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

Question

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.

Answer 1

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.