4.3.1. Non-Equilibrium Thermodynamics of Heat

Kinematics of heat transfer can be thought of as kinematics of phonons, and kinematics of phonons has been successfully described by Boltzmann-like dynamics, where the distribution function of phonons plays the role of state variable, see e.g., book [31]. By the reduction from the kinetic theory to fluid mechanics, see e.g., [1,14], kinematics of phonons can be expressed in terms of the entropy density and momentum related to entropy transport, the kinematics of which is expressed by a hydrodynamic-like Poisson bracket (see [32]). Subsequent transformation to density of matter *ρ*, total momentum of matter and phonons **m**, entropy density *s* and conjugate entropy flux **w**, which is equal to the ratio of phonon momentum and entropy density, leads to bracket

$$\begin{aligned} \left\{ F, G \right\}^{\left( \text{Cat} \right)} &= & \left\{ F, G \right\}^{\left( \text{FM} \right)} \Big|\_{\mathbf{u} = \mathbf{m}} + \int \text{d} \mathbf{r} \big( G\_{\text{w}\_{i}} \partial\_{i} F\_{\text{s}} - F\_{\text{w}\_{i}} \partial\_{i} G\_{\text{s}} \big) \\ &+ \int \text{d} \mathbf{r} w\_{j} \left( \partial\_{i} F\_{\text{w}\_{i}} G\_{\text{m}\_{j}} - \partial\_{i} G\_{\text{w}\_{i}} F\_{\text{m}\_{j}} \right) \\ &+ \int \text{d} \mathbf{r} \left( \partial\_{i} w\_{j} - \partial\_{j} w\_{i} \right) \left( F\_{\text{w}\_{i}} G\_{\text{m}\_{j}} - G\_{\text{w}\_{i}} F\_{\text{m}\_{j}} \right) \\ &+ \int \text{d} \mathbf{r} \frac{1}{s} \left( \partial\_{i} w\_{j} - \partial\_{j} w\_{i} \right) F\_{\text{w}\_{i}} G\_{\text{w}\_{j}}. \end{aligned} \tag{58}$$

expressing kinematics of matter and heat—the Cattaneo Poisson bracket. The name Cattaneo is due to the implied hyperbolicity of heat transport [19]. The Poisson bracket (58) generates reversible evolution equations

$$\begin{aligned} \frac{\partial \rho}{\partial t}\_{i} &= -\partial\_{k}(\rho E\_{m\_{k}})\_{i} \\\\ \frac{\partial m\_{i}}{\partial t}\_{i} &= -\partial\_{k}(\rho E\_{m\_{k}})\_{i} - \partial\_{k}(\rho E\_{m\_{k}})\_{i} - \partial\_{k}E\_{m\_{k}} - \rho \nabla\_{k}E\_{m\_{k}} - \rho \nabla\_{k}E\_{m\_{k}} \end{aligned} \tag{59a}$$

$$\begin{split} \frac{\partial \mathcal{E}^{\rm m\_{\parallel}}}{\partial t} &= -\partial\_{\dot{\jmath}}(m\_{i}E\_{\rm m\_{\dot{\jmath}}}) - \partial\_{\dot{\jmath}}(w\_{i}E\_{\rm w\_{\dot{\jmath}}}) - \rho \partial\_{\dot{\imath}}E\_{\rho} - m\_{\dot{\jmath}}\partial\_{i}E\_{\rm w\_{\dot{\jmath}}} - s\partial\_{i}E\_{s} \\ &- w\_{k}\partial\_{\dot{\imath}}E\_{\rm w\_{k}} + \partial\_{\dot{\imath}}(E\_{\rm w\_{k}}w\_{k}), \end{split} \tag{59b}$$

$$\frac{\partial \mathbf{s}}{\partial t}\_{\mathbf{j}} = -\partial\_{\mathbf{k}} \left( s E\_{m\_{\mathbf{k}}} + E\_{w\_{\mathbf{k}}} \right), \tag{59c}$$

$$\frac{\partial \stackrel{\cdots}{w\_k}}{\partial t}\_{\cdot} = -\partial\_k E\_s - \partial\_k (w\_j E\_{w\_j}) + (\partial\_k w\_j - \partial\_j w\_k) \left( E\_{w\_j} + \frac{1}{s} E\_{w\_j} \right). \tag{59d}$$

These evolution equations express reversible dynamics of fluid mechanics and conjugate entropy flux **<sup>w</sup>**. Note that, for the heat flux, i.e., flux of energy, the usual relation **<sup>q</sup>** = *EsEwk* = *<sup>T</sup>***J**(*s*) holds true, see [14] for more details.

Local dissipation is enforced by adopting an algebraic dissipation potential, the simplest of which is

$$\Xi(\mathbf{w}^\*) = \int \mathrm{d}\mathbf{r} \, \frac{1}{2} \frac{1}{\tau} (\mathbf{w}^\*)^2 = \int \mathrm{d}\mathbf{r} \, \frac{1}{2} \frac{1}{\tau} \left(\frac{1}{s^\dagger} \mathbf{w}^\dagger\right)^2 \,, \tag{60}$$

where the last equality follows from the transformation between energetic and entropic representation. Then, the irreversible terms generated by this dissipation potential are added to the reversible evolution, Equations (59),

$$\frac{\partial \rho}{\partial t} = -\partial\_k (\rho m\_k^\dagger)\_\prime \tag{61a}$$

$$\frac{\partial m\_{\rm i}}{\partial t} = -\partial\_{\dot{j}}(m\_{\rm i}m\_{\dot{j}}^{\dagger}) - \partial\_{\dot{j}}(w\_{\rm i}w\_{\dot{j}}^{\dagger}) - \rho \partial\_{\dot{i}}\rho^{\dagger} - m\_{\dot{j}}\partial\_{\dot{i}}m\_{\dot{j}}^{\dagger} - s\partial\_{\dot{i}}s^{\dagger} \tag{61b}$$

$$-w\_k \partial\_i w\_k^\dagger + \partial\_i (w\_k^\dagger w\_k)\_\prime$$

$$\frac{\partial \mathbf{s}}{\partial t} = -\partial\_k \left( s m\_k^\dagger + w\_k^\dagger \right) + \frac{1}{\tau (s^\dagger)^2} (\mathbf{w}^\dagger)^2 \,, \tag{61c}$$

$$\begin{split} \frac{\partial w\_k}{\partial t} &= -\partial\_k \mathbf{s}^\dagger - \partial\_k (w\_j m\_j^\dagger) + (\partial\_k w\_j - \partial\_j w\_k) \left( m\_j^\dagger + \frac{1}{s} w\_j^\dagger \right) \\ &- \frac{1}{\tau s^\dagger} w\_k^\dagger \,. \end{split} \tag{61d}$$

These are the GENERIC equations for fluid mechanics with hyperbolic heat conduction.
