Random Walks

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\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \def\lya{Ly\alpha}

\section{Random Walks}

If $\vec\phi=\phi_1,\phi_2,\dots$ is a random variable consisting of a series of random values, then a random walk $\vec w$ is a sum over this random variable: \begin{equation} w_i=\sum_{n=1}^{i}\phi_n \end{equation}

Random walks have the property that \begin{equation} |w_n|\propto\sqrt{n}, \end{equation} or more generally: \begin{equation} |w_n-w_m|\propto\sqrt{n-m} \end{equation}


A fun and useful example of random walks that anyone can try is with a few volunteers and perfume. Have the volunteers stand equally spaced in a line, then go to one end of the line and spray perfume in the air. Record the distances of the volunteers from the epicenter and time how long it takes for each one to smell the perfume. You will find that the time until the perfume is detected goes with the square of the distance, so that if someone at 1 meter takes 10 seconds to smell the perfume, someone at 3 meters will take 90 seconds.

This is because the particulates in the air carrying the perfume smell all start in the same place and get jostled in random directions by air molecules. Therefore, they all spread from their starting point through random walks, so the absolute distance they travel scales with the square root of number of steps they take (and therefore the time). In other words, the distance squared is proportial to the time.

The exact same phenomenon can be seen with Plinko Boards, spilling salt on a table, the stock market, and numerous other examples.

\section{Random walks inside a star}

Radiative energy transport inside stars can be approximated as a random walk, where a photon gets repeatedly absorbed and re-emitted in a random direction as it tries to make it way out of through the stellar interior. The step length of this random walk is equal to the mean free path $\lambda$. For the sun-type star, $\lambda \sim 10^{-2}$ cm. The number of steps required to reach the stellar radius r is, $N = (r/\lambda)^{2}$. For the sun, $r \sim 10^{11}$ cm, so $N \sim 10^{26} $ steps. In each step, light travels approximately a distance $\lambda$. Therefore by the time the photon reaches the surface it has traveled $\sim N \times \lambda = 10^{24}$ cm. At the speed of light, that would take $\sim 10^{7}$ years!

The figure below shows a simulation of a random walk process inside a hypothetical star of radius 10 cm. Why such a small star? If my (non-super) computer runs through $10^{6}$ random walk steps in a second, then it would take $10^{20}$ seconds to complete the random walk of a single photon moving through the entire sun's radius. That is twice the age of the universe! I doubt I can get an extension on this project for that long.

Random walk of three photons inside a star.

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