"Random walk" is a term that refers to a commonly-used statistical model, of a journey consisting of N steps where all steps are the same length and the direction of each step is chosen randomly....

Extracted text: "Random walk" is a term that refers to a commonly-used statistical model, of a journey consisting of N steps where all steps are the same length and the direction of each step is chosen randomly. Consider a 1-dimensional random walk, where each step (of length I) can go either forwards (+) or backwards (-). The statistical model is similar to a paramagnet, where each spin can be either up (+) or down (-). The equivalent of a “microstate" is defined by specifying whether each step in the walk is + or -. The equivalent of a "macrostate" is defined by specifying N. (how many of the N steps in the walk are +). The most probable macrostate is N. = N/2, meaning at the end of a long random walk you are most likely to end up back at your starting point. (a) The multiplicity function 2(N, N.) is very highly peaked around N. = N/2. Using the change-of-variable x = N. – N/2, find the multiplicity function 2(N, x) in the vicinity of the peak (x = 0). Your answer should have the form of a Gaussian. [Hint: Use Stirling's Approximation. You may find it easiest to then work with In(2) under the assumption x < n, neglect term(s) that are much smaller than the other terms, and exponentiate.] (b) as a function of n and i, approximately how far from your starting point would you reasonably expect to end up, if "reasonably" is defined as “number of forward steps is within a half-width of the peak of the gaussian distribution"? [hint: recall the half-width of a gaussian is where it falls to 1/e of its peak value.] (c) a molecule diffusing through a gas roughly follows a "random walk", where i is the mean free path. using the "random walk" model and pretending the molecules can only move in 1 dimension, estimate the "reasonable" net displacement of an air molecule in one second, at room temperature and atmospheric pressure (mean free path = 150 nm, average time between collisions = 3 x 10-10 s). n,="" neglect="" term(s)="" that="" are="" much="" smaller="" than="" the="" other="" terms,="" and="" exponentiate.]="" (b)="" as="" a="" function="" of="" n="" and="" i,="" approximately="" how="" far="" from="" your="" starting="" point="" would="" you="" reasonably="" expect="" to="" end="" up,="" if="" "reasonably"="" is="" defined="" as="" “number="" of="" forward="" steps="" is="" within="" a="" half-width="" of="" the="" peak="" of="" the="" gaussian="" distribution"?="" [hint:="" recall="" the="" half-width="" of="" a="" gaussian="" is="" where="" it="" falls="" to="" 1/e="" of="" its="" peak="" value.]="" (c)="" a="" molecule="" diffusing="" through="" a="" gas="" roughly="" follows="" a="" "random="" walk",="" where="" i="" is="" the="" mean="" free="" path.="" using="" the="" "random="" walk"="" model="" and="" pretending="" the="" molecules="" can="" only="" move="" in="" 1="" dimension,="" estimate="" the="" "reasonable"="" net="" displacement="" of="" an="" air="" molecule="" in="" one="" second,="" at="" room="" temperature="" and="" atmospheric="" pressure="" (mean="" free="" path="150" nm,="" average="" time="" between="" collisions="3" x="" 10-10="">