Physical Cost of Observer-Information Entropy Production: From Stochastic Thermodynamics to Fisher–Shannon Information Dissipation

Abstract

Observer-information entropy production quantifies the loss of recoverable information about a physical source when an already formed observation undergoes source-independent irreversible degradation. The statistical second-law statement \[ \Sigma_{\mathrm{obs}}[s,t] =I(X;R_s)-I(X;R_t) =I(X;R_s\mid R_t)\ge 0 \] follows from data processing, but by itself it does not identify a physical energetic or thermodynamic cost. This paper establishes such a connection within an explicit and deliberately restricted stochastic-thermodynamic regime. We consider a finite observer memory $R_t$ coupled to a single heat bath at temperature $T$. A source label $X$ is fixed during the passive relaxation, the transition generator of $R_t$ is independent of $X$, and the memory dynamics satisfies detailed balance with respect to a strictly positive equilibrium law $q$. For the joint source--memory ensemble and for the unconditioned memory marginal, we define the standard stochastic-thermodynamic entropy-production rates. Using the relative-entropy chain rule and the detailed-balance $H$-theorem, we prove the exact correlation-dissipation decomposition \[ \frac{\dot S_{\mathrm{tot}}^{XR}}{k_B} = \sigma_{\mathrm{obs}} + \frac{\dot S_{\mathrm{tot}}^{R}}{k_B}. \] Consequently, \[ \dot S_{\mathrm{tot}}^{XR}\ge k_B\sigma_{\mathrm{obs}}, \qquad \Delta S_{\mathrm{tot}}^{XR}[s,t]\ge k_B\Sigma_{\mathrm{obs}}[s,t]. \] At fixed temperature the same result is an excess-free-energy dissipation bound, \[ \Fex_{XR}(s)-\Fex_{XR}(t) \ge k_BT\Sigma_{\mathrm{obs}}[s,t]. \] Equality holds when the memory marginal remains at equilibrium, in which case all thermodynamic irreversibility is attributable to the destruction of source--record correlation. A symmetric two-state memory provides an exactly solvable saturation example. The result is not a universal Landauer principle. It constrains total entropy production and loss of nonequilibrium free energy during passive decorrelation; it does not assert a universal heat or work cost for arbitrary information loss. Driven memories, source-dependent transition rates, active measurement, feedback, multiple reservoirs, and nonequilibrium steady states require the broader information-thermodynamic bookkeeping developed in stochastic thermodynamics. We then place the thermodynamic result beside two Fisher--Shannon bridges. Under Gaussian observation heat flow $Y_t=X+B_t$, observer-information dissipation obeys the exact identity \[ \sigma_{\mathrm{obs}}(t) = \frac12\left(\frac{d}{t}-J(Y_t)\right) = \frac{1}{2t^2}\operatorname{mmse}(X\mid Y_t), \] while pairwise statistical distinguishability obeys \[ \frac{d}{dt}\KL(p_t\|q_t) =-\frac12\Jrel(p_t\|q_t). \] Recent temporal-Fisher speed-limit results provide a complementary route from physical entropy production to the speed of probability evolution. Together these results separate three objects that are often conflated: thermodynamic irreversibility, Shannon information loss about the physical source, and Fisher-geometric statistical motion. The paper thereby supplies a physical-cost layer for observer information theory without identifying information with energy or thermodynamic entropy.