 So, continuing we will see that the 2 to the power n terms will cancel and here n square minus n square I take the n out. So, n square will when you take it out it will come out as n here and 1 minus m square by n square will come here. Similarly, this term also is n square minus n square. So, when I take out n it will be n to the power n n it will be it will contribute here when I take out to the n then and n will cancel and as a result I can write in a simplified way the denominator as equal to pi that n will come then it will have e to the power minus n will come and as I mentioned n to the power n also will come then the terms are square root of 1 minus n square by n square will enter here then we will have 1 minus m square by n square to the power n by 2 and here you will have 1 plus m by n divided by 1 minus m by n to the power m by 2. This is going to be my expression and we have not still reached asymptotic form. So, to proceed further of course, we said we will take the limit n tends to infinity, but that is not just sufficient. As we have seen as the number of steps increases his mean dispersity also is increases that means, the walker is likely to be more and more farther away at sites of m closer to one standard deviation which we know is proportional to root n. So, since distance of m bar not really m bar m square bar to the power half is of the order of root n, we must also make m tend to infinity in order to get an expansion. So, our limit should involve n tends to infinity and m also tending to infinity and there needs to be a rule and that rule is such that m square by n is held fixed. We have to impose this criteria in order to arrive at some meaningful or tangible result because this result this choice is hinted by the understanding that the site m also increases rather his occupancy to larger further sites increases and we are looking at around that point. So, the since the standard deviation is proportional to root n this could be the choice. So, in other words as n tends to infinity and m tends to infinity, m square by n is fixed and denote that fixed quantity by quantity x square like this which basically means a limit n tends to infinity and m tends to infinity, but x is some fixed quantity. So, we take now the asymptotic approximation subject to this constraint note that which means x equal to m by root n. Basically the we can even say that it is a scaling method we force see that the scaling of m should be done with respect to the mean dispersity that is root n. So, then we can see that wherever m occurs we should write which means wherever m comes it is x root n or whenever m by n comes which we need in. So, this is going to be x divided by root n and wherever m square by n square comes n square is going to be m square by n square can be written as x square because m square by n is x square. So, it is going to be x square by n. So, with this understanding we can write the expression denominator as d n equal to pi n e to the power minus n n to the power n square root of 1 minus x square by n and next it is same thing, but 1 minus let us separate this definitions 1 minus x square by n to the power n by 2 and here it is going to be 1 plus x by root n divided by 1 minus x by root n to the power m by 2. So, m by 2 is going to be x by 2 root n. Now, we have to carefully take the limit. So, we must then use the definition of e to the power x as a limit n going to infinity. Of course, the first term first few terms up to n to the power n they will remain as such we want we do not want to take the absolute limit of n becoming infinity, but n going towards infinity. Now, the term here is just a square root of a 1 minus x square by n and x is held fixed as we said then x square by n will tend to 0 it is just a square root of 1. So, it will remain as 1. Here it is tricky because it will become as n tends to infinity the second term will go to 0, but 1 to the power infinity it will become and that is what always is to be carefully done e to the power and. So, is it going to be here also as n goes to infinity since x is fixed this will end up as 1 to the power infinity. So, we should take these limits properly and make use of the fact that limit n tends to infinity 1 minus x square by n to the power n by 2 will be e to the power minus x square by 2 the definition of e to the power x itself basically we have made we made use of property 1 plus x by n some x or anything else let us use another notation z by n to the power n equal to e to the power z limit n tends to infinity we made use of this property. So, here z is minus x square. So, you are getting minus n and then of course, whole to the power half. So, you are going to get x square by 2. Similarly, we note that 1 plus x by root n to the power root n into x by 2 that was another term that was hanging there and this is going to be e to the power x square by 2 again, but plus interestingly the denominator term 1 minus x by root n to the power root n and again minus x by 2 it was because it is always to the same power as the second term here this also will become e to the power x square by 2. So, the sum and substance is all the 3 important quantities that is the 1 minus x square by n to the power n by 2 1 plus x by root n to the power here as well as 1 minus x by root n to the power minus all the 3 have been now given their asymptotic forms and when we substitute it systematically we are going to have the denominator as pi n e to the power minus n n to the power n e to the power there are 3 terms. So, e to the power minus x square by 2 e to the power plus x square by 2 and again e to the power plus x square by 2. So, one of them will cancel eventually giving rise to pi n e to the power minus n n to the power n e to the power minus x square by 2 sorry e to the power plus x square by 2. So, this is as n tends to infinity. Now, we go back and look at the numerator we have seen that the numerator we have defined whose asymptotic approximation as n tends to infinity is going to be n to the power n e to the power minus n root 2 pi n. So, if we bring it forward we have our final expression W n limit n tends to infinity W n m will now have W n x and which will have the form numerator divided by denominator that will be root 2 pi n n to the power n e to the power minus n divided by pi n n to the power n e to the power minus n and e to the power x square by 2. When we carefully simplify all the terms cancel out n to the power n will cancel e to the power minus n will cancel. So, here this is a root 2 pi is extends include n. So, the 1 by root 2 will remain, but 1 by root pi n will come. So, you will have basically it will be square root of 2 by pi n e to the power minus x square by 2. This will go up as minus x square, but we can re express in terms of the original variable it will be m square by 2 n and this is an asymptotic limit n tends to infinity m tends to infinity. So, we have arrived at a very neat looking expression which of course, is Gaussian or normal distribution Gaussian distributed which again we had seen from central limit theorem to be true. Only difference being that now we note that that parity constraint is still valid this distribution is 0 whenever the right even odd combination does not occur that is whenever n equal to m whenever n is odd m should be odd otherwise the probability at that point will be 0. So, we should remember. So, it is basically giving you distribution for any step at alternate points that has to be conserved. It is quite interesting that if we actually calculate this the probability occupancy probability with the exact solution as well as is asymptotic distribution at various values of n steps. If they sort of closely agree for n sort of exceeding 4 or 5 percentage errors become negligibly small although we have made this assumption n tending to infinity that asymptotic level is reached fairly quickly. So, that is where the problem of exact solution is valid when we want to understand the behavior in a very few steps early steps or in extremely confined situations where one has to be able to predict the stochastic progress of the stochastic phenomena. Otherwise in a sufficiently open system when the large number of steps are involved the continuum approximation that we have achieved via the asymptotic mod asymptotic approximations is quite valid. To carry forward and connect this distribution to reality in terms of a continuous distribution we have to do us mapping of a site to location and that is what we do now that is converting we will take a first page converting to continuous distribution. So, to do that let us note let us be on the only one side of the axis to safe space and let us locate sites 1, 2, 3, say 4, 5 and let us say 6. So, let us consider an E 1 event let us say n is E 1 example. So, this is W occupancy probability W n say if n is E 1 let us say yes is an E 1 case then the probabilities will exist only for 2, 4 and 6 and of course, 0 also. Now we will have to distribute to be continuous we will have to distribute this probability mass which are pure probabilities to neighboring sites and this is true because when it is E odd it is going to be completely shifted to the odd. So, in either case there are always 2 numbers adjacent to the probability mass which are not occupied and that can be done by ensuring that the total probability that we have indicated is conserved and we distribute it over a length of 2 units this length is 1 and 3. So, it is 2 units similarly here the 4 we distribute between 3 and 5 6 we distribute between 5 and 7 let us say. So, this way now we are assigning probability at every point continuously the what was earlier a probability is now distributed. So, locally if you see it is going to be probability density. So, with that convention we can convert that discrete distribution to a continuous distribution here we have to now define a continuous quantity and let us say as we began let the each of these distances is L. So, which means at the mth site the position is going to be m into L. Similarly, let us assume that the random walker takes a step at regular intervals of time tau. So, so is the step n equal to some time by tau we will use this for replacing steps with the time similarly here we will replace m equal to x by L. So, x is some continuous variable say position similarly here tau t continuous time. So, with this we have to just replace m and n in the asymptotic approximation that we obtain, but that is not just sufficient we must note that since we are spreading the total probability and total probability has to be conserved my probability density W of now x and t that multiplied by the total length because this is height is going to be the probability density representation. So, height into the width that should be the total area under the curve. So, that is a twice L and that should be actual probability mass functions or occupancy probabilities that we already derived thereby implying that my probability density function therefore, will be 1 by 2 L of W n m this will only conserve the total probabilities. So, with this crucial assumptions and from the fact that we have derived W n m equal to root of 2 by pi n e to the power minus m square by 2 n. So, if we carefully represent this is going to be root 2 by pi and n is t by tau. So, it will be t by tau similarly here m is going to be x by L. So, it is going to be x square by 2 L square and it is again t by n is t by tau. So, we will arrive at this expression here we introduce a very important concept of the concept of diffusion coefficient for the process is a definitional, but it is very important. So, diffusion coefficient we define at the moment as L square by 2 tau and here we have to also add 1 by 2 L by virtue of mass conservation. So, that will enter here. So, when we eliminate L square by tau everywhere you will finally, end up with the expression W x t is going to be 1 by square root of 4 pi d t e to the power minus x square by 4 d t. It is easy to see because L square by tau is going to be 2 d. So, you will have 2 into 2 d is 4 d and then time will remain 4 d t. Similarly, here when you take L inside it is going to be L square and then 2 it will going to be 4 when you square. So, that will also lead to 4 pi d t. So, this is a completely normalized probability density distribution function for the evolution of the position as a function of time for a random walker obtained by a model called the random walking model. At this point it is a right opportunity to note that one can have several models not necessarily the one we have adopted to arrive at this form of a distribution function. So, it is a kind of degenerate because many processes whichever may be the actual transition probability distribution eventually end up giving us a Gaussian distribution in the symptotic limit. The difference therefore, essentially lies only in small time or small step limits where individual memories will be held whether he has been only a nearest neighbor walker random walker or he has been performing length jumps to other sites or not. Just to complete this we note that if you plot it as a function of x at a different times originally it could be a delta function which eventually will broad as I mentioned with the variance which is given by sigma square will be 2 d t or sigma will be square root of 2 d t which is the famous Einstein result. This is consistent with that we introduce ourselves now to an extension or going a step beyond the symmetric random walk. Symmetric random walk assumed that random walker had no preference to move either left or right. Same random walk model the fundamental concept and formulation that we made can be applied to a situation where there could be a slight bias in moving backward or forward. To give you an example a small fine dust particle which settles under gravity if it is of course, very large it will be so much affected by gravity that you would not expect it to have a random walk it becomes a very deterministic motion. On the other hand if the particle is extremely small then because of viscous drag the it does not feel gravity as much, but it feels its Brownian motion very strongly. So, there will be a significant random walk, but however gravity must act at some level so there will be a some bias in that particular particle taking either an upward jump or a downward jump. Random walk formulation allows us to handle this kind of situations and that is called as random walk with bias. Without having to worry about what actually is the physics that governs this bias in our model it is very easy to introduce this as symmetric motion by merely saying that the probability of right jump and that of the left jump need not be half now. We introduce a symmetric by saying that if p is the probability of right jump then it is just 1 minus p is the probability of left jump, but p need not be half. So, with this slight generalization one obtains fairly interesting expression with much wider applications in many many several problems which we are going to study in next couple of lectures. Thank you.