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\title{Stochastic Modeling in Physics and Microbiology\\
MATH 6960--1 -- Spring 2013 \\
Homework 1}
\date{Due Friday, February 22 at 2:00 PM}
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\begin{document}
\maketitle
This homework has 130 points plus 40 bonus points available but, as always, homeworks are graded out of 100 points. Full credit will generally be awarded for a solution only if it is both correctly and efficiently presented using the techniques covered in the lecture and readings, and if the reasoning is properly explained. If you used software or simulations in solving a problem, be sure to include your code, simulation results, and/or worksheets documenting your work. If you score more
than 100 points, the extra points do count toward your homework total.
\section{Brownian Pinball Wizard}
Another way in which we might formulate a discrete model for Brownian motion, rather than discretizing time into regular intervals as in class, is to propose that the particle moves always at a constant speed $ v $ but in a random direction. Moreover, the particle moves along a given direction for a random amount of time $ T $ according to a prescribed probability density function $ \pT (t) $. At the end of this time, the particle moves in a new random direction, independent of all previous motion, for another random amount of time given by the PDF $ \pT (t) $, and so on. Each time a random direction is chosen, it is to be uniformly distributed. The idea is that the particle is moving freely at a given speed until it collides into some other object; the times between these collisions are assumed to be randomly distributed according to PDF $ \pT (t) $, and each collision causes the particle to move in an entirely new random direction. You may work with a two-dimensional model, or for some bonus credit, with a three-dimensional model (which requires some extra care in how the random direction is handled.)
\begin{enumerate}
\item (\textbf{15 plus 15 bonus points}) Compute the mean and variance of the particle position along a coordinate direction after $ n $ collisions. Your answers should be expressed for general probability density functions $ \pT (t) $ satisfying some mild conditions. For bonus credit, compute also the covariance of the particle position along different coordinate directions.
\label{part:meanvar}
\item (\textbf{10 points}) The diffusivity of a particle along a given coordinate direction is generally understood to be half the rate at which the variance of the position of the particle (along that coordinate direction) grows with time, over a sufficiently long time period. Write down a corresponding mathematical expression for the diffusivity of the particle in the present model, and compute how it relates to the governing parameters of the problem. \label{part:ensemble}
\item (\textbf{15 points}) Suppose now we consider a limit in which the time between collisions becomes very short. Given an arbitrary PDF $ \pT (t) $, we can describe this limit by rescaling the PDF as $ \pTeps (t) \equiv \epsilon^{-1} \pT (t/\epsilon) $ and considering $ \epsilon $ approaching zero. Explain as precisely as possible whether the dynamics of the particle position $ \bX(t) $ under the model with rescaled collision time PDF $ \pTeps (t) $ can be described by a stochastic differential equation of the form
\begin{equation*}
\difd \bX (t) = \sqrt{2 \kappa} \, \difd \bW (t)
\end{equation*}
for some constant $ \kappa $, where $ \bW (t) $ is the vector-valued Wiener process (Brownian motion). If the rescaled model does not converge to such a stochastic differential equation, explore whether such a stochastic differential equation model can result if the velocity $ v $ of the particle is also suitably rescaled together with the particle collision time PDF. \label{part:limit}
\item (\textbf{15 points plus 5 bonus points}) Now approach the model through an Eulerian approach. Begin by writing down a discrete-time forward Kolmogorov equation to relate the probability density function for the position of a particle after a collision in terms of the probability density function for the position of a particle at the previous collision.
\item (\textbf{20 points}) Next, assume the collision time is short (but still random!) to derive a partial differential equation for the evolution of the probability density function for the position of the particle. Express any coefficients directly in terms of the speed $v $ and probability density function $ \pT (t) $. If you need to rescale the speed $ v $ of the particle to obtain a nontrivial limit in the limit of short correlation time, explain precisely what you need to assume.
\item (\textbf{15 points}) Develop a numerical simulation for the model described in this problem, and plot some sample trajectories. Choose any probability distribution you wish for the time between changes in direction. \label{part:numerical}
\item (\textbf{10 points}) Now use the results of your numerical simulation to validate your analytical results for the mean and variance of the particle position from Part~\ref{part:meanvar} and diffusivity from Part~\ref{part:ensemble}. The proper way to numerically estimate the statistics of a random variable $ Y $ from numerical simulations is as follows: Simulate an ensemble of realizations $ \{Y^{(j)}\}_{j=1}^N $ of the random variable with $ N $ reasonably large, then estimate the mean by:
\begin{equation*}
\hat{\mu}_{Y} \equiv \frac{1}{N} \sum_{j=1}^N Y^{(j)}
\end{equation*}
and the variance by
\begin{equation*}
\hat{\sigma}_{Y}^2 \equiv \frac{1}{N-1} \sum_{j=1}^N (Y^{(j)}-\hat{\mu}_Y)^2
\end{equation*}
\label{part:statistical}
\item (\textbf{10 points}) For ergodic systems, diffusivity (along coordinate direction $ i $) is often defined in terms of a long time trajectory, without any ensemble averaging:
\begin{equation*}
\Differg_i \equiv \lim_{t \rightarrow \infty} \frac{(X_i (t) - X_i (0))^2}{2t}.
\end{equation*}
We'll call this ``single-trajectory diffusivity'' because it refers only to a single trajectory, rather than an ensemble of trajectories.
Explain how you would estimate the single-trajectory diffusivity from your numerical simulations in Part~\ref{part:numerical}.
Apply carefully the concepts from Part~\ref{part:statistical} to show the numerical estimate so obtained for the single-trajectory diffusivity agrees in fact with your analytical calculations for the diffusivity defined as in Part~\ref{part:ensemble}.
\label{part:singletraj}
\item (\textbf{20 bonus points}) Prove theoretically for your original model (not assuming a short time between collisions) that the single-trajectory diffusivity expression in Part~\ref{part:singletraj} agrees with the definition used in Part~\ref{part:ensemble} for the diffusivity defined via an ensemble average.
\item (\textbf{10 points}) Use your numerical simulation to check whether the motion of the particle in different directions is correlated and discuss your findings in light of your above theoretical work.
\item (\textbf{10 points}) Use your numerical simulation to generate a histogram for the position of the particle projected along an arbitrary axis. Describe the shape of this histogram generated over a large number of realizations (experiments) after $ n=5$, $ n=10 $, $n=100 $, and $ n=1000 $ collisions. Examine also how this histogram behaves in the limit described in Part~\ref{part:limit}, and discuss how your observations relate to the theory you developed.
\end{enumerate}
\end{document}