Holographic Kolmogorov-Sinai entropy and the quantum Lyapunov spectrum

Georg Maier,Andreas Schäfer,Sebastian Waeber

doi:10.1007/jhep01(2022)165

Abstract

In classical chaotic systems the entropy, averaged over initial phase space distributions, follows a universal behavior. While approaching thermal equilibrium it passes through a stage where it grows linearly, while the growth rate, the Kolmogorov-Sinai entropy (rate), 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 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, should be given by πT times the thermal state’s von Neumann entropy. Strongly coupled, large N theories with black hole duals should saturate the bound. To test this we study 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 compute 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 and given by λi = ±2πT. This fits to the behavior of classical systems with plus/minus symmetric Lyapunov spectra, a symmetry which implies the validity of Liouville’s theorem.

Highlights

JHEP01(2022)165 entropy as the Kolmogorov-Sinai entropy rate, since it is not an entropy, but rather an entropy growth rate
We argue that the MaldacenaShenker-Stanford bound on quantum Lyapunov exponents 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, should be given by πT times the thermal state’s von Neumann entropy
Large N theories with black hole duals should saturate the bound. To test this we study a large number of isotropization processes of random, spatially homogeneous, far from equilibrium initial states in large N, N = 4 Super Yang Mills theory at strong coupling and compute the ensemble averaged growth rate of the dual black hole’s apparent horizon area

Summary

Introduction

JHEP01(2022)165 entropy as the Kolmogorov-Sinai entropy rate, since (contradicting its name often found in the literature) it is not an entropy, but rather an entropy growth rate. More and more phase space cells (used to evaluate the coarse grained entropy S) are required to cover its shape, while Liouville’s theorem ensures that its total volume stays unchanged Those directions in phase space for which the Lyapunov exponents are positive, will contribute to the growth of the number of needed cells. In general only the ensemble averaged entropy, where we average over a large ensemble of initial phase space configurations that are far from equilibrium, will allow us to determine the sum over all positive Lyapunov exponents, as demonstrated in [21]. In [3] the authors speculated that the Maldacena-Shenker-Stanford (MSS) bound implies an upper bound on entropy growth This bound should be saturated for conformal theories with Einstein gravity duals in the bulk, which implies that black holes are the fastest entropy generators with dS/dt = i,λi>0 2πT. We determine the number of Lyapunov exponents from the number of degrees of freedom of the equilibrated black hole, which is taken to be (albeit in all generality not proven to be) the same as its Bekenstein-Hawking entropy Seq

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