Self-diffusion of fluids in narrow cylindrical pores

Fluids under stochastic dynamics in narrow cylindrical pores exhibit a dynamical transition from single-file diffusion (SFD) to Fickian bulk diffusion. For long time, the mean square displacement will change as the pore size increases, with a transition from SFD, ${\mathrm{\sim t}}^{\text{1/2}},$ to Fickian, $\mathrm{\sim t,}$ while the diffusion coefficient ${\mathrm{(D}}_{\mathrm{xx}})$ increases from zero. We present a theory of this important process in terms of a hopping time, ${\tau }_{\mathrm{hop}},$ leading to $D_{\mathrm{xx}}{\mathrm{\propto (\tau }}_{\mathrm{hop}}{)}^{\text{−1/2}}$ which is verified with simulation. While the crossover is to be expected, the simple form is a priori unanticipated and is likely to be universal.