Finite-Temperature de Bruijn Identities: Fisher Information as the Spectral Gap of Blahut--Arimoto Dynamics

We uncover a finite-temperature extension of de Bruijn’s identity – the classical relation $\frac{d}{dt}h(X+\sqrt{t}Z)=\frac{1}{2}J(X)$ connecting differential entropy and Fisher information. Our framework is the spectral theory of Blahut–Arimoto (BA) dynamics, recently developed by Wang~\cite{Wang2026} for the analysis of rate-distortion optimization. The central observation is elementary yet profound: for Gaussian sources, the spectral gap $\text{lam}$ of the BA relaxation kernel $\text{G}$ satisfies $\text{lam}=1\mathrm{/}(2\beta {\sigma }^2)$~\cite{Wang2026}, while the Fisher information of the source is $J=1\mathrm{/}{\sigma }^2$. Hence $$\text{lam}=\frac{J}{2\beta }$$ for all inverse temperatures $\beta >1\mathrm{/}(2{\sigma }^2)$. This identifies the BA spectral gap as a \emph{finite-temperature regularization of Fisher information}. From this observation we derive an exact finite-temperature de Bruijn identity: $$\frac{\mathrm{\partial }{F}_{\beta }}{\mathrm{\partial }{\sigma }^2}=\frac{1}{2\beta {\sigma }^2}=\text{lam},$$ where $F_{\beta }$ is the BA free energy. This identity holds for all finite $\beta$ without any limit procedure. The classical de Bruijn identity follows as the exact consequence $\beta \mathrm{\partial }{F}_{\beta }\mathrm{/}\mathrm{\partial }{\sigma }^2=J\mathrm{/}2$. The significance is structural: classical de Bruijn is not an isolated fact about Gaussian convolutions, but the $\beta \to \mathrm{\infty }$ shadow of a one-parameter family of exact identities living in the spectral geometry of rate-distortion optimization. We discuss implications for the entropy power inequality, the ${\chi }^2$-dissipation structure of BA dynamics, and the geometric unification of information inequalities.