Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, Netherlands.
Institute for Theoretical Physics, Utrecht University, 3584 CC Utrecht, Netherlands.
Phys Rev Lett. 2018 May 18;120(20):201604. doi: 10.1103/PhysRevLett.120.201604.
Holographic theories with classical gravity duals are maximally chaotic; i.e., they saturate the universal bound on the rate of growth of chaos [J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 106JHEPFG1029-847910.1007/JHEP08(2016)106]. It is interesting to ask whether this property is true only for leading large N correlators or if it can show up elsewhere. In this Letter, we consider the simplest setup to tackle this question: a Brownian particle coupled to a thermal ensemble. We find that the four-point out-of-time-order correlator that diagnoses chaos initially grows at an exponential rate that saturates the chaos bound, i.e., with a Lyapunov exponent λ_{L}=2π/β. However, the scrambling time is parametrically smaller than for plasma excitations, t_{}∼βlogsqrt[λ] instead of t_{}∼βlogN^{2}. Our result shows that, at least in certain cases, maximal chaos can be attained in the probe sector without the explicit need of gravitational degrees of freedom.
全息理论与经典引力对偶具有最大混沌性；也就是说，它们饱和了混沌增长率的普遍上限[J. Maldacena、S. H. Shenker 和 D. Stanford，J. High Energy Phys. 08（2016）106JHEPFG1029-847910.1007/JHEP08(2016)106]。有趣的是，我们想知道这个性质是否仅适用于主导大 N 关联函数，或者它是否可以在其他地方出现。在这封信中，我们考虑了最简单的设置来解决这个问题：一个布朗粒子与热系综耦合。我们发现，诊断混沌的四点点外时间顺序相关函数最初以饱和混沌边界的指数速率增长，即 Lyapunov 指数 λ_{L}=2π/β。然而，混乱时间比等离子体激发的混乱时间小一个参数，t_{}∼βlogsqrt[λ]，而不是 t_{}∼βlogN^{2}。我们的结果表明，至少在某些情况下，在没有引力自由度的情况下，探针领域可以达到最大混沌。
