Drude current closure and collisionless limit

Use a material rest frame: an inertial frame in whose coordinates each modelled material element has fixed spatial position $x$ and history $t\mapsto (ct,x)$, as defined by the framework's earlier material-closure formulation. In that frame, let $E,P:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be the electric field and polarization, and let $V:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be a free-carrier current density. At every fixed $x\in \mathrm{\Omega }$, require $E(x,\cdot )\in C^0(I;{\mathbb{R}}^3)$ and $P(x,\cdot ),V(x,\cdot )\in C^1(I;{\mathbb{R}}^3)$, so the displayed time derivatives are classical. These are time-regularity conditions only; this postulate imposes no spatial regularity, which must be stated separately wherever a consuming field-equation result requires it. In SI, $E$ is V m${}^{-1}$, $P$ is C m${}^{-2}$, and $V$ is A m${}^{-2}$. Adopt

${\mathrm{\partial }}_{t}P(x,t)=V(x,t),{\mathrm{\partial }}_{t}V(x,t)+\gamma V(x,t)=aE(x,t),$

$D(x,t)={\epsilon }_{\mathrm{\infty }}E(x,t)+P(x,t),B(x,t)=\mu H(x,t)\mathrm{.}$

The constant scalar parameters satisfy ${\epsilon }_{\mathrm{\infty }}>0$, $\mu >0$, $a>0$ and $\gamma \ge 0$. Their units are respectively F m${}^{-1}$, H m${}^{-1}$, F m${}^{-1}$ s${}^{-2}$ and s${}^{-1}$. The fields $V$ and $P$ have units A m${}^{-2}$ and C m${}^{-2}$. All coefficients are independent of $x$ and $t$, and the equation at $x$ depends only on the state and electric field at $x$; these clauses define the homogeneous, linear, spatially local closure used here. To identify these constitutive quantities with the framework's macroscopic definitions, stipulate $P_{\mathrm{t}\mathrm{o}\mathrm{t}}=({\epsilon }_{\mathrm{\infty }}-{\epsilon }_0)E+P$ and $M_{\mathrm{t}\mathrm{o}\mathrm{t}}=({\mu }_0^{-1}-{\mu }^{-1})B$. Then $D={\epsilon }_{0}E+P_{\mathrm{t}\mathrm{o}\mathrm{t}}={\epsilon }_{\mathrm{\infty }}E+P$ and $H={\mu }_0^{-1}B-M_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mu }^{-1}B$. This is a constitutive identification within the model, not a microscopic decomposition.

At an initial time $t_0\in I$, $P(x,t_0)$ and $V(x,t_0)$ are state data. A constant polarization offset is fixed with the Gauss-law data and is not determined by the current equation alone. The branch $\gamma =0$ is the collisionless model; it has no decaying stationary impulse kernel, and its zero-frequency response is not part of this postulate. This is a phenomenological closure, not a derivation of carrier mass, density, collision statistics or any actual material response.

Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch

The macroscopic Maxwell equations in terms of D, H and the free sources

The electric and magnetic fields as fixed-time spatial vector fields

Polarization, magnetization, bound and free sources, and the fields D and H

Framework-local SI units, the vacuum constants and the exact/measured status of their values