 So, far we have been speaking about fluctuations of random walker and of Brownian particle from purely a theoretical perspective. I thought we should get a feeling of how exactly we do a simulation, how exactly we see the random components emerging out of underlying fluctuations. So, we do a very simple numerical simulation and I demonstrate it to you today. So, before that let us understand the elementary fluctuations in velocity coming from the fluctuations in the underlying acceleration term A t. First we should realize that A t is occurring in extremely small time scales. The particle being massive is undergoing changes in much larger time scales. So, there is basically two time scales involved. The microscopic time scale has to be averaged out for any perceptible change to occur in the macroscopic time scale. And that averaging is sometimes called as some coarse graining. So, we have to talk of changes in velocities occurring in macroscopic time scales by which time large number of collisions, events or microscopic things have occurred. So, in other words if I have to show a time diagram and between some time t and another elementary time delta t, this is a macroscopic time of width delta t. The number of elementary fluctuations the particle is supposed to be very large. So, these are microscopic collisions let us say. So, this is macroscopic time because of these differences although the Langevin equation that we wrote down here in this video, you can see this is the Langevin equation we wrote down. Although this is theoretically valid at every microscopic instant of time, you cannot solve it by numerical integration because you do not have a detailed character of A term at those levels of refinement. All that we have are some kind of an integrated behavior of A. So, it is therefore necessary for numerical treatment and for simulation first to carry out a certain coarse graining of this equation and that is what we demonstrate. So, when we say coarse graining, we assume that the time scale we are talking is actually much larger than those microscopic time scales. So, in that sense I could write my dv by dt as delta v by delta t in those time scales as minus beta v and if there is some external force f by m plus 8. So, now if I do an ensemble averaging I can say that change in velocity on an average that occurs is going to be minus beta v that instantaneous velocity in that coarse grained time scale plus f by m all are smooth quantities into delta t plus of course, 0 since because since A we make use of it. So, this is the average change in velocity because of the systematic effects basically it includes the instantaneous velocity plus that external field if it exists because A effect of course cancels in this. However, around this mean velocity there is going to be a fluctuation. We can even denote also sometimes we will just call it as the delta v bar also same thing same notation will be used. So, let us go to one level higher to be able to understand to be able to develop a formula for how does the fluctuation around delta v exist what form it exists. So, for that we go back to our old derivation in the previous lecture where we showed that v square bar it of course decays very rapidly at a time scales much larger than 1 by beta. However, it does not decay to 0 because it eventually has to attain thermal equilibrium, but it relaxes via a function 1 minus e to the 4 minus 2 beta t. Now, we see that this was original v bar if there is no now let us consider a special case f equal to 0 no external force. So, then this term is basically v bar t whole square. Hence, we have v square bar minus v bar square is essentially gamma by 2 beta into 1 minus e to the power minus 2 beta t. If we denote the sigma square in a small time delta v. So, sigma square of delta v that will be in fact v square bar minus v bar square taken in the limit of delta t going to 0 the small change that we will see in the squared velocity and that is nothing, but gamma by 2 beta into 1 minus e to the power minus 2 beta t in the limit t becoming delta t and delta t going to 0. So, very small it is not actually going to 0, but small delta t. So, that is going to be simply gamma into delta t. So, we get 2 important results by this that is at any instant of time the mean velocity can be minus beta v bar delta t it is proportional to that small time at for a change in time delta t the change in average velocity is expected to be this. If you have external force then it f by m will be there this is when f equal to 0 case and correspondingly the underlying variance around this mean in velocity fluctuation is gamma delta t. Thereby we get at least 2 important characters on the velocity fluctuation somewhere when we started we made a hypothesis about the character of A. We said the A is a Gaussian white noise. We later explained what the white noise part is because the white noise refers to the frequency spectra contained in those instantaneous momentum transfers all frequencies with equal strength. Gaussian because those integrated momentum transfers over the impulses integrated over the basically the momentum transfers taking place per collision are Gaussian distributed. They were coming from in our model we built it was coming by the velocity of host gas which is Maxwellian distributed or Gaussian distributed. Hence, assumption that eventually the underlying stationary cause of fluctuations is Gaussian distributed and the particle also attains equilibrium with it implies that the strength of that underlying fluctuations should be also gamma delta t. So, gamma delta t therefore, refers to the variance of the underlying momentum or well e velocity because we are talking of now velocity fluctuations. From the mean and variance that we obtained for delta v bar we can construct a probability distribution for the velocity delta v in a time delta t as for example, will be 1 by square root of 2 pi gamma delta t e to the power minus delta v minus minus of the delta v bar whole square divided by 2 gamma delta t. And as the time approaches infinity delta v bar tends to 0 if there is no external force if f is 0 this is as t tends to infinity then it attains completely in equilibrium with the host gas. So, you will have 2 pi gamma delta t e to the power minus delta v square by 2 gamma delta t. So, armed with this understanding that the underlying gas imparted velocity fluctuation is ultimately reflected in the velocity distribution of the Brownian particle we now proceed to how to employ this in actual solve actually solving the Langevin equation. So, for that we note that if we do a small integration of the Langevin equation for a small time v t plus delta t which will be let us say v t you can always show that that if you integrate the Langevin equation between t and t plus delta t this whole thing will be t to t plus delta t a t prime d t prime for a short time. Since we could not have been able to solve the differential equation written in its original form in that slightly integrated form over delta t is much larger than microscopic timescales, microscopic timescales, but it is much smaller than macroscopic times. So, this is the important assumption time of fluctuations to be very specific time of Brownian motion. So, the time delta t is both large and small it is large when we are talking of a t a t prime here. So, we have to carry out this integral even though delta t is small we cannot simply say it is a t prime delta t it has no meaning. So, when we integrate we note that integration of an acceleration over a finite time is actually like an impulse integrated and that is a momentum transferred. So, we call any integration from over a small time t to t plus delta t a t prime d t prime equal to some small momentum transferred we call it as delta b which is a function of delta t no doubt, but it is not a function of t it is not a function of t because the host gas is completely stationary and its state does not change with time and accordingly the strength of fluctuation in the team parts does not depend on time. Once we have that the Langevin equation assumes a form which is you can be used for computing its instantaneous velocity at different times via this equation delta delta t. So, this equation is velocity at the next instant velocity previous instant the acceleration or velocity from fluctuations this is a course grain velocity we should say that course grain integrated to solve this equation we must note that this is a statistical or stochastic equation still because delta b is still statistical quantity. However, since we know that the Brownian particle has its variance governed by gamma delta t in equilibrium the host gas its velocity fluctuations it is imparting should correspond to that. So, we carry out an integration of this we now solve the modified Langevin equation of the form v at the next instant t plus delta t equal to v t e to the power minus beta delta t plus delta b which of course is a function of delta t under the rule that delta b average is 0 and delta b square average is given by gamma delta t. To do that I will now use software which many of you should having access will use the Mathematica software which is very easy I am sure any other software should be as good where you have random number generation. So, I will just illustrate to you how this problem is solved in Mathematica. So, very simple few lines equation and then show the kind of result we get and how do we do that. So, we start with a very simple case that is we say that the particle starts with let us say t equal to 0 then let us say beta that is what we called as beta equal to we assign some value 1 just one example. In Mathematica you have to separate them by colons then we define the tau relax which is 1 by beta. Beta is you are free to set any value, but we consider value 1 units arbitrary units all times are now measured in terms of 1 by beta basically. Then we give a time delta t we call it as delta t that is going to be tau relax. Let us say we divide this time scale because Brownian particles characteristic time scale is tau relax you divide by 100 you can choose any width you want, but to get fine results finer delta t the better. So, somewhere we decided 100 we need to assign some value to this gamma that is the strength of fluctuations and we make use of the theorem that it is related to the rms velocity. So, let us measure all velocities call it as v rms as our characteristic velocity. So, we take as 1 then the strength of fluctuations gamma I call it as gm here and that is we use the property that it is twice beta v rms square can write this command. So, therefore, the sigma we call it as a sinc in velocity sinc v that is going to be square root of gamma square root of gamma into delta t. So, since we know it is gamma into delta t the variance is gamma into delta t. Therefore, you can put a star here or you need not put a star in Mathematica, but for your convenience I put stars for multiplications although you do not need it the gap will do. So, with this we just develop a do loop and then see how the velocity evolves, but then one critical thing is we must know how to sample from the distribution of our interest. So, let us start with the Brownian particle with its initial velocity v 0 equal to let us say 10 units it is quite higher compared to the rms velocity that is idea. Then we set at my time as let us say some. So, we do a do loop simulations my time has to be measured in terms of some in terms of I delta t I delta. So, we need a for looping here at this point that is we say for v 0 equal to 10 t equal to I delta t before that of course, we should define before I starting with 0 some I equal to 0 colon and instantaneous velocity let us say v t is to start with it is v 0 and we also have to define the macroscopic time is some I times its elementary part I delta t and let us say my I I take some 400 that number can be 100 or anything, but when I say when I is 400 it is a time is 400 times delta t which is 4 times tau relax. So, we expect that it will come to tau relax and then I do I plus plus which means continue until I reaches 400. Then I ask him to print time with some gap and velocity of course, it is still doing the looping first time it will just print to the original values, but after the print I give it del b equal to a random number generation command random in a mathematical within bracket if you define the distribution and we define normal distribution with the character mean 0 and standard deviation defined by sig v we defined above random is close. So, now the logic of this statement is it samples from a normal distribution with the mean 0 and standard deviation sig v which is coming from the gamma delta t types randomly it gives a number which is equivalent to delta b which is the imparted momentum to the particle. Hence in the next step my velocity of the particle should be the previous velocity e to the power minus beta delta t beta delta t t back at close e to the power I should you can write like this, but I prefer x here maybe I will rewrite the whole thing. So, it is v t star x minus beta star del t plus del b because it just adds the fluctuation in front and then I close my equation. So, you need do not need this. So, I close I start in some closing brackets I do. So, the as you can see it first assigns all these values then gets into a do loop here. So, for starts with the i equal to 0 assigns initial values first prints the very first values that we started with then randomly evaluates del b from a random number generator from a normal distribution. In fact, one can first do some trial with this random command itself and after collecting let us say hundreds of data just group them together and plot you should get the normal distribution that you wanted and one has to confirm that it is generating a normal distribution with these characteristics. After having done that you simply add it. So, every time the counter moves it gives you a new velocity and that gets printed here when it comes next. So, you get then a velocity at various times including fluctuation as well as systematic motion and that is what we have we show here the essential of that result. So, it is a illustration of a velocity fluctuation of Brownian particle obtained through numerical simulation of the Langevin equation modified Langevin equation. So, that we do by this is the same command that I wrote for you just typed out and shown you would notice an e to the power beta del t by 2 term, but it is not really necessary this was because it is almost 0 because my del t is so small that you can take it as e to the power 0 as 1, but it is a nicety that is introduced. So, the rest of the things is same thing as what I wrote and this is the result one gets. So, you can see that there is a 5 such simulations were done each of them can be tabulated now as a table. Mathematica allows you to plot as well as tabulate, but in this case I tabulated and then took it to some other software origin and then plotted you can take it to excel and plot also where you can manipulate and obtain many other features, but these are the 5 individual simulations you can see that the velocity decays, but with lots of fluctuations coming from del B and ultimately it reach as it approaches 0 it however does not subside because now the natural fluctuations take over. So, you can see the bold curve is an average that is taken which tends to 0, but it is only 5 simulations. So, it is still not exactly going to 0, but it should eventually otherwise average should go to 0 it is almost approaching, but because fluctuations are large and time probably is not enough it would take longer time. So, this is just an illustration and with time permits we may do more such simulations to understand the character or how actually to implement this. However, it is important that one familiarizes we oneself with random number generation either through a software like Mathematica or Matlab or whatever else that one has and one can do it for random walks also. With this we move over to the next part of our remaining lecture on Brownian motion. Thank you.