime delta of t prime minus t double prime dt prime dt double prime. So, this part we call it as the first term there is nothing more to do here this whole part we call it as the second term. So, we first focus on the second term because first term is just a simple term. So, the second term when we look at we have the second term by definition it was gamma e to the power minus beta t 1 plus t 2 and a double integral 0 to t 1 and 0 to t 2 e to the power beta t prime plus t double prime and it has to be integrated over a delta function which involves the difference between t prime and t double prime and one important property of delta function we should never forget that it selects the value when its argument is 0 providing the integral passes through 0 somehow. So, that means, if I am integrating over t double prime if I am integrating here over t double prime from 0 to t 2 then that t double prime when it becomes t prime only it will have a value which means t double prime when it runs it should run beyond 0 beyond t prime actually. Now, since we have not so far ordered t 1 and t 2 we cannot say. So, for the next step we make an assumption that let t 2 be greater than t 1 this is just an assumption because t 1 and t 2 are arbitrary let us say we have selected t 2 greater than t 1. So, if we do that then the inner integral t 2 which goes beyond t 1 can be split as 1 up to t 1 and the next one between t 1 and t 2. So, with this assumption the second term can be written as gamma e to the power minus t 1 plus t 2 the first integral is up to t 1. So, we can take that coefficient that is beta t prime will be out the dt prime is the first integral. So, the second integral if I put a curly bracket it is I split it as 0 to t 1 first of whatever is there that is e to the power beta t double prime delta t prime minus t double prime dt double prime and the second part will be t 1 to t 2 because t 2 is larger. So, it will be still a positive integral it will be e to the power beta same thing double prime delta t prime minus t double prime dt double prime and this curly bracket closes here. So, now let us look at the first integral and the second integral again further. So, if you look at this integral I call it as a and this as integral b. So, we have already selected 2 and in the second one now we look at a and we look at b. If you look at a the integral a here the time interval for between both for t double prime as well as t prime they run between 0 to t 1. You can see the upper limits are t 1 which means that when I am first integrating over t double prime somewhere it will it will have to cross t prime since t prime also runs between 0 to t 1 which means t double prime is allowed now to select the remaining part of the function corresponding to the value t prime since they cross each other that is why this diagram is drawn. So, we can see that the a integral therefore, the a integral will be simply 0 to t 1 e to the power beta t prime dt prime and t double prime is going to be t prime. So, it will be e to the power beta again t prime only. So, we have this part of the integral can be easily evaluated because it is e to the power 2 beta t prime. So, it is integral 0 to t 1 e to the power 2 beta d prime dt prime which is going to be e to the power x dx. So, it is e to the power 2 beta t 1 minus 1 divided by 2 beta. So, the a part of the integral is evaluated. Let us look at the b part of the integral. Now, for b part of the integral if you look at this here the inner integral varies from t 1 to t 2 and we have assumed that our t 2 is larger than t 1 whereas, the outer integral varies from 0 to t 1 only. So, you can see that the t prime varies from 0 to t 1 and t double prime varies from t 1 to t 2. So, what does it mean that the in the variable of the inner integral that is the t double prime never crosses t prime they are exclusive intervals. Hence in b t double prime does not cross t prime. Hence that whole integral b will be 0. So, we have put together the various terms now the 2 integral therefore, is only the a part with the of course, this pre factor. So, we write neatly the second integral therefore, second integral or second term we called is going to be here gamma e to the power minus gamma e to the power minus beta t 1 plus t 2 into the a which is e to the power 2 beta 1 minus 1 by 2 beta. So, we write it down as e to the power minus beta t 1 plus t 2 will go back gamma of course, being outside e to the power 2 beta t 1 minus 1 by 2 beta. This can be simplified as the 2 term we take this e to the power minus beta t 1 plus t 2 we can take inside. So, this will lead to gamma by 2 beta here there is 2 beta t 1 and here there is a minus beta t 1. So, it is going to be e to the power since t 2 is larger we keep in that form it will be minus beta into t 2 minus t 1 which is a positive quantity. So, it will be plus t 1 and minus t 2 that is maintained and the second one of course, e to the power minus beta t 1 plus t 2 here it does not matter which is larger and we got it under the assumption that t 2 is greater than t 1. Now, if we had if you go back and see the assumption we made that is t 2 greater than t 1 and then followed the steps we got this result. But supposing we had started with the t 1 greater than t 2 then the whole thing would have been in the converse form that is the integral would have been then t 2 to t 1 here and the same logic would have applied the difference being that if t 1 was greater than t 2 then this term this term would have been only changed because t 1 plus t 2 does not change here it would have been minus beta into t 1 minus t 2 this term would have become that that is the only difference. So, on the whole whether t 1 is greater than t 2 or t 2 is greater than t 1 all that matters is there a difference should be positive which is possible when you call it a mod function. Hence, in general for any combination of t 1 t 2 the second term will be gamma by 2 beta e to the power minus beta modulus of say t 2 minus t 1. Now, it does not matter of course, minus e to the power minus beta t 1 plus t 2 for all t 1 t 2 regardless of which is greater. So, we can now put the first and the second terms together you can see here the first term is simply v naught square e to the power minus beta into t 1 plus t 2. So, we combining all that we have we can write v t 1 v t 2 bar will be we can write it this way v naught square minus gamma by 2 beta multiplying e to the power minus beta t 1 plus t 2 and another term gamma by 2 beta into e to the power minus beta mod t 2 minus t 1. This is only a rearrangement you can note it from the combination of the previous result. See here you have a term minus gamma by 2 beta e to the power minus beta t 1 plus t 2 and in the first you had v naught square e to the power minus beta t 1 plus t 2. So, the exponential term being common we can subtract that and this modulus term can be kept separately. So, that is what we have done. Now let us say let us say that our times are large both t 1 t 2 are large. So, of course, when it is large we must give a unit and that implies what we meant is both beta t 1 is and beta t 2 are greater than much greater than 1. Now we precisely know what we mean by saying large t 1 and t 2 are much larger than the relaxation time of the system that is 1 by beta or the relaxation relaxation rate into time is much larger than 1. However, it is possible that the difference beta of t 2 minus t 1 can still be of the order 1 it could become parable t 2 can be large supposing t 1 is 100 units or beta t 1 is let us say 10 beta t 2 is 11 then both are large, but their difference 11 minus 10 may not be large it is comparable to beta. So, under that condition we can note from this expression that the first term t 1 and t 2 beta t 1 beta t 2 are very large. So, this exponential decays and we are left with the second term which depends only on the difference of the time which need not be large. So, under this assumption so as t 1 tends to infinity t 2 tends to infinity, but t 2 minus t 1 finite we have the autocorrelation function v t 1 v t 2 bar will be gamma by 2 beta e to the power minus beta mod t 2 minus t 1. It is a very important result that the velocity autocorrelation function is finite it is not it is a exponentially decaying function. We can write some in a way sometimes this is written in a better way or in a more transparent way that if t 2 is t 1 plus tau and t 1 we call as t then often it is written as the autocorrelation between velocities at time t and t plus tau is a simply gamma by 2 beta e to the power minus beta mod tau. So, if tau is always positive then we do not need to give mod, but it because of the modness it basically implies the velocities you measure at time t plus tau and correlate with the velocity at time t decays e to the as e to the power minus beta into tau if tau is positive or negative that is the prior or later correlations necessarily have the same functional form both of them decay exponentially, but with the mod argument. So, we can represent such a function in the following fashion supposing tau is my time and this is the autocorrelation v t v t plus tau bar then it will have a form a cusp like behavior because both are exponentials. So, there will be a nondifferentiable property here. So, this is an exponential e to the power minus beta plus tau and this we can say e to the power beta plus tau if tau is 0 here. So, this is an exponentially decaying correlation. In stochastic mechanics literature correlation between quantities if they decay in any form other than through a delta function it is called as a colored noise color noise problem that is the velocity correlations have the form of a color noise as opposed to the acceleration correlation which we started which was basically white noise. To understand the meaning of this color perhaps it is instructive to understand the meaning of white. When we say white what we mean is the autocorrelation is basically delta correlated and if you look at the frequency spectra we contribute to the time domain and if you convert it into the frequency domain via Fourier transforms and we know that e to the power i k t of delta t is 1 if t goes from minus infinity to infinity. This is basically the Fourier spectrum with respect to k the conjugate variable and the conjugate in time is the frequency. Conjugate in space is the kind of a reciprocal length whereas, a conjugate in time. So, k now will be having the dimension of a frequency and this says that the correlation since it is a delta correlated it is made up of all frequencies with the equal intensity equal strength and that is what is supposed to be a white color. When you say white although strictly it is not equal intensity, but it contains all colors in a mix. So, carrying forward that terminology a situation when the Fourier spectrum has all frequency components it is called as the white noise. From that perspective the color noise emerges as that correlation in which it is not a delta correlated, but there is some persistence of correlation for certain period of time. Delta correlation means the force force imparted by the molecule at one point in time is not correlated to that at the next instance. But to velocity the particle if you measure velocity now there is a persistence of velocity because of viscous drag and other things in spite of the noise for a certain period of time. But after times of the order of much larger than 1 by beta that correlation will decay and particle will have another velocity. So, that is the implication of the velocity autocorrelation function. Interestingly the whatever we derived for tau equal to 0 case implies that v t 1 v t plus tau will be simply v square average and that comes back to gamma by 2 beta which we already proved. This is already proved by thermodynamics very important relationship. I now remember to tell you that in statistical physics literature this relationship relating the strength of random fluctuations to the decay constant is called as the first fluctuation dissipation theorem. First fluctuation dissipation theorem because it relates the fluctuating force to a dissipation velocity. So, there is a connection between fluctuations and dissipations via thermal equilibrium because we just said to v square equal to v square average equal to k t by m. We now delve a little more into it obtain more instructive and deeper results especially the so called Stokes-Einstein relationship between the diffusion coefficient and the friction coefficient by continuing with these arguments. Thank you.