The authors discuss a generalized transport model including memory and nonlocal effects, which aims to describe the transition of heat transport from the diffusive regime to the ballistic regime. By using an effective thermal conductivity depending on the Knudsen number, they describe in a single equation the behavior of conductivity in terms of the system size and a reduction in the limit flux through nanoscale devices.

A thermodynamic approach to rapid phase transformations within a diffuse interface in a binary system is developed. Assuming an extended set of independent thermodynamic variables formed by the union of the classic set of slow variables and the space of fast variables, we introduce finiteness of the heat and solute diffusive propagation at the finite speed of the interface advancing. To describe transformations within the diffuse interface, we use the phase-field model which allows us to follow steep but smooth changes of phase within the width of the diffuse interface. Governing equations of the phase-field model are derived for the hyperbolic model, a model with memory, and a model of nonlinear evolution of transformation within the diffuse interface. The consistency of the model is proved by the verification of the validity of the condition of positive entropy production and by outcomes of the fluctuation-dissipation theorem. A comparison with existing sharp-interface and diffuse-interface versions of the model is given.

We use phonon hydrodynamics with a surface slip flow as a simplified macroscopic model accounting for a reduction in lateral thermal conductivity in nanosystems. For high Knudsen numbers, the corresponding effective thermal conductivity decreases linearly with the radius or the width, in contrast with the quadratic dependence predicted by usual phonon hydrodynamics. The linear dependence is accounted for by the surface slip flow. The difference in the expressions for the surface tangential flow in the hydrodynamic and the diffusive regimes is commented on and the influence of boundary conditions on the form of the effective thermal conductivity is explored.