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Imagine I have a set of $(s_1,...,s_N) \in S$ Brownian particles in a box of sidelength $L$, each with the same coefficient of diffusion $D$. We fix one particle at the center of the box, and draw a sphere of radius $r$ about it. Equivalently, we draw a sphere of the same radius around one of the diffusing particles without bothering to fix the particle in place. Fick's law can be used to tell us the diffusive flux through this sphere as a function of time.

Let $V$ be some duration of time in seconds. What is the probability that the sphere only contains the one Brownian particle at its center for a continuous stretch of $q \leq R$ seconds over this time interval?

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  • $\begingroup$ Where do the particles start? What happens when they hit the boundary of the box? $\endgroup$
    – Nate Eldredge
    Commented Nov 11, 2013 at 16:39
  • $\begingroup$ How is fixing a particle at the center of the box equivalent to letting the sphere move with a particle? Even without boundary effects, subtracting one Brownian motion from multiple independent Brownian motions gives you dependent Brownian motions. $\endgroup$
    – Douglas Zare
    Commented Nov 12, 2013 at 2:55

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