Fractional-order theory of heat transport in rigid bodies

• A new theory of fractional-order heat transport is introduced in the paper.
• Fractional operators involve non-local thermal exchanges in the conductor.
• Non-local effects depend of generalized measures of relative temperatures.
• Thermodynamic consistency has been assessed for diffusive/ballistic heat transfer.
• Numerical simulations have been reported in the paper.

The non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53], Mongioví and Zingales, 2013 [54]) that thermal energy transport is due to two phenomena: (i) A short-range heat flux ruled by a local transport equation; (ii  ) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law decay of the distance yielding a novel temperature equation in terms of αα-order Marchaud fractional-order derivative (0⩽α⩽1)(0⩽α⩽1). Thermodynamical consistency of the model is provided in the context of Clausius–Plank inequality. The effects induced by the boundary conditions on the temperature field are investigated for diffusive as well as ballistic local heat flux. Deviations of the temperature field from the linear distributions in the neighborhood of the thermostated zones of small-scale conductors are qualitatively predicted by the used fractional-order heat transport model, as shown by means of molecular dynamics simulations.