On Entropic Characterization of Symmetry Breaking in Dynamical Systems I: Spontaneous Symmetry Breaking

Abstract

Symmetry breaking is a central organizing principle of nonlinear dynamics, yet its information-theoretic signatures remain poorly understood. In this work, we develop an entropic framework for analysing spontaneous symmetry breaking in equivariant dynamical systems, that distinguishes between local and global mechanisms of symmetry loss and shows that each requires a different analytical lens. For local spontaneous symmetry breaking, where a symmetric equilibrium loses stability through bifurcation, we show that critical slowing down broadens the regularized invariant density, increases Shannon entropy, and sharply amplifies steady-state directional information transfer. For global spontaneous symmetry breaking, where the invariant measure itself undergoes qualitative reorganization, we derive a general entropy law for invariant-density restructuring. The broken-phase entropy decomposes into an internal sector entropy and a symmetry-label information term, with classical invariant-set splitting arising as the disjoint-sector limit. Consequently, global symmetry breaking may either increase or decrease entropy depending on how probability is redistributed across symmetry-related sectors. Analytical results are illustrated through canonical examples, providing a quantitative bridge between symmetry, bifurcation structure, invariant measures, and information theory in dynamical systems.