The form and the role of the entropy flux in the thermodynamic analysis of the transport equations are essentially open questions in nonequilibrium thermodynamics. In particular, nonlocal heat-transport equations at nanoscale may exhibit some peculiar behaviors which seem to violate well-known statements of the second law of thermodynamics. Here we examine one of these behaviors in axial heat transport from the perspective of a generalized entropy flux, i.e., ${\mathbf{J}}^{\left(s\right)}=\mathbf{q}/T+\mathbf{k}$, and show that such a generalization allows it to be consistent with the second law. In contrast with previous formal analyses, this paper provides an explicit form for the nonclassical part of the entropy flux, that is, $\mathbf{k}={\ell }^2/\left(\lambda T^2\right)\nabla {\mathbf{q}}^{\mathrm{T}}·\mathbf{q}$ and links it to a concrete physical phenomenon which is accessible to current experimental possibilities for systems with sufficiently long mean-free path $\ell$, whereas for short enough $\ell$ the classical results are recovered. The derivation of the nonclassical part of the entropy flux is obtained within the frame of extended irreversible thermodynamics from two different perspectives, namely, a 13-field theory with higher-order fluxes and a 4-field theory with higher-order gradients.

• Received 29 July 2012

DOI:https://doi.org/10.1103/PhysRevB.87.054302