$t = $ time
$\langle p^i \rangle = $ momentum distribution
$E^i = $ electric field
$B^i = $ magnetic field
$\epsilon_{ijk} = $ Levi-Civita connection
$\delta^i_j = $ Kronecker delta
$q = $ particle charge
$n = $ number density
$m = $ particle mass
$\tau = $ mean free time between ionic collisions

Drude:
$\frac{d}{dt} \langle p^i \rangle = q E^i + (\frac{q}{m} {\epsilon^i}_{jk} B^k - \frac{1}{\tau} \delta^i_j) \langle p^j \rangle $
$J^i = \frac{n q^2 \tau}{m} E^i$

$\langle p^i \rangle = q \tau E^i$
$\langle p^i \rangle = m \langle v^i \rangle$
$J^i = n q \langle v^i \rangle$
...therefore...
$\langle p^i \rangle = \frac{m}{nq} J^i$
...therefore...
$\frac{d}{dt} (q \tau E^i) = q E^i + (\frac{q}{m} {\epsilon^i}_{jk} B^k - \frac{1}{\tau} \delta^i_j) (q \tau E^j) $
...for constant $q$ and $\tau$...
$\frac{d}{dt} E^i = \frac{q}{m} {\epsilon^i}_{jk} B^k E^j $

Maxwell:
$\nabla \cdot D = \rho$
$\nabla \cdot B = 0$
$\nabla \times E = -\partial_t B$
$\nabla \times H = J + \partial_t D$
$H = \frac{1}{\mu_0} B - M$
$D = \epsilon E + P$
$D = \epsilon_0 E + \chi E = \epsilon_0 (1 + \chi) E = \epsilon_0 \epsilon_r E$ as a linear approximation... but $\epsilon$ can be a matrix ...

...as IVP...
$\partial_t B + \nabla \times E = 0$
$\partial_t D - \nabla \times H = -J$
$\partial_t (\epsilon E + P) - \nabla \times H = -J$