Noise-Induced Restoration of Entropy Concavity Near Gaussian Equilibrium
Abstract
We study a quantitative form of noise-induced restoration of entropy concavity for one-dimensional nonequilibrium states relaxing toward a Gaussian thermal equilibrium. A partially thermalized state is modeled by Xq=√q X+√(1−q) G, where G is Gaussian equilibrium noise and q is the retained signal fraction. For two independently prepared copies we ask when the entropy of the mixed state h(√(1−t) Xq+√t Yq) is concave in the mixing parameter. The problem is motivated by entropy geometry along Ornstein--Uhlenbeck/Fokker--Planck relaxation, where the relative score is the nonequilibrium thermodynamic force and relative Fisher information is entropy-production rate. We derive an exact symmetric endpoint-curvature formula, separating a nonnegative Ornstein--Uhlenbeck restoring term from a fourth-cumulant correction, and extend it to nonsmooth generalized-Gaussian states e−a|x|p for p>3/2. For half thermalization of a symmetric log-concave input, posterior-variance identities give a quantitative outer-region theorem confining any endpoint obstruction to two explicitly bounded near-equilibrium sectors. We also give spectral and MMSE reformulations of the auxiliary inequality A≥2δ, prove it in an explicit nonlinear neighborhood of Gaussian equilibrium, and show how it sharpens the endpoint bounds. The main all-mixing result concerns the leading symmetric non-Gaussian Hermite mode: for pa=phiG(1+aHe4), 0<a≤10−2, we prove strict entropy concavity for every 0<t<1 and every 0<q≤5/8, hence throughout the currently unresolved symmetric range 0<q≤4/7 for this explicit nonequilibrium family. Numerical searches over broader log-concave families are reported only as supporting evidence.
Related
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BibTeX
@misc{abuajamieh2026noiseinduced,
title = {{Noise-Induced Restoration of Entropy Concavity Near Gaussian Equilibrium}},
author = {Fayez Abu-Ajamieh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22386},
eprint = {2610.22386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22386},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noise-Induced Restoration of Entropy Concavity Near Gaussian Equilibrium
AU - Fayez Abu-Ajamieh
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.22386
UR - https://arxiv.org/abs/2610.22386
AB - We study a quantitative form of noise-induced restoration of entropy concavity for one-dimensional nonequilibrium states relaxing toward a Gaussian thermal equilibrium. A partially thermalized state is modeled by X_q=√q X+√(1−q) G, where G is Gaussian equilibrium noise and q is the retained signal fraction. For two independently prepared copies we ask when the entropy of the mixed state h(√(1−t) X_q+√t Y_q) is concave in the mixing parameter. The problem is motivated by entropy geometry along Ornstein--Uhlenbeck/Fokker--Planck relaxation, where the relative score is the nonequilibrium thermodynamic force and relative Fisher information is entropy-production rate. We derive an exact symmetric endpoint-curvature formula, separating a nonnegative Ornstein--Uhlenbeck restoring term from a fourth-cumulant correction, and extend it to nonsmooth generalized-Gaussian states e^−a|x|^p for p>3/2. For half thermalization of a symmetric log-concave input, posterior-variance identities give a quantitative outer-region theorem confining any endpoint obstruction to two explicitly bounded near-equilibrium sectors. We also give spectral and MMSE reformulations of the auxiliary inequality A≥2δ, prove it in an explicit nonlinear neighborhood of Gaussian equilibrium, and show how it sharpens the endpoint bounds. The main all-mixing result concerns the leading symmetric non-Gaussian Hermite mode: for p_a=phiG(1+aHe_4), 0<a≤10^−2, we prove strict entropy concavity for every 0<t<1 and every 0<q≤5/8, hence throughout the currently unresolved symmetric range 0<q≤4/7 for this explicit nonequilibrium family. Numerical searches over broader log-concave families are reported only as supporting evidence.
ER -