Single-File Diffusion of Atomic and Colloidal Systems: Asymptotic Laws

In this work we present a general derivation of the non-Fickian behavior for the self-diffusion of identically interacting particle systems with excluded mutual passage. We show that the conditional probability distribution of finding a particle at position $x_{t}$ after time $t$, when the particle was located at $x_{0}$ at $t=0$, follows a Gaussian distribution in the long-time limit, with variance $2W(t)\sim t^{1/2}$ for overdamped systems and with variance $2W(t)\sim t$ for classical systems. The asymptotic behavior of the mean-squared displacement, $W(t)$, is shown to be independent of the nature of interactions for homogeneous systems in the fluid state. Moreover, the long-time behavior of self-diffusion is determined by short-time and large scale collective density fluctuations.