Thermal Emission from Semiclassical Dynamical Systems

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.

Figure 1

The sketch of the trajectories of the incoming particles in the classical mechanics (solid lines) and the quantum corrections (broken lines) near the hyperbolic fixed point $(x,p)=(0,0)$ in the phase space. The dotted lines are the separatrices ($E=0$).

Figure 2

The sketch of the relation between the two trajectories (classical and quantum) and the probability ratio. The ratio of the probability of taking the classical trajectory to that of the quantum one is 1 to $\exp (-{\beta }_L|E|)$, where $E$ is the energy of the particle. They can be regarded as the two level system and the quantum trajectory can be interpreted as an excited state.

Figure 3

The classical droplet of the Fermi fluid in the phase space. We consider the incoming right moving Fermions up to energy level $E_0$.