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

We propose a novel framework to analyze symmetry breaking in dynamical systems through the lens of entropy and information transfer. Information transfer quantifies the directional exchange of entropy between observables, allowing us to anticipate the onset of symmetry breaking. For local symmetry breakings, namely, local Spontaneous Symmetry Breaking (SSB) and Dynamical Symmetry Breaking (DSB), we show that as a system loses symmetry, its trajectories exhibit a pronounced slowdown accompanied by an increase in Shannon entropy. This establishes a direct link between symmetry loss, dynamical slowing down, and entropy growth. We also extend the analysis to global symmetry breaking and characterize its associated entropy change. Finally, we demonstrate the efficacy of the proposed framework using representative examples, showing that information theoretic quantities can serve as reliable precursors and diagnostics of symmetry breaking transitions.