Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity

Abstract

Anomalous heat transport and divergent thermal conductivity have attracted increasing attention in recent years. The linearized Boltzmann transport equation (BTE) proposed by Goychuk is discussed in superdiffusive and ballistic heat conduction, which is characterized by super-linear growth of the mean-square displacement (MSD) Δx2, namely, Δx2∼tγ with 1<γ⩽2. We show that this fractional-order BTE predicts a fractional-order constitutive equation and divergent effective thermal conductivity κeff. In the long-time limit, the divergence obeys a power-law type κeff∼tα, while the asymptotics of Δx2 reads γ=α+1. This connection between κeff and Δx2 coincides with previous investigations such as the linear response and Lévy-walk model. The constitutive equation from Goychuk’s model is compared with a class of fractional-order models termed generalized Cattaneo equation (GCE). We show that Goychuk’s model is more appropriate than other models of the GCE class to describe superdiffusive and ballistic heat conduction.