CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Equal number density, mean velocity, and scalar energy do not determine covariance stress

Statement refuted

For solutions in the setting of Exact kinetic moment balances and unclosed fluxes, any two phase-space number densities with the same number-density field $n$ , mean-velocity vector $u$ , and internal kinetic-energy density $e$ have the same covariance stress $Π$ .

Facts & Assumptions

Given: a particle mass $m > 0$ , number density $n_{0} > 0$ , and positive velocity-squared parameters $α, β, γ$ with $α \neq β$ .

[F1]

Kinetic density, velocity, covariance stress and heat flux: $u = n^{- 1} \int v F d v$ , $Π = m \int (v - u) \otimes (v - u) F d v$ , and $e = \frac{1}{2} tr Π$ .

[F2]

Exact kinetic moment balances and unclosed fluxes: the local moment equations apply to periodic, sufficiently decaying $F$ solving the displayed kinetic equation with velocity-independent acceleration and a source whose number, momentum, and energy moments vanish.

[F3]

Regularity and invariance of the cutoff Gaussian-mixture class: for $0 < a < b$ and $Σ \in K_{a, b}$ , the normalized Gaussian is $γ_{Σ, u} (v) = (2 π)^{- 3 / 2} (\det Σ)^{- 1 / 2} \exp (- (v - u)^{T} Σ^{- 1} (v - u) / 2)$ .

[F4]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ : $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ .

[F5]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative measurable product integrals may be evaluated iteratively on the stated sigma-finite factors.

Counterexample

1.1F2F3F4F5algebra

Define $Σ_{1} : = diag (α, β, γ)$ and $Σ_{2} : = diag (β, α, γ)$ . Set $a_{*} : = \min {α, β, γ}$ and $b_{*} : = \max {α, β, γ}$ . Since $α, β, γ > 0$ and $α \neq β$ , one has $0 < a_{*} < b_{*}$ and $Σ_{1}, Σ_{2} \in K_{a_{*}, b_{*}}$ . On any flat three-torus with positive periods and on $I = R$ , take $a = 0$ and $Q = 0$ . Define $F_{j} (r, v, t) = n_{0} γ_{Σ_{j}, 0} (v)$ using [F3]. Each $F_{j}$ is smooth and rapidly decreasing in velocity, is constant in $r, t$ , and solves the kinetic equation with zero acceleration and zero source. Gaussian normalization follows from [F4] by scaling each coordinate, and [F5] permits the product integration.

2.1F1F4step 1.1algebra∎

Each density has number density $n_{0}$ and mean-velocity vector $u = 0$ by normalization and oddness. In one coordinate with variance $s^{2}$ , differentiating $x \exp (- x^{2} / (2 s^{2}))$ and integrating over $R$ gives $\int x^{2} e^{- x^{2} / (2 s^{2})} d x = s^{2} \int e^{- x^{2} / (2 s^{2})} d x$ ; the boundary term vanishes because the Gaussian decays to zero. Applying this to each coordinate gives covariance matrices $Σ_{1}$ and $Σ_{2}$ . Thus both have the same scalar internal energy density $e = m n_{0} (α + β + γ) / 2$ , while $Π_{1} = m n_{0} diag (α, β, γ)$ and $Π_{2} = m n_{0} diag (β, α, γ)$ . Since $α \neq β$ , $Π_{1} \neq Π_{2}$ , contradicting determination of $Π$ by $(n, u, e)$ .

Equal number density, mean velocity, and scalar energy do not determine covariance stress

Statement refuted

Facts & Assumptions

Counterexample

Depends on

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