Fractional-order heat conduction models from generalized Boltzmann transport equation

Philos Trans A Math Phys Eng Sci. 2020 May 29;378(2172):20190280. doi: 10.1098/rsta.2019.0280. Epub 2020 May 11.


The relationship between fractional-order heat conduction models and Boltzmann transport equations (BTEs) lacks a detailed investigation. In this paper, the continuity, constitutive and governing equations of heat conduction are derived based on fractional-order phonon BTEs. The underlying microscopic regimes of the generalized Cattaneo equation are thereafter presented. The effective thermal conductivity κeff converges in the subdiffusive regime and diverges in the superdiffusive regime. A connection between the divergence and mean-square displacement 〈|Δx|2〉 ∼ tγ is established, namely, κefftγ-1, which coincides with the linear response theory. Entropic concepts, including the entropy density, entropy flux and entropy production rate, are studied likewise. Two non-trivial behaviours are observed, including the fractional-order expression of entropy flux and initial effects on the entropy production rate. In contrast with the continuous time random walk model, the results involve the non-classical continuity equations and entropic concepts. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Keywords: Boltzmann transport equation; constitutive equation; continuity equation; fractional-order derivative; heat conduction.