Anomalous kinetics and transport from 1D self-consistent mode-coupling theory

We study the dynamics of long-wavelength fluctuations in one-dimensional (1D) many-particle systems as described by self-consistent mode-coupling theory. The corresponding non-linear integro-differential equations for the relevant correlators are solved analytically and checked numerically. In particular, we find that the memory functions exhibit a power-law decay accompanied by relatively fast oscillations. Furthermore, the scaling behaviour and, correspondingly, the universality class depend on the order of the leading non-linear term. In the cubic case, both viscosity and thermal conductivity diverge in the thermodynamic limit. In the quartic case, a faster decay of the memory functions leads to a finite viscosity, while the thermal conductivity exhibits an even faster divergence. Finally, our analysis puts on a firmer basis the previously conjectured connection between anomalous heat conductivity and anomalous diffusion.