From Response Laws to Dissipative Structures: A First-Principles Derivation of the Physically Admissible Response Function Set

We derive, from first physical principles, a mathematically closed set of admissible response functions for natural systems obeying three fundamental laws: finite energy capacity, non-zero causal delay, and low-order stability. By approximating the delay kernel with a Padé-(1,1) rational function, we show that all such systems reduce to a universal two-pole cumulative response form with strict concavity. We further prove that this form uniquely minimizes a quadratic entropy-production functional under the Prigogine dissipative framework. Classical response curves from biology, neuroscience, and behavior—such as Michaelis-Menten, Hill functions, and hyperbolic discounting—are shown to be projections or limits of this set. Our results reveal that dissipative structures are not arbitrary but obey a tight, thermodynamically constrained grammar of admissible response curves.