Extended irreversible thermodynamics

Remarks

Example: for heat conduction in a rigid solid, take $s(u, q) = s_{eq}(u) - \frac{\tau}{2 \rho k T^2} \abs{q}^2$ with relaxation time $\tau > 0$, density $\rho$, conductivity $k > 0$ and temperature $T$. Requiring non-negative entropy production in the simplest linear way gives the Maxwell–Cattaneo law $$\tau \frac{\partial q}{\partial t} + q = -k \nabla T,$$ which reduces to Fourier's law when $\tau = 0$. Non-example: classical linear irreversible thermodynamics with Fourier's law fails clause 1, since there $q$ is a function of $\nabla T$, not an independent variable.