Anomalous heat diffusion from fractional Fokker-Planck equation

Anomalous heat diffusion, which is commonly characterized by the nonlinear growth of mean square of displacement (MSD), <vertical bar Delta x vertical bar(2)> similar to t(beta) (0 < beta <= 2), is usually paired with a length-dependence of effective thermal conductivity kappa(e)(ff), namely, kappa(e)(ff) similar to L-alpha with L the system length. In this work, a generic time and length-dependence of kappa(e)(ff) is obtained based on the fractional Fokker-Planck equation (FFPE) with orders (gamma, mu) is an element of R-2, namely, kappa(e)(ff) proportional to t(gamma-1) L-mu. Two existing paradigmatic results, kappa(e)(ff) proportional to t(beta-1) and kappa(e)(ff) proportional to L2-2/beta, are first unified in our work, which reflect memory effects and nonlocality in energy fluctuations, respectively. We formulate the effective thermal conductivity in terms of entropy generation, which does not rely on the local-equilibrium hypothesis. (C) 2019 Elsevier Ltd. All rights reserved.