The Operator Theory: A Yardstick for Complexity from Quarks to Memons—Relationships with Evolution and Thermodynamics

Thermodynamic theory predicts that the universe develops towards maximum energy dispersal. Meanwhile, complex systems continue to form. The search for an explanation of these seemingly opposing trends has inspired many scientists. The theory of nonequilibrium thermodynamics brought much progress, allowing subsystems to become more complex at the costs of external energy gradients. But energy gradients may not tell the whole story, because, while they explain the existence of cells, gradients alone cannot explain the existence of complex organisms such as plants, tigers, or humans. Contributing to our understanding of the relationships between complexity and thermodynamics, this study focuses on a hierarchical subset of all complex systems. The systems in this subset have formed, in a step-by-step way, through a series of “dual-closure” processes. Every system produced through dual closure is called an “operator,” and their stringent complexity hierarchy is called the “operator hierarchy.” It is demonstrated that the operators can be grouped into three major classes with fundamentally different thermodynamics: (1) abiotic operators resulting from condensation reactions, (2) organisms resulting from contained autocatalysis and competition, and (3) neural network organisms driven by autocatalysis, learning, and competition. To these three groups a fourth group of rapidly evolving systems that are not operators can be added: “artifacts” made by organisms, notably humans. While normally being viewed as the result of self-organization, the design of artifacts may in fact be the product of “allo-organization.”