Spatiotemporal properties of diffusive systems with a mobile imperfect trap

We study analytically a one-dimensional system initially uniformly filled with diffusing particles $A$, and a single imperfect mobile trap $T$ initially located at the origin, $x=0\mathrm{}$. For arbitrary values of diffusion constants $D_A$ and $D_T,$ and any trapping rate constant $V$, we calculate exactly the total rate of trapping as well as the asymptotic concentration of $A$’s at $x=0\mathrm{}$. For $D_A{=D}_T$ we also analytically derive the local rate of trapping and the concentration of $A$’s at any point $x$. Characteristic length scales and extensions to higher dimensions are also discussed.