# A bound on chaos

## Abstract

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

- Authors:

- Princeton Univ., NJ (United States)
- Stanford Univ., CA (United States)

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1434623

- Grant/Contract Number:
- SC0009988

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2016; Journal Issue: 8; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 1/N Expansion; Black Holes; AdS-CFT Correspondence

### Citation Formats

```
Maldacena, Juan, Shenker, Stephen H., and Stanford, Douglas.
```*A bound on chaos*. United States: N. p., 2016.
Web. doi:10.1007/JHEP08(2016)106.

```
Maldacena, Juan, Shenker, Stephen H., & Stanford, Douglas.
```*A bound on chaos*. United States. doi:10.1007/JHEP08(2016)106.

```
Maldacena, Juan, Shenker, Stephen H., and Stanford, Douglas. Wed .
"A bound on chaos". United States. doi:10.1007/JHEP08(2016)106. https://www.osti.gov/servlets/purl/1434623.
```

```
@article{osti_1434623,
```

title = {A bound on chaos},

author = {Maldacena, Juan and Shenker, Stephen H. and Stanford, Douglas},

abstractNote = {We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.},

doi = {10.1007/JHEP08(2016)106},

journal = {Journal of High Energy Physics (Online)},

issn = {1029-8479},

number = 8,

volume = 2016,

place = {United States},

year = {2016},

month = {8}

}

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