 Hello students, modeling of stochastic phenomena can take place at many levels. This is true of Brownian motion as well. In the models that we developed in Brownian motion, particles randomly jumped either to the left or to the right along a one dimensional path. This was basically called the random walk model of Brownian motion. When doing this modeling, we did not ask the question how exactly particles randomly change their direction of jump. This calls for exploring some underlying mechanisms responsible for the change or random change in the direction of jumps. Such a mechanistic theory is provided by a higher level theory called the theory based on Langevin dynamics. In a couple of lectures from now, we will explore this approach from the perspective of providing deeper understanding of the phenomenon of Brownian motion. Let us consider the case of a Brownian particle which is immersed in a fluid. We have a fluid, let us say it is a stationary fluid and a Brownian particle is immersed. Let us say that the so called Brownian particle is actually a colloidal particle. It has a certain velocity v along the positive side of x direction. Now, if you sit on the particle and the construct the coordinate system on the particle, then it would look like the fluid is moving in the direction of minus v. So, from that perspective the fluid motion or fluid streamlines will look like this is approximately shown here. So, this is our particle colloidal particle. This problem of describing the velocity profile of the fluid which is flowing around a sphere is a very well discussed from the discussed in fluid mechanics. Fundamentally, a thin layer of the fluid now sticks on the surface of the particle because of the property of sticking. On the other hand, the fluid at large distances from the particle should have the velocity let us say u infinity which is equal to of course, minus of v in this case. So, it is required to find how the velocity now changes from 0 on the surface of the particle to u infinity as you go away from the particle. The this profile for the velocity is calculated by using what is called as the Navier-Stokes equation along with what is called as the continuity equation. Navier-Stokes equation is basically momentum balance equation for the fluid and continuity equation is basically the conservation of mass. Because of the existence of a velocity profile in the neighborhood of the particle and because of the property of viscosity certain tangential as well as normal stresses get developed onto the particle as a result of which it experiences a force upon it and this force is often called as the drag force. So, whenever there is a particle which moves with a certain velocity in a fluid it experiences what is known as a drag force. From the Navier-Stokes equation which we will not get into the details here one can estimate or one can derive an expression for this drag force. Generally, one solves the Navier-Stokes equation in what is called as the Creepy flow limit or it is equivalently viscous limit. So, in this limit the particle Reynolds number particle Reynolds number which is defined as the let us say the characteristic velocity in this case the particle velocity itself the size or diameter of the particle which is of the order of 2 into a if a is the radius of the particle divided by the kinematic viscosity of the fluid this should be less than 1. So, when that condition is met the inertial term of the Navier-Stokes equation can be neglected and in a steady state equation for velocity profile can be set up which can be solved by using continuity equation these are derived in standard fluid mechanics books. So, we will not get into the details here what we would like to see is whether this possibility exists that is the Reynolds number being much less than 1 for a typical tiny particle immersed in a fluid. Let us consider for example, a typical case of particle diameter of the order of 100 nanometers a colloidal particle which is of the order of 10 to the power minus 7 meters and let us say it is moving with some velocity of the order of 0.1 meter per second very slowly moving and kinematic viscosity for water like substance is of the order of 10 to the power minus 6 meter per square per second this is kinematic viscosity particle diameter. So, then we find that the Reynolds particle Reynolds number the order of 10 to the power minus 1 and 10 to the power minus 7 meters divided by 10 to the power minus 6 which is of the order of 10 to the power minus 2 which is much less than 1. Hence a very slowly moving colloidal particle definitely satisfies this condition which means one can use the solutions obtained from the assumption of a creeping flow that is viscous limit for calculating the drag force on this particle. We merely code the result do not go into the details of Navier-Stokes equation as I mentioned. So, this drag force F d denoted by F d is calculated as minus 6 pi eta a into v where eta viscosity of the fluid a radius of the particle and v velocity relative to the fluid particle velocity relative to fluid. Here we have taken the fluid to be at rest, but otherwise one can subtract that velocity also. What this physically what this expression means is that we can highlight this expression. It implies that when a fluid when a particle moves in a fluid with the certain velocity let us say in the positive direction then a force is imparted to it in the opposite direction. So, as to oppose this motion basically. So, this force will eventually try to stop the particle from moving. And in fact, this is called the Stokes formula of drag and this forms the basis of calculating the so called critical settling velocity or terminal velocity terminal velocity of a particle in a fluid even under gravity instead of accelerating the particle attains a steady velocity and that is because there is a drag force which is opposing the motion and the balance between the drag force and the gravitational force gives you the terminal settling velocity. So, these are all well known classical knowledge which one studies in the course of fluid mechanics. Now, let us denote this force fundamentally as a steady force. So, this is inevitably going to be there on a particle whenever it is immersed in a fluid and if it has a motion. However, we know that if you resolve the force experienced by the particle at very very tiny time scales the collisions from the molecules would be causing it certain fluctuating motion in the system. So, apart from a steady velocity it will be also having certain zigzag random component of velocity and this comes from the molecular collisions. In other words, if one were to therefore, characterize the force of a part force experienced by a particle F say as a vector at any time t very finely resolved let us say that would be the F drag that we discussed and it must have therefore, this forces coming by molecular collisions occurring very rapidly, but in a highly time resolved sense that collision forces will be there let us say is a function of time plus of course, there could be externally imposed forces imposed by say gravity it could be electric forces it could be shear forces it could be many other forces acting on the particle. So, this is this force is random and this is a steady this is external. So, because the drag force is steady and it comes from macroscopic fluid mechanics often the total force is resolved as systematic part here as well as of course, this is also systematic part and fluctuating part. Actual Brownian motion one observes because of the fluctuating part of the force experienced by the particle. What we do is we assume for the moment or for the sake of convenience the case of a free particle that is no imposed or impressed forces no imposed force let us say gravity electric force etcetera neglected which means F i equal to 0 and we therefore, write the total force experienced by the particle therefore, will be F drag we denote by D and the fluctuating force of the random force we simply call it as F r which is a function of time is a force experienced at any time. So, once we have formally written down the forces we can precisely calculate the particle velocity if we have a machine to detect it at that fine time scales that we are talking about via Newton's law of motion. So, we can use Newton's law to write down the equation of motion for the velocity that is m rate of change of momentum if m is the mass of the colloidal particle then m dv by dt should be equal to all the forces experienced by the particle external forces and this is F drag if V is a vectorially F drag plus F random which could be a function of t this is for a free particle here m mass of the particle V velocity we can divide throughout by m and write the above equation as dv by dt equal to F d by m plus F r by m. Now, F d is minus 6 pi a eta v this quantity is called the friction coefficient F. So, it is written as minus F into v where F equal to 6 pi a eta is called as the friction coefficient and we denote formally F r the random force by m equal to an some random acceleration term which is a function of time here this is random acceleration induced by collisions induced by collisions. So, with that we can write the law for the rate of change of velocity as dv by dt equal to minus F by m v plus a d a further simplification is written as minus beta v plus a t and we remind ourselves that here where beta equal to friction coefficient divided by mass explicitly it is in the present context 6 pi a eta by mass which has the dimension of rate say second per minus. So, it is called the relaxation rate of the particle and a t once again very important quantity we have introduced is called the random acceleration force by mass. While the first term in Newton's equation is has been used in all problems mechanical problems introduction of a time dependent random acceleration is the original contribution done by Paul Langevin to the Newtonian dynamics of a Brownian particle and that is why this equation stands out and more we go through the understanding of this random acceleration more we realize the uniqueness represented by this equation. So, this is actually what we can call as Langevin equation which simply says the rate of change of velocity equal to that coming from drag force plus that coming from collisions or random forces coming from the molecular collisions. We have not really developed any solution so far, but we have formulated an equation perhaps which has the capability to predict how the velocity primarily is going to change and there is a molecular basis for this change. And once we have understood the velocity changes then we can understand the position changes and that therefore, should form a kind of a higher level theory of Brownian motion. In fact, the former process the random walk process that we discussed in the previous lectures was called Wiener process because it is a quenched model the particle instantly takes a jump. The model that we are building based on Langevin's equation is called as the Uhlenbeck process as opposed to as different from the Wiener process. Langevin went one step further he postulated that the random acceleration about which we know very little because molecular dynamics is a very complex subject. However, some broad general properties can be extracted. One is that supposing we define an ensemble average of a properties of a a t. He assumed that the ensemble average of a that is a t ensemble average that should be 0 that is because on an average the particle should not experience any force other than that is given by the drag force. If that is not true then drag force itself will be will be further corrected. Since microscopically we get drag force from fluid mechanics the microscopic fluctuations should necessarily have the property that the other ensemble average should be 0. Another important property that he assigned is that the random accelerations should be independent of each other. However, now we are talking of a continuous time process and when we say they are independent of each other what it means that the force experienced at some time t random force will not be correlated with the force that it will experience at some other time t prime. However, of course, when t equal to t prime the correlation should exist and for discrete jump variables we always did it via chronicle delta functions right in the earlier examples we have done that. So, for continuous variable this property that the accelerations are delta correlated that is the ensemble of the product a t and at some other time t prime the ensemble average that should be equal to some constant gamma to delta t minus t prime this is a assumption which requires certain explanation and understanding. So, first of course, when I say ensemble averaging we should know what we mean. So, all that it says is supposing I start with a Brownian particle colloidal particle I follow its timeline like this. So, it will undergo velocity fluctuations like this. So, a average the forces it has received with respect to a replica of such particles this is the fluid is a stationary and its temperature is constant. So, it is imparting on an average same types of random collisions and therefore, the ensemble average therefore, represents all possible varieties of collisions that you it would have and when we do this is an ensemble. So, when average over ensemble. So, these are all average over ensemble. So, when such averages are taken they should have a delta function dependence for the auto correlation function. So, this is also called auto correlation function for the acceleration. The property that it is delta correlated basically comes from independence of molecular collisions and in the next lecture we try to understand how exactly such situations arise. How can we understand the origin of these random fluctuations by building a simple model? Thank you.