Abstract
The Brownian description of heat conduction proposed by Razi-Naqvi and Waldenstrøm (2005) is generalized into non-Brownian anomalous heat conduction. It is found that there exists an entropic relation between the heat equation and Fokker–Planck equation (FPE) of energy fluctuations. Based on the entropic relation and fractional Brownian motion (FBM), we propose an anomalous heat equation (AHE), which is able to perform Brownian and non-Brownian long-time asymptotics of the mean-square displacement (MSD), Δx2∝tβ. The AHE predicts a power-law length-dependence of the effective thermal conductivity κeff connected to the MSD, namely, κeff∝Lβ−1 with L denoting the system length. This scaling connection has been observed in the Lévy Walk (LW) model and linear response theory. Due to the coincidences with existing studies, the AHE can be considered as phenomenological models for anomalous heat conduction.
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