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) 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 ${\lambda }_L=2\pi /\beta$. However, the scrambling time is parametrically smaller than for plasma excitations, $t_{*}\sim \beta \log \sqrt{\lambda }$ instead of $t_{*}\sim \beta \log N^2$. 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.

• Received 26 December 2017
• Revised 12 March 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.201604

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Published by the American Physical Society