Permutation entropy of indexed ensembles: quantifying thermalization dynamics

We introduce 'PI-Entropy' Pi((rho) over tilde) (the Permutation entropy of an Indexed ensemble) to quantify mixing due to complex dynamics for an ensemble rho of different initial states evolving under identical dynamics. We find that Pi((rho) over tilde) acts as an excellent proxy for the thermodynamic entropy S(rho) but is much more computationally efficient. We study 1-D and 2-D iterative maps and find that Pi((rho) over tilde) 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, Pi((rho) over tilde) behaves like the uniformly hyperbolic 2-D 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 Pi((rho) over tilde) adapts to experimental work and its general utility in quantifying how complex systems change from a low entropy to a high entropy state.