Holographic Kolmogorov-Sinai entropy and the quantum Lyapunov spectrum

Abstract

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.