Derivation of fluctuating hydrodynamics and crossover from diffusive to anomalous transport in a hard-particle gas

A recently developed non-linear fluctuating hydrodynamics theory has been quite successful in describing various features of anomalous energy transport. However the diffusion and the noise terms present in this theory are not derived from microscopic descriptions but rather added phenomenologically. We here derive these hydrodynamic equations with explicit calculation of the diffusion and noise terms in a one-dimensional model. We show that in this model the energy current scales anomalously with system size $L$ as $\sim L^{-2/3}$ in the leading order with a diffusive correction of order $\sim L^{-1}$. The crossover length $\ell_{c}$ from diffusive to anomalous transport is expressed in terms of microscopic parameters. Our theoretical predictions are verified numerically.