**Appendix B. Fluid Mechanics**

The reversible part of fluid mechanics, where the state variables are density, momentum density and entropy density (per volume), (*ρ*, **m**,*s*), is generated by the Poisson bracket of classical hydrodynamics (see, e.g., [42,43]),

$$\begin{array}{lcl}\{A,B\} = & \int \operatorname{d\mathbf{r}} \rho \left(\partial\_{\bar{i}} A\_{\rho} B\_{m\_{i}} - \partial\_{\bar{i}} B\_{\rho} A\_{m\_{i}}\right) \\ & + \int \operatorname{d\mathbf{r}} \operatorname{m}\_{\bar{i}} \left(\partial\_{\bar{j}} A\_{m\_{i}} B\_{m\_{\bar{j}}} - \partial\_{\bar{j}} B\_{m\_{i}} A\_{m\_{\bar{j}}}\right) \\ & + \int \operatorname{d\mathbf{r}} \boldsymbol{s} \left(\partial\_{\bar{i}} A\_{\bar{s}} B\_{m\_{i}} - \partial\_{\bar{i}} B\_{\bar{s}} A\_{m\_{i}}\right) .\end{array} \tag{A.3}$$

Supplied with energy, e.g.,

$$E = \int \mathrm{d}\mathbf{r} \, \frac{\mathbf{m}^2}{2\rho} + \varepsilon(\rho, s), \tag{A4}$$

the reversible evolution of an arbitrary functional of the hydrodynamic variables *A*(*ρ*, **m**,*s*) is

$$
\dot{A} = \{A, E\}.\tag{A5}
$$

By rewriting the time-derivative of the functional as

$$\dot{A} = \int \mathrm{d}\mathbf{r} \, A\_{\theta} \partial\_{l} \rho + A\_{\mathcal{W}\_{l}} \partial\_{l} \mathcal{W}\_{l} + A\_{s} \partial\_{l} s\_{s} \tag{A6}$$

and comparing with Equation (A5), one can read the reversible part of evolution equations for the hydrodynamic fields,

$$
\partial\_t \rho\_{\!\!\!/} = \ -\partial\_{\!\!\!/} (\rho E\_{\!\!\!/ \!/}) = \ -\partial\_{\!\!\!/} \mathcal{m}\_{\!\!\!/} \tag{A7a}
$$

$$
\partial\_l m\_i = -\partial\_j \left( m\_i E\_{m\_j} \right) - \rho \partial\_i E\_p - m\_j \partial\_l E\_{m\_j} - s \partial\_l E\_s = -\partial\_j \left( \frac{m\_i m\_j}{\rho} \right) - \partial\_i p\_s \tag{A7b}
$$

$$
\partial\_l s \quad = \quad -\partial\_i (s E\_{\mathfrak{m}\_i}) = -\partial\_i \left( \frac{s m\_i}{\rho} \right),
\tag{A7c}
$$

where *p* = −*ε*(*ρ*,*s*) + *ρερ* + *sε<sup>s</sup>* is the pressure. The reversible evolution of hydrodynamic fields with energy (A4) is of course the Euler equations.
