Stochastic Processes

Hide code cell source

 
import mmf_setup; mmf_setup.nbinit()
import os, sys
from pathlib import Path
FIG_DIR = Path(mmf_setup.ROOT) / '../Docs/_build/figures/'
os.makedirs(FIG_DIR, exist_ok=True)
import logging; logging.getLogger("matplotlib").setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt
try: from myst_nb import glue
except: glue = None

from matplotlib.animation import FuncAnimation
from phys_581.contexts import FPS

This cell adds /home/docs/checkouts/readthedocs.org/user_builds/wsu-phys-581-computation/checkouts/latest/src to your path, and contains some definitions for equations and some CSS for styling the notebook. If things look a bit strange, please try the following:

  • Choose "Trust Notebook" from the "File" menu.
  • Re-execute this cell.
  • Reload the notebook.

Stochastic Processes#

Stochastic processes and stochastic differential equations play a key role in extending classical and quantum mean-field models to include thermal and/or quantum fluctuations. The general strategy is to include a stochastic (noise) term with specific statistical properties.

References#

These notes are currently just a stub, and will be fleshed out in future versions of this course, but here are some references to get you started:

  • [Higham, 2001]: A fantastic short and practical introduction to the numerical simulation of stochastic differential equations.

  • [Evans, 2013]: A very short but quite complete mathematical introduction.

  • [van Kampen, 2007]: A longer but more gentle introduction to the theory.

Example: Brownian Motion#

To whet your appetite, consider the problem of Brownian motion, starting in 1D. The discretized process can be described by stochastic difference equation (using the notation from [Higham, 2001]):

\[\begin{gather*} W_{j} - W_{j-1} = \d{W}_{j} = \sqrt{\delta t} \eta, \qquad \eta \sim N(0, 1). \end{gather*}\]

Here \(\eta \sim N(\mu=0, \sigma=1)\) is a normally distributed random variable with mean \(\mu = 0\) and standard deviation \(\sigma = 1\).

Consider now the limit of \(\delta t \rightarrow 0\). Does the this limit exist? If so, we might write this as a stochastic differential equation:

\[\begin{gather*} \dot{W} = \xi. \end{gather*}\]

What type of random variable is \(\xi\)?

Contents