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

We develop an entropy based framework for analyzing symmetry breaking in dynamical systems. Information transfer, which measures the directional exchange of entropy between observables, provides a quantitative early indicator of symmetry loss. For local spontaneous symmetry breaking (SSB), we show that as a symmetric equilibrium approaches instability, trajectories exhibit pronounced critical slowing down accompanied by a rise in Shannon entropy. This establishes a direct link between symmetry loss, slowing down, and entropy growth. We further characterize the entropy discontinuity associated with global symmetry breaking (GSSB) through an ergodic decomposition viewpoint. Numerical examples illustrate that entropy and information transfer measures serve as reliable precursors and diagnostics of symmetry breaking transitions.