Anomalous heat transport in harmonic long-range chains

In recent years there has been a remarkable rise of interest in the study of heat transport in long-range systems, i.e, systems in which the interaction between constituents scales as a power-law of their distance. The results present in the literature suggest that heat transport in long-range systems can be anomalous, but the understanding of transport properties in these systems remains a difficult and open problem, especially if one wants to obtain analytical results. We thus decided to analyze heat transport properties in linear long-range chains in one dimension, and this work contains the results of this analysis. We begin by considering the simplest long-range system: the fully-connected chain coupled to two external baths. In this case we are able to analytically extract the scaling of the heat-flux with the system's size: this system thus proved a reference result for more complex interactions. We then consider the case of a power-law long-range chain coupled to two baths: in this case we show numerically that the heat flux scales as non-trivial power of system's size and we provide an analytical estimate of the exponent. Finally, we consider a power-law long-range ring with stochastic collisions: in this case we are able to exactly compute the scaling exponent of the thermal conductivity using the Green-Kubo formula, and we find good agreement between our results and numerical simulations.