Short-time diffusion behavior of Brownian particles in confining potentials

Diffusion behavior of Brownian particles in confined spaces was studied for the displacements notably shorter than the confinement size. The confinements, resembling structure of porous solids, were modeled using a spatially-varying potential field with an infinitely large potential representing the solid part and zero potential in the void space. Between them, a smooth transient mimicking the interaction potential of the tracer molecules with the pore walls was applied. The respective Smoluchowski equation describing diffusion of tracer particles in the thus created force field was solved under certain approximations allowing for a general analytical solution. The time-depended diffusion coefficient obtained was found to agree with that obtained earlier using the Fick's diffusion equation, but with a different numerical constant. Numerical solution of the Smoluchowski equation with selected interaction potentials was used to clarify the origin of this discrepancy. The conditions under which the solutions obtained within these two approaches converge were established.