# First passage time of diffusing particle with partially absorbing boundary

Given the solution to the spatiotemporal evolution of a single particle on a 1-D surface $P(x,t)$ a nice result (that I gleaned elsewhere on physics.SE) is that for a boundary at $x=0$ where $x_0>0$,

1) a reflecting boundary condition at $x=0$ for the particle is represented exactly by

$$P_{ref}(x,t|x_0)|_0^\infty=\frac{1}{\sqrt{4 \pi D t}}e^{-\frac{(x-x_0)^2}{4Dt}}+\frac{1}{\sqrt{4 \pi D t}}e^{-\frac{(x+x_0)^2}{4Dt}}$$

This imposes the condition $\frac{\partial P}{\partial x}|_{x=0}=0$, i.e., "no flux".

2) an absorbing boundary at $x=0$ (i.e., all collisions with the boundary are lost to the system) is represented exactly by

$$P_{abs}(x,t|x_0)|_0^\infty=\frac{1}{\sqrt{4 \pi D t}}e^{-\frac{(x-x_0)^2}{4Dt}}-\frac{1}{\sqrt{4 \pi D t}}e^{-\frac{(x+x_0)^2}{4Dt}}$$

Unlike the reflecting case, the absorbing case tracks surviving particles such that $\int_0^\infty P(x,t|x_0) dt$ corresponds to $P(\text{no collision with boundary at time } t)$, i.e., "first passage".

I'm interested in the intermediate case (survival probability) where a particle that collides with the the boundary has probability $p_r$ in reflecting and $1-p_r$ in being absorbed. This corresponds to a type of Robin (radiative) boundary condition. It is tempting to think that combining the image approach above would solve this problem generically, that is:

$$P_{rad}(x,t|x_0)|_0^\infty=\frac{1}{\sqrt{4 \pi D t}}(e^{-\frac{(x-x_0)^2}{4Dt}}+p_r e^{-\frac{(x+x_0)^2}{4Dt}}-(1-p_r)e^{-\frac{(x+x_0)^2}{4Dt}})$$

In fact, some of the properties of $P_{rad}$ at the boundary suggest this might be true despite the unprincipled derivation. For example, the flux of probability density lost in the full absorbing case ($P_{abs}$, or $p_r=0$) is:

$$J_{abs}=-\frac{x_0}{2t^{3/2} \sqrt{\pi D}} e^{-\frac{x_0^2}{4 D t}}$$

This is one extreme case, if every collision at the boundary led to absorption. The other extreme is the reflecting case where where $J_{ref}=0$. Computing $J_{rad}$ for arbitrary $p_r$ I get an ugly mess which I won't show here, but renormalizing the flux over the maximum flux I find:

$$\frac{J_{rad}}{J_{abs}} = 1-p_r$$

In other words, the rate of absorption out of the system linearly increases from $p_r=1$, where flux is zero, and all collisions are reflected, to $p_r=0$ and $J_{rad}=J_{abs}$.

Question: Is $P_{rad}$ a solution to the actual problem, i.e., what is the survival density for a particle diffusing between a boundary at $x=0$ and infinity where the boundary absorbs a colliding particle with probability $1-p_r$ and reflects otherwise? I.e., is $\int_0^\infty P_{rad}(x,t) dx$ equal to $P(\text{particle has not been absorbed at time } t)$?

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