4.4.2. DynMaxEnt to the Displacement Field—Passage to MHD

Let us apply the DynMaxEnt reduction to the displacement field in Equations (73) so that we approach the magnetohydrodynamics (MHD). The reversible part of the MHD equations can be obtained easily by projection to variables (*ρ*, **m**,*s*, **B**) as in [14], but we wish to also obtain the irreversible part of the equations.

Assuming energy quadratic in **D**, the MaxEnt value is **D**˜ = 0, which erases all terms containing the displacement field (but not **D**†) from the equations. Except for the evolution of entropy, **D**† only remains in Equation (74d), which can be solved and gives

$$
\hat{D}\_i^{\dagger} = \frac{s^{\dagger}}{\sigma} \varepsilon\_{ijk} \partial\_j B\_k^{\dagger} \,. \tag{75}
$$

Introducing the *curl* of Equation (75) into Equation (74d) yields a dissipative evolution equation for the magnetic field. The complete set of equations after this reduction reads

$$\begin{cases} \frac{\partial \rho}{\partial t} = -\partial\_k(\rho m\_k^\dagger), \\ \end{cases} \tag{76a}$$

$$\frac{\partial m\_i}{\partial t} = -\partial\_j(m\_i m\_j^\dagger) - \rho \partial\_i \rho^\dagger - m\_j \partial\_i m\_j^\dagger - s \partial\_i s^\dagger - B\_j \partial\_i B\_j^\dagger + \partial\_j (B\_j B\_i^\dagger),\tag{76b}$$

$$\frac{\partial \mathbf{s}}{\partial t} = -\partial\_{\mathbf{k}} \left( s m\_{\mathbf{k}}^{\dagger} \right) + \frac{1}{\sigma} (\nabla \times \mathbf{B}^{\dagger})^2,\tag{76c}$$

$$\frac{\partial B\_{\rm i}}{\partial t} = -\partial\_{\rangle} \left( B\_{\rm i} m\_{\slash}^{\dagger} - m\_{\text{i}}^{\dagger} B\_{\rm j} \right) - m\_{\text{i}}^{\dagger} \partial\_{\slash} B\_{\rm j} - \varepsilon\_{\text{ijk}} \partial\_{\not\choose j} \left( \frac{\text{s}^{\dagger}}{\sigma} \varepsilon\_{klm} \partial\_{l} B\_{\rm m}^{\dagger} \right), \tag{76d}$$

which is compatible with [36]. The first terms on the right-hand side of Equation (76d) can be further simplified for constant *σ*/*s*†, provided that the contribution of the magnetic field to the energy is **<sup>B</sup>**<sup>2</sup> *μ*0 . We can then write

$$\frac{\partial \mathbf{B}}{\partial t} = \frac{\mathbf{s}^{\dagger}}{\mu\_0 \sigma} \Delta \mathbf{B} - \nabla \times \left( \mathbf{B} \times \mathbf{m}^{\dagger} \right) \,, \tag{77}$$

where *∂iBi* = 0 was finally used. Keeping in mind that **m**† = **v**, Equation (77) is the advection–diffusion equation for the magnetic field, *cf.* ([37] Equation 2.15) or [38]. The coefficient *<sup>s</sup>*† *<sup>μ</sup>*0*<sup>σ</sup>* is referred to as the magnetic diffusivity.

In summary, after the relaxation of free charge density, one gets the electroneutral continuum coupled with Maxwell equations. Further dissipation in the displacement field then leads by the DynMaxEnt reduction to the MHD equations including magnetic diffusivity.
