Anomalous heat diffusion from fractional Fokker–Planck equation

Abstract

Anomalous heat diffusion, which is commonly characterized by the nonlinear growth of mean square of displacement (MSD), |Δx|2∼tβ (0<β≤2), is usually paired with a length-dependence of effective thermal conductivity κeff, namely, κeff∼Lα with L the system length. In this work, a generic time- and length-dependence of κeff is obtained based on the fractional Fokker–Planck equation (FFPE) with orders γ,μ∈R2, namely, κeff∝tγ−1L−μ. Two existing paradigmatic results, κeff∝tβ−1 and κeff∝L2−2∕β, 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.