On diffusion in an external field and the adjoint source problem

If diffusion in an external field is described by $\partial \rho /\partial t = D{\nabla ^2}\rho - \rho /\tau - \nabla \cdot \left ( {F\left ( r \right )\rho } \right )$, the function $\gamma \left ( {{r_0}} \right )$ describing the probability that a particle at ${r_0}$ will reach a collector surface before decaying or being absorbed by other surfaces satisfies the equation $D{\nabla ^2}\gamma - \gamma /\tau + F\left ( r \right ) \cdot \nabla \gamma = 0$. This equation has no singularity to disturb any geometric symmetry available. Boundary conditions on $\gamma \left ( r \right )$ at the collector surface and other influencing surfaces are derived and shown to be independent of the external field. The boundary conditions at the secondary surfaces are homogeneous. The collector surface boundary condition is inhomogeneous.