 Let us continue with the differential equation formulation of the stochastic phenomena. An extended random walk model in the continuum space and continuous time domain after working with the Chapman Kolmogorov version of the Markov process by Taylor expanding the occupancy probability W x t, we arrived at a Taylor expanded version of the C k equation Chapman Kolmogorov equation which read like this tau dw by dt partial equal to minus d by dx xi bar x tau W x t plus half d 2 by dx square xi square bar W x t plus of the order of tau cube of the order of xi bar cube where is the xi's averages xi is the displacement there the averages of the transition probability defined as xi to the power n p say x prime or any point let say it could be x p x xi tau d xi over all values of xi xi to the power n principle is a function of x and tau and p was a transition probability whose meaning is that it is the probability that a system transits by a length xi at the point x during a time tau. So, this is the moment of the transition probability distribution and it is normalized automatically when you integrate with respect to xi to proceed further with this equation we have to ascribe some properties of the distribution function first we will discuss what is it that we expect then we will establish some kind of a underlying nature of the distribution which can result in that expectation. So, for that we start let us say that xi bar the first moment which is a function of x and tau has a property that it is some u x into tau. So, what does it mean it is the first moment of the transition probability function if the random walk was symmetric equal probability for over left and right there is no bias then obviously xi bar should have been 0 there is no net length the existence of a xi bar implies that there is some bias in the random walk. So, that bias can come in by some kind of a velocity which makes the system to be transiting with the little preference in one direction it is like a flow or a drift what it is called and if that velocity let us say is it exists it is a background force and it is u then in a small time tau tau is the time taken for a jump the this the net displacement that it would suffer is the measure of that bias. So, that is why phenomenon phenomenologically we write xi bar equal to some velocity into the time between two successive steps. This next you write the second property that we write is that the xi square bar the second moment of the jump length distribution function will be we will anticipating from a dimensional perspective it should have a dimension of let us say length of square by time if xi represents a length and then again that is also proportional to tau this is a very important assumption. We will of course justify this later, but right now let us make this assumption that there exists a constant of the system called d which could depend on the position of the random walker and the 2 is just a factor it is a way of defining. So, if you are working in real spatial dimensions it will have the dimension of meter square per unit time. If you are working in a configurational setting where the random walk could be taking place on some other variable let us say the size of a particle or it could be random walk in the dollar space or rupee space then it will have corresponding definition. But basically the key point to note here is the second moment also we assume to be proportional to time and we further say that the higher moments xi to the power n averages they will be of the order of tau square and higher for all n greater than or equal to 3. So, with this assumption we can substitute and rewrite equation and I mean the differential equation that we derived we can now substitute these xi bar and xi square bar in these equations and then we arrive at tau d w by dt w is a function of x n time plus it is an order tau square. So, we will write it as tau square we can write it as an order tau square will be minus now tau I will write out minus tau d by dx of it will be u x w xt and in the second one also it will be 2 tau by 2 d 2 by dx square into the x w xt and here it will be of the order now tau cube and higher to be safe we can even say it is tau square and higher because we said it is the order of tau square. Now divide by tau and take the limit divide by tau and take limit of tau tending to 0. So, we get d w by dt plus this will now go to 0 when we take the limit tau going to 0 because it is our order of tau square and here it will become d by dx of u x w xt and again tau will cancel here. So, we will be left with d 2 by dx square dx w xt and all higher order terms will become strictly 0 because they are of the order of tau square and higher. We can combine this and write it in a standard format we can write it in a standard form that is d w by dt will be equal to d by dx of minus u x plus d prime x of w plus dx d w by dx. This term we call it minus u effective x so that the equation in question will have the form minus of d by dx u effective x w plus d by dx of dx d w by dx. The last form has been written in a way that you can understand and connect to classical transport theory. If you now interpret w right now in w is a probability density, but if we interpret it as a concentration or a volume or a mass fraction then left hand side represents the rate of change of that mass fraction at any point and that is governed by two terms. The first term is because of the convection term convective transport and the second term is the diffusive transport and we have made a relevance of the possibility that the coefficients of convective transport as well as that of diffusive transport could depend on the space point that is we have we can identify the equation with the transport equation or mass transport equation or convective diffusion equation. You could note that the equation that we derived minus d by dx of u effective w plus d by dx of diffusion coefficient into d w by dx we arrived at it without taking recourse to an empirical law like Fick's law. We merely started with a transition probability function and assigned certain properties to its moments and those properties are very generic and we will now establish why that is possible, but in a way if we interpret this whole thing as d by dx of some flux term d by dx of minus u effective w plus dx d w by dx and if this is whole thing is a flux term j we notice that the second term is equivalent of a Fick's law the flux is proportional to the gradient of d by dx this will be actually minus flux term we should say let us say minus term and then the first term on the right hand side this term is actually a drift or a convective term. So, in in random walk or stochastic literature the drift term the convective term is often called as the drift term. So, this is equal to the drift term plus diffusion term and u will be the drift velocity. Let us now move forward to understand how one can have those special properties that we saw the ones we wrote here the property that the mean is proportional to tau mean square also proportional to tau. Let us establish from broad principles proof of it is not a proof, but some kind of an arrival or understanding, but let us say proof of proof of moments of transition probability. Supposing we postulate that the transition probability for example has a shape like this a Gaussian all that we are saying is the probability that the random walker transits by a length l length xi is distributed according to a Gaussian whose variance is proportional to time. So, this is the sigma and this is the xi bar on an average there is a preference it is not a symmetric it is a symmetric function around xi bar. So, that is if I can postulate a distribution p of let us say x xi tau the transition probability has 1 by sigma root 2 pi e to the power minus xi minus u tau whole square divided by 2 4 dx where of course, sigma is equal to square root of 2 dx tau. Supposing we postulate a Gaussian then we know that its moment first moment xi bar will be integral xi p d xi and that will be 1 by sigma root 2 pi integral xi e to the power minus xi minus u tau whole square by 4 we can call it as sigma square. So, we will call this whole thing as 2 sigma square since we have already defined sigma this will be 2 sigma square d xi. If we make a variable transformation equal to xi minus u tau by sigma then xi goes over to t d xi will be sigma d t and the integral for xi bar will be xi bar will be 1 by sigma root 2 pi integral minus infinity to infinity. Now xi will be xi is going to be sigma t plus u tau sigma t plus u tau sigma t plus u tau d xi will be just sigma d t e to the power minus t square by 2. Now we see that sigma cancel here. So, this will be 1 by root 2 pi and the first integral first integral will be integral minus infinity to infinity sigma t e to the power minus t square by 2 d t. This integral will be 0 because t is an odd function e to the power minus t square by 2 is an even function. So, the integral of an odd function on both sides is 0 and the second term therefore will be u tau into 1 by root 2 pi minus infinity to infinity e to the power minus t square by 2 d t. And this value we know it is a Gaussian integral is root 2 pi and this integral is 0. Hence xi bar is going to be u tau, but u is a function of x. So, that is a first result by postulating in the transition probability a certain shift from the mean in the from the mean 0 in the distribution function of a Gaussian form we can get this. Same exercise if we continue xi square bar is going to be 1 by sigma root 2 pi minus infinity to infinity xi square e to the power minus xi minus u x tau whole square divided by 2 sigma square where d xi and again transform t equal to xi minus u x tau divided by sigma implies xi will be the sigma t plus u x tau and d xi will be sigma d t. So, when we do the substitution we will have xi square bar is going to be 1 by sigma root 2 pi integral minus infinity to infinity and xi square. So, xi square is going to be the square of this quantity square of this quantity here. So, it will be sigma square t square plus u square tau square plus 2 sigma t u x tau. So, it will be 3 terms it will be sigma square t square plus u x square tau square plus 2 sigma sigma is also a function of x, but keep it 2 sigma u x t tau e to the power minus t square by 2 into this will be sigma d t. So, this sigma cancels and we are left with the and this term is now odd term because it is a integral of t. So, this term will vanish. So, it will be left with 1 by root 2 pi sigma square minus infinity to infinity t square e to the power minus t square by 2 d t plus 1 by root 2 pi u square u square and they do not depend on t u square x tau square into minus infinity to infinity e to the power minus t square by 2 d t. Since, t square average is variance and now the variance is unity. So, this will be simply sigma square plus u square x tau square and since our since sigma square is 2 d tau that is the way we have defined xi square average is going to be 2 d x tau plus u square x tau square. You would notice that our xi square average has a term other than 2 d x tau, but since it is of a higher order when we take finally, the limit in the limit tau tends to 0 u square x tau square by tau tends to 0, but 2 d x tau by tau tends to 2 d x. Hence, we tend to we neglect the second term. Hence, we write xi square average is 2 d x tau. So, this sort of justifies why we made the proposal to have these moments having dependence on time tau I mean time interval the elementary time not the macroscopic, but the elementary time in a linear fashion. This we demonstrated for a Gaussian distribution. Actually, we can demonstrate that it could be true for any form of the distribution providing it has the moments many moments and is symmetrical with respect to its argument. We will see it in just as a general understanding to assure ourselves that we need not necessarily start with the Gaussian. This differential equation can be arrived at by having a transition probability which need not necessarily be of the shape of a Gaussian. Thank you.