A Physical Framework for Algorithmic Entropy

This paper does not aim to prove new mathematical theorems or claim a fundamental unification of physics and information, but rather to provide a new pedagogical framework for interpreting foundational results in algorithmic information theory. Our focus is on understanding the profound connection between entropy and Kolmogorov complexity. We achieve this by applying these concepts to a physical model. Our work is centered on the distinction, first articulated by Boltzmann, between observable low-complexity macrostates and unobservable high-complexity microstates. We re-examine the known relationships linking complexity and probability, as detailed in works like Li and Vitányi’s An Introduction to Kolmogorov Complexity and Its Applications. Our contribution is to explicitly identify the abstract complexity of a probability distribution $\mathrm{K}\left(\rho \right)$ with the concrete physical complexity of a macrostate $\mathrm{K}\left(M\right)$. Using this framework, we explore the “Not Alone” principle, which states that a high-complexity microstate must belong to a large cluster of peers sharing the same simple properties. We show how this result is a natural consequence of our physical framework, thus providing a clear intuitive model for understanding how algorithmic information imposes structural constraints on physical systems. We end by exploring concrete properties in physics, resolving a few apparent paradoxes, and revealing how these laws are the statistical consequences of simple rules.