Beyond phonon hydrodynamics: Nonlocal phonon heat transport from spatial fractional-order Boltzmann transport equation

Spatially convoluting formulations have been used to describe nonlocal thermal transport, yet there is no related investigation at the microscopic level such as the Boltzmann transport theory. The spatial fractional-order Boltzmann transport equations (BTEs) are first applied to the description of nonlocal phonon heat transport. Constitutive and continuity equations are derived, and two anomalous behaviors are thereafter observed in one-dimensional steady-state heat conduction: one is the power-law length-dependence of the effective thermal conductivity, κeff∝Lβ with L as the system length, and the other is the nonlinear temperature profile, Tx−Tx=0∼x1+η. A connection between the length-dependence and nonlinearity exponents is established, namely, β=−η. Furthermore, we show that the order of these BTEs should be restricted by the ballistic limit. In minimizing problems, the nonlocal models in this work give rise to different results from the case of Fourier heat conduction, namely that the optimized temperature gradient is not uniform.