Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations

A dynamical nonequilibrium temperature has been proposed to describe relaxational equations for the heat flux. This temperature provides an alternative description to the Maxwell-Cattaneo equation. In the linear regime and in bulk systems both descriptions are equivalent but this is not so when nonlinear effects are included. Here we explore the influence of nonlinear terms on the phase speed of heat waves in nonequilibrium steady states in both theoretical models and we show that their predictions are different. This could allow to explore which description is more suitable, when experiments on these situations will become available. Furthermore, we have analyzed a nonlinear and nonlocal constitutive equation for the heat flux and we have shown its analogy with the Navier-Stokes equation in the regime of phonon hydrodynamics in nanosystems. This analogy allows one to define a dimensionless number for heat flow, analogous to the Reynolds number, and to predict a critical heat flux where nonlinear effects could become dominant.