"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 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).
"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 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).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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!["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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0a416aa-90b7-478f-8c7c-e07ad14a269e%2Fadc4be7e-20a3-4b3e-972c-b385eb4d0123%2Fgr175z_processed.jpeg&w=3840&q=75)
Transcribed Image 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).
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