Permutation entropy of indexed ensembles: quantifying thermalization dynamics

Abstract We introduce ‘PI-Entropy’ Π ( ρ stretchy="false">˜ stretchy="false">) (the Permutation entropy of an Indexed ensemble) to quantify mixing due complex dynamics for ensemble ρ different initial states evolving under identical dynamics. find that acts as excellent proxy the thermodynamic $S(\rho)$?> S but is much more computationally efficient. study 1-D and 2D iterative maps distinguish a variety system time scales track global loss information relaxes equilibrium. There universal S-shaped relaxation equilibrium generally chaotic systems, this characterized by shuffling timescale correlates with system’s Lyapunov exponent. For Chirikov Standard Map, mixed phase space where chaos grows nonlinear kick strength K , we high behaves like uniformly hyperbolic Cat Map. low see periodic behavior envelope resembling those regime, frequencies depend on size location in well . discuss how adapts experimental work its general utility quantifying systems change from state.