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

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) approximately t(1/2) for overdamped systems and with variance 2W(t) approximately 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.