In many biological and small scale technological applications particles may transiently bind to a cylindrical surface. In between two binding events the particles diffuse in the bulk, thus producing an effective translation on the cylindrical surface. We here derive the effective motion on the surface allowing for additional diffusion on the cylindrical surface itself. We find explicit solutions for the number of adsorbed particles at one given instant, the effective surface displacement, as well as the surface propagator. In particular sub- and superdiffusive regimes are found, as well as an effective stalling of diffusion visible as a plateau in the mean squared displacement. We also investigate the corresponding first passage problem.
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In our problem we deal with a truncated Cauchy distribution, therefore the normalization factor in Eq. (70) is actually different from 1/π. However, as the truncation is accomplished by the rapidly decaying compressed Gaussian, the correction to 1/π is expected to be small, and the Cauchy part contains almost the entire probability,since t ≪ tκ.
\[\frac{2}{\pi }\int _0^{\sqrt{D_bt}}\frac{\kappa t}{z^2+\kappa ^2 t^2}dz= \frac{2}{\pi }\arctan \sqrt{\frac{t_{\kappa }}{t}}\approx 1,\]
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2011
American Institute of Physics
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