Question

1. "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 size and the direction of each step is chosen randomly. Consider a 1-dimensional random walk, where each step (of length l) 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 ?(N, N+) is very highly peaked around N+ = N/2. Derive a formula for the multiplicity ?(N, x), where x = N+ – N/2, in the vicinity of this peak. Your formula should have the form of a Gaussian. [Hint: Use Stirling's Approximation. You may find it easiest to then work with ln(?) under the assumption x << N, neglect term(s) that are much smaller than the other terms, and exponentiate.] (b) Suppose you take a random walk of 10,000 steps, with l = 1 m. Approximately how far from your starting point would you reasonably expect to end up, if "reasonably" is defined as "within a half-width 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 l 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). (d) Show that the "reasonable" net displacement in part (c) increases in proportion to T^3/4 (where T is temperature), assuming pressure is kept constant. [Hint: at everyday temperatures, recall the average molecular speed of a gas increases proportionally to ?T.]

          1. "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 size and the direction of each step is chosen randomly. Consider a 1-dimensional random walk, where each step (of length l) 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 ?(N, N+) is very highly peaked around N+ = N/2. Derive a formula for the multiplicity ?(N, x), where x = N+ – N/2, in the vicinity of this peak. Your formula should have the form of a Gaussian. [Hint: Use Stirling's Approximation. You may find it easiest to then work with ln(?) under the assumption x << N, neglect term(s) that are much smaller than the other terms, and exponentiate.]

(b) Suppose you take a random walk of 10,000 steps, with l = 1 m. Approximately how far from your starting point would you reasonably expect to end up, if "reasonably" is defined as "within a half-width 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 l 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).

(d) Show that the "reasonable" net displacement in part (c) increases in proportion to T^3/4 (where T is temperature), assuming pressure is kept constant. [Hint: at everyday temperatures, recall the average molecular speed of a gas increases proportionally to ?T.]

1. "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 size and the direction of each step is chosen randomly. Consider a 1-dimensional random walk, where each step (of length l) 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 ?(N, N+) is very highly peaked around N+ = N/2. Derive a formula for the multiplicity ?(N, x), where x = N+ – N/2, in the vicinity of this peak. Your formula should have the form of a Gaussian. [Hint: Use Stirling's Approximation. You may find it easiest to then work with ln(?) under the assumption x << N, neglect term(s) that are much smaller than the other terms, and exponentiate.]

(b) Suppose you take a random walk of 10,000 steps, with l = 1 m. Approximately how far from your starting point would you reasonably expect to end up, if "reasonably" is defined as "within a half-width 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 l 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).

(d) Show that the "reasonable" net displacement in part (c) increases in proportion to T^3/4 (where T is temperature), assuming pressure is kept constant. [Hint: at everyday temperatures, recall the average molecular speed of a gas increases proportionally to ?T.]

Added by Alfredo H.

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