“…Thus, even though Onsager relations are satisfied, the coefficient's dependence on nonequilibrium state variables and external fields, indicates that any variational principle associated with the entropy production [132,134,135,140,141], is not applicable to the present case [69,142] In the absence of the BGK reactive term, null magnetic field, and for the strictly stationary inhomogeneous bath case, we recover the inhomogeneous media advection-diffusion equation, presented by [96] and given by ∂n(x,t) ∂t = ∇ (λ(x) [n(x,t)∇U (x) + n(x,t)∇T (x) + T (x)∇n(x,t)]) (62) and for the rigid conductor (constant particle density n), no reactive term and null external fields, we retrieve the inertial corrections to Fourier's law, known as the Maxwell-Cattaneo equation [137]- [139] (of the hyperbolic type, known also as the telegrapher's equation)…”