Diffusive-Time Self-Averaging via a Degree-Four Standard-Pair Estimate

Diffusive-Time Self-Averaging via a Degree-Four Standard-Pair Estimate Overview This preprint studies energy transport in a pinned harmonic chain driven by a fast deterministic chaotic magnetic field. The main question is whether the empirical energy profile follows a deterministic macroscopic heat equation on diffusive time scales, rather than merely having an expectation that satisfies such an equation. At each site of a periodic one-dimensional chain, the oscillator position and momentum are two-dimensional vectors. The dynamics combine harmonic interactions, pinning, and a rapidly evolving deterministic forcing generated by an expanding map. The forcing is scaled so that its cumulative effect remains nontrivial in the macroscopic limit. The analysis considers a fixed-anisotropy shell ensemble and the regime in which the microscopic forcing time scale becomes sufficiently small relative to the system size. Main result Under the hypotheses stated in the manuscript, the principal result is a diffusive-time law of large numbers for the empirical energy profile. The paper establishes convergence in the stated mean-square and probabilistic senses to a deterministic heat profile for the specified initial ensemble and scaling regime. The principal technical issue is that control of the expected energy profile does not by itself imply that the random empirical profile concentrates around its deterministic limit. The principal technical difficulty is that conditional spherical concentration controls fluctuations around a conditional mean, while that mean remains random through the initial field. The proof therefore requires control of environment-induced correlations in addition to concentration of the initial shell ensemble. The proof therefore requires control of correlations generated by the environment, in addition to concentration of the initial shell ensemble. Proof strategy The argument combines four main components: An explicit calculation of the initial energy profile and estimates adapted to the anisotropic shell ensemble. Conditional spherical concentration estimates for fluctuations of quadratic observables. A rank-four covariance estimate for a stochastic proxy in which two replicas evolve under shared environmental noise. A paired, degree-four standard-pair comparison transferring the proxy estimates to the original deterministic chaotic dynamics. The paired estimate controls the two-replica observable under the same fast field. Its proof uses local block expansions, derivative and coefficient bounds, cell freezing, conditioned transfer estimates, and decay of off-site correlations in the delayed fast-time variable. The accumulated estimates yield the sufficient scale-separation condition that epsilon_N is at most N raised to the power minus alpha, for any fixed alpha greater than 12. This exponent is a sufficient condition derived by the proof; the manuscript does not claim that it is optimal. Relation to previous work The closest predecessor is the work of Canestrari, Liverani and Olla, Heat equation from a deterministic dynamics, which establishes an expectation-level heat limit for the same deterministic magnetic-forcing model. The present manuscript addresses the distinct fluctuation problem needed for a law of large numbers for the empirical energy profile. It does not claim novelty for the microscopic model, the expectation-level heat equation, standard-pair technology, concentration inequalities, or the general Green–Kubo principle. The manuscript also places its result within the established literature on deterministic homogenization, transfer operators, harmonic-chain hydrodynamics, and energy diffusion. Work proving energy-profile laws of large numbers for stochastic velocity-flip models is discussed as a related but distinct setting. A dedicated normalization section derives the Green–Kubo coefficient convention used in this manuscript, compares it with the displayed formulas in the closest predecessor, and provides an ordered-time calculation and scalar consistency check. Scope and limitations The result concerns the specified deterministic-forcing model, fixed-anisotropy shell ensemble, and sufficient scaling regime described in the paper. It does not establish a law of large numbers for arbitrary initial distributions, nor does it claim that the sufficient exponent greater than 12 is sharp. Files and reproducibility This record includes the PDF manuscript and its LaTeX source. The source is provided to facilitate inspection of the assumptions, derivations, theorem statements, and bibliography. AI assistance This work was produced with substantial assistance from an AI system (ChatGPT, Claude, Gemini). The proofs have not yet been independently checked by a human specialist. Corrections are welcome and encouraged.