^{*}

^{3}.

Recently the bound on the Lyapunov exponent

Understanding quantum gravity is one of the most important problems in theoretical physics. Through developments in gauge-gravity correspondence

Recently the idea of the maximal Lyapunov exponent was proposed by Maldacena, Shenker, and Stanford, and it might capture one of the essences of gauge-gravity correspondence

Apart from the interest in the gauge-gravity correspondence, the possible existence of the bound on the Lyapunov exponent is interesting in its own right. Especially, we can rewrite the bound [Eq.

Such a possibility of the emergence of thermal natures in nonthermal classical systems reminds us of the black hole thermodynamics. Although black holes are just classical solutions of general relativity, they behave as the thermal baths through the Hawking radiation in the semiclassical regime

Note that the original proposal of the bound [Eq.

The aim of this Letter is to pursue this possibility in Hamiltonian dynamical systems described by particle(s). If the system is chaotic, then, typically, a hyperbolic fixed point exists, and we investigate particle motions near this fixed point. We will see that indeed energy emission that obeys a Boltzmann distribution is induced around the fixed point quantum mechanically. We will also argue that this thermal emission is related to acoustic Hawking radiation

We first consider particle motions in chaotic systems in classical mechanics. Many chaotic behaviors arise through the two ingredients: a hyperbolic fixed point and a broken homoclinic orbit

We will show that the hyperbolic fixed point plays the key role for the emergence of thermal behaviors in the quantum chaotic systems. The particle motions near the hyperbolic fixed point might be effectively captured by the one-dimensional particle motions in an inverse harmonic potential

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

Suppose a particle with energy

Here we consider the quantum corrections to this classical particle motion by solving the Schrödinger equation with the Hamiltonian

In the case of

In this way, the quantum corrections may change the classical trajectories of the particle motions to the new ones, which are forbidden in classical mechanics. The probabilities of taking the quantum trajectories in the two cases can be combined into the single form

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

This interpretation of the two level system may be clarified by considering the energy transportation through these processes. In the case of

Therefore the particle motion near the hyperbolic fixed point

So far we have seen that, if the particle with energy

The emergence of thermal emission in such a semiclassical system reminds us of Hawking radiation. Here we argue that this energy flow is indeed related to acoustic Hawking radiation in quantum fluid

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

On the other hand, since we are considering many Fermions, we can treat them as a one-dimensional Fermi fluid. It is known that the Fermi fluid composed of the nonrelativistic free Fermions classically obeys the following continuity equation and the Euler equation with pressure

By considering the classical fluid flow corresponding to Fig.

We have argued for the mechanism of the emergence of the thermal emission in chaotic systems by investigating the particle motions near the hyperbolic fixed point through the inverse harmonic potential model [Eq.

The idea that every chaotic system with finite Lyapunov exponents cannot be zero temperature quantum mechanically based on the bound [Eq.

One important question is about the temperature bound [Eq.

Another important point is that the particle motion in the inverse harmonic potential [Eq.

Finally we discuss a possible connection to the original conjecture of the bound on chaos in quantum many-body systems

The author would like to thank Koji Hashimoto, Pei-Ming Ho, Satoshi Iso, Shoichi Kawamoto, Manas Kulkarni, Gautam Mandal, Yoshinori Matsuo, Joseph Samuel, and Asato Tsuchiya for valuable discussions and comments. The work of T. M. is supported in part by Grant-in-Aid for Young Scientists B (No. 15K17643) from JSPS.

Although the definition of the Lyapunov exponent in quantum chaotic systems has not been established, the classical Lyapunov exponent would be well defined semiclassically within the Ehrenfest’s time.

Semi-classical thermal radiations from inverse harmonic potentials have been discussed in several contexts. See for example Refs.

In chaotic systems, it would be hard to find the energy levels. Here we consider just the inverse harmonic potential and implicitly introduce the IR cutoff so that the Fermions are confined, and obtain the energy levels.