Fluctuation–dissipation cross-exponent test (physical frontier)

[Matesax]

Motivation

Recovering a single relaxation law (e.g. Omori) from a memory kernel is consistency, not validation. By Bernstein’s theorem, a completely monotone kernel can represent essentially any relaxation curve, so matching one exponent constrains nothing.

A genuine test must be selective: it must relate two independently measured quantities in a way the theory forces.

Here, the candidate selective structure is the constitutive law

$$ K = \psi(G^{-1}F), $$

linking:

• information geometry (via $G^{-1}F$), and
• coherence loss (via the generator $K$),

with $\psi$ a constitutive function (e.g. $\psi(\lambda)=\kappa/\lambda$, $\psi(\lambda)=\kappa/\lambda^p$, etc.).

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Open question

Part 1 — Theory

• Goal: Derive a fluctuation–dissipation cross‑exponent relation for the linear‑stochastic model that links:

  • the memory‑kernel decay (from the dissipative part $K(\tau)$), and
  • the fluctuation spectrum of the residual force $\langle F(t)F(s)^\top\rangle$ (from the null covariance),

  under a constitutive law $K = \psi(G^{-1}F)$.

• Task: Reduce this to a checkable relation between:

  • a relaxation exponent (coherence‑loss / kernel decay), and
  • an independently measurable fluctuation exponent (information content / residual‑force spectrum),

  with explicit dependence on $\psi$ (e.g. $\psi(\lambda)=\kappa/\lambda$, $\psi(\lambda)=\kappa/\lambda^p$).

Part 2 — Data

• Goal: Test the derived cross‑exponent relation on a real relaxation system where both exponents are separately measurable.

• Candidate systems:

  • aftershock sequences (Omori decay + independent fluctuation statistics),
  • dielectric relaxation,
  • viscoelastic creep,
  • controlled RC/RL bath or similar lab system.

• Task: Measure:

  • independent coherence‑loss (relaxation exponent),
  • independent information content / fluctuation spectrum (fluctuation exponent),

  and compare:

  • predicted exponent relation (from $K = \psi(G^{-1}F)$),
  • measured exponent relation (from data).

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Deliverables

• Theory:

  • Sharp statement of the cross‑exponent relation for the linear model, explicitly showing how $\psi$ constrains the relation between relaxation and fluctuation exponents.
  • Constitutive‑law analysis: which choices of $\psi$ (e.g. $\kappa/\lambda$, $\kappa/\lambda^p$) are compatible with the derived relation.

• Protocol:

  • Preregistered test protocol:
    • what is measured (coherence‑loss exponent, fluctuation exponent),
    • tolerance bands,
    • null hypothesis (e.g. “no fixed relation between exponents”),
    • thresholds fixed before looking at outcomes.

• Data:

  • Application to ≥ 1 real dataset, with:
    • relaxation exponent estimated independently,
    • fluctuation exponent estimated independently,
    • constitutive prediction computed from $\psi$.

• Verdict:

  • Clear statement:
    • support: relation holds within preregistered tolerance → experimental support for the chosen constitutive law $K = \psi(G^{-1}F)$;
    • refutation: relation fails → the linear‑stochastic modelling or the specific $\psi$ is ruled out for that system.

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Definition of done

A report containing:

• the derived cross‑exponent relation (theoretical part),
• the preregistered protocol (measurement and thresholds),
• at least one honest data confrontation (real system),
• a clear verdict on whether the constitutive law $K = \psi(G^{-1}F)$ is experimentally supported or refuted for that system.

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Honesty requirements

• Calibration assumptions: State all calibration and baseline assumptions explicitly (e.g. null‑baseline condition from Pillar 1, stationarity assumptions, noise model).
• Negative results: Report negative results in full; a clean refutation of a constitutive law or of the linear‑stochastic model is a first‑class outcome.
• No fake confirmation: Do not present mere consistency (a fitted power law or a post‑hoc exponent match) as confirmation. Only preregistered, cross‑exponent agreement counts as support.

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Contributor profile

• Theory: Statistical physics, information geometry, operator theory, control theory.
• Data: Time‑series analysis, experimental data analysis, seismology / condensed matter / electrical circuits / relaxation phenomena.