 So, in all the previous lectures while discussing discrete space random walk phenomena, most of our focus has been on nearest neighbor transitions only. By that what we meant to recapitulate was that if I have a site M and next site M plus 1, site behind it is M minus 1, then at every step the transition is limited to occurring to the nearest neighbors either M to M plus 1 with the probability P or M to M minus 1 with probability Q. We did not think I mean we did not consider any multiple or multiple site transitions, but it is possible to extend very easily to let us say instead of the nearest neighbor let us say 2 neighbor transitions. So, I have a random walker at site M and he can have an upward transition to M plus 1 and also to M plus 2 with the different probabilities. Similarly, he can transit downwards to M minus 1 or to M minus 2 with the different probabilities. All that we have to do is to postulate a probability P 1 for this and a probability P 2 for this. Similarly, a probability Q 1 for this and a probability Q 2 for this. Subject to the kind of constraints we need depending on whether it is a symmetric random walk or a biased random walk. To be simple if it is symmetric for symmetric random walk which means total upward and downward probability should be equal. Symmetric case let us say P 1 plus P 2 should be half should be equal to Q 1 plus Q 2 at each step transition should occur with total probabilities as equal. In which case one can then write that the random walk equation for example, it will be a bit longer. So, it will be something like W n plus M it will we can start from P for example, it will be with probability P 2 W M minus 2 it will come from W M minus 2 or with the probability P 1 of course, at the nth step at the nth step probability P 1 M minus 1 or the it can for example, it can start with the probability Q 2 W n M plus 2 it can have a reverse jump from M plus 2 and with the probability Q n Q 1 from M plus 1. So, this is a 2 site jump or transition random walk. One can now visualize that you can actually similarly construct random walk with for multiple site jumps with various tendencies various propensities or strengths. So, in general we can visualize that transition probability could exist from a given site to any site infinite in extent. The intensities may decrease of course, such that the total probability is conserved in which case our random walk equation can be just generalized something like this. For example, I could have W n plus 1 M it would now be a infinite sum of coming from all possible states let us say M prime varying from minus infinity to infinity W M M prime and a transition probability P 2 M from M prime. So, we are actually back to our Markov definition, but this time specifically established for a lattice random walk. So, what is the physical meaning of this? The probability of finding the random walker at site M at the n plus 1 step is the probability that in the previous step he was at M time M prime and in that next one step he jumped from M prime to M for any M prime that is possible. If you notice we did not postulate a P and Q which is not really necessary. We did it for convenience in nearest neighbor random walk, but in this formulation since my M prime can be any site there is no backward or forward difference all that we need to postulate is a transition probability number P. If M prime is more than M it will turn out to be a backward transition and if M prime is less than M then it will turn out to be a forward. So, it is only a superficial distinction mathematically all that can be covered by a single expression called a PMM prime. What is the property of PMM prime that has to be preserved that is sigma PMM prime over all M this should be conserved. The random walker at every step has to transit somewhere from for every possible site for all sites. So, with this now it is a matter of method of finding a solution to this problem. This analogy of transition occurring to multiple sites at a given step can be gradually taken over to the continuous form that is what we now attempt to do. When we were discussing Markov processes we briefly introduced the kind of a Markov criteria for a continuous transitions. One important thing we should remember which was also introduced in the previous lectures is the W's that we are talking about here in the case of discrete are probabilities masses the pure probabilities, but when we move over to continuous variables they become probability densities. So, accordingly although we may use the same notation W we should be understood as the probability of a finding or probability intensity of the strength or of a finding the particle let us say of the system at x between x and x plus dx for example, will be W dx. So, this way it has to be interpreted. A simple extrapolation to continuous formulation can be almost carried out by inspection continuous space and time formulation. We have been seeing and we have been doing it quite often that to passage to a continuous form of a solution is achieved by replacing the site index m with the space index x through by postulating a certain length m into l equal to x. If you remember x was defined as m into l and similarly the real time the continuous time was defined as step into some time for transition. When we do random walk by step method there is no concept of time you can wait for as long as you please and perform the next simulation, but however once we have to find an analogy with the continuous time the step width or step length has to be defined delta t should be of the order of it should be constant. Every step has to be performed then only it can be linked to time. So, we postulate the existence of a time scale tau. So, with this a continuous form of analogy takes the form W x t plus tau we will explain the meaning of this, but first we write down this will be x let us say at we will seek the limits soon that is the probability or probability density that the random walker existed at a point y between y and y plus d y and heat transited by a probability p to x y in the next interval of time tau probability that it translated from y to between x and x plus d x that is a correct way to write and for all intermediary sides of y and if we taking a one dimensional system y can take up any value from minus infinity to infinity and this will be d x here. So, we can make it more spacious we can say W x t plus tau d x is equal to minus infinity to infinity W y t d y p x slash y tau d x. So, to repeat the meaning of this is very simple the probability of finding the random walker at x between x and x plus d x in a time t plus tau is the sum or the integral of all transitions first occurring from y to x the probability of finding the particle at y at a time t and then in the next tau because it is t plus tau here. So, in the remaining time tau you should have translated from y to x between x and x plus d x and for all such values of y. So, this is a chain law exactly the Markov criterion, but this is a special name it is called the Chapman Kolomogorov equation. In some literature it is also called Smolyshevsky equation we start from this and apparently an integral equation integral functional equation looks quite complicated because tau looks an arbitrary number and it is a difference W y t and W x t plus tau different functions are involved and it is an integral. So, it looks formidable as such we find a way to arrive at a differential formulation from this before that we can note here that now I can cut d x on both sides and we get the probability density satisfying Chapman Kolomogorov equation y t p x y tau d y. So, as we mentioned this is probability at x in time t plus tau at the moment let us focus on this probability at x this is probability at y time t and transits from y to x in here you write probability at x in time t plus tau. So, this is the verbal interpretation of the equation we have written it is quite self-evident of course, self-evident from after learning that we have used the Markovian property in this Markovian property to recapitulate simply says that the transition in the next step depends only on the state present and may be the future state, but not on the history of the path that the system has taken so far and that is what we have used. What matters is the probability density at y and the transition probability between x and y that is this is the two quantities. We note that basically in defining this transition probability this way we have still continuing to use the conditional probability logic to x from y or given y probability of being at x given y this is the concept. We want we will refine this notation slightly without of course, changing the meaning meaning remains the same this is for convenience. Firstly note that if a random walker transits from y to x all that is happening is he has executed a jump by a length of x minus y that is he has arrived at x from y by executing a length x minus y. So, this is a simple algebraic equality keeping this concept in mind P x slash y can be a different notation can be used. If we define x minus y as a new notation xi let us say xi equal to x minus y and replace y with xi then this integral will be over xi it takes the form then y will be this implies y will be x minus xi then the equation takes the form integral equation w x p plus tau it will be now integrated over xi will be minus infinity to infinity w x minus xi wherever y is there at time t and p transition to x from a point x minus xi in time tau integrated with respect to xi this is perfectly valid formulation given the definition here. So, xi is the new variable. So, dy will be minus d xi, but the limit again will be it is adjusted the negative sign is adjusted it is a standard transformation and we get this formulation. So, here now the transition probability is probability of transiting from x minus xi to x in time tau. So, we have consistently we have to keep note of the exact definition of p it is a probability of transiting in a time tau from x minus xi to x how does it go from x minus xi to x by just picking up a length xi x minus xi plus xi becomes x. So, keeping that in mind notationally I can I could have as well written the transition probability as dependent on the first site where it starts that is x minus xi and the length by which it is undergoing a transition that is xi. So, I can refine my notation p x x minus xi tau by a notation p of x minus xi xi tau I am using the same p for because p we want to maintain for transition probability, but maybe I can call it a small lower case p just to remind ourselves that it may be looking like a different function, but the concept it captures. So, what is the concept transition from x minus xi to x is what the transition probability is which is defined as transition from a point x minus xi by picking up a length xi in time tau exactly then naturally it will go to x from x minus xi when it picks up a length xi it goes to x. So, once we have this definition we can rewrite the Chapman Kolmogorov equation as same xi minus infinity to infinity this will be w x minus xi t and transition probability. Now, with the new notation being understood as a transition from a site x minus xi by picking up a length xi transition from x minus xi to x of course, but by picking up a length xi which were explicitly indicating as a variable tau d xi. We work with this form of the equation for the remaining part of this talk this is again is the same as Chapman Kolmogorov equation rewritten. Now, let us look at the quantity in the integral examine we have w x minus xi t and p x minus xi and s another argument tau. So, this is the first argument and this is another argument we expand this function around x in the Taylor form and while doing so, we will later use the properties of certain moments of the transition probability as of now we have only used the property that integral minus infinity to infinity p of from any point x prime tau d xi should be 1 that is p x prime xi tau is the probability transition probability transition probability from x prime to x prime plus xi in time tau that is all it means and you should transit somewhere from x prime that is all. So, this conservation law we have used. So, we look at this quantity and develop expansion. So, this will be expanded with respect to the first argument x minus xi around x. So, we can write for example, w x minus xi t p x minus xi xi tau will be w x t p x xi tau for xi equal to 0 in the first argument plus or it will keep the minus because it is minus xi it will be minus xi d w by dx same thing d d by dx of we will write like this d by dx of w x t p x xi tau the next term will be xi square by 2 d 2 by dx square of w x t p x xi tau of the order of xi q it will be 3 factorial this is what our term is. So, with that we now go back to our equation we see that on the left hand side we have t plus tau right hand side we have t and then tau of course, if we want to develop an equation for t we have to do some Taylor expansion here also, but not with respect to x, but with respect to t. So, if we do that partial derivative there and then combine the two things together we will have an equation like this expand the LHS of the C k equation with respect to tau with respect to t that is it is going to be w x t plus tau d w by dt x t plus of the order tau square and the right hand side considering this expansion that we wrote down this expansion we will have to insert it inside the integral that will be we can while inserting it inside the integral we may note a few things. Our integral if you see here in this equation our integral is with respect to the xi variable. So, the functions which do not depend on xi can be kept out of the integral. So, we have to just integrate this whole thing with respect to xi. So, the first w x t will stay out. So, it will be integral p x xi tau with respect to xi. So, that of course, is unity. So, we will have we have that advantage. So, it will be only w x t integral minus infinity p x xi tau d xi x xi tau d xi which of course, is unity, but we will retain it now and the next term is minus. Now, I am integrating with respect to xi and there is a xi outside and there is a xi inside, but however, the derivative is with respect to x. So, I can easily take the xi inside and integrate at this point only because the integral over xi w does not depend on xi. So, I could then write the second term as minus of d by dx w x t let us let us keep a bracket here right once again it will be d by dx of bracket w x t and then an integral over xi and the transition probability p x xi tau d xi that confirms this second term. And then we will have the same logic for xi square we can take the xi square inside the x square second derivative term and then perform the integral here again w x t stands out the integral. So, that then takes the form plus write it separately plus d 2 by dx square and of course, there is a half coming from the Taylor expansion and that again we will have the form here we will complete the bracket here. So, the form w x t integral xi square p x xi tau d xi and it will go on we will not write the next term, but merely indicate that that will be of the order of an integral of the order of xi cube term. So, let me integral of the order xi cube before we proceed further we can simplify a few things. Since this term is equal to 1 since minus infinity to infinity p x xi tau d xi is going to be 1 transition must occur from x to some site the w w's cancel and left hand side therefore, is left with tau d w by dt its argument is x and t only and here of the order of tau square and the right hand side one term has gone. So, it is minus of d by dx. So, to sort of cancel this here this is equal to 1 and then this whole term will cancel with this. So, reverting back we are going to have minus this derivative and this integral we call it as xi bar. So, d by dx of w x t xi bar and it has every reason to be a function of x because p could be a function of x. So, we leave it at this and the next term in the same manner we can write it as d 2 by dx square w x t it will be xi square bar again it has every reason to be a function of x then of course, higher order term we will simply call it as h o t higher order term where we complete the definition where we define xi bar is integral minus infinity to infinity xi p xi bar as a function of x. So, let us keep that x xi it is a function of tau also is d xi and xi square bar function of x will be minus infinity to infinity xi square p x xi tau d xi it will be a function of tau, but tau is a parameter it is not a variable that is the reason why we are not including it explicitly here, but strictly speaking tau we should be suffixed with tau. So, with this formal stage having we have reached a formal stage, but very important assumptions very physically meaningful assumptions have to be made from now on for a arriving at a differential equation formulation this we will take up in our next lecture. Thank you.