3.3.4. Relation to GENERIC

Evolution equations for the damped particle (11) can be seen as a particular realization of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) [4,14,23,24], with the following building blocks

$$\mathbf{L} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \qquad \text{and} \qquad \Xi^{(\mathbf{p})} = \frac{1}{2} \frac{1}{\tau} (\mathbf{p}^\*)^2,\tag{27}$$

where the former expression defines a Poisson bivector while the latter dissipation potential (Irreversible evolution generated by a dissipation potential is also called gradient dynamics. It is in tight relation to the Steepest Entropy Ascent [25,26], which is essentially equivalent to the formulation of GENERIC with dissipative brackets [4].). The dissipation potential is naturally formulated in terms of **p**∗, which can be interpreted as derivative of entropy. The evolution equation consists of reversible part **L** · d*E* and irreversible Ξ**p**<sup>∗</sup> |**p**∗=*S***<sup>p</sup>** . In order to use only derivative w.r.t. energy, we recall the relation among the two representations **<sup>p</sup>**<sup>∗</sup> <sup>=</sup> <sup>−</sup>**p**†/*s*†, see [14,27] for more details (Indeed, on the Gibbs–Legendre manifold, where conjugate variables are identified with derivatives of thermodynamic potentials, one has *<sup>∂</sup><sup>s</sup> ∂pi e* = − *∂e ∂pi s ∂e ∂s* −1 **<sup>p</sup>** and *<sup>∂</sup><sup>e</sup> ∂s* **p** = *∂s ∂e* −1 **<sup>p</sup>** .) The equations are then explicitly

$$
\frac{\mathbf{d}}{\mathbf{d}t} \begin{pmatrix} \mathbf{r} \\ \mathbf{p} \\ s \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} \mathbf{r}^{\dagger} \\ \mathbf{p}^{\dagger} \\ s^{\dagger} \end{pmatrix} + \begin{pmatrix} 0 \\ -\frac{1}{\overline{\tau}} \frac{\mathbf{p}^{\dagger}}{s^{\dagger}} \\ \frac{1}{\overline{\tau}} \left(\frac{\mathbf{p}^{\dagger}}{s^{\dagger}}\right)^{2} \end{pmatrix},\tag{28}
$$

which is the same as Equations (11). The equations thus possess the GENERIC structure.

After the relaxation of the momentum **p**, expressed by Equation (15), evolution of positions becomes irreversible, given by Equation (16). Is this equation compatible with GENERIC? In other words, is there a dissipation potential generating that evolution? The answer is affirmative and actually easy to find at least in the case of the damped particle. Indeed, by evaluating dissipation potential Ξ(**p**) at the constitutive relation (15), we obtain dissipation potential for the lower level

$$\left. \Xi^{(\mathbf{r})}(\mathbf{r}^\*) = \Xi^{(\mathbf{p})} \right|\_{\mathbf{p}^\* = -\tau \mathbf{r}^\dagger = \tau \frac{\mathbf{r}^\*}{\varepsilon^2}} = \frac{\tau}{2} \left( \frac{\mathbf{r}^\*}{\varepsilon^\*} \right)^2,\tag{29}$$

where *e*<sup>∗</sup> = *Se* = 1/*s*†, *e* being total energy of the particle. Derivative of this dissipation potential w.r.t. **p**∗ reads

$$\dot{\mathbf{r}} = \boldsymbol{\Xi}\_{\mathbf{r}^\*}^{(\mathbf{r})} = \boldsymbol{\tau} \frac{\mathbf{r}^\*}{(e^\*)^2} = -\tau \mathbf{s}^\dagger \mathbf{r}^\dagger,\tag{30}$$

which is the reduced evolution equation (16). In other words, at least in this particular case the reduced evolution equation for **r** is generated by a dissipation potential constructed from the original dissipation potential for **p** by a projection, see also [21]. At least in the case of damped particle, the reduced evolution is a particular realization of GENERIC, and the reduced dissipation potential is obtained from the original dissipation potential.
