Finite Lorentz oscillator closure

Work in 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. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be the material region in that frame and $I\subseteq \mathbb{R}$ a nonempty open time interval. The electric field $E:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$, polarization fields $P_{\mathrm{\ell }}:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$, and magnetic fields $B,H:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ are real spatial-vector fields in that frame. At every fixed $x\in \mathrm{\Omega }$, require $E(x,\cdot )\in C^0(I;{\mathbb{R}}^3)$ and $P_{\mathrm{\ell }}(x,\cdot )\in C^2(I;{\mathbb{R}}^3)$ for each $1\le \mathrm{\ell }\le N$, 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 in V m${}^{-1}$, $P_{\mathrm{\ell }}$ in C m${}^{-2}$, $B$ in T and $H$ in A m${}^{-1}$.

For a fixed finite integer $N\ge 1$, adopt at every material point $x$ the constitutive equations

$D(x,t)={\epsilon }_{\mathrm{\infty }}E(x,t)+{\sum }_{\mathrm{\ell }=1}^N{P}_{\mathrm{\ell }}(x,t),B(x,t)=\mu H(x,t),$

${\mathrm{\partial }}_t^2{P}_{\mathrm{\ell }}(x,t)+{\gamma }_{\mathrm{\ell }}{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}(x,t)+{\omega }_{\mathrm{\ell }}^2{P}_{\mathrm{\ell }}(x,t)=a_{\mathrm{\ell }}E(x,t)(1\le \mathrm{\ell }\le N)\mathrm{.}$

Here ${\epsilon }_{\mathrm{\infty }}>0$ is a constant scalar with units F m${}^{-1}$, $\mu >0$ is a constant scalar with units H m${}^{-1}$, $a_{\mathrm{\ell }}>0$ are constant parameters with units F m${}^{-1}$ s${}^{-2}$, ${\omega }_{\mathrm{\ell }}>0$ are constant angular frequencies with units s${}^{-1}$, and ${\gamma }_{\mathrm{\ell }}\ge 0$ are constant damping rates with units s${}^{-1}$. The parameters are independent of position and time; this is the meaning of homogeneous in this model. The model is linear because all displayed field and oscillator relations are linear with fixed coefficients, and spatially local because the oscillator equation at $x$ uses only the field and oscillator state at that same $x$. The magnetic law has only the constant scalar coefficient $\mu$ and no independent magnetic polarization variable; this is the nonmagnetic oscillator model adopted here.

The word nonmagnetic here means that the oscillator sector has no dynamic magnetic response; the fixed scalar coefficient $\mu$ can still encode a separate static constitutive response. To identify the displayed quantities with the framework's macroscopic definitions, stipulate $P_{\mathrm{t}\mathrm{o}\mathrm{t}}=({\epsilon }_{\mathrm{\infty }}-{\epsilon }_0)E+{\sum }_{\mathrm{\ell }=1}^N{P}_{\mathrm{\ell }}$ and $M_{\mathrm{t}\mathrm{o}\mathrm{t}}=({\mu }_0^{-1}-{\mu }^{-1})B$. Then $D={\epsilon }_{0}E+P_{\mathrm{t}\mathrm{o}\mathrm{t}}={\epsilon }_{\mathrm{\infty }}E+\sum P_{\mathrm{\ell }}$ and $H={\mu }_0^{-1}B-M_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mu }^{-1}B$. This is a constitutive identification in the model, not a microscopic decomposition.

At an initial time $t_0\in I$, the oscillator state consists of $P_{\mathrm{\ell }}(x,t_0)$ and ${\mathrm{\partial }}_t{P}_{\mathrm{\ell }}(x,t_0)$. For a selected causal zero-past branch, choose an onset $\tau \in I$ and require $E(x,t)=0$ and $P_{\mathrm{\ell }}(x,t)={\mathrm{\partial }}_t{P}_{\mathrm{\ell }}(x,t)=0$ for every $t\in I$ with $t\le \tau$. Continuity gives $E(x,\tau )=0$, and the equation gives ${\mathrm{\partial }}_t^2{P}_{\mathrm{\ell }}(x,\tau )=0$, consistent with the $C^2$ zero extension through onset. For $t\ge \tau$ this branch is the retarded initial-value equation with those data, so its value at time $t$ uses the applied field only on $[\tau ,t]$. If ${\gamma }_{\mathrm{\ell }}=0$, no decaying stationary impulse response is assumed; in particular, this is not an absolutely integrable convolution for arbitrary bounded histories. This postulate does not assert that any actual material realizes the equations.

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