Permutation entropy of indexed ensembles: quantifying thermalization dynamics

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