# Residency time of a spherical Brownian particle in a cylindrical container with another spherical particle at a fixed position

I place two spherical particles, $P_1$ and $P_2$ (with radii $r_1$ & $r_2$), into a cylindrical container of radius $r_c$ ($r_1$ & $r_2$ $\leq \frac{1}{2}r_c$) and height $h$. While $P_1$ is immobilized at the centerpoint of the cylindrical container, $P_2$ has a coefficient of diffusion $D$, and can freely diffuse throughout the container and across its walls (i.e. the boundaries of the cylindrical container are non-reflecting).

We randomly position $P_1$ and $P_2$ somewhere inside the cylindrical container. Can we derive an expression for the mean residency time of $P_2$ as a function of the relative sizes of the particles and the cylindrical container? Is it a fair approximation to simply estimate the residency time of $P_2$ in a cylindrical container resized to subtract the volume of $P_1$?

Update - $P_1$ is now fixed at the centerpoint of the cylindrical container, and $r_1$ & $r_2$ are defined to be $\leq \frac{1}{2}r_c$, s.t. $P_1$ cannot block access, say, to half the cylindrical container.

Update 2 - In practice, $r_1$, $r_2$, and $r_c$ will be within one or two orders of magnitude of one another, i.e. it is not the case that $r_1$ & $r_2$ $<< \frac{1}{2}r_c$). Also, all collisions between particles, like the walls of the container, are fully reflecting.

Update 3 - I'd be perfectly happy calling the cylindrical container a "spherical container" with the same radius, $r_c$. Similarly, I'd be happy to not pin $P_1$ at any particular point, and instead to simply make the walls of the sphere reflecting specifically for this particle.

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You haven't said anything about the relative sizes of the particles and the cylindrical container. Clearly, in one extreme case (P1's radius is as large as the radius of the cylinder so it completely blocks access to part of the cylinder) P1 matters. Just as clearly, P1 doesn't matter in the other extreme case (e.g. the cylinder has a diameter measured in light years and P1 and P2 are the size of small molecules.) You need to tell us something more about what you're actually trying to model... – Brian Borchers Jul 8 '11 at 18:21
@Brian, you're absolutely right. Hopefully the added clarifications will better address your concerns. – Rob Grey Jul 8 '11 at 19:56
Not really. If r_1 and r_2 are far smaller than r_c, then the additional particle shouldn't have any significant effect on the escape time. The problem only becomes interesting if r_1 is big enough. You also haven't told us what happens if particle 1 and particle 2 interact with each other. Does particle 2 just bounce off particle 1? – Brian Borchers Jul 8 '11 at 23:28
Rob, there's an hour to go on this bounty, and one answer which is very reasonable. Is there a problem with the answer? – Nilima Nigam Jul 20 '11 at 1:23

## 1 Answer

You can use the Feynman-Kac formula to get the Moment Generating Function of the time it takes the particle to leave.

I will consider the case where you fix $P_1$ and let $P_2$ move. Whatever the geometry of your problem, you can get an equivalent problem with a point particle diffusing in some region in space, where there is one wall that it reflects off (let's call it $\Gamma_0$) and another where it is absorbed (let's call it $\Gamma_1$.) Let $X(t)$ be the position of the particle at time $t$. Let $T$ be the time at which the process first hits $\Gamma_1$. Let $f(x,\lambda) =E [ e^{\lambda T} | X(0)=x]$. Then Feynman-Kac gives you that $f$ satisfies $\nabla^2 f/2 + \lambda f =0$ with $\frac{\partial f}{\partial n} = 0$ on $\Gamma_0$ and $f=1$ on $\Gamma_1$. This is the Helmholtz equation on your domain with mixed boundary conditions. $f(x,\lambda)$ is the moment generating function of $T$ with $X(t)=x$, evaluated at $\lambda$. So now you can compute the mean of $T$, etc.

One geometry for your problem for which you can get an exact solution for each $\lambda$ is if you have one sphere fixed at the middle of the big sphere and you are tracking a point particle that diffuses, bounces off the small sphere and is absorbed by the big sphere. In that case the solutions to the PDE are spherical Bessel functions.

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