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Separate entropy-only radiation state maps — Examples

1 · Prerequisites

2 · Summary

The example verifies an exact underdamped periodic Cattaneo mode by direct substitution. It is a mathematical solution of the stated linear subsystem, not a reported measurement or claim about a material.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Damped Cattaneo Fourier mode

Example

Choose $C, κ, τ, T_{0}, L > 0$ and an integer $n \geq 1$ . Set $k : = 2 π n / L$ and assume $k^{2} > C / (4 τ κ)$ . Let $Θ \in R$ with $0 < ∣ Θ ∣ < T_{0}$ and

$ω : = \sqrt{\frac{κ k^{2}}{C τ} - \frac{1}{4 τ^{2}}}, ϕ (t) : = Θ e^{- t / (2 τ)} \cos (ω t) .$

On $T_{L}^{3}$ , let $e_{1} = (1, 0, 0)$ be the first standard basis vector of $R^{3}$ and define

$θ (t, x) : = ϕ (t) \cos (k x_{1}), q (t, x) : = - \frac{C}{k} ϕ^{'} (t) \sin (k x_{1}) e_{1} .$

This is a nonzero, smooth, spatially periodic solution for every $t \in R$ of the linear Cattaneo subsystem

$C θ_{t} + div q = 0, τ q_{t} + q = - κ \nabla θ .$

Under the conventional SI interpretation of [F1], $L$ is measured in m, $k$ in m $^{- 1}$ , $ω$ in s $^{- 1}$ , and $Θ, ϕ, θ, T_{0}$ in K; the units of $C, κ, τ, q$ are those stated in [F1]. These unit labels do not assert a material realization.

For $t \geq 0$ , the proxy field $T_{0} + θ$ remains positive, and its oscillation envelope decays as $e^{- t / (2 τ)}$ . The physical-temperature interpretation is limited to the interval where $T_{0} + θ > 0$ .

Facts & Assumptions

Given: The positive parameters, integer $n$ , wave number $k$ , strict underdamping condition, and real amplitude $Θ$ with $0 < ∣ Θ ∣ < T_{0}$ in the Example.

[F1]

Fixed-coefficient Cattaneo entropy and propagation defines the fixed-coefficient Cattaneo system and its effective entropy and quadratic Lyapunov identities. The current carrier is draft; the explicit mode below is checked directly against the displayed PDE.

Verification

1.1givenalgebra

The strict inequality makes $ω > 0$ , and $ϕ (t) = Re (Θ e^{(- 1 / (2 τ) + i ω) t})$ solves $τ ϕ^{''} + ϕ^{'} + (κ k^{2} / C) ϕ = 0$ , since $- 1 / (2 τ) \pm i ω$ are the roots of $τ λ^{2} + λ + κ k^{2} / C = 0$ .

2.1givenstep 1.1algebra

Differentiating $q$ in $x_{1}$ gives $div q = - C ϕ^{'} (t) \cos (k x_{1})$ , so $C θ_{t} + div q = 0$ . Also $τ q_{t} + q = - \frac{C}{k} (τ ϕ^{''} + ϕ^{'}) \sin (k x_{1}) e_{1} = κ k ϕ \sin (k x_{1}) e_{1} = - κ \nabla θ$ , where step 1.1 supplies the second equality and $\nabla θ = - k ϕ \sin (k x_{1}) e_{1}$ . Thus both PDE hold.

3.1givenF1step 2.1∎

The fields are smooth and periodic because $k = 2 π n / L$ and $ϕ$ is smooth; $Θ \neq 0$ makes the mode nonzero. For $t \geq 0$ , $∣ θ (t, x) ∣ \leq ∣ Θ ∣ e^{- t / (2 τ)} < T_{0}$ , so $T_{0} + θ > 0$ . The mode's amplitude decays with the displayed exponential envelope. The entropy-production and Lyapunov identities are the exact identities of [F1], applied to this verified solution.