 While setting up the Langevin equation for a Brownian particle, we introduced a term called the random acceleration or random force. To understand the origin of this random force, in addition to the existing viscous force from the same fluid, we pursued a gas kinetic approach. And in that we showed that the momentum transfer that takes place in an individual collision, momentum transfer to the particle from the gas molecule was derived to be say delta p i in the ith collision was shown to be twice mg into g i minus v. This was based on a simple one-dimensional model which is basically a model meant for educational purposes. It differs from the exact solution only by a small factor that we saw yesterday. But for the present purpose, this will suffice for us to let us know how exactly the both the random as well as the systematic force originate from random collisions only. So, here g i is the random velocity of the gas molecule. So, this is the random part of a collision event because my g i is going to fluctuate. And this was the systematic part of the momentum transfer which comes from the viscous or equivalent which is represented as the viscous term from the Navier-Stokes perspective. And this is change in particle momentum per collision particle momentum per collision say ith collision. From this when we took the ensemble average ensemble average with respect to all possible velocities of the molecules whose average is 0 because the gas is stationary we find that the systematic part gets reproduced and we decided to call it as equivalent of a friction term. So, we introduced a gas kinetic friction coefficient F 1. Now, let us look at the random part. So, we therefore, separately work with the random part now because that is the most critical thing to understand how that it can be defined. Now, supposing there is a situation I plot this collision events taking place let us say between two time intervals 0 and t second 0 at arbitrarily sometime t. So, it is a stationary gas temperature is maintained molecules are continuing to strike the particles and at starting from time 0 one would expect if the y axis is let us say the force imparted on the particle and x axis time then you would see that there will be strikes like this along one direction and what these peaks basically represent are forces with the certain area the area under the curve delta p i this area we can say delta p i if let us say a molecule as a struck the particle at time t i. And the actual rise and fall that we see is a is an impulsors delivered to the particle by virtue of collisions and we therefore, use the concept of the so called force as a impulse term and we know that impulses are represented by delta functions. The reason why we are able to do it is because the number of collisions that the particle suffers and the time for each collision is so short and the number of collisions are so large that on an average in the time scale that we are talking about large number of collisions have occurred and each collision occupies very little time essentially one can treat it as a delta function. So, with this picture we can say that the equivalent force represented by a collision the equivalent of force. So, we call it as delta f say i imparted to the particle from collision can be written as delta p i then the direct delta function t minus t i. So, this delta f is t this is instantaneous force due to one collision and this is of course, direct delta function. If we integrate this equation with respect to time the integral of the force is a momentum because force is rate of change of momentum and we get back the delta p i as the total momentum transferred to the particle to which we have an expression already. So, with this now we ask a question what is the total random force that the particle has suffered in a time t how to represent it. We can add up delta f i in the entire time interval t. Now we must say something about the time that we have chosen that time t is a macroscopic time as opposed to the collision time which is a microscopic time and we have taken it as almost a delta function equivalent. So, this t is so large although it could still be very small like for example, t could be in microseconds whereas, the collision times could be in picoseconds. So, t is then large compared to the collision time very large number of collisions would have taken place and in this time n is the number of number of molecules which have collided in time t. So, this number is the mean collision rate nu c which we have defined earlier number of molecules colliding the particle in time per unit time into t. So, we say almost on an average there will be n collision in time t. So, whenever I say time t n is proportional to it. Now we can say the net result of this n collisions is to impart a force term which is a sum of all the individual forces. So, I write that the f random will denote t can be written as sum i equal to 1 to n delta f i t and more explicitly it can be written as i equal to 1 to n delta pi and direct delta t minus ti summed over all the pulses that means, in our time t 0 to t there were n pulses randomly and each of them let us say the ith pulse here has transferred some momentum delta pi and we are summing over all these corresponding forces represented by individual delta functions to represent the total force. So, that is what we have done it is a. So, we introduce the delta function to be again to represent a very sharp instantaneous collision which looks quite physically reasonable. From that we sum up these delta functions to represent the time dependence of the force experienced by the particle as a result of some n collisions. Later we will take the asymptotic limit, but at the moment some n collisions have taken place in time t and this is the force expression that we write. So, by more specific form we can write the f random f r we will represent experienced by the particle in time t is equal to twice mg summed over i equal to 1 to n the molecular velocity g i delta t minus ti. This is the random component we have calculated the systematic component anyway that was multiplying to v multiplying new to v new c to v we get the systematic force which anyway we have set aside because it is a linear combination of the two that we have proved from a single collision hypothesis itself. Now, this expression given the property of g i given g i average equal to 0 this tells us that f r t ensemble average and we have defined ensemble average last time it is average over replicas of this collision sequences. So, there are two types of ensemble averages one with respect to the velocity and the second one with respect to t i the collision times they also could be random. So, a particle could strike randomly with some velocity randomly at some time i ti. Hence whenever we speak of an ensemble average we have to carry out averages with respect to both. So, in this sense when we do and these two are independent the velocity that a particle picks up is entirely uncorrelated with respect to the time when that molecule decides to strike. So, with that this becomes 2 mg and this ensemble average i equal to 1 to n will involve an average over g i and an average over delta of t minus t i. We will describe how to take the second average in the following page, but right now we do not need it because we know that g i its average itself is 0 and hence the overall average of the mean force of the random force equal to 0. It is a very important result which of course, we anticipated that is the way we define this random component that it does not contribute on the average. So, then what is its contribution that is where we go for the correlation function between two time correlation functions and average of the two time correlation function defined in the ensemble sense as we did here. So, this is the ensemble as well as time averaged. So, we now take up the next question of correlation two time correlation of random forces that is F random at some time t how does it correlate with the F random at some other time t prime and this averaging to be consistent with what we said involves double average ensemble averaging one over the random velocities of molecules and another over the time at which a particular event has occurred. So, as I mentioned we know how to take the average over the molecules velocities because it comes from the equivalent of a Maxwell Boltzmann distribution which is simply a Gaussian distribution. So, we average over the velocities over a Gaussian distribution. So, that part is fine how do we take the average over the impinging times for that we note that if you look at our time sequence a particular molecules impinged at time ti and it could have impinged anywhere between 0 to t all that we know that n particles are hit which means a given particle should have hit between 0 to t. We can therefore, treat it as a uniform distribution it could have hit any time and hence the probability this implies probability that the particle that the molecule has a hit the particle between times ti and ti plus dti that probability given by pti dti would be equal to dti by t. This is true for every one of those molecules which is striking because each of these molecules would have struck anytime between 0 to t. Hence, 1 by t is the normalizing factor this is valid between 0 to any time t prime lying between 0 to t or ti any ti lying between 0 to t. So, we use this uniform distribution this is basically a uniform distribution of striking times. So, with this probability density pti we can now estimate the ensemble averages with respect to time sequence as well. So, with this our expression we can say Fr we now construct the two time correlation function Fr t first we write down Fr t Fr t prime. So, t prime is another time and let us say that t prime is greater than t. So, we just extend this to another time t prime. So, we are seeking correlations between this. So, there would have been some collisions again consistent with the fact that the average density nu c has to be maintained. So, Fr t Fr t prime can be written as given the previous expression for here given Fr t is a 2 mg g i delta t minus ti. So, when I am doing Fr t prime it is again the same sum, but just to distinguish i is a dummy index I can write it as j equal to 1 to n prime because it is corresponding to another time. So, I would write it as a double sum i equal to 1 to n and another sum j equal to 1 to say n prime of 4 mg square that 2 mg occurs twice then g i then there will be g j delta t minus ti delta of t minus t j where n equal to nu t and n prime equal to nu c t prime. So, it maintains this average consistency with the average collision rate. Let us say that t prime is greater than t just it is a it is not necessary t can be greater than t prime or t prime can be greater than t, but for convenience right now let us say that t prime is greater than t this implies that let us say n prime is more than n. So, writing more clearly the then we take the expectation. So, we can now take the expectation or or ensemble average f r t f r t prime ensemble averaged in the double sense again to specify this is with respect to velocity the inner one this is velocity of gas molecule and this is time ti. So, this can be written as 4 mg square which we can take out and it will be g i g j i equal to 1 to n j equal to 1 to n prime g i g j average and another average coming from delta t minus t i delta t prime minus t j. At this point we note another property of g i that is the correlation between two different molecules impinging I should say this should be g j the correlation between two different molecules impinging is given by some g square average let us consistently use the notation g square average delta i comma j this is chronicle delta that is the velocities have autocorrelated the correlate with themselves g i with g i alone, but there is no correlation between two different impingements one molecule strikes with some velocity which is the no relation with the velocity which is not correlated with the velocity of another molecule. So, this we had assumed right in the beginning which is in the nature of random collisions of molecules which means we can now split this term f r t f r t prime as mg square all terms were i and j i equal to j terms then there will be i equal to j there will be g i square average and here delta t minus t i delta t prime minus t i because we are taking i equal to j. So, when i equal to j one terms all disappear I mean we do not we put i equal to j and take the sum of all the terms occurring with that condition then the remaining terms however are those plus those which are in those for which i is not j. So, i 1 to n j 1 to n prime subject to the constraint that j not equal to i because that already we have kept then we will have the whole thing g i g j delta t minus t i delta t prime minus t j ensemble average. Clearly we see that the second construction here is 0 g i g j should be g j is 0. So, we are now left with only the first term this comes from the property that we saw here because of this the second term will disappear. So, that leaves us with the term f r at time t f r at time t prime is going to be 4 mg square g square bar sigma i equal to 1 to n. In fact, whichever is the smaller number here we said t prime is larger. So, n will be smaller than n prime and that is what we are retaining time average of delta t minus t i delta t prime minus t i and this averaging has to be done with respect to t i and we note that given the definition of p t i this quantity the so called averaging process will be something like this t minus t i delta t prime minus t i this process basically is an integral in the time interval up to t with that for each of the i's p t i delta t minus t i t prime minus t i d t i by t this is the way we. So, we will put that t later which turns out to be this is of course, right now p t i turns out to be 1 by t stays out and integral will be just delta t minus t i delta t prime minus t i d t i and since t i is all smaller than t, t is the interval that contains all the collisions for every one of those i's. So, this integral first delta function can be evaluated let us say that happens where the t i equal to t I can just evaluate. So, this leaves us one delta function of the two delta functions one of them gets killed by this integration process. So, we are left with delta t minus t prime or it can be t prime minus t that does not matter delta function is an even function. So, whichever way you write it is to be understood in the mod mod sense. So, that means, we have now the next expression for the two point correlation function of the random forces and assembled averaged with respect to molecular velocities and the strike times as 4 mg square g square average and this sum has delta t minus t prime stands out now it no longer contains t 1 by t and this is summing of n i equal to 1 to n of unity and this is nothing, but n itself. So, we can show that it will become finally, lead to the result since a since sigma i equal to 1 to n of unity is n we have the ensemble averaged force f r t f r t prime will be 4 mg square g square average n by t delta t minus t prime we will write it as delta t minus t prime. Now, we know that n by t is nothing, but the collision rate nu c. So, this leads to 4 mg square g square average into nu c into delta t minus t prime. You can note here that from an expression that involved a series of delta functions for the individual force it is a two point correlation two time correlation of the random force finally, got reduced to a single delta function and that is the very important result and we developed a kind of a heuristic model for understanding this. I would not say this is a very rigorous model in the sense of a rigor of the pre factors, but it captures the concept that individual molecular collisions ultimately can lead to both the systematic components and random components and the random component having a two time correlation function which is proportional to the delta function involving the difference between these two times. To complete this argument we now construct correlation for the acceleration instead of the force which is just dividing the force by mass of the particle. So, a t a t prime we can for ease we remove the double bracket that was introduced specifically for concept, but on the whole we can say it is nothing, but ensemble averaged which includes averaging over the microscopic times as well as averaging over all randomness basically both velocity randomness as well as the striking time randomness that now becomes 4 mg square g square average divided by m particle square into nu c delta t minus t prime. There is a convention to denote this entire pre factor by a parameter called gamma and since we obtained it from a gas kinetic elementary perspective from a from a illustrative perspective and not a rigorous perspective I will distinguish it as gamma 1 into delta t minus t prime a t a t prime. So, ensemble average of the two time correlation function of the acceleration of the particle is given by a pre factor gamma and a delta function where the pre factor will retain the expression for the pre factor as gamma equal to or gamma 1 equal to 4 mg square g square bar collision rate divided by m p square. This is the dimension we will see later as meter square per second cube. This dimension follows very easily from here also after all the left hand side this left hand side here are product of two accelerations. Each acceleration is meter per second square meter per second square. So, this should be meter per second to the power 4 delta has already 1 by second in it always because we know that delta has a dimension of whatever is reciprocal of what is in the argument. So, this has 1 by second. So, this has to be therefore, meter square per second cube and we can also convince that from the pre factor that we have defined here. Let us express it in a slightly more standard form. If you remember we derived the expression for the gas kinetic friction coefficient as 2 mg mu c. This was the expression for gas kinetic friction coefficient. Friction coefficient is a pre factor between the drag force and the velocity that is the way to understand it. So, then we can express and the gamma factor the pre factor for the random velocity correlation the gamma factor as in terms of f 1 I will skip a few steps. We can further use the result that the average velocity square the so called mean square velocity of a molecule along any direction has to be kT by m and this G square is of the gas. So, it should be kT by mg this is mean squared velocity of gas molecule along say x direction it is true for all directions. We can assume this to be a part of so called equipartition law in a gas the kinetic energy per degree of freedom should be equal to half kT. So, from that you can easily obtain this. So, combining these two results we can write gamma 1 after substituting and cancelling several terms as 2 kT beta 1 by m p where beta 1 the relaxation rate is always the friction coefficient divided by the mass of the particle. So, that is another definition. So, this is the relaxation rate of the particle due to collisions on a macroscopic scale. This relaxation is the macroscopic time scale relaxation multiple collisions are taking. So, how does each collision the impact of each collision how long does it last. So, beta decides that rate. So, this is an expression that we have derived which from a microscopic argument for the coefficient of velocity of acceleration correlation in terms of the friction coefficient or the relaxation rate. Thank you.