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

摘要

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) Delta x(2), namely, Delta x(2) similar to t(gamma) with 1 gamma = 2. We show that this fractional-order BTE predicts a fractional-order constitutive equation and divergent effective thermal conductivity kappa(eff). In the long-time limit, the divergence obeys a power-law type kappa(eff) similar to t(alpha), while the asymptotics of Delta x(2) reads gamma = alpha + 1. This connection between kappa(eff) and Delta x(2) coincides with previous investigations such as the linear response and Levy-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. (C) 2019 Elsevier Ltd. All rights reserved.