A gradient system for a higher-gradient generalization of Fourier’s law of heat conduction

Abstract

We derive a generalized heat conduction problem for a rarefied gas at slip regime from a gradient system where the driving functional is the entropy. Specifically, we construct an Onsager system [Formula: see text] such that the associated evolution of the system is given by [Formula: see text], where the Onsager operator, [Formula: see text], contains higher-gradients of the absolute temperature [Formula: see text]. Moreover, through Legendre–Fenchel theory, we write the Onsager system as a classical gradient system [Formula: see text] with an induced gradient flow equation, [Formula: see text]. We demonstrate the usefulness of the approach by modeling scale-size thermal effects in periodic media that have been recently observed experimentally.