Résumé |
Knowing the position of the Earth today does not enable us to predict its
position 10 Myrs from now, yet, the planetary orbits in the Solar System are
stable for the next 5 Gyrs [1]. This is a typical feature of classical systems
whose Hamiltonian slightly differs from an integrable one -- their Lyapunov time
is orders of magnitude shorter than their ergodic time. This puzzling fact may
be understood by considering the simple situation of an integrable system
perturbed by a weak, random noise: there is no Kolmogorov-Arnold-Moser (KAM)
regime and the Lyapunov instability can be shown to happen almost tangent to the
invariant tori. I will extend this analysis to the quantum case, and show that
the discrepancy between Lyapunov and ergodicity times still holds, where the
quantum Lyapunov exponent is defined by the growth rate of the 4-point Out-of-
Time-Order Correlator (OTOC) [2]. Quantum mechanics limits the Lyapunov regime
by spreading wavepackets on a torus. Still, the system is a relatively good
scrambler in the sense that the ratio between the Lyapunov exponent and kT/\hbar
is finite, at a low temperature T [3]. The essential characteristics of the
problem, both classical and quantum, will be demonstrated via a simple example
of a rotor that is kicked weakly but randomly.
[1] J. Laskar, Chaotic diffusion in the solar system, Icarus 196, 1 (2008).
[2] T. Goldfriend and J. Kurchan, Quasi-integrable systems are slow to
thermalize but may be good scramblers, arXiv:1909.02145.
[3] J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, JHEP 08, 106
(2016). |