 So, we consider the random work problem with bias and with the certain probability of pausing at each step and by using a logic of considering only the so called successful steps successful meaning those which resulted in jumps and using a binomial probability for them we could obtain an analytical expression for the occupancy probability of a most general case of a random work with the bias and pausing. Now, we set out to also obtain the generating function for this case the advantage being that the generating function can be sometimes conveniently used for extracting many properties including the mean variance etcetera in a simple way. So, for that we start with the expression where which we used at the beginning of derivation. So, this is generating function for bias plus pausing. So, when we set out to derive an expression we started with a formula the probability of occupancy at nth step at site m with this general case paus and bias together is equal to the sum the probability the binomial probability of S successful steps out of total of n steps multiplied by the occupancy probability for these successful steps being at the site m considering bias. So, pausing is in p and the bias occupancy is in w and from this we proceeded further explicitly using the formula for both p and w and arrive at a solution. For generating function right now we start with this and then as usual define the generating function g let us say nth step for the case of p plus b. Now, as a function of the parameter z as always is the sum over sites that is m it is an m sum meaning site sum minus infinity to infinity w n p plus b m z to the power m and if you call this as equation 1 and this is equation 2. So, here we have to now multiply equation 1 by z to the power m sum over all m and we note that the sum over m and the sum over s could be interchanged. So, the sum over m can be taken inside just before the term w s b m. So, that would then be g n p plus b z. So, s equal to 0 to n p n s this does not have any n terms m m dependent terms. So, the sum could be taken inside over all m minus infinity to infinity w s with the bias m z to the power m. So, it is a s step occupancy probability at site m due to bias random walk it is a. So, this expression is going to be a generating function for the s step probability. So, one can write this as s equal to 0 to n p n s g s z with the bias just we will keep that superscript to remind ourselves that this expression corresponds to the expression with the bias cases. Now, if you revert back to the case where we considered bias without pausing it is a bias without pausing we can see that the generating function g for any step say s step with bias z it will corresponds since there is no pausing here. So, it must be the left and right probability is a sum to unity the left and right jump probability is sum to unity. So, in the present context they are denoted by p prime and q prime. So, it would then be p prime z plus q prime divided by z to the power s where p prime plus q prime equal to 1 these are normalized probabilities normalized with respect to just to remind ourselves p prime equal to p divided by 1 minus delta. They are normalized with respect to the time or the probability that is remaining apart from weighting. So, non weighting probability so normalized with respect to that similarly for q prime q divided by 1 minus delta. So, p plus q is 1 minus delta. So, p prime plus q prime will be 1. So, once this is done we can go back to the previous expression here in this say we call it equation 3. So, we can go back to equation 3 or let us call this one as equation 3. So, here we can substitute that expression to gs bz. So, we arrive at the generating function for the n step in the p plus b case as a function of z is going to be sum s equal to 0 to n the binomial distribution of obtaining s steps out of n into p prime z plus q prime by z to the power s and if we go back again and see what was the expression that we would we had for pns pns this probability was binomial that is all s values it would be this probability is a binomial probability. So, that is given by ncs 1 minus delta is the non pausing probability. So, success. So, 1 minus delta to the power s and delta to the power delta is the equivalent of a non success. So, delta to the power n minus s it will be. Hence when we put it back in say equation 4 we have gn p plus b of z will now be sum. So, hence s equal to 0 to n ncs we write delta to the power n minus s first and 1 minus delta to the power s and here we have p prime z plus q prime divided by z to the power s. So, I can take 1 minus delta inside. So, it is going to be 1 minus delta p prime z plus 1 minus delta q prime divided by z whole to the power s. So, all that I have done here is to take 1 minus delta inside and we note that p prime into 1 minus delta is going to be p. So, this expression becomes s equal to 0 to n ncs delta to the power n minus s and p z plus q divided by z to the power s. So, if this is again of the type a plus b to the power n each of the terms will have a to the power s and b to the power n minus s multiplied by ncs. So, this is here the term corresponding to s is this and n minus this is this. So, if this you the all terms corresponding to the power s if you call a is a to the power s and this is b to the power n minus s ncs. So, it will be a plus b to the power n finally. So, a here is p z plus q by z and b is simply delta all of a it to the power n. So, we have very close form and a neat expression for the generating function for the complete problem of pause and bias put together. We may recall that if delta equal to 0 it goes back to our standard expression if delta equal to 0 g of b p plus b becomes g b that is all g of p plus b goes over to g of b only that is quite consistent. Interestingly same expression can also be derived in order to bring out a certain fact that there are many ways of arriving at the same result many arguments are used and this internal consistency between arguments is what finally, throws some insight into the way of solving problems or into the mechanisms. So, here for example, you could have arrived at the same expression for the generating function by the random walk equation itself. So, we now do proof of generating function by the random walk equation with pause and bias. So, let us set up the equation. So, as earlier p is the right jump probability q left jump delta pausing or non-jumping. So, that p plus q plus delta equal to 1. So, pictorially at any site m m plus 1 m minus 1 this is q this is p and of course, turning to itself is delta. One can write down the random walk equation at the nth plus n plus 1th step the probability w to be complete we will write it as with the bias and pause together at the n plus 1th step at any site m should be coming because of 3 reasons. Now, one there was a right jump at with the probability p from a state of m minus 1 at the nth step of course, this is just a index. Super script is just to denote the case we are discussing and this must be from m minus 1 plus q w n of course, b plus p from m plus 1 plus delta of w n b plus p m itself because pausing is a situation when a particle does not occupy any newer site it just jumps on to itself. So, from the previous step if it had remained at m that would continue to remain at the n plus 1 step at n. So, this is basically the equation and we can have an initial condition that w 0 m is delta k m w 0th step m pause plus pass equal to delta m 0 that is its origin. So, now we can use generating function for this this equation you know how to do that same definition. So, when we do we get the it is very easy to see that if you if you substitute multiply by z to the power m multiply by z to the power m and sum over m then using the logic that we developed earlier in the case of symmetric random walk or even the biased random walk by replacing m minus 1 by m prime and m plus 1 also by m prime which showed how the whole thing some reduces over to a kind of a binomial power of a z plus 1 by z type. So, here similarly we can argue that g and z for this bias plus pause case is going to be it will be p z plus q by z plus delta to the power n it just comes very easily. So, one can use this method also which is exactly the same same as that obtained by considering the binomial probabilities. It is an alternative approach. Now that we have the generating function we can consider a very interesting special case that is a special case corresponding to delta equal to 50 percent half delta equal to half and symmetric random walk p equal to 1 fourth and q equal to also 1 fourth. So, it is a case in which there is a 25 percent probability of jumping to the left at each step and each side and 25 percent probability of jumping to the right and 50 percent probability of not jumping at each step. So, this is one special case and then the corresponding generating function for the case symmetric. So, it is not bias, but just pause special case turns out to be. So, 1 fourth z by 4 plus 1 by 4 z plus delta is half half can be written as 2 by 4 whole to the power n. So, if you take 4 out this becomes 1 by 4 to the power n into z plus 1 by z plus 2 to the power n. The same thing can be written by anticipating this kind of terms as square root of z by 1 by square root of z to the power 2 n. Here we note here it follows from the expression we note that root z plus 1 by root z whole square is equal to z plus 1 by z plus 2. So, the entire expression is therefore, can be written as a square of root z by root z plus 1 by root z whole square. So, now you can carry out the binomial expansion as usual. So, this is going to be 1 by 4 to the power n sum of for example, r equal to 0 to n root 0 to 2 n now because it is to the power 2 n. So, it is going to be root z to the power r. So, z to the power r by 2 we should write the binomial coefficient first. So, it is 2 n c r z to the power r by 2 and then here z to the power minus of half. So, it is going to be z to the power minus of. So, we could write it as r minus 2 n by 2. So, it will be this. So, when we can simplify this expression r by 2 r by 2 is r. So, r minus 2 n. So, 2 n by 2 is n. So, the whole thing easily sums up to a form which is will be 1 by 4 to the power n. So, it will be 1 by 4 to the power n we can write it r equal to 0 to 2 n 2 n c r z to the power r minus n. Now, the binomial the coefficients of z to the power m they are the occupancy probabilities. So, if you call r minus n as m and do the identify the coefficient we will get for this case the probability W and m for pause with the probability 0.5 that is delta with the pause with probability here delta equal to 0.5 is going to be 1 by 4 to the power n 2 n c n plus m. So, far simpler expression is obtained for this case. So, it is somewhat useful to work with this formula sometimes to illustrate. So, this is special case delta equal to half p equal to q equal to 1 4. It is quite a useful formula to illustrate the pausing problem. The generating function that we developed is quite useful for estimating the mean and let us see how the mean behaves. So, to go back to the general expression we showed that g n of z for the case of p plus b was p z plus q by z plus delta to the power n. So, and we also have noted that there is a bias factor gamma which conveniently represents the tendency to move one way, tendency to be asymmetric as opposed to the symmetric random walk. So, it is a kind of an indicator of the degree of asymmetry. So, if you use this notation we can obtain the mean can denote it as m bar or m over bar that is g n prime 1 by definition. And you can easily see by differentiating this that because of the symmetry it will have 1 minus delta into n gamma when you put differentiate and then put z equal to 1 we are going to get this. You may recall that if delta is 0 this reduces to the case of biased random walk where the mean is n into gamma that is the mean continuously shifts towards right if gamma is positive with the same speed as n. So, it is proportional to n. However, the presence of pausing that is delta non-zero retards this motion basically certain amount of time is sort of wasted in pausing that is the reason in a way it retards. So, there is a tendency to retard pausing retards drift or yeah this can be called as a drift motion because it increases proportional to n. Similarly, we can work out the variance which is g n double prime 1 we have derived this earlier and used it many times g n prime 1 minus g n prime square 1. One special thing is we have to take the second derivative which is a bit elaborate expression I am skipping that step. So, giving you the result then you can show that it comes to n into 1 minus delta into 1 minus 1 minus delta square gamma square. So, we can see that the variance also is retarded by the same factor as the drift as far as the m n is concerned variance is also proportional to n it is proportional to n into 1 minus delta, but there is an additional term. So, that term depends on the strength of gamma. So, if gamma is fairly strong then the variance is more than retarded by the pausing part it even affects the asymmetry factor it is almost like it enhances the degree of I mean it mutes it it reduces the effect of asymmetry by being a multiplier of value less than unity. However, in the first order approximation this expression can be written as n into 1 minus delta that is assuming plus we can say order gamma square and if we neglect the terms of the order gamma square and higher then it comes to just n into 1 minus delta, but still it includes the effect of retardation. The interesting thing to note which is physically quite obvious that when we discussed random walk without pausing it was always the case that m has to be pausing m has to be even when n is even and m has to be odd when n is odd and we illustrated it by actual examples. However, this is need not be the case when you have pausing. For example, without pausing when the first jump is done the particle shifts either to plus 1 and minus 1 and the probability of its being at the origin is 0 in the case of non pausing in the in the non pausing situations. However, when a certain probability of pausing exists at the first jump particles of course, would be found at plus 1 minus 1 as well as at the origin. This looks or shows or gives an apparent perception of a kind of an oscillatory behavior. Supposing we plot as a function of steps at some site say origin let us say this is W n 0 with the pause then it would look like it starts with the some probability here it will come down may go up like this. So, the probability of being at the origin might decay, but it will move in a oscillatory way depending on the value of delta. So, this is the contrasting and interesting pattern that we should see we should remember as opposed to the case when there was no pausing. After sufficient number of steps of course, it attains a smooth behavior. So, here the lesson is that pausing allows persistence at sites persistence sites and shows oscillatory behavior. It does not disturb the symmetry of distribution in an unbiased case for example, one can say that the the so called symmetric distribution that one would obtain in the case of a symmetric random walk. Supposing this is without pause the distribution with the pausing this is with the pause would be always narrower as compared to that without pausing because it retards the dispersion itself. Of course, it it is possible that it will extend beyond we can redraw it to by conserving the total probability is just a schematic to indicate. So, it could just be having a higher probability at the starting point because of the pausing effect. So, that the overall area is conserved, but smaller dispersity as compared to non pausing situation. For a special case we can show special case that was when delta was half p equal to 1 by 4 q equal to 1 by 4 we have the result that variance with the pause equal to n by 2 that is half the variance half of variance without pause. So, one can plot variance alone and if you plot variance alone with the time with the steps the variance without pause let us say it goes like this then the variance would be seen as this is variance without pause variance with the pausing. So, these are some points to note the variance is less. So, this with this we sort of have understood the various possible situations in a one dimensional random walk and the one dimension that we considered was almost a lattice extending from minus infinity to infinity. We now take up special cases in the next video. Thank you.