Observer-Information Entropy Production and a Second-Law Principle for Passive Observation: Shannon Information Decay, Fisher Dissipation, and the Physical Acquisition Balance

Source abstract

Classical data processing states that post-processing cannot increase information about an upstream variable, but this principle does not by itself distinguish passive degradation of an already formed observation from active acquisition of new physical evidence. This distinction becomes fundamental in observer-mediated systems whose sensing configuration, state record, calibration, and processing chain participate in information formation. We introduce an observer-information entropy-production framework in which a physical source $X$ generates a record $R_s$ under an observer state, and subsequent source-independent Markov degradation produces a coarser record $R_t$. For any prior, the observer-information entropy production is defined by Σobs[s,t]:=I(X;Rs)−I(X;Rt). For a passive degradation family $X\to R_s\to R_t$, we prove non-negativity, additivity, the conditional-mutual-information identity $\Sigma_{\mathrm{obs}}[s,t]=I(X;R_s\mid R_t)$, and, for discrete $X$, the posterior-uncertainty identity $\Sigma_{\mathrm{obs}}[s,t]=H(X\mid R_t)-H(X\mid R_s)$. Under absolute continuity in a degradation parameter, this yields a second-law principle $\sigma_{\mathrm{obs}}(t):=-\frac{d}{dt}I(X;R_t)\ge 0$. Equality is characterized by Bayesian sufficiency; in finite experiments with a full-support prior, zero production is equivalent to recoverability of the pre-degradation experiment through a parameter-independent reverse kernel.The passive law is complemented by an exact acquisition--dissipation balance. If a new physical interaction produces fresh evidence $A_n$ and a subsequent irreversible processing stage generates $R_{n+1}$, then I(X;Rn+1)−I(X;Rn)=I(X;An∣Rn)−Σobs,n. Thus active acquisition and passive information entropy production are distinct non-negative terms. A strong-data-processing coefficient gives a quantitative lower bound on entropy production. For Gaussian heat-flow degradation $Y_t=X+B_t$, we derive the exact bridge σobs(t)=12(dt−J(Yt))=12t2mmse⁡(X∣Yt) in nats per unit noise variance, thereby connecting Shannon information decay, de Bruijn's identity, Fisher information, and I--MMSE in one observer-degradation law. For pairwise hypotheses, Gaussian smoothing gives ddtDKL(pt‖qt)=−12Jrel(pt‖qt), showing that relative Fisher information is the instantaneous dissipation rate of statistical distinguishability.We finally separate this statistical second-law principle from physical entropy production. In a classical electromagnetic observation chain, the invariant propagation speed $c$ constrains when new evidence can become causally available, while $\Sigma_{\mathrm{obs}}$ measures irreversible loss after information has been physically instantiated in the record. No universal identity between observer-information entropy production and thermodynamic entropy production is asserted; Landauer-type bounds require an explicit physical memory-erasure model. The resulting framework provides a precise observer-mediated distinction among physical propagation, information acquisition, reversible representation transport, irreversible quality degradation, and thermodynamic dissipation.