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

TL;DR

This work introduces an entropy-based framework to analyze symmetry breaking in dynamical systems, linking the approach to instability with Shannon entropy growth and critical slowing down. It shows that local SSB induces a monotone increase in entropy as a symmetric equilibrium loses stability, while global SSB produces a discontinuous entropy jump tied to ergodic decomposition and a reorganization of invariant measures. The authors further connect entropy dynamics to directional information transfer, proving monotonic amplification under linear-Gaussian assumptions and validating the approach with Duffing and Stommel models as concrete demonstrations. Collectively, the framework provides theoretical and computational tools for early-warning diagnostics and mechanistic insights into symmetry-breaking transitions, with potential applicability to a broad class of dynamical systems via the Perron–Frobenius and Koopman formalisms.

Abstract

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.

Figures (9)

• Figure 1: A circle is perfectly symmetric—its structure is preserved under rotation—whereas its deformed counterpart lacks symmetry and appears more irregular. This contrast in apparent order reflects the connection between symmetry and structure.
• Figure 2: Hierarchy of symmetry breaking in dynamical systems. Local SSB alters the equilibrium structure in state space, DSB corresponds to transient symmetry loss along trajectories, and global SSB reorganizes the invariant measure itself.
• Figure 3:
• Figure 4: (a) Stability-losing SSB. (b) Multi-stability-driven SSB.
• Figure 5: $\mathbb{Z}_2$ symmetry-breaking bifurcation: for $\mu < \mu_c$, a symmetric equilibrium $x^\star(\mu)$ exists; for $\mu > \mu_c$, the symmetric branch becomes unstable and two asymmetric equilibria $x_1(\mu), x_2(\mu)$ emerge, breaking the symmetry.

Theorems & Definitions (37)

• Definition 1: Equivariant Dynamical System stewart_bookdynamical_symmetry_book
• Example 2
• Remark 3
• Definition 4: Isotropy Set
• Definition 5: Isotropy subgroup of an invariant set
• Definition 6: Fixed-point set in state space
• Definition 7: Invariant densities under group action
• Definition 8: Pushforward of a measure
• Remark 9: Notation for measures and densities
• Definition 10: Spontaneous Symmetry Breaking stewart_bookdynamical_symmetry_book