A bound on energy dependence of chaos

Koji Hashimoto, Keiju Murata, Norihiro Tanahashi, Ryota Watanabe

We conjecture a chaos energy bound, an upper bound on the energy dependence
of the Lyapunov exponent for any classical/quantum Hamiltonian mechanics and
field theories. The conjecture states that the Lyapunov exponent $\lambda(E)$
grows no faster than linearly in the total energy $E$ in the high energy limit.
In other words, the exponent $c$ in $\lambda(E) \propto E^c \,(E\to\infty)$
satisfies $c\leq 1$. This chaos energy bound stems from thermodynamic
consistency of out-of-time-order correlators (OTOC's) and applies to any
classical/quantum system with finite $N$ / large $N$ ($N$ is the number of
degrees of freedom) under plausible physical conditions on the Hamiltonians. To
the best of our knowledge the chaos energy bound is satisfied by any
classically chaotic Hamiltonian system known, and is consistent with the
cerebrated chaos bound by Maldacena, Shenker and Stanford which is for quantum
cases at large $N$. We provide arguments supporting the conjecture for generic
classically chaotic billiards and multi-particle systems. The existence of the
chaos energy bound may put a fundamental constraint on physical systems and the
universe.