Single-file Diffusion with Random Diffusion Constants

The single-file problem of N particles in one spatial dimension is analyzed, when each particle has a randomly distributed diffusion constant D sampled in a density $\rho(D)$. The averaged one-particle distributions of the edge particles and the asymptotic ($N\gg 1$) behaviours of their transport coefficients (anomalous velocity and diffusion constant) are strongly dependent on the D-distribution law, broad or narrow. When $\rho$ is exponential, it is shown that the average one-particle front for the edge particles does not shrink when N becomes very large, as contrasted to the pure (non-disordered) case. In addition, when $\rho$ is a broad law, the same occurs for the averaged front, which can even have infinite mean and variance. On the other hand, it is shown that the central particle, dynamically trapped by all others as it is, follows a narrow distribution, which is a Gaussian (with a diffusion constant scaling as $N^{-1}$) when the fractional moment $<D^{-1/2}>$ exists and is finite; otherwise ($\rho(D)\propto D^{\alpha-1}, \alpha\le\case12$), this density is, far from the origin, a stretched exponential with an exponent in the range $]0, 2]$; then the effective diffusion constant scales as $N^{-\beta}$, with $\beta = 1/(2\alpha)$.