Thermodynamic Optimization of Sensory Adaptation via Game-Theoretic Path Integrals

Biological sensory systems, from \textit{E.~coli} chemotaxis to sensory neurons in \textit{C.~elegans}, achieve reliable adaptation over wide dynamic ranges despite operating in strongly noisy and overdamped regimes. Here, we present a field-theoretic framework in which sensory adaptation emerges from a variational free-energy principle, formulated as a stochastic differential game between an organism and its environment. Using an Onsager--Machlup path-integral formalism, we show that the resulting adaptive dynamics are mathematically equivalent to a class of model reference adaptive control schemes and can be interpreted as a dynamic renormalization of the system's Green's function. Within this framework, the phasic overshoot commonly observed in sensory responses arises naturally from an effective inertia ($m^* \approx \tau \gamma$) generated by memory-dissipation coupling, rather than from biochemical fine-tuning. Quantitative fits to experimental data across species yield $R^2 > 0.88$, and indicate that adaptive sensory processing operates within a narrow thermodynamically optimal regime bounded by signal-to-noise and stability constraints.