Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Augmented energy and passive absorption

Physical scope

Conditional local energy identities and fixed-volume passive-energy balances for the finite Lorentz oscillator closure and the Drude current closure, in one homogeneous material rest frame under the stated smoothness and finite-energy Maxwell hypotheses. The undriven nonincrease result is for the closure-defined PEC branch. No existence theorem, microscopic explanation, empirical material claim, or energy law outside these constitutive branches is asserted.

Statement

Let $I \subseteq R$ be a nonempty open interval and let $Ω \subseteq R^{3}$ be a nonempty bounded Lipschitz polyhedron in one material rest frame. Use real SI fields: $E, H, D, B : Ω \times I \to R^{3}$ , a free current density $J_{f} : Ω \times I \to R^{3}$ , polarization $P : Ω \times I \to R^{3}$ , and, when present, carrier current $V : Ω \times I \to R^{3}$ . 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 $ε_{\infty} > 0$ and $μ > 0$ 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

$curl E = - \partial_{t} B, curl H = J_{f} + \partial_{t} D$

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 $C^{\infty}$ pointwise and satisfy $E, H \in C^{1} (I; L^{2} (Ω; R^{3})) \cap C (I; H (curl; Ω))$ ; $J_{f}$ and each material-current term belong to $C (I; L^{2})$ . Here $H (curl; Ω)$ 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 $P_{ℓ}$ jointly $C^{\infty}$ and in $C^{2} (I; L^{2})$ , or Drude fields $P, V$ jointly $C^{\infty}$ 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:

Finite Lorentz branch. For some finite $N \geq 1$ , $D = ε_{\infty} E + \sum_{ℓ = 1}^{N} P_{ℓ}$ , $B = μ H$ , and $\partial_{t}^{2} P_{ℓ} + γ_{ℓ} \partial_{t} P_{ℓ} + ω_{ℓ}^{2} P_{ℓ} = a_{ℓ} E,$ where $P_{ℓ}$ is jointly $C^{2}$ , $a_{ℓ} > 0$ in F m $^{- 1}$ s $^{- 2}$ , $ω_{ℓ} > 0$ in s $^{- 1}$ , and $γ_{ℓ} \geq 0$ in s $^{- 1}$ .
Drude branch. $D = ε_{\infty} E + P$ , $B = μ H$ , $\partial_{t} P = V$ , and $\partial_{t} V + γ V = a E$ , where $P, V$ are jointly $C^{1}$ , $a > 0$ in F m $^{- 1}$ s $^{- 2}$ , and $γ \geq 0$ in s $^{- 1}$ .

For the Lorentz branch define

$u_{L} : = \frac{ε_{\infty} ∣ E ∣^{2} + μ ∣ H ∣^{2}}{2} + \sum_{ℓ = 1}^{N} \frac{∣ \partial_{t} P_{ℓ} ∣^{2} + ω_{ℓ}^{2} ∣ P_{ℓ} ∣^{2}}{2 a_{ℓ}} .$

For the Drude branch define

$u_{D} : = \frac{ε_{\infty} ∣ E ∣^{2} + μ ∣ H ∣^{2}}{2} + \frac{∣ V ∣^{2}}{2 a} .$

The Lorentz density is positive definite in $(E, H, (P_{ℓ}, \partial_{t} P_{ℓ})_{ℓ = 1}^{N})$ . 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:

$\partial_{t} u_{L} + div (E \times H) = - E \cdot J_{f} - \sum_{ℓ = 1}^{N} \frac{γ_{ℓ}}{a_{ℓ}} ∣ \partial_{t} P_{ℓ} ∣^{2},$

$\partial_{t} u_{D} + div (E \times H) = - E \cdot J_{f} - \frac{γ}{a} ∣ V ∣^{2} .$

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 $U (t) = \int_{Ω} u_{L} (x, t) d x$ or $U (t) = \int_{Ω} u_{D} (x, t) d x$ 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.

Statement references

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 ( ${A C}_{ω}$ )

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

Facts & Assumptions

Given: Countable Choice, the bounded-domain, time-regularity, SI, and closure-defined PEC conventions of the finite-energy boundary model; either the finite Lorentz branch or the Drude branch and its stated coefficients; the pointwise Maxwell curl equations and local regularity in the Statement.

[A1]

The Lorentz closure is the real, homogeneous, spatially local constitutive model $D = ε_{\infty} E + \sum P_{ℓ}$ , $B = μ H$ , $P_{ℓ, t t} + γ_{ℓ} P_{ℓ, t} + ω_{ℓ}^{2} P_{ℓ} = a_{ℓ} E$ , with $a_{ℓ}, ω_{ℓ}, ε_{\infty}, μ > 0$ , $γ_{ℓ} \geq 0$ , and oscillator initial state $(P_{ℓ}, P_{ℓ, t})$ . Its macroscopic identification is $P_{t o t} = (ε_{\infty} - ε_{0}) E + \sum P_{ℓ}$ and $M_{t o t} = (μ_{0}^{- 1} - μ^{- 1}) B$ (Finite Lorentz oscillator closure).

[A2]

The Drude closure is $P_{t} = V$ , $V_{t} + γ V = a E$ , $D = ε_{\infty} E + P$ , $B = μ H$ , with $a, ε_{\infty}, μ > 0$ , $γ \geq 0$ and the initial polarization state specified separately from $V$ . Its macroscopic identification is $P_{t o t} = (ε_{\infty} - ε_{0}) E + P$ and $M_{t o t} = (μ_{0}^{- 1} - μ^{- 1}) B$ (Drude current closure and collisionless limit).

[A3]

Countable Choice is assumed, as required by the weak Maxwell energy supplier's real $H (curl)$ convention (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

In the finite-energy boundary model, $H (curl; Ω)$ consists of $L^{2}$ vector fields whose distributional curl is represented by an $L^{2}$ field; the PEC condition is the graph-norm closure definition of $H_{0} (curl; Ω)$ , not an unproved identification with a trace kernel (Finite-energy PEC and impedance boundary formulation).

[F2]

For $C^{1}$ real vector fields, $div (E \times H) = H \cdot curl E - E \cdot curl H$ (The divergence and curl of a cross product).

[F3]

The Euclidean product is a symmetric positive-definite real bilinear form, so $∣ X ∣^{2} = ⟨ X, X ⟩ \geq 0$ and its componentwise pairing obeys the product rule (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

For differentiable real scalar functions, sums, products and scalar multiples obey their usual derivative rules; these apply componentwise to the field squares and oscillator energies (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L1]

For smooth macroscopic material fields, $div D = ρ_{f}$ , $div B = 0$ , $curl E = - \partial_{t} B$ , and $curl H = J_{f} + \partial_{t} D$ (The macroscopic Maxwell equations in terms of D, H and the free sources, clause (b)).

[L2]

For the finite-energy Maxwell model with time-independent bounded symmetric uniformly positive coefficients, $C^{1} (I; L^{2}) \cap C (I; H (curl))$ fields, $J \in C (I; L^{2})$ and the two $L^{2}$ evolution identities, the closure-defined PEC branch satisfies $W_{f} (t) - W_{f} (s) = - \int_{s}^{t} \int_{Ω} E \cdot J d x d r$ , where $W_{f} = \frac{1}{2} \int_{Ω} (E \cdot ε E + H \cdot μ H) d x$ (Energy balance and zero-data forward uniqueness for admissible weak Maxwell fields, Statement).

Proof

technique · derive the local Maxwell work balance, dot each material equation with its velocity/current, then apply the exact weak Poynting result only to the PEC fixed-volume conclusion

1.1givenA1A2F2F3F4L1

In either branch put $J_{t o t} = J_{f} + \sum_{ℓ = 1}^{N} \partial_{t} P_{ℓ}$ for Lorentz, and $J_{t o t} = J_{f} + V$ for Drude. By [L1], $D_{t} = ε_{\infty} E_{t} + \sum P_{ℓ, t}$ or $D_{t} = ε_{\infty} E_{t} + V$ and $curl H = J_{f} + D_{t}$ ; also $B_{t} = μ H_{t} = - curl E$ . Thus $ε_{\infty} E_{t} = curl H - J_{t o t}$ , and [F3]–[F4] give $\partial_{t} ((ε_{\infty} ∣ E ∣^{2} + μ ∣ H ∣^{2}) / 2) = E \cdot curl H - E \cdot J_{t o t} - H \cdot curl E$ . By [F2], this is $- div (E \times H) - E \cdot J_{f} - \sum E \cdot P_{ℓ, t}$ in the Lorentz branch, and $- div (E \times H) - E \cdot J_{f} - E \cdot V$ in the Drude branch.

1.2givenA1F3F4

In the Lorentz branch, dot the $ℓ$ th oscillator equation with $P_{ℓ, t} / a_{ℓ}$ . By [F3]–[F4], the product rule gives $E \cdot P_{ℓ, t} = \partial_{t} ((∣ P_{ℓ, t} ∣^{2} + ω_{ℓ}^{2} ∣ P_{ℓ} ∣^{2}) / (2 a_{ℓ})) + (γ_{ℓ} / a_{ℓ}) ∣ P_{ℓ, t} ∣^{2}$ . The division is valid since $a_{ℓ} > 0$ ; the $ω_{ℓ}^{2} P_{ℓ} \cdot P_{ℓ, t}$ term is the derivative of the potential-energy term and the damping term is nonnegative.

1.3givenA2F3F4

In the Drude branch, dot $V_{t} + γ V = a E$ with $V / a$ . By [F3]–[F4] and $a > 0$ , $\partial_{t} (∣ V ∣^{2} / 2) = V \cdot V_{t}$ and $E \cdot V = \partial_{t} (∣ V ∣^{2} / (2 a)) + (γ / a) ∣ V ∣^{2}$ . The last term is nonnegative because $γ \geq 0$ .

2.1step 1.1step 1.2step 1.3algebra

Substituting step 1.2 into the Lorentz balance of step 1.1 cancels every $E \cdot P_{ℓ, t}$ term and yields the displayed identity for $u_{L}$ . Substituting step 1.3 into the Drude balance of step 1.1 cancels $E \cdot V$ and yields the displayed identity for $u_{D}$ . These are pointwise identities under the stated local regularity; they do not require a boundary condition.

2.2givenA1A2A3F1L2step 1.2step 1.3

For the Lorentz fixed-volume conclusion, the fields in step 1.1 satisfy the weak model of [L2] with coefficient maps $ε (x) = ε_{\infty} I_{3}$ , $μ (x) = μ I_{3}$ and current $J = J_{t o t}$ . These are static symmetric uniformly positive maps, and $J_{t o t} \in C (I; L^{2})$ by the hypotheses. Apply [L2] to obtain the field-energy change $- \int_{s}^{t} \int_{Ω} E \cdot J_{t o t} d x d r$ . Integrating step 1.2 over $Ω \times [s, t]$ adds $+ \int_{s}^{t} \int_{Ω} E \cdot \sum P_{ℓ, t} d x d r - \sum \int_{s}^{t} \int_{Ω} (γ_{ℓ} / a_{ℓ}) ∣ P_{ℓ, t} ∣^{2} d x d r$ ; the polarization work cancels the same term in $J_{t o t}$ . Hence $U_{L} (t) - U_{L} (s) = - \int_{s}^{t} \int_{Ω} E \cdot J_{f} d x d r - \sum_{ℓ = 1}^{N} \int_{s}^{t} \int_{Ω} (γ_{ℓ} / a_{ℓ}) ∣ P_{ℓ, t} ∣^{2} d x d r$ . The same argument in the Drude branch uses $J_{t o t} = J_{f} + V$ , applies [L2] with the same constant positive coefficients, and adds the integral of step 1.3, giving $U_{D} (t) - U_{D} (s) = - \int_{s}^{t} \int_{Ω} E \cdot J_{f} d x d r - \int_{s}^{t} \int_{Ω} (γ / a) ∣ V ∣^{2} d x d r$ . The PEC boundary contribution is zero by the exact PEC branch in [L2].

3.1step 2.2A1A2F3algebra∎

In either branch, $ε_{\infty}, μ, a_{ℓ}, ω_{ℓ}$ (or $a$ ) are strictly positive, so [F3] makes the displayed densities nonnegative. The Lorentz density vanishes only when $E, H, P_{ℓ}, \partial_{t} P_{ℓ}$ all vanish; the Drude density vanishes exactly when $E, H, V$ vanish, without constraining a constant polarization offset. With $J_{f} = 0$ , every term on the right sides in step 2.2 is nonpositive because each $γ_{ℓ}, γ \geq 0$ . Thus $U_{L} (t) \leq U_{L} (s)$ or $U_{D} (t) \leq U_{D} (s)$ for $s < t$ in $I$ , and both are nonnegative energies. This is a conditional model deduction, not an empirical claim.

Augmented energy and passive absorption

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources