Diffusional escaping from the well. Simple model and qualitative results

A simple two-state model of diffusional escaping from a potential well is proposed. The two states correspond to particles inside and outside the well. Time evolution of both states is described by distribution functions (DFs). In the first state (inside the well), DF is assumed quasiequilibrium (thermal) during the escaping process. In the second state (outside the well), the evolution of DF is governed by the free diffusion equation. The escaping/capture process is approximated by a simple kinetic coupling between these two states. In the limit of high escaping and capture rates, the model reproduces almost all expressions obtained earlier for escaping kinetics by rigorous solution of the Smoluchowsky equation (SE). The model is applied to 1D escaping from the well in the presence of some other wells. The effect of interaction anisotropy on escaping kinetics is also discussed. As an example, recombination of radicals with strongly anisotropic (depending on relative orientation of radicals) interaction is considered. The kinetics is shown to be very close to exponential in this process. Magnetic field effects (MFEs) in the presence of the well are discussed briefly within the two-state model. This simple model reproduces rigorous results obtained for MFEs by the SE approach and permits some generalizations of the results.