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

Drude:
$\frac{d}{dt}⟨p^i⟩=q{E}^i+(\frac{q}{m}{{ϵ}^i}_{jk}{B}^k-\frac{1}{\tau }{\delta }_j^i)⟨p^j⟩$
$J^i=\frac{n{q}^2\tau }{m}{E}^i$

$⟨p^i⟩=q\tau E^i$
$⟨p^i⟩=m⟨v^i⟩$
$J^i=nq⟨v^i⟩$
...therefore...
$⟨p^i⟩=\frac{m}{nq}{J}^i$
...therefore...
$\frac{d}{dt}(q\tau E^i)=q{E}^i+(\frac{q}{m}{{ϵ}^i}_{jk}{B}^k-\frac{1}{\tau }{\delta }_j^i)(q\tau E^j)$
...for constant $q$ and $\tau$...
$\frac{d}{dt}{E}^i=\frac{q}{m}{{ϵ}^i}_{jk}{B}^k{E}^j$

Maxwell:
$\mathrm{\nabla }\cdot D=\rho$
$\mathrm{\nabla }\cdot B=0$
$\mathrm{\nabla }\times E=-{\partial }_{t}B$
$\mathrm{\nabla }\times H=J+{\partial }_{t}D$
$H=\frac{1}{{\mu }_0}B-M$
$D=ϵE+P$
$D={ϵ}_{0}E+\chi E={ϵ}_0(1+\chi )E={ϵ}_0{ϵ}_{r}E$ as a linear approximation... but $ϵ$ can be a matrix ...

...as IVP...
${\partial }_{t}B+\mathrm{\nabla }\times E=0$
${\partial }_{t}D-\mathrm{\nabla }\times H=-J$
${\partial }_t(ϵE+P)-\mathrm{\nabla }\times H=-J$