2021/07/02 by Georg Maier, Andreas Schäfer, Sebastian Waeber · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Entropy (arrow of time) #Hilbert space #Lyapunov exponent #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear system #Phase space #Physics #Quantum #Quantum many-body systems #Quantum mechanics #Statistical physics #Upper and lower bounds #Von Neumann entropy #gr-qc #hep-th #nlin.CD #nucl-th
paper · pdf · doi:10.1007/jhep01(2022)165
published in Journal of High Energy Physics 2022(1) (Springer Nature) · 20 pages, 6 figures
arxiv created 2021/07/02 · openalex publication_date 2022/01/01 · arxiv updated 2022/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A bstract 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 , N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 = ±2 πT . This fits to the behavior of classical systems with plus/minus symmetric Lyapunov spectra, a symmetry which implies the validity of Liouville’s theorem.