Past exam of the soft condensed matter course of the University of Cambridge / 2022 / Section B / 5 / d / Solution

Put $k=\beta F$ and $q=k\delta$. The absorbing boundary condition for a diffusion gives $A=j_0{e}^q/(Dk)$, henceThis is nonnegative throughout the allowed interval. Normalizing the probability density givesThereforeEquivalently the normalized density isThe denominator is positive for $q>0$, since $e^q>1+q$. The dimensions are consistent: $j_0$ has units of inverse time, $A$ and $p$ of inverse length, and $q$ is dimensionless.