# Thermal Emission from Semiclassical Dynamical Systems

Morita, Takeshi (Department of Physics, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan and Graduate School of Science and Technology, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan)

13 March 2019

Abstract: Recently the bound on the Lyapunov exponent ${\lambda }_{L}\le 2\pi T/\hslash$ in thermal quantum systems was conjectured by Maldacena, Shenker, and Stanford. If we naïvely apply this bound to a system with a fixed Lyapunov exponent ${\lambda }_{L}$, it might predict the existence of the lower bound on temperature $T\ge \hslash {\lambda }_{L}/2\pi$. Particularly, it might mean that chaotic systems cannot be zero temperature quantum mechanically. Even classical dynamical systems, which are deterministic, might exhibit thermal behaviors once we turn on quantum corrections. We elaborate this possibility by investigating semiclassical particle motions near the hyperbolic fixed point and show that indeed quantum corrections may induce energy emission, which obeys a Boltzmann distribution. We also argue that this emission is related to acoustic Hawking radiation in quantum fluid. Besides, we discuss when the bound is saturated, and show that a particle motion in an inverse harmonic potential and $c=1$ matrix model may saturate the bound, although they are integrable.

Published in: Physical Review Letters 122 (2019)