Anomalous transport in low-dimensional systems: A pedagogical overview

This manuscript contains the notes of the lectures given in July 2021 at the Summer School “Fundamental Problems in Statistical Physics XV”, Bruneck, Italy. A pedagogical approach has been adopted by first providing the students a summary of the basic concepts of non-equilibrium statistical mechanics associated with transport processes. A detailed account about anomalous heat transport in anharmonic lattices is the first research topic described in these lecture notes. The interest of these results for material science is then highlighted, in order to point out the applicative potentials of anomalous transport in nanostructures, polymers etc. Further details about anomalous transport phenomena are reported in the presence of a magnetic field or of long-range interactions. A large section of these lecture notes is devoted to the theoretical approaches adopted for obtaining a consistent description of anomalous transport phenomena, including recent achievements that point out the crucial importance of finite-size effects on theoretical predictions and, accordingly, on experimental verifications. The final section is devoted to a summary about coupled transport in low-dimensional systems and the consequences of the presence of anomalous diffusion processes in this complex scenario. The appendices provide additional information about transport in basic 1d models (harmonic chains and rotor model), in 2d lattices and about integrable nonlinear models.