Universality and non-universality of mobility in heterogeneous single-file systems and Rouse chains

We study analytically the tracer particle mobility in single-file systems with distributed friction constants. Our system serves as a prototype for non-equilibrium, heterogeneous, strongly interacting Brownian systems. The long time dynamics for such a single-file setup belongs to the same universality class as the Rouse model with dissimilar beads. The friction constants are drawn from a density $\varrho (\xi )$ and we derive an asymptotically exact solution for the mobility distribution $P[{\mu }_0(s)]$, where ${\mu }_0(s)$ is the Laplace-space mobility. If $\varrho$ is light-tailed (first moment exists) we find a self-averaging behaviour: $P[{\mu }_0(s)]=\delta [{\mu }_0(s)-\mu (s)]$ with $\mu (s)\propto s^{1\mathrm{/}2}$. When $\varrho (\xi )$ is heavy-tailed, $\varrho (\xi )\simeq {\xi }^{-1-\alpha }(0<\alpha <1)$ for large $\xi$ we obtain moments $⟨[{\mu }_s(0){]}^n⟩\propto s^{\beta n}$ where $\beta =1\mathrm{/}(1+\alpha )$ and no self-averaging. The results are corroborated by simulations.