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 $X_q=\sqrt{q}X+\sqrt{1-q}G$, 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 $h(\sqrt{1-t}{X}_q+\sqrt{t}{Y}_q)$ 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 $A\ge 2\delta$, 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 $p_a=\phiG(1+a\He_4)$, $0<a\le 10^{-2}$, we prove strict entropy concavity for every $0<t<1$ and every $0<q\le 5/8$, hence throughout the currently unresolved symmetric range $0<q\le 4/7$ for this explicit nonequilibrium family. Numerical searches over broader log-concave families are reported only as supporting evidence.

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