 So, when we derive this diffusion equation, if you remember if you recall what we said was that I draw my step lengths. So, I wrote down an evolution for the position of the random walker at time t plus delta t plus some step l of t, but the step sizes were drawn from a distribution which is 0 mean and some constant variance right. The 0th moment was 1, the first moment was 0 and the second moment was a square. Now, you could have a particle undergoing diffusion even when this mean is nonzero ok. And if this mean is nonzero what you would have is a net drift term. So, if the mean is nonzero you would have a net drift term, but in addition to sort of doing random things overall if you look for large enough the particle is moving net in some direction it has some drift velocity. So, it has some drift velocity. So, it could happen if these individual steps of the random walk have a nonzero mean or for example, if you apply some sort of an external force. So, for if you have charged ions or charged proteins or whatever and you apply an electric field then that electric field will tend to push the ion in certain direction depending on where how you apply the electric field. So, if you have the net force then again you will have some sort of a net drift. And I can write this drift velocity let us call this V d this drift velocity is in general proportional to the force that you apply. And the constant of proportionality you can write either in terms of a mobility or in terms of a friction coefficient. So, mu is called the mobility gamma is called the friction coefficient. So, you apply an external force that causes the particle to move in some direction with certain velocity which is called the drift velocity and the drift velocity is proportional to the force. Is this what you generally encounter in a general setup is the velocity proportional to the force no right. So, this is a slightly special case and we will do deal with that over the next lectures when we do hydrodynamics. So, this is a special regime of low Reynolds numbers low Reynolds numbers which may or may not make sense to you, but which we will define. So, this happens when you can neglect inertial forces. So, when you can neglect inertial forces. So, the acceleration term is not important in Newton's laws. Generally, that is the regime in which biological systems operate. We will calculate how to quantify the biological medium in which this objects these proteins these organisms are undergoing random walks. And that characterization happens to this quantity called the Reynolds number. And we will see in what sort of regimes is this is this approximation that the velocity is proportional to the force valid. But for the time being let us take this as a given that the drift velocity is proportional to the force the larger the force you apply the larger the velocity. So, if this is my velocity v d then in a time t it has moved in time t in time t it has moved an amount which is let us say delta t in time delta t it has moved an amount v d delta t right. So, given this I will now try to rederive the diffusion equation. So, this is my setup I have a net drift velocity due to some force that causes the particles to take in time delta t to be shifted by an amount v d times delta t. And I will now try to rederive the diffusion equation. So, I will start off with this x of t plus delta t is x of t plus l of t. And the else are still going to be drawn from a distribution chi, except now the moments will change somewhat. What is the 0th moment going to be? Will this change? No right, because that is just normalization that is always true. Will the first moment change? Yes, what will it be? Mu, no y mu yes you are saying v into t that is the distance it has moved on an average in a single step. And the second moment is still assumed to be a square. So, it is still constant. So, that is the difference from this unbiased random work. So, this is what is called the bias random work. So, now I do the same thing as before. So, I still write my rho of x what was I writing x t plus delta t right. And I will expand this and then I will substitute for these moments. So, I will just write it rho of x comma t integral chi z d z. So, minus del rho del x integral z chi z d z right and plus half del 2 rho del x 2 integral z square chi z. So, this is exactly what we did last class for the unbiased diffusion, except in the last class we are thrown away this term because the first moment was 0. We took unbiased random works where the mean was the first moment was 0. In this case I have some finite first moment. So, I need to take into account. And this of course, so this is 1 this is 1 this is v d times delta let us just call it v whatever delta t and this is a square. So, again if I take this term to this side rho of x t plus delta t minus rho x t I divide by delta t what I get is del rho del t what I get is del rho del t. Then what I have over here is equal to minus v del rho del x plus d del 2 rho del x 2. So, when you have a nonzero mean for these step lengths in addition to the standard diffusion term d del 2 rho del x 2 here this additional will drift term which is minus v times del rho del x ok. So, this is called the drift diffusion equation. So, this is called the drift diffusion equation or that Vexian diffusion equation it has multiple means. So, just for sake of interest you can derive this in multiple ways the drift diffusion equation or the diffusion equation. I will just show you one more way just because I will use this formalism. So, let us say I take a 1 d random work right. So, let us say I take a 1 d random work. So, you have a line and you have a random worker which can only move left or right on this line. Let us say that the probability it takes a step and let us say I have some step length which I will call as delta x. So, it takes equal steps. So, steps of length delta x right each step has size delta x it can hop to the right with a probability p and it can hop to the left with the probability q that is only two possibilities that it can have which means that I my p plus q is equal to 1 at every time instant it can must either hop to the right or it must hop to the left. So, I can now write the time evolution of the probabilities that what is the probability for this random worker to be at position x at time t plus delta t. So, if it needs to be at position x at time t plus delta t at time t that is one step before it could have been in only one of two positions right x minus delta x or x plus delta x. So, it could have been in x minus delta x comma t or it could have been in x plus delta x comma t. If it was here if it was at x minus delta x then with what probability will it come to x p right because it has to take a hop to the right. So, p times this plus if it was at this position it would have to take a hop to the left which means q. So, it can be either here x minus delta x or x plus delta x if it was here it hops to the right probability if it was here it hops to the left with that probability ok. Now, let me expand these terms in Taylor series. So, p x comma t plus delta t is equal to p of x comma t that is in some writing delta let me just write it plus delta t del p delta t p of x minus delta x comma t is p of x comma t minus delta x del p del x plus half delta x square del 2 p delta x 2 right and similarly for p x plus delta x comma t that is p of x comma t plus delta x del p del x plus half delta x square del 2 p. Now, this x minus delta x term comes with a p small p. So, let me write the small p throughout small p small p small p this one comes with a q. So, q q q and now let me put this back into this equation this p x of t which is on the left hand side and this p plus q into p x of t will cancel of course, because p plus q is 1. So, there will be no p x of t. So, on the left hand side I will have a delta t del p del t is equal to this is a minus sign that is a plus sign. So, minus let me call this p minus q delta x del p del x and then a plus this is again a p plus q which is 1. So, plus a half delta x square del 2 p del x 2 I bring this delta t down. So, this is what I have. Now, this is exactly the same form as the drift diffusion equation that I have once I identify the proper forms of the velocity and the diffusion coefficient. So, the velocity is obviously, from this term. So, the velocity is p minus q delta x by delta t the diffusion constant is again length square over time. So, delta x square by 2 delta t. So, the moment you have a random work where it is more likely to hop in one direction versus the other that translates automatically into a velocity a net velocity which is proportional to the difference between these two to the difference between these two probabilities ok. If p was equal to q you would have a unbiased random work the velocity would be 0 and I would come back to my standard diffusion equation in the absence of drift. This is just another alternate way of thinking about this from the language of random works once I have this. So, once I have So, once I do that of course, I come back to this drift diffusion equation. And since I have this let me just write down what is therefore, the flux or the current what is the conserved current in this case then what is j. Remember we talked about the continuity equation which says that in the absence of any sources or sinks del rho del t is equal to minus divergence of j right which means in a 1 d case it is minus del j del x in 1 d. So, given that what is the current in this case then this is v times rho right the current is v times rho minus d del rho del x such that you still satisfy the continuity equation which is good to write down the expression for the current as well good. So, I had the diffusion equation last class now we have derived the drift diffusion equation del rho del t is equal to minus v del rho del x plus d del 2 rho