The Information Theory of Self-Organization Phenomena in Thermal Systems

This paper revisits Brownian motion from the perspective of Information Theory, aiming to explore the connections between Information Theory, Thermodynamics, and Complex Science. First, we propose a single-particle discrete Brownian motion model (SPBM). Within the framework of the maximum entropy principle and Bayesian inference, we demonstrate the equivalence of prior information and constraint conditions, revealing the relationship between local randomness and global probability distribution. By analyzing particle motion, we find that local constraints and randomness can lead to global probability distributions, thereby reflecting the interplay between local and global dynamics in the process of information transfer. Next, we extend our research to multi-particle systems, introducing the concepts of "Energy as Encoding" and "Information Temperature" to clarify how energy distribution determines information structure. We explore how energy, as not only a fundamental physical quantity in physical systems but also an inherently informational one, directly dictates the prior probability distribution of system states, thus serving as a form of information encoding. Based on this, we introduce the concept of "Equilibrium Flow" to explain the self-organizing behavior of systems under energy constraints and Negative Information Temperature. By proving three theorems regarding Equilibrium Flow systems, we reveal the criticality of Self-Organization, energy-information conversion efficiency, and the characteristic that event occurrence probabilities follow the Fermi-Dirac distribution. Through theoretical analysis and theorem proofs, we offer new perspectives for understanding the dynamics of Complex Systems, enriching the theoretical framework of Information Theory, Thermodynamics, and Complex Science, and providing a new theoretical basis for further research in related fields.