Causal oscillator media and passivity

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

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

Let $I\subseteq \mathbb{R}$ be a nonempty open interval and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be a nonempty bounded Lipschitz polyhedron in one material rest frame. Use real SI fields: $E,H,D,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$, a free current density $J_f:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$, polarization $P:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$, and, when present, carrier current $V:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$. Their units are V m${}^{-1}$, A m${}^{-1}$, C m${}^{-2}$, T, A m${}^{-2}$, C m${}^{-2}$ and A m${}^{-2}$, respectively. Let ${\epsilon }_{\mathrm{\infty }}>0$ and $\mu >0$ be constant scalar coefficients in F m${}^{-1}$ and H m${}^{-1}$. Each $u_L,u_D$ is in J m${}^{-3}$ and the flux $E\times H$ is in W m${}^{-2}$.

Assume Countable Choice, the smooth macroscopic Maxwell material model, and its curl equations

$\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}H=J_f+{\mathrm{\partial }}_{t}D$

hold pointwise. The smooth macroscopic material model includes continuous free sources and the total polarization/magnetization identifications specified by the Lorentz or Drude closure. Assume $E,H$ are jointly $C^{\mathrm{\infty }}$ pointwise and satisfy $E,H\in C^1(I;L^2(\mathrm{\Omega };{\mathbb{R}}^3))\cap C(I;H(\mathrm{curl};\mathrm{\Omega }))$; $J_f$ and each material-current term belong to $C(I;L^2)$. Here $H(\mathrm{curl};\mathrm{\Omega })$ is the real $L^2$ vector fields whose distributional curl is represented by an $L^2$ vector field, with the graph norm. Take Lorentz fields $P_{\mathrm{\ell }}$ jointly $C^{\mathrm{\infty }}$ and in $C^2(I;L^2)$, or Drude fields $P,V$ jointly $C^{\mathrm{\infty }}$ with $V\in C^1(I;L^2)$, so that the stored energies and current terms are finite. Take either one of the following constitutive branches:

1. Finite Lorentz branch. For some finite $N\ge 1$, $D={\epsilon }_{\mathrm{\infty }}E+{\sum }_{\mathrm{\ell }=1}^N{P}_{\mathrm{\ell }}$, $B=\mu H$, and ${\mathrm{\partial }}_t^2{P}_{\mathrm{\ell }}+{\gamma }_{\mathrm{\ell }}{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}+{\omega }_{\mathrm{\ell }}^2{P}_{\mathrm{\ell }}=a_{\mathrm{\ell }}E,$ where $P_{\mathrm{\ell }}$ is jointly $C^2$, $a_{\mathrm{\ell }}>0$ in F m${}^{-1}$ s${}^{-2}$, ${\omega }_{\mathrm{\ell }}>0$ in s${}^{-1}$, and ${\gamma }_{\mathrm{\ell }}\ge 0$ in s${}^{-1}$.
2. Drude branch. $D={\epsilon }_{\mathrm{\infty }}E+P$, $B=\mu H$, ${\mathrm{\partial }}_{t}P=V$, and ${\mathrm{\partial }}_{t}V+\gamma V=aE$, where $P,V$ are jointly $C^1$, $a>0$ in F m${}^{-1}$ s${}^{-2}$, and $\gamma \ge 0$ in s${}^{-1}$.

For the Lorentz branch define

$u_L:=\frac{{\epsilon }_{\mathrm{\infty }}\mathrm{\mid }E{\mathrm{\mid }}^2+\mu \mathrm{\mid }H{\mathrm{\mid }}^2}{2}+{\sum }_{\mathrm{\ell }=1}^N\frac{\mathrm{\mid }{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}{\mathrm{\mid }}^2+{\omega }_{\mathrm{\ell }}^2\mathrm{\mid }{P}_{\mathrm{\ell }}{\mathrm{\mid }}^2}{2a_{\mathrm{\ell }}}\mathrm{.}$

For the Drude branch define

$u_D:=\frac{{\epsilon }_{\mathrm{\infty }}\mathrm{\mid }E{\mathrm{\mid }}^2+\mu \mathrm{\mid }H{\mathrm{\mid }}^2}{2}+\frac{\mathrm{\mid }V{\mathrm{\mid }}^2}{2a}\mathrm{.}$

The Lorentz density is positive definite in $(E,H,(P_{\mathrm{\ell }},{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}{)}_{\mathrm{\ell }=1}^N)$. The Drude density is positive definite in $(E,H,V)$; a constant polarization offset is not part of this energy.

Then the respective local identities hold pointwise:

${\mathrm{\partial }}_t{u}_L+\mathrm{div}(E\times H)=-E\cdot J_f-{\sum }_{\mathrm{\ell }=1}^N\frac{{\gamma }_{\mathrm{\ell }}}{a_{\mathrm{\ell }}}\mathrm{\mid }{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}{\mathrm{\mid }}^2,$

${\mathrm{\partial }}_t{u}_D+\mathrm{div}(E\times H)=-E\cdot J_f-\frac{\gamma }{a}\mathrm{\mid }V{\mathrm{\mid }}^2\mathrm{.}$

If, in addition, the fields satisfy the exact closure-defined PEC branch of the finite-energy boundary model and $J_f=0$, then the total stored energy $U(t)={\int }_{\mathrm{\Omega }}{u}_L(x,t)dx$ or $U(t)={\int }_{\mathrm{\Omega }}{u}_D(x,t)dx$ is nonnegative and nonincreasing. More generally the corresponding fixed-volume balance includes the work $-\int E\cdot J_f$ and the displayed nonnegative material-loss terms; no claim of conservation without these terms is made.

Finite-energy PEC and impedance boundary formulation

Finite Lorentz oscillator closure

Drude current closure and collisionless limit

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

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Energy balance and zero-data forward uniqueness for admissible weak Maxwell fields

Use the finite Lorentz oscillator closure in its homogeneous material rest frame. Let $N\ge 1$ be finite, with ${\epsilon }_{\mathrm{\infty }}>0$ in F m${}^{-1}$, $a_{\mathrm{\ell }}>0$ in F m${}^{-1}$ s${}^{-2}$, ${\omega }_{\mathrm{\ell }}>0$ in s${}^{-1}$ and ${\gamma }_{\mathrm{\ell }}\ge 0$ in s${}^{-1}$ as in that model. Assume Countable Choice; for the strictly damped integrable-kernel dispersion clause also assume Dependent Choice, as required by the causal-response supplier. For the time convention $e^{-i\omega t}$, let $z$ denote complex angular frequency in s${}^{-1}$ and define the absolute oscillator susceptibility, with units F m${}^{-1}$, by

${\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}(z):={\sum }_{\mathrm{\ell }=1}^N\frac{a_{\mathrm{\ell }}}{{\omega }_{\mathrm{\ell }}^2-z^2-i{\gamma }_{\mathrm{\ell }}z},\epsilon (z):={\epsilon }_{\mathrm{\infty }}+{\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}(z)\mathrm{.}$

The dimensionless transfer function in the causal-response theorem is ${\chi }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}(z):={\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}(z)\mathrm{/}{\epsilon }_0$, so that it has the theorem's convention $P={\epsilon }_0{\chi }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}E$ for harmonic histories.

The function ${\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}$ is analytic for $\mathrm{Im}z>0$ for every ${\gamma }_{\mathrm{\ell }}\ge 0$. For real angular frequency $\omega >0$ in s${}^{-1}$ away from undamped poles, ${\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}(\omega )$ is the scalar multiplier for a harmonic particular solution: if $E(t)=\mathrm{Re}(e{e}^{-i\omega t})$ and $P_{\mathrm{\ell }}(t)=\mathrm{Re}(p_{\mathrm{\ell }}{e}^{-i\omega t})$, then $p_{\mathrm{\ell }}=a_{\mathrm{\ell }}e\mathrm{/}({\omega }_{\mathrm{\ell }}^2-{\omega }^2-i{\gamma }_{\mathrm{\ell }}\omega )$ and $p={\sum }_{\mathrm{\ell }}{p}_{\mathrm{\ell }}={\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}(\omega )e$. This coefficient calculation is not by itself a zero-past response to an eternal harmonic input. In the undamped branch the retarded onset solution is specified by initial data and the real poles are excluded; when every ${\gamma }_{\mathrm{\ell }}>0$, the separately stated integrable retarded kernels realize the same harmonic multiplier.

$\mathrm{Im}\epsilon (\omega )={\sum }_{\mathrm{\ell }=1}^N\frac{a_{\mathrm{\ell }}{\gamma }_{\mathrm{\ell }}\omega }{({\omega }_{\mathrm{\ell }}^2-{\omega }^2{)}^2+{\gamma }_{\mathrm{\ell }}^2{\omega }^2}\ge 0.$

When every ${\gamma }_{\mathrm{\ell }}>0$, each causal impulse-response kernel is exponentially decaying, satisfies the integrable-kernel hypotheses of the causal-response theorem, and $\epsilon -{\epsilon }_{\mathrm{\infty }}$ obeys its unsubtracted Hilbert-transform relations. When ${\gamma }_{\mathrm{\ell }}=0$, that oscillator's poles lie at $z=\pm {\omega }_{\mathrm{\ell }}$ on the real boundary; the transfer function has no pointwise value at those poles, and no pointwise dispersion formula there is asserted. The theorem makes no universal group-velocity bound.

Finite Lorentz oscillator closure

Specified causal material response: transfer function, dispersion relations and the passivity condition

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain

Assume Countable Choice, the hypothesis carried by the finite-energy Maxwell supplier. Work in a fixed homogeneous material rest frame in SI and standard right-handed Cartesian coordinates with orthonormal basis $\widehat{x},\widehat{y},\widehat{z}$. Let ${\epsilon }_{\mathrm{\infty }}>0$ be in F m${}^{-1}$, $\mu >0$ in H m${}^{-1}$, $a>0$ in F m${}^{-1}$ s${}^{-2}$, and $\gamma \ge 0$ in s${}^{-1}$ as in the adopted Drude current closure. For a spatially uniform harmonic electric field $E(t)=\mathrm{Re}(e{e}^{-i\omega t})$, with $\omega >0$ in s${}^{-1}$ and $e$ in V m${}^{-1}$, consider the harmonic particular solution of the Drude equations, written $V(t)=\mathrm{Re}(v{e}^{-i\omega t})$ and $P(t)=\mathrm{Re}(p{e}^{-i\omega t})$, where $v$ is in A m${}^{-2}$ and $p$ is in C m${}^{-2}$. The frequency-domain response below refers to this harmonic particular solution; arbitrary initial data may add homogeneous solutions, which are not included in these amplitudes. The conductivity $\sigma (\omega )$ is the complex scalar mapping $e$ to $v$, in S m${}^{-1}$, and the absolute permittivity $\epsilon (\omega )$ is the scalar mapping $e$ to the displacement amplitude in $D={\epsilon }_{\mathrm{\infty }}E+P$, in F m${}^{-1}$.

For every $\omega >0$,

$\sigma (\omega )=\frac{a}{\gamma -i\omega },\mathrm{Re}\sigma (\omega )=\frac{a\gamma }{{\gamma }^2+{\omega }^2}\ge 0,\epsilon (\omega )={\epsilon }_{\mathrm{\infty }}-\frac{a}{\omega (\omega +i\gamma )}\mathrm{.}$

The carrier energy density and the loss term in the smooth Maxwell energy balance are $\mathrm{\mid }V{\mathrm{\mid }}^2\mathrm{/}(2a)$ and $\gamma \mathrm{\mid }V{\mathrm{\mid }}^2\mathrm{/}a$, respectively; the energy density has units J m${}^{-3}$ and the loss density has units W m${}^{-3}$.

In the collisionless branch $\gamma =0$, set ${\omega }_p:=\sqrt{a\mathrm{/}{\epsilon }_{\mathrm{\infty }}}>0$ (so ${\omega }_p$ is in s${}^{-1}$). For a source-free nonzero transverse bulk plane wave with real $k\in {\mathbb{R}}^3\setminus \{0\}$ in m${}^{-1}$, frequency $\omega >0$ in s${}^{-1}$, and nonzero complex electric amplitude $e$ satisfying $k\cdot e=0$, the Maxwell amplitude equations imply