  We introduce `PI-Entropy' $\Pi(\tilde{\rho})$ (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 $\Pi(\tilde{\rho})$ 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 $\Pi(\tilde{\rho})$ 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 \emph{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
$K$, $\Pi(\tilde{\rho})$ 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
$\Pi(\tilde{\rho})$ adapts to experimental work and its general utility in
quantifying how complex systems change from a low entropy to a high entropy
state.
