Anomalous transport in the Fermi-Pasta-Ulam-Tsingou model: a review and open problems

This review provides an up-to-date account of energy transport in Fermi-Pasta-Ulam-Tsingou (FPUT) chains, a key testbed for nonequilibrium statistical physics. We discuss the transition from the historical puzzle of thermalization to the discovery of anomalous heat transport, where the effective thermal conductivity $\kappa$ diverges with system size $L$ as $\kappa \propto L^\delta$. The article clarifies the distinction between two universality classes: the FPUT-$\alpha \beta$ model, characterized by $\delta = 1/3$ and linked to Kardar-Parisi-Zhang (KPZ) physics, and the symmetric FPUT-$\beta$ model, where numerical and theoretical evidence support $\delta = 2/5$. We investigate how finite-size effects - unavoidably induced by the thermostatting protocols - can disguise the asymptotic scaling. Additionally, we analyze the role of conservative noise in preserving hydrodynamic properties and examine how proximity to integrable limits leads to long-lived quasi-particles and, thereby, to diffusive regimes over intermediate spatial scales.