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Thermal Conductivity Under Asymmetric Momentum Noise

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
For oscillator coordinates $q_x$ and momenta $p_x$, the Hamiltonian dynamics has Liouville generator $A$. Define $D_x=p_{x+1}\partial_{p_x}-p_x\partial_{p_{x+1}}$; the source's asymmetric momentum-noise generator is $S=\sum_x(D_x^2+p_xp_{x+1}D_x)$. The combined dynamics adds this stochastic generator to the harmonic Hamiltonian generator. Thermal conductivity is the linear-response coefficient relating energy current to a small temperature gradient, or its source Green--Kubo thermodynamic-limit formulation.
Source scope and qualifications.
This noise does not conserve total momentum, as the report explicitly says. The catalogue literature assessment quotes results for a different energy-and-momentum-conserving noise. Those results do not answer this asymmetric-noise question; pinning, dimension, equilibrium, and limit conventions must remain those of the source model.
Catalogue mathematical target.
Catalogue wording retained for traceability.
For harmonic oscillators, add to the Hamiltonian dynamics the asymmetric momentum noise $S=\sum_x[(p_{x+1}\partial_{p_x}-p_x\partial_{p_{x+1}})^2+p_xp_{x+1}(p_{x+1}\partial_{p_x}-p_x\partial_{p_{x+1}})]$. Is the resulting thermal conductivity finite?
Saved literature context.
The supplied catalogue's assessment dated 2026-08-21 records: The harmonic energy-momentum conserving model has an explicit answer: conductivity diverges in dimensions one and two and is finite in dimension at least three or with pinning; for general anharmonic interactions and disorder the question remains open. The assessment is retained as dated provenance; it is not a new comprehensive literature or open-status certification.

Short English statement

Determine whether the specified asymmetric momentum noise makes the harmonic chain's thermal conductivity finite.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-221-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-221

Thermal Conductivity Under Asymmetric Momentum Noise

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
For oscillator coordinates $q_x$ and momenta $p_x$, the Hamiltonian dynamics has Liouville generator $A$. Define $D_x=p_{x+1}\partial_{p_x}-p_x\partial_{p_{x+1}}$; the source's asymmetric momentum-noise generator is $S=\sum_x(D_x^2+p_xp_{x+1}D_x)$. The combined dynamics adds this stochastic generator to the harmonic Hamiltonian generator. Thermal conductivity is the linear-response coefficient relating energy current to a small temperature gradient, or its source Green--Kubo thermodynamic-limit formulation.
Source scope and qualifications.
This noise does not conserve total momentum, as the report explicitly says. The catalogue literature assessment quotes results for a different energy-and-momentum-conserving noise. Those results do not answer this asymmetric-noise question; pinning, dimension, equilibrium, and limit conventions must remain those of the source model.
Catalogue mathematical target.
Catalogue wording retained for traceability.
For harmonic oscillators, add to the Hamiltonian dynamics the asymmetric momentum noise $S=\sum_x[(p_{x+1}\partial_{p_x}-p_x\partial_{p_{x+1}})^2+p_xp_{x+1}(p_{x+1}\partial_{p_x}-p_x\partial_{p_{x+1}})]$. Is the resulting thermal conductivity finite?
Saved literature context.
The supplied catalogue's assessment dated 2026-08-21 records: The harmonic energy-momentum conserving model has an explicit answer: conductivity diverges in dimensions one and two and is finite in dimension at least three or with pinning; for general anharmonic interactions and disorder the question remains open. The assessment is retained as dated provenance; it is not a new comprehensive literature or open-status certification.

Short English statement

Determine whether the specified asymmetric momentum noise makes the harmonic chain's thermal conductivity finite.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Checks energy conservation, failure of momentum conservation, periodic Gaussian invariance and quartic generation by the exact asymmetric noise.
Sources and literature check.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Context and investigated scope.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Exact generator checks.
Use a periodic chain with at least three sites, write $D_x=p_{x+1}\partial_{p_x}-p_x\partial_{p_{x+1}}$, and $K=\tfrac12\sum_xp_x^2$. Since $D_xK=0$, the stated noise satisfies $SK=0$. For total momentum $P=\sum_xp_x$, however, $D_xP=p_{x+1}-p_x$ and $D_x^2P=-(p_x+p_{x+1})$, so no corresponding conservation follows; for uniform nonzero momenta the drift contribution vanishes and $SP=-2P\ne0$.
Stationarity and a nonclosing hierarchy.
Under the centered product Gaussian measure, $D_x$ is skew-adjoint: its vector field is divergence-free and tangent to the quadratic Gaussian exponent. The adjoint of $a_xD_x$, $a_x=p_xp_{x+1}$, sends $1$ to $-D_xa_x=-(p_{x+1}^2-p_x^2)$. Summing around the ring telescopes to zero. Thus $S^*1=0$, confirming Gaussian invariance in this finite periodic setting. Yet \[ S(p_x^2)=2(p_{x-1}^2+p_{x+1}^2-2p_x^2) +2p_x^2(p_{x+1}^2-p_{x-1}^2). \] The cubic drift creates quartic observables from a quadratic one, so the usual closed quadratic covariance method for a linear harmonic-noise model does not apply directly.
First gap.
Energy conservation and Gaussian stationarity are not bounds on the time-integrated current correlation. One needs decay or resolvent estimates uniform in system size for this asymmetric generator. The distinct momentum-conserving noise discussed in the saved literature context cannot supply them without a new argument. The finite-conductivity question for the specified source model remains unresolved.
Verification and adversarial audit.
Checked periodic telescoping, generator adjoints, uniform-momentum countertest and quadratic-to-quartic coefficients; kept the source noise distinct from the cited different model. Model self-audit only; no independent human proof checking. Prompts A and B were applied to this canonical record and its qualified source scope. Checked periodic telescoping, generator adjoints, uniform-momentum countertest and quadratic-to-quartic coefficients; kept the source noise distinct from the cited different model. Retained unresolved status and zero original-target progress; Uniform-in-volume decay/resolvent estimates for its current correlation are missing.

Coordinator checked energy conservation, failure of momentum conservation, the telescoping adjoint drift and the quartic terms in the quadratic-observable generator. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 was used for file preparation and structural checks; no numerical experiment or theorem prover was used. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Uniform-in-volume decay/resolvent estimates for its current correlation are missing.

Completion Estimate

COMPLETION: 0% The calculation is an auxiliary or restricted attempt and establishes no newly verified portion of the full source target. Uniform-in-volume decay/resolvent estimates for its current correlation are missing.

Sources

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