4.4.1. Non-Equilibrium Thermodynamics of Charged Mixtures

Dynamics of charged mixtures is governed by the Maxwell equations interacting with fluid mechanics of the species, see ([14] Section 6.4). Let us start with a Poisson bracket for the binary mixture of oppositely charged species endowed with a single entropy, total momentum density, displacement field and magnetic field, that is,

$$\{F,\mathcal{G}\}^{\text{(EM-D2)}}\left(\rho\_{+},\rho\_{-},\mathbf{m},s,\mathbf{D},\mathbf{B}\right) = \left\{F,\mathcal{G}\right\}^{\text{(CIT-2)}}\left(\rho\_{+},\rho\_{-},\mathbf{m},s\right) \tag{69}$$

$$\{F,\mathcal{G}\}^{\text{(EM)}}\left(\mathbf{D},\mathbf{B}\right) \left\{ \begin{array}{c} + \int \text{d}\mathbf{r}\left(F\_{D\_{i}}\odot\_{ijk}\partial\_{j}G\_{R\_{i}} - G\_{D\_{i}}\varepsilon\_{ijk}\partial\_{j}F\_{B\_{i}}\right) \\ + \int \text{d}\mathbf{r} D\_{i}\left(\partial\_{j}F\_{D\_{i}}G\_{m\_{j}} - \partial\_{j}G\_{D\_{i}}F\_{m\_{j}}\right) \\ + \int \text{d}\mathbf{r}\partial\_{j}D\_{j}\left(F\_{m\_{i}}G\_{D\_{i}} - G\_{m\_{i}}F\_{D\_{i}}\right) \\ + \int \text{d}\mathbf{r} D\_{j}\left(F\_{m\_{i}}\partial\_{j}G\_{D\_{i}} - G\_{m\_{i}}\partial\_{j}F\_{D\_{i}}\right) \\ + \int \text{d}\mathbf{r} D\_{i}\left(\partial\_{j}F\_{B\_{i}}G\_{m\_{j}} - \partial\_{j}G\_{B\_{i}}F\_{m\_{j}}\right) \\ + \int \text{d}\mathbf{r} \partial\_{j}B\_{j}\left(F\_{m\_{i}}G\_{B\_{i}} - G\_{m\_{i}}F\_{B\_{i}}\right) \\ + \int \text{d}\mathbf{r} D\_{j}\left(F\_{m\_{i}}\partial\_{j}G\_{B\_{i}} - G\_{m\_{i}}\partial\_{j}F\_{B\_{i}}\right) \end{array} \right.$$

where the CIT2 (binary classical irreversible thermodynamics [1,34]) bracket stands for

$$\begin{split} \{F, G\}^{\text{(CT-2)}}(\rho\_{+}, \rho\_{-}, \mathbf{m}, s) &= \int \mathbf{d} \mathbf{r} \, \rho\_{+} \left(\partial\_{i} F\_{\rho\_{+}} \mathbf{G}\_{m\_{i}} - \partial\_{i} \mathbf{G}\_{\rho\_{+}} F\_{m\_{i}}\right) \\ &+ \int \mathbf{d} \mathbf{r} \, \rho\_{-} \left(\partial\_{i} F\_{\rho\_{-}} \mathbf{G}\_{m\_{i}} - \partial\_{i} \mathbf{G}\_{\rho\_{-}} F\_{m\_{i}}\right) \\ &+ \int \mathbf{d} \mathbf{r} \, m\_{i} \left(\partial\_{j} F\_{m\_{i}} \mathbf{G}\_{m\_{j}} - \partial\_{j} \mathbf{G}\_{m\_{i}} F\_{m\_{j}}\right) \\ &+ \int \mathbf{d} \mathbf{r} \, s \left(\partial\_{i} F\_{\delta} \mathbf{G}\_{m\_{i}} - \partial\_{i} \mathbf{G}\_{\delta} F\_{m\_{i}}\right), \end{split} \tag{70}$$

cf. bracket (A3).

This system is additionally required to satisfy the Gauß laws for electric and magnetic charge, respectively. We have

$$
\partial\_i D\_i = \eta \epsilon\_0 \left( \frac{\rho\_+}{m\_+} - \frac{\rho\_-}{m\_-} \right) \qquad \text{and} \qquad \partial\_i B\_i = 0 \; , \tag{71}
$$

where the right-hand side of Equation (71)left is the free charge density.

Total density *ρ* = *ρ*<sup>+</sup> + *ρ*<sup>−</sup> and the free charge density can be used for the description instead of *ρ*<sup>+</sup> and *ρ*−. Such transformation allows for the projection to the state variables without free charge density (i.e., where free charge density is relaxed) by letting the functionals depend only on (*ρ*, **m**,*s*, **D**, **B**). Consequently, bracket (69) transforms into

$$\begin{aligned} \{F, G\}^{\text{(EMHD)}}\left(\rho, \mathbf{m}, \mathbf{s}, \mathbf{D}, \mathbf{B}\right) &= \{F, G\}^{\text{(FM)}}\left(\rho, \mathbf{m}, s\right) + \{F, G\}^{\text{(EM)}}\left(\mathbf{D}, \mathbf{B}\right) \\ &+ \{F, G\}^{\text{(SP)}}\left(\mathbf{D}, \mathbf{m}\right) + \{F, G\}^{\text{(SP)}}\left(\mathbf{B}, \mathbf{m}\right) \,, \end{aligned} \tag{72}$$

and it is equipped with the updated constraint on the displacement field (given by the relaxed value of free charge density, typically zero) and magnetic field. Bracket (72) describes reversible evolution of electroneutral continuum coupled with the Maxwell equations.

Although the continuum described by bracket (72) is electroneutral, it can conduct electric current. This can be seen as a dissipation of the displacement field as suggested in [35]. Let us define dissipation potential

$$\mathbb{E}(\mathbf{D}^\*) = \int \mathrm{d}\mathbf{r} \frac{\sigma}{2} \left(\mathbf{D}^\*\right)^2 = \int \mathrm{d}\mathbf{r} \frac{\sigma}{2} \left(\frac{\mathbf{D}^\dagger}{s^\dagger}\right)^2 \,. \tag{73}$$

The reversible evolution generated by (72) and the irreversible evolution due to (73) give together

$$\frac{\partial \rho}{\partial t} = -\partial\_k (\rho m\_k^\dagger),$$
 
$$(74a)^\gamma \tag{74a}$$

$$\begin{split} \frac{\partial m\_{i}}{\partial t} &= -\partial\_{\rangle}(m\_{i}m\_{\neq}^{\dagger}) - \rho \partial\_{i}\rho^{\dagger} - m\_{\neq}\partial\_{i}m\_{\neq}^{\dagger} - s\partial\_{i}s^{\dagger} - D\_{\neq}\partial\_{i}D\_{\neq}^{\dagger} - B\_{\neq}\partial\_{i}B\_{\neq}^{\dagger} \\ &+ \partial\_{\rangle}(D\_{j}D\_{i}^{\dagger} + B\_{j}B\_{i}^{\dagger}), \end{split} \tag{74b}$$

$$\frac{\partial \mathbf{s}}{\partial t} = -\partial\_k \left( s m\_k^\dagger \right) + \frac{\sigma}{(s^\dagger)^2} (\mathbf{D}^\dagger)^2,\tag{74c}$$

$$\frac{\partial D\_{\rm i}}{\partial t} = \varepsilon\_{\rm ijk} \partial\_{\rm j} B\_{\rm k}^{\dagger} - \partial\_{\rm j} \left( D\_{\rm i} m\_{\rm j}^{\dagger} - m\_{\rm i}^{\dagger} D\_{\rm j} \right) - m\_{\rm i}^{\dagger} \partial\_{\rm j} D\_{\rm j} - \frac{\sigma}{s^{\ddagger}} D\_{\rm i}^{\dagger},\tag{74d}$$

$$\frac{\partial B\_{\dot{l}}}{\partial t} = -\varepsilon\_{\dot{l}\bar{\beta}\dot{l}}\partial\_{\dot{j}}D\_{k}^{\dagger} - \partial\_{\dot{j}}\left(B\_{\dot{l}}m\_{\dot{j}}^{\dagger} - m\_{\dot{l}}^{\dagger}B\_{\dot{j}}\right) - m\_{\dot{i}}^{\dagger}\partial\_{\dot{j}}B\_{\dot{j}}.\tag{74e}$$

The conjugate displacement field **D**† is interpreted as electric intensity **E**, and the irreversible (last) term in Equation (74d) then tells that the electric current is **J** = *<sup>σ</sup> <sup>s</sup>*† **E**. Dissipation in the **D** field can be thus seen as Ohm's law (Alternatively, Ohm's law can be derived from the relaxation of matter in the presence of the electric field, see [14].).
