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

Assume the memory branch of the material-closure postulate Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in a material rest frame, and assume the hypotheses of the mathematical suppliers Causal convolution response, transfer function and dispersion relations and Dispersion relations for a finitely subtracted analytic response: Countable Choice for the smooth comparison route and, in the integrable branches, Dependent Choice (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). Let $E:\mathbb{R}\to {\mathbb{R}}^3$ be the electric-field history in the material rest frame, read componentwise in its spatial basis (The electric and magnetic fields as fixed-time spatial vector fields), and let $P$ be the polarisation response declared by the model.

(a) [Causal response] If the kernel of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(c) satisfies $K\in L^1(0,\mathrm{\infty };{\mathbb{R}}^{3\times 3})$, then for every bounded continuous history $E$ the response $P(t)={\epsilon }_0{\int }_0^{\mathrm{\infty }}K(s)E(t-s)ds$ is well defined for every $t$, bounded and continuous, linear in $E$, and causal: histories equal on $(-\mathrm{\infty },t_0]$ produce responses equal on $(-\mathrm{\infty },t_0]$.

(b) [Transfer function and dispersion relations] If $K$ is smooth with compact support in $(0,\mathrm{\infty })$, the transfer function $\chi (z)={\int }_0^{\mathrm{\infty }}K(s)e^{izs}ds$ is entire, satisfies $\chi (-\omega )=\overline{\chi (\omega )}$ for real $\omega$, obeys $\mathrm{\mid }{\chi }_{jk}(z)\mathrm{\mid }\le \mathrm{\parallel }{K}_{jk}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^1}\mathrm{\mid }z{\mathrm{\mid }}^{-2}$ for $z\mathrm{\ne }0$ with $\mathrm{Im}z\ge 0$ (with the whole-plane bound carrying the exponential factor $e^{b\max (0,-\mathrm{Im}z)}$ when $\mathrm{supp}K\subseteq (0,b)$), and its entries satisfy the principal-value dispersion relations of Causal convolution response, transfer function and dispersion relations(b). If $K$ is locally absolutely continuous on $[0,\mathrm{\infty })$ with $K,K^{\mathrm{\prime }}\in L^1$ and $sK\in L^1$, then $\chi$ is holomorphic on the open upper half-plane, continuous with continuous first derivative on the real boundary, obeys the stated growth bound, and satisfies the same pair of relations; an instantaneous real part ${\chi }_{\mathrm{\infty }}$ may be added to the model, and the relations apply to ${\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-{\chi }_{\mathrm{\infty }}$ for ${\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\chi }_{\mathrm{\infty }}+\chi$.

(c) [Finitely-subtracted and polynomial branches] For each specified integer $n\ge 1$ for which the full hypotheses of Dispersion relations for a finitely subtracted analytic response hold for the declared response transfer function, its $n$-subtracted relations apply entrywise. Any transfer-only physical branch additionally requires $\chi (-\omega )=\overline{\chi (\omega )}$ entrywise for every real $\omega$, including real $\chi (0)$, as in Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(c); scalar transfers act as scalar matrices. These hypotheses include upper-half-plane holomorphy, continuity on the closed upper half-plane, the stated boundary $C^1$ regularity, holomorphy near the real subtraction point ${\omega }_0$, and the order-specific growth bound $\mathrm{\mid }{\chi }_{jk}(z)\mathrm{\mid }\le C_{jk}(1+\mathrm{\mid }z{\mathrm{\mid }}^{n-1})$ outside a compact set. The Taylor polynomial and residual are

$P_n(z)={\sum }_{j=0}^{n-1}\frac{{\chi }^{(j)}({\omega }_0)}{j!}(z-{\omega }_0{)}^j,f_n(z)=\frac{\chi (z)-P_n(z)}{(z-{\omega }_0{)}^n},$

with removable values and the boundary $C^1$ hypothesis read entrywise as in the supplier. A growth bound for one order $n_0$ permits every order $n\ge n_0$ under the same local analytic and boundary hypotheses; it does not justify lower orders for which the required growth fails. The unsubtracted $n=0$ case is included only under its own decay and boundary hypotheses: $P_0=0$, $f_0=\chi$, growth $O(1\mathrm{/}\mathrm{\mid }z\mathrm{\mid })$, and factor $(\omega -{\omega }_0{)}^0=1$ with the supplier's principal-value convention. The separate constitutive-polynomial branch declares the normalized polarisation $P\mathrm{/}{\epsilon }_0={\sum }_{j=0}^m{A}_j{E}^{(j)}+K\ast E$, with real scalar coefficients (acting as scalar matrices) or real $3\times 3$ matrices $A_j$ whose entries carry ${\mathrm{s}}^j$, and $K$ satisfying the supplier's integrable causal hypotheses, is causal on histories with bounded continuous derivatives through order $m$. Its transfer is $Q(z)+{\chi }_K(z)$ with $Q(z)={\sum }_{j=0}^m{A}_j(-iz{)}^j$. Subtracting the declared $Q$ leaves the proved relations for ${\chi }_K$, without assuming that $Q$ is the Taylor polynomial $P_n$.

(d) [Monochromatic work, instantaneous work and passivity] Let $\omega >0$, $e\in {\mathbb{C}}^3$ and $E(t)=\mathrm{Re}(e{e}^{-i\omega t})$. Write ${\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}}(\omega )$ for the full transfer matrix used in the response: it is $\chi (\omega )$ without an instantaneous term and ${\chi }_{\mathrm{\infty }}+\chi (\omega )$ when the arbitrary real instantaneous matrix ${\chi }_{\mathrm{\infty }}\in {\mathbb{R}}^{3\times 3}$ is present. For a transfer-function-only branch adopt $P(t)=\mathrm{Re}({\epsilon }_0{\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}}(\omega )e{e}^{-i\omega t})$ as the harmonic response convention; in a kernel or normalized polynomial branch this follows by substitution. Here $e$ has units $\mathrm{V}{\mathrm{m}}^{-1}$, ${\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is dimensionless, $e^{\ast }$ denotes conjugate transpose, $(C^{\ast }{)}_{jk}=\overline{C_{kj}}$ for a complex matrix $C$, a matrix $H$ is Hermitian when $H^{\ast }=H$ and is positive semidefinite when $v^{\ast }Hv\ge 0$ for every $v\in {\mathbb{C}}^3$, in the convention of Causal convolution response, transfer function and dispersion relations(d), and $⟨g⟩:=T^{-1}{\int }_0^{T}g(t)dt$ with $T=2\pi \mathrm{/}\omega$. The period-averaged work per unit volume, in $\mathrm{W}{\mathrm{m}}^{-3}$, is

$⟨E\cdot {\mathrm{\partial }}_{t}P⟩=\frac{{\epsilon }_0\omega }{2}\mathrm{Im}(e^{\ast }{\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}}(\omega )e)\mathrm{.}$

It is nonnegative for every $e$ if and only if the full Hermitian dissipative part

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}(\omega )=\frac{{\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}}(\omega )-{\chi }_{\mathrm{t}\mathrm{o}\mathrm{t}}(\omega {)}^{\ast }}{2i}$

is positive semidefinite. This frequencywise monochromatic passivity is an additional model condition, not a consequence of causality or dispersion.

For the instantaneous part set $S=({\chi }_{\mathrm{\infty }}+{\chi }_{\mathrm{\infty }}^{\mathsf{T}})\mathrm{/}2$ and $A=({\chi }_{\mathrm{\infty }}-{\chi }_{\mathrm{\infty }}^{\mathsf{T}})\mathrm{/}2$, so $S^{\mathsf{T}}=S$ and $A^{\mathsf{T}}=-A$. For any real $C^1$ history its exact instantaneous work split is

$E\cdot {\mathrm{\partial }}_t({\epsilon }_0{\chi }_{\mathrm{\infty }}E)=\frac{d}{dt}(\frac{{\epsilon }_0}{2}E\cdot SE)+{\epsilon }_{0}E\cdot A{\mathrm{\partial }}_{t}E\mathrm{.}$

The symmetric part therefore supplies the quadratic storage term; no positivity of that term is asserted unless $S$ is additionally positive semidefinite. The antisymmetric part has $E\cdot AE=0$ but can supply nonzero work over a cycle. Its monochromatic contribution is ${\epsilon }_0\omega \mathrm{Im}(e^{\ast }Ae)\mathrm{/}2$, and

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}(\omega )=M_K(\omega )+A\mathrm{/}i,M_K(\omega )=\frac{\chi (\omega )-\chi (\omega {)}^{\ast }}{2i}\mathrm{.}$

Thus only a symmetric real instantaneous matrix contributes purely this storage term for all histories; for an arbitrary real instantaneous matrix the passivity test must include its antisymmetric contribution. For example, ${\chi }_{\mathrm{\infty }}=\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right)$, $K=0$ and $e=(1,i,0{)}^{\mathsf{T}}$ give $E=(\cos \omega t,\sin \omega t,0)$, zero quadratic storage and averaged work ${\epsilon }_0\omega >0$. The opposite circular amplitude $(1,-i,0{)}^{\mathsf{T}}$ gives averaged work $-{\epsilon }_0\omega$, so this instantaneous response is causal but fails the all-amplitude passivity condition.

(e) [Scope] The response is declared in the material rest frame of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(a); boosting the material changes the fields and the rest observer, and no frame-independent instantaneous response law is asserted. The framework does not claim that any particular material realises these branches, no microscopic mechanism is derived, and no empirical statement is made.