Springer Berlin Heidelberg
Berlin/Heidelberg
Springer
13130
10.1007/13130.1029-8479
1029-8479
32745009
Journal of High Energy Physics
J. High Energ. Phys.
Physics
Elementary Particles, Quantum Field Theory
Quantum Field Theories, String Theory
Classical and Quantum Gravitation, Relativity Theory
Quantum Physics
Physics and Astronomy
2022
2022
12
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154
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The Author(s)
2022
17657
2107.01300
10.1007/JHEP01(2022)165
165
165
Holographic Kolmogorov-Sinai entropy and the quantum Lyapunov spectrum
Regular Article - Theoretical Physics
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2022
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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (
CC-BY 4.0
), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
13130
2022
2022
1
1
Georg
Maier
georg.maier@physik.uni-regensburg.de
Andreas
Schäfer
andreas.schaefer@physik.uni-regensburg.de
Sebastian
Waeber
swaebe@uw.edu
grid.7727.5
0000 0001 2190 5763
Institute of Theoretical Physics
University of Regensburg
D-93040
Regensburg
Germany
grid.6451.6
0000000121102151
Department of Physics
Technion
32000
Haifa
Israel
grid.34477.33
0000000122986657
Department of Physics
University of Washington
Seattle
WA
98195-1560
USA
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,
$\mathcal{N}$
$$ \mathcal{N} $$
= 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 λ
i
= ±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.
Keywords
Black Holes
AdS-CFT Correspondence
Gauge-Gravity Correspondence
Holography and quark-gluon plasmas
ArXiv ePrint:
2107.01300