  In classical chaotic systems the entropy, averaged over initial phase space
distributions, follows an universal behavior. While approaching thermal
equilibrium it passes through a stage where it grows linearly, while the growth
rate, the Kolmogorov-Sinai entropy, is given by the sum over all positive
Lyapunov exponents. A natural question is whether a similar relation is valid
for quantum systems. We argue that the Maldacena-Shenker-Stanford bound on
quantum Lyapunov exponents $\lambda$ implies that the upper bound on the growth
rate of the entropy, averaged over states in Hilbert space that evolve towards
a thermal state with temperature $T$ and entropy $S_{eq}$, should be given by
$S_{eq} \pi T =\sum_{\lambda >0}2 \pi T$. Strongly coupled, large $N$ theories
with black hole duals should saturate the bound. By studying a large number of
isotropization processes of random, spatially homogeneous, far from equilibrium
initial states in large $N$, $\mathcal{N}=4$ Super Yang Mills theory at strong
coupling and computing the ensemble averaged growth rate of the dual black
hole's apparent horizon area, we find both an analogous behavior as in
classical chaotic systems and numerical evidence that the conjectured bound on
averaged entropy growth is saturated granted that the Lyapunov exponents are
degenerate $\lambda = \pm 2 \pi T$. This fits to the behavior of classical
systems with plus/minus symmetric Lyapunov spectra, a symmetry which implies
the validity of Liouville's theorem.
