Scaling of relaxation exponent with island volume fraction in Kicked Harmonic Net

Determine whether the observed scaling relation between the relaxation-time exponent η and the volume fraction ε occupied by islands of regularity, established for the kicked rotor and Zaslavsky web map, also holds for the Kicked Harmonic Net; specifically, establish whether η varies systematically with ε in the Kicked Harmonic Net or remains approximately constant across parameter regimes.

Background

The authors measure relaxation timescales τ for regularization under bath coupling and find a power-law τ ∝ b^{-η}. In the kicked rotor, η increases as the island-of-regularity volume fraction ε decreases, suggesting a potential universal relation η(ε).

They confirm that η(ε) roughly holds for the web map but report that they could not reproduce this dependence in the many-body Kicked Harmonic Net, where preliminary experiments suggest η ≈ 5 across parameters. Clarifying this behavior is important for understanding many-body self-organization and potential universality.