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Assignment 11: Brownian Motion (Random Walks)

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import mmf_setup; mmf_setup.nbinit()
import os
from pathlib import Path
import logging
logging.getLogger("matplotlib").setLevel(logging.CRITICAL)
logging.getLogger('numba').setLevel(logging.CRITICAL)
logging.getLogger('OMP').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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Assignment 11: Brownian Motion (Random Walks)#

For details about Brownian motion and random walks, please see the following (part of my Standard Model course):

The assignment asks you to consider Brownian motion where the steps \(x_n\) are independent and identically and independently distributed (iid) with some probability density function bracket (PDF) \(p(x)\). The assignment asks you to consider two cases:

  1. A binary random walk with integer steps \(x_{n} = \pm 1\) with probability \(p\) and \(1-p\) for \(x_{n} \in \{+1, -1\}\) respectively. I.e., choosing the steps by flipping a coin (biased if \(p \neq 0.5\)):

    \[\begin{gather*} p(x) = p \delta(x-1) + (1-p)\delta(x+1). \end{gather*}\]

  2. A gaussian random walk with normally distributed steps of mean \(\mu\) and standard deviation \(\sigma\):

    \[\begin{gather*} p(x) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left(\frac{-(x-\mu)^2}{2\sigma^2}\right). \end{gather*}\]

Consider the trajectory \(\vect{X}\) where the particle is at position \(X_{N}\) after \(N\) steps:

\[\begin{gather*} X_{n} = \sum_{n=1}^{N}x_n. \end{gather*}\]

  1. What is the PDF \(p_N(X)\) for the position after \(N\) steps? You are asked to empirically check the mean \(\mu_N\) and standard deviation \(\sigma_N\) of \(X_N\).

  2. For what values of \(\mu\) and \(\sigma\) does the gaussian random walk have the same statistical properties as the binary random walk with probability \(p\) in the limit of large \(N\)?

  3. What are the average maximum and minimum excursions after \(N\) steps? I.e., take a large number of random walks of length \(N\), and take the average of the maximum \(\max_{n \leq N} X_n\) and minimum \(\min_{n \leq N} X_n\).

    Warning

    I do not think these are trivial to compute analytically as Schroeder [Schroeder, 1991] mentions in footnote 6 on page 146 of Chapter 6, §Counterition Runs Rampand in Random Runs:

    Surprisingly, a certain statistics professor, who shall remain nameless, was not able to derive the formula for the probability of ruin for the finite-duration game which he had posed as an exercise for the students at the beginning of the summer semester in 1948. During the entire semester his recurrent refrain was “we’ll tackle that one next week.” But he never did tackle it.

    Show how \(\braket{\max \vect{X}_N}\) where \(\vect{X}_{N}\) are walks of length \(N\) is related to the probability of ruin for a finite-duration game.

  4. What is the probability of return? I.e. what is the probability that a random walk will return to \(X_N = 0\)?