The closing act on Thursday, the 11th of July is the American blackers UADA, who'll perform at the Barbablu (!!!) stage.
On Thursday, the dark mists of UADA will engulf Devilstone's Barbablu stage. This heavily touring four-piece act from Portland, USA has released two albums so far and has proven to be a noteworthy creative force in the vein of the third wave of black metal.
UADA's fierce and melodic sound merges the best feats of the 90's black metal tradition with the current trends in the genre. Their music finds a perfect balance between the atmospheric and the extreme, the experiment and the rule, the dynamic and the aggressive, making UADA an interesting find in comparison to the European black metal and the current USBM scene.
We're sure that UADA will win the hearts of black metal fans at Devilstone.


nojirokku guidance|Screw | Screw | Chiba | Abiko-shi | Matsumoto Sangyo | From general size to super microscrews less than 1 millimeter to Matsumoto Sangyo of the screw.
The materials of "the Noji lock" can be chosen from various material including titanium and the stainless steel let alone iron.
"The Noji lock" transformed a part of the thread, and effective aperture grew big in comparison with non-deformity by establishing the protrusion on the frank plane and realized the high coherency with the internal thread.
When an axis power hangs over the screw tightening back, the protrusion of the Frank aspect transforms the Frank aspect of the internal thread and cuts into an internal thread. Because I am transformed so that gills stretch at the same time towards a slack direction, I melt and am opposed to the torque of the direction.
I remove it and can get an equal slack prevention effect even if I use it repeatedly.
Because a screw of the use of bond application and the screw including Class washer are expensive, cost increases. Bond processing is unnecessary, and "the Noji lock" resists the slack, and a large cost cut is realized.
As a slack-response screw, a screw including Class washer, a bond processing screw are folded into the errand, but there is a trouble factor to be clogged up in the supply pipe of the automatic conclusion machine as for the screw including a washer group, and, as for the bond application screw, it follows that it being in the factor of the trouble "that it is hard to screw" when I am applied to the screw tip thick, (2) bond itself lower productivity and social nature in the things that are not good for a chemical substance in environment again without being applied in the case of (1) bond processing uniformly. I greatly contribute to improvement of the productivity without being hit by the trouble mentioned above in the case of "Noji lock". In addition, when you have a problem with the falling off of the screw, I prevent it from falling off and am available as a screw for temporarily fastening it repeatedly.
(1) As for the lower hole diameter, there are big merits such as the improvement of the productivity improvement (3) tap life of the (2) tap process that I can increase. Furthermore, I can use it for difficulty cut materials and am available without choosing the partner material.
●The conclusion is enabled without letting a soft pledge sink.
●I temporarily fasten it at any point, and the conveyance to the next process is enabled.
●I show an effect more by stopping playing with it, and using the prevention head together.
I recommend that I have you perform the following test to optimize a slack stopping effect of "the Noji lock".
Why is a test necessary?
The difference by the following conditions is to the internal thread put together to male screw side "Noji lock".
Therefore which type of "the Noji lock" performs matching of the effect of the slack stopping screw "Noji lock" to your internal thread to come enough or strongly recommends that I have you perform a test because it is necessary to show an effect more, or to ascertain it.
(1) Using "Noji lock" of the middle size CS type, please conclude it by your last clamping torque first.
(2) After having tightened it by the next last torque, it can relax 180 ゜ from 90 ゜.
(3) Please do a test, evaluations such as a vibration examination or the drop test in your company in a state of (2).
If the slack is in condition not to progress, the matching with "Noji lock" and the internal thread is good, and, as a result of above-mentioned test, it is thought that it is the most effective.
On orders such as the case that a sample is not effective or the test trust, I hope that sorry for your inconvenience, but I contact us.


The World Bank's political risk insurance arm has issued a $48.8m guarantee to help the bamboo plantations manager expand its Nicaraguan operations.
The Multilateral Investment Guarantee Agency (MIGA), the political risk insurance arm of the World Bank Group, has doubled its guarantee for Nicaraguan bamboo producer EcoPlanet Bamboo to $48.8 million.
The guarantee will enable EcoPlanet to expand its project in Nicaragua and purchase more degraded land for conversion into sustainable bamboo plantations. The cover is for up to 15 years against the risks of expropriation, war and civil disturbance.
The guarantee follows a $8.6 million deal with EcoPlanet's South African operation in February.
As well as helping companies move away from harvesting natural forests for their timber needs, EcoPlanet has also developed over 8,500 acres of degraded land, received validation of 1.5 million tons of atmospheric carbon dioxide and has created hundreds of direct and indirect jobs in the region, according to a World Bank release.
"MIGA is thrilled to expand its work with EcoPlanet Bamboo Group as they revolutionise the timber industry to combat climate change – creating jobs and adhering to high corporate social responsibility standards in the process," said MIGA's chief executive Keiko Honda.
The fibre produced is exported for use in pulp production, laminates, activated carbon and renewable energy production.


Dumb waiters are used for carrying small parcels, laundry, plates of food and/or dishes between floors.
They can be manufactured in a range of finishes to suit both your budget and your décor.
The doors are electrically interlinked to the lift so they will not open unless the lift is at that particular level.
A landing light system can also be installed so a waiter or bar tender is aware the lift has arrived.
An intercom system can be installed for communication between the two floors.
Our dumb waiters are hydraulically powered for increased reliability and minimal maintenance costs.
A simple two button operation, call and send.
These pictures are of a Dumb Waiter in a demo shaft.


text:
Q: Display text for 5 seconds after destroying object I'm making mobile game in which I need to show text message for 5 seconds after destroying object. I tried with yield waitforseconds but doesn't work.
This is my code now:
void OnTriggerEnter2D (Collider2D other)
    {

        if (other.gameObject.CompareTag ("bomb")) {
            other.gameObject.SetActive (false);
            Destroy (this.gameObject);
            scoretext.SetActive (true); //this text need to be displayed for 5 seconds after destroying game object
            SceneManager.LoadScene ("__Main1");

        }

I hope that someone can help with this. Thanks.

A: Distroy() method has another version with 2 arguments:
Destroy(scoretext, 5f); //destroys scoretext in 5 seconds

A: You can only yield return in coroutines.
Try
void OnTriggerEnter2D (Collider2D other)
{
    if (other.gameObject.CompareTag ("bomb"))
    {
        other.gameObject.SetActive (false);

        StartCoroutine(DeferredDestroy());
    }
}

IEnumerator DeferredDestroy()
{
    this.gameObject.SetActive(false);

    yield return new WaitForSeconds(5.0f);

    Destroy (this.gameObject);
    scoretext.SetActive (true);

    SceneManager.LoadScene ("__Main1");
}

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}

Understanding the quantum gravity is the one of the most important problem in theoretical physics.
Through the developments in the gauge/gravity correspondence \cite{Maldacena:1997re, Itzhaki:1998dd}, many people expect that large-$N$ gauge theories may illuminate the natures of the quantum gravity.
However only special classes of the large-$N$ gauge theories which possess certain properties may describe the gravity, and understanding what are the essential properties in the gauge theories to have their gravity duals is a crucial question.

Recently the idea of the maximal Lyapunov exponent was proposed, and it might capture the one of the essence of the gauge/gravity correspondence \cite{Maldacena:2015waa}.
The authors of \cite{Maldacena:2015waa} conjectured that thermal many-body quantum systems have an upper bound on the Lyapunov exponent:
\begin{align}
\lambda_L \le \frac{2\pi T}{\hbar}  ,
\label{L-bound}
\end{align}
 where $T$ is temperature of the system and $\lambda_L$ is the Lyapunov exponent. (We take $k_B=1$ in this article.)
Particularly, if a field theory at a finite temperature has the dual black hole geometry, the gravity calculation predicts that the field theory should saturate this bound $\lambda_L = 2\pi T/ \hbar$ \cite{Shenker:2013pqa, Shenker:2013yza}.
(This would be related to the conjecture that the black hole may provide the fastest scrambler in nature \cite{Sekino:2008he}.)
This is called maximal Lyapunov exponent, and the properties of this bound is actively being studied.
One remarkable example is the SYK model \cite{Sachdev:1992fk, Kitaev:2015}.
We can show that this model saturates the bound from the field theory calculation, and now people are exploring what is the dual gravity of this model \cite{Almheiri:2014cka,Maldacena:2016hyu,Jensen:2016pah,Maldacena:2016upp,Engelsoy:2016xyb,Cvetic:2016eiv,Blake:2016jnn,Mandal:2017thl,Stanford:2017thb,Mezei:2017kmw,Forste:2017kwy,Das:2017pif,Taylor:2017dly,Mertens:2017mtv,Grumiller:2017qao,Sarosi:2017ykf,Das:2017hrt, Das:2017wae, Haehl:2017pak,Forste:2017apw}.

The purpose of this article is understanding the mechanism of the appearance of the bound on the Lyapunov exponent in a semi-classical regime through a simple model.
Although, in the recent studies of the chaos bound \cite{Maldacena:2015waa,Kurchan:2016nju,Tsuji:2017fxs,Scaffidi:2017ghs,Liu:2018}, the Lyapunov exponents defined through the out-of-time-ordered correlator (OTOC) \cite{1969JETP...28.1200L} are mainly investigated, we focus on classical chaos and consider the Lyapunov exponent defined by
\begin{align}
\delta q(t) \sim \delta q(0) \exp(\lambda_L t)
\label{chaos}
\end{align}
Here $q(t)$ is the value of an observable at time $t$ and $\delta q(t)$ is the deviation of  $q(t)$  through the change of the initial condition by $\delta q(0)$.
Such an exponential sensitivity of the initial condition is a characteristic behavior of chaos, and it appears ubiquitously in nature.
Since many studies on chaos have been developing in classical systems, it would be valuable to investigate the chaos bound in the semi-classical regime.

The simplest dynamical model in which we can observe the exponential sensitivity of the initial condition (\ref{chaos}) would be the classical motion of a point particle in an inverse harmonic potential
\begin{align}
m \ddot x(t) = -V'(x), \qquad V(x)=-\frac{\alpha }{2}x^2.
\label{inverse-EOM}
\end{align}
Here the solution of the motion is given by
\begin{align}
x(t)=c_1 e^{\sqrt{\alpha/m}t} + c_2 e^{-\sqrt{\alpha/m}t},
\label{particle-chaos}
\end{align}
where $c_1$ and $c_2$ are constants determined by the initial condition, and it shows the sensitivity of the initial condition (\ref{chaos}) with Lyapunov exponent
\begin{align}
\lambda_L=\sqrt{\frac{\alpha}{m}}.
\label{Lyapunov}
\end{align}
Of course the motion (\ref{particle-chaos}) is too simple and not chaotic.
However, if even such a simple model shows the chaos bound (\ref{L-bound}), we may gain some important insights into the bound in the actual chaotic systems.
For this reason, we investigate this model.

If we apply the bound on the Lyapunov exponent (\ref{L-bound}) to the model (\ref{inverse-EOM}), we obtain the relation
\begin{align}
T \ge \frac{\hbar}{2\pi} \sqrt{\frac{\alpha}{m}}.
\label{T-bound}
\end{align}
from (\ref{Lyapunov}).
Since the right hand side is just a constant determined by the parameters in the model, this relation predicts the existence of the lower bound on temperature in this system,
\begin{align}
T_L := \frac{\hbar}{2\pi} \sqrt{\frac{\alpha}{m}}.
\label{T-mini}
\end{align}
Although this temperature becomes zero in the strict classical limit $\hbar \to 0$, it might be meaningful in a semi-classical regime.

We will argue that indeed the lowest temperature $T_L$ plays an important role.
To see it clearly, we consider $N$ non-relativistic free fermions in the inverse harmonic potential (\ref{inverse-EOM}).
This model is related to a one-dimensional matrix quantum mechanics called $c=1$ matrix model which describes a two dimensional gravity through the non-critical string theory (see, e.g. the review
of \cite{Klebanov:1991qa,Ginsparg:1993is,Polchinski:1994mb}).

\begin{figure}
\begin{center}
\includegraphics[scale=1.0]{string.eps}
\caption{The sketch of the fermions in the inverse harmonic potential which corresponds to the bosonic non-critical string theory. $-\mu$ is the fermi energy.
(RIGHT) The configuration of the fermions in the phase space.
The region occupied by the fermions is called ``droplet".
}
\label{Fig-string}
\end{center}
\end{figure}

We consider the configuration of the fermions which describes the bosonic non-critical string theory (figure \ref{Fig-string}).
(In the following arguments, the correspondence to the string theory is not important.)
Although this configuration is stable in the strict classical limit, it is unstable via the quantum tunneling of the fermions.
We will show that this instability causes a thermal flux with the temperature $T_L$ (\ref{T-mini}) in the asymptotic region $x \to - \infty$ which flows from around $x=0$.
This radiation is an analogous of the Hawking radiation \cite{Hawking:1974sw, Hawking:1974rv} in the sense that the system is not thermal classically whereas a thermal spectrum is induced quantum mechanically.
Therefore the inverse harmonic potential can be regarded as a black body with the temperature $T_L$ for the observer in the asymptotic region, and the temperature bound (\ref{T-bound}) may be saturated.

We will also argue that this radiation is related to the acoustic Hawking radiation in a supersonic fluid \cite{Unruh:1980cg}.
(A related study in the context of condensed matter physics has been done in \cite{Giovanazzi:2004zv} and developed in \cite{Giovanazzi:2006nd,Parola:2017qpn}.)
The collective motion of the $N$ fermions are described as a fermi fluid, and it classically obeys the hydrodynamic equations \cite{Dhar:1992rs,Dhar:1992hr,Mandal:2013id}.
Hence, if the fluid velocity exceeds the speed of sound, acoustic Hawking radiation may be emitted.
We will argue how the supersonic region can be realized in the $c=1$ matrix model, and show that the acoustic event horizon arises only at the tip of the inverse harmonic potential $x=0$ if the flow is stationary.
Thus the radiation may arise from $x=0$ when we turn on the quantum corrections.
We will show that the temperature of this acoustic Hawking radiation is always given by the lowest temperature $T_L$, and argue the connection to the instability of the bosonic non-critical string theory.

Note that related works on the particle creations in the $c=1$ matrix model have been done in the context of the non-critical string theory \cite{Takayanagi:2003sm,Douglas:2003up}.
They include time dependent background and tachyon condensation \cite{Karczmarek:2004ph,Das:2004hw,Mukhopadhyay:2004ff,Karczmarek:2004yc}, and high energy scattering and black hole physics \cite{Martinec:2004qt, Friess:2004tq,Karczmarek:2004bw, Banks:2015mya, Betzios:2016yaq}.

The organization of this article is as follows.
In section \ref{sec-c=1}, we argue the appearance of the thermal radiation with the temperature $T_L$ (\ref{T-mini}) in the $c=1$ matrix model in the configuration of the fermions corresponding to the bosonic non-critical string theory.
In section \ref{sec-acoustic-HR}, we show the connection between this radiation and acoustic Hawking radiation.
A brief review on the derivation of the acoustic Hawking radiation is included.
Section \ref{sec-discussion} is discussions and future problems.

\section{The $c=1$ matrix model and the temperature bound }
\label{sec-c=1}

We consider the $c=1$ matrix model which is equivalent to $N$ non-relativistic free fermions.
The fermions classically obey the equation of motion (\ref{inverse-EOM}), and the
Hamiltonian for the single fermion is given by
\begin{align}
\hat H =  -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} - \frac{\alpha}{2} x^2.
\label{H-c=1}
\end{align}
This Hamiltonian is unbounded from below, and usually we introduce suitable boundaries  so that the fermions are confined.
These boundaries are irrelevant as far as we focus on phenomena around $x=0$.
In this section, we introduce infinite potential walls at $x=\pm L$.

We consider the configuration of the fermions which describes the two dimensional bosonic non-critical string theory.
We put all the fermions on the left of the inverse harmonic potential, and tune the parameters $N,\alpha,m$ and $L$ so that $N \to \infty$ and the fermi energy $-\mu$ ($\mu>0$) is close but below zero.
We sketch this configuration in figure \ref{Fig-string}.

Now we show that this configuration involves thermal radiation with the temperature $T_L$ defined in (\ref{T-mini}).
Obviously the above configuration cannot be stationary.
The fermions will leak to the right side of the potential through the quantum tunneling.
Such a tunneling effect is significant for the fermions close to the fermi surface.
For a single fermion with energy $E$ $(E \le - \mu)$, the tunneling probability $P_T(E)$  can be calculated as \cite{Moore:1991zv},
\begin{align}
 \quad P_T(E)= \frac{1}{\exp\left( -\frac{2\pi}{\hbar} \sqrt{\frac{m}{\alpha}} E \right)+1}.
\label{PT}
\end{align}
The derivation is shown in appendix \ref{app-QM}.
This formula is suggestive.
If we regard $-E$ as energy, $P_T(E)$ is identical to the Fermi-Dirac distribution with the  temperature
\begin{align}
T = \frac{\hbar}{2\pi}\sqrt{\frac{\alpha}{m}}
\label{T-mini-2}
\end{align}
which precisely agrees with the lowest temperature $T_L$ (\ref{T-mini})\footnote{This tunneling probability is for a single particle, and the appearance of the Fermi-Dirac distribution is nothing to do with the fact that we are considering the fermions.}.
Thus the tunneling of the fermions might cause thermal radiation.

In order to establish this expectation, we evaluate the contributions of the tunneling to physical quantities and ask whether they are thermal or not.
We will investigate the local density of an observable $O(\hat{x},\hat{p})$.
First we will compute this quantity in the classical limit ($\hbar \to 0$).
Then we will calculate the same quantity including the quantum effect and, by comparing these two results, we will evaluate the contributions of the tunneling.

Let us start from the calculation in the classical limit $\hbar \to 0$.
In this limit, the fermion with negative energy $E$ is confined in the left region $-L \le x \le - x_0(E)$,
where $x_0(E):= \sqrt{-2E/\alpha} $ is the classical turning point. (See figure \ref{Fig-string}.)
The WKB wave function for such a fermion is given by
\begin{align}
\phi_E(x)= \frac{C(E)}{\sqrt{p(E,x)}} \frac{1}{\sqrt{2}} \left( e^{\frac{i}{\hbar} \int_{-L}^x dy \, p(E,y) }-e^{-\frac{i}{\hbar} \int_{-L}^x dy \, p(E,y) } \right)
\label{WKB-wave}
\end{align}
where $p(E,x)$ is the classical momentum for the particle,
\begin{align}
p(E,x)=\sqrt{2m\left(E+\frac{\alpha}{2}x^2\right)},
\label{moment}
\end{align}
and $C(E)$ is the normalization factor satisfying
\begin{align}
|C(E)|^2 = \left( \int_{-L}^{-x_0(E)} \frac{dx}{ |p(E,x)| } \right)^{-1} ,
\end{align}
so that $\phi_E(x)$ is normalized to 1.
Through the boundary condition $\phi_E(-L)=\phi_E(-x_0(E))=0$, energy is quantized as
\begin{align}
\int_{-L}^{-x_0(E)} dx\,| p(E,x)| = \pi  \hbar n, \qquad (n=1,2,\cdots),
\label{BS-cond}
\end{align}
where we have ignored the fragment terms.
By using this WKB wave function, we evaluate the expectation value  of the observable $O(\hat{x},\hat{p})$ for the $N$ fermions as
\begin{align}
\langle O(\hat{x},\hat{p}) \rangle_N =& \sum_{i=1}^N \langle i| O(\hat{x},\hat{p}) |i \rangle= \int_{E_G}^{-\mu} dE \, \rho(E)  \int_{-L}^{-x_0(E)} dx \, |\phi_E (x)|^2
O(x,p(E,x))
\label{VEV-O-all}
\end{align}
where $E_G:=- \alpha L^2/2 $ is the ground state energy and $\rho(E)$ is the energy density given by
\begin{align}
\rho(E) = \frac{\partial n}{\partial E}= \frac{m}{\pi \hbar } \int_{-L}^{-x_0(E)} \frac{dx}{| p(E,x)| }
\label{E-density}
\end{align}
through the quantization condition (\ref{BS-cond}).
Here we have assumed that $N$ is sufficiently large and the spectrum is continuous.
From (\ref{VEV-O-all}), the density of the observable $O(\hat{x},\hat{p})$ at position $x$ is given by
\begin{align}
 \int_{E_0(x)}^{-\mu} dE \, \rho(E)   |\phi_E (x)|^2
O(x,p(E,x))=
 \frac{m}{\pi \hbar}
 \int_{E_0(x)}^{-\mu} \frac{d E}{|p(E,x)|} \, O(x,p(E,x)).
 \label{O-classical}
\end{align}
Here $E_0(x):=-\alpha x^2/2$ is the minimum energy at given $x$. (See figure \ref{Fig-string}.)

Now we evaluate the quantum correction to the density (\ref{O-classical}).
In the WKB wave function (\ref{WKB-wave}),
$\exp\left(\frac{i}{h} \int^x dy \, p(y)\right)$ can be regarded as the incoming wave toward the inverse harmonic potential, while $\exp\left(-\frac{i}{h} \int^x dy \, p(y)\right)$  can be regarded as the reflected wave.
Then through the tunneling effect (\ref{PT}), a part of the incoming wave will penetrate to the right side of the potential, and
the reflected wave will be decreased by the factor $\sqrt{1-P_T(E)}$.
Hence the WKB wave function (\ref{WKB-wave}) will be modified as
\begin{align}
\phi_E(x) \to \frac{C(E)}{\sqrt{p(E,x)}} \frac{1}{\sqrt{2}} \left( e^{\frac{i}{\hbar} \int_{-L}^x dy \, p(E,y) }- \sqrt{1-P_T(E)} e^{-\frac{i}{\hbar} \int_{-L}^x dy \, p(E,y) } \right), \quad (-L \ll x \ll -x_0(E)).
\end{align}
Note that this result is not reliable near the turning point $ -x_0(E)$, since the WKB approximation does not work.
Also we have assumed that the boundary $x=-L$ is sufficiently far, and we can ignore the further reflections there.
This correction modifies the left hand side of (\ref{O-classical}), and the density of the observable $O(\hat{x},\hat{p})$ becomes,
\begin{align}
 \frac{m}{\pi \hbar}
 \int_{E_0(x)}^{-\mu}  \frac{dE}{|p(E,x)|}  \,
\left(1- \frac{1}{2} P_T(E)\right) O(x,p(E,x))
 \end{align}
By subtracting the classical result (\ref{O-classical}) from this result,
we obtain the contributions of the quantum tunneling to the density  of the observable $O(\hat{x},\hat{p})$ as
\begin{align}
- \frac{m}{2\pi \hbar} \int_{E_0(x)}^{-\mu} \frac{dE}{|p(E,x)|}  \,   \frac{ O(x,p(E,x))}{e^{ -\beta_L E }+1} = - \frac{m}{2\pi \hbar} \int_\mu^\infty   \frac{d E}{|p(-E,x)|}  \frac{ O(x,p(-E,x))}{e^{ \beta_L E }+1}
 \label{VEV-O-semi}
\end{align}
where $\beta_L$ denotes the inverse temperature $1/T_L$ and we have approximated $E_0(x) \approx  -\infty$ by assuming that we are considering the density in a far region\footnote{In this region, the tunneling of the particle with the energy $E_0(x)$ is sufficiently suppressed.} $-L \ll  x \ll  - (\hbar^2/m \alpha)^{1/4}$.
We also flipped $E \to - E$ in the last step.

We can interpret the result (\ref{VEV-O-semi}) as follows.
Due to the tunneling, the holes with positive energy $E$ $(E \ge \mu)$ appear in the fermi sea.
They move to the left and contribute to the quantity $O(x,p)$.
Importantly, (\ref{VEV-O-semi}) shows that these holes are thermally excited and obey the Fermi-Dirac distribution at the temperature $T_L$.
Therefore the quantum tunneling indeed induces thermal radiation.

For example, if we choose $O(\hat{x},\hat{p})=\hat H$ in (\ref{VEV-O-semi}), we obtain the energy density at $x$ in the far region,
\begin{align}
- \frac{m}{2\pi \hbar} \int_0^\infty   \frac{d E}{|p(-E,x)|}
 \frac{-E }{e^{ \beta_L E }+1} = \frac{1}{ (- x)} \frac{T_L}{48}\left(1 + O\left(\frac{T_L}{\alpha x^2}\right) \right)
\end{align}
Here we have taken $\mu=0$ for simplicity.
The obtained energy density is positive reflecting the fact that the holes carry the positive energies.

Our result indicates that the inverse harmonic potential plays a role of a black body at the temperature $T_L$.
This result reminds us the Hawking radiation \cite{Hawking:1974sw, Hawking:1974rv} in which the thermal excitation arises through the quantum effect even though the system is not thermal in the classical limit.
Indeed we will see that the thermal excitation in the matrix model is related to the acoustic Hawking radiation in quantum fluid.

Notice that, since the thermal radiation at the temperature $T_L$ arises,
 the temperature bound (\ref{T-bound}) may be saturated in this system\footnote{This interpretation is subtle.
Since the $c=1$ matrix model is free and integrable, normal thermalization may not occur.
(In integrable systems, thermalization to the Generalized Gibbs ensemble (GGE) \cite{PhysRevLett.98.050405, Mandal:2013id} may occur.
It may be interesting to consider the role of the temperature $T_L$ in terms of GGE.)
Another issue is that, at late times, the system will settle down into the state in which the half of the fermions are in the left side of the potential and the other half are in the right.
There the formula (\ref{VEV-O-semi}) would not work.
Thus the argument of the temperature bound  would be valid only for the initial stage of the time evolution starting from the meta-stable configuration sketched in figure \ref{Fig-string} such that the formula (\ref{VEV-O-semi}) is reliable.
}.

\section{Thermal radiation in the $c=1$ matrix model and acoustic Hawking radiation}
\label{sec-acoustic-HR}

We will show that the thermal radiation in the $c=1$ matrix model argued in the previous section is related to the acoustic Hawking radiation \cite{Unruh:1980cg}.
(Several arguments in this section have been done by \cite{Giovanazzi:2004zv} in the context of condensed matter physics.)

\subsection{Acoustic Hawking radiation in the $c=1$ matrix model}
\label{sec-review-acoustic}

As is well known, the fermions in the $c=1$ matrix model compose a two dimensional ideal fluid.
The fermion density field $\rho(x,t)$ and velocity field $v(x,t)$ classically obey the following the continuity equation and  the Euler equation with pressure $p=\hbar^2 \pi^2 \rho^3/3 m^2$ \cite{Dhar:1992rs,Dhar:1992hr,Mandal:2013id},
\begin{align}
&\partial_t \rho + \partial_x(\rho v)=0, \qquad
 \partial_t v + \partial_x \left(\frac{1}{2}  v^2  +  \frac{ \hbar^2 \pi^2}{2m^2} \rho^2 + \frac{1}{m}  V(x)\right)=0.
\label{euler}
\end{align}
Here $V(x)=- \alpha x^2/2$.  (Many of the arguments in this section will work for general $V(x)$.)
We ignore the infinite potential walls at $x=\pm L$ in this section.
Hence we can apply the story of the acoustic Hawking radiation to this system.

Here we briefly show the derivation of the acoustic Hawking radiation \cite{Unruh:1980cg}.
(See e.g. a review article \cite{Visser:1997ux}.)
We consider the following small $\epsilon$ expansion
\begin{align}
\rho(x,t)=& \rho_0(x,t) + \epsilon \rho_1(x,t) + \epsilon^2 \rho_2(x,t) + \cdots, \nonumber \\
v(x,t)=& v_0 (x,t) + \epsilon v_1 (x,t) + \epsilon^2 v_2 (x,t) + \cdots,
\label{expansion}
\end{align}
Here we regard that $\rho_0$ and $v_0$ describe a background current, and $\rho_1$ and $v_1$ describe the phonons propagating on it, and $\rho_n$ and $v_n$ ($n\ge 2$) are the higher corrections.
We will see that when we turn on the quantum effect, $\rho_1$ and $v_1$ may show a thermal excitation if there is a supersonic region in the background.

\subsubsection{Acoustic geometry for phonons}
\label{sec-acoustic-GR}

In order to derive the acoustic Hawking radiation, we consider the equations for the phonons.
By substituting the expansion (\ref{expansion}) into the hydrodynamic equation (\ref{euler}) and expanding them with respect to $\epsilon$, we will obtain the equations for $\rho_n$ and $v_n$ order by order.
Especially, at order $\epsilon$, we obtain the equations for $\rho_1$ and $v_1$,
\begin{align}
&\partial_t \rho_1 + \partial_x\left(\rho_0 v_1+v_0 \rho_1 \right)=0, \qquad
 \partial_t v_1 + \partial_x\left( v_0 v_1  +  \frac{\hbar^2 \pi^2 \rho_0}{m^2}   \rho_1 \right)=0.
\label{eq-v1}
\end{align}
Here we introduce the velocity potential $\psi$ such that $v_1=- \partial_x \psi$.
Then from the second equation, we obtain
\begin{align}
\rho_1= \frac{m^2}{\hbar^2 \pi^2 \rho_0} \left( \partial_t \psi + v_0 \partial_x \psi \right).
\end{align}
By substituting this result to the first equation of (\ref{eq-v1}), we obtain the wave equation for the velocity potential $\psi$ as
 \begin{align}
0=& \partial_\mu \sqrt{-g} g^{\mu\nu} \partial_\nu \psi
\label{wave-phonon}
\end{align}
where $g_{\mu\nu}$ ($\mu,\nu=t,x$) is given by
\begin{align}
g_{\mu\nu} = \frac1{ \gamma \rho_0}
\begin{pmatrix}
c^2-v_0^2 & v_0 \\
v_0 & -1
\end{pmatrix},
\label{g_mn}
\end{align}
and $c$ denotes the speed of sound defined by
 \begin{align}
 c(x,t) :=  \sqrt{\frac{\partial p}{\partial \rho}}= \frac{ \hbar \pi }{m} \rho_0 (x,t).
 \label{sound speed}
 \end{align}
Here $p$ is the pressure defined above (\ref{euler}).
Note that $\gamma$ in $g_{\mu\nu}$ is an arbitrary constant, which cannot be fixed \footnote{In two dimension, the wave equation (\ref{wave-phonon}) is invariant under a scale transformation, and we have the ambiguity $\gamma$.}.
This wave equation is identical to the wave equation of a scalar field on a curved manifold with the metric (\ref{g_mn}).
For this reason, $g_{\mu\nu}$ is called acoustic metric.

Importantly, if the fluid velocity $|v_0|$ is increasing along $x$-direction and exceeds the speed of sound $c$  at a certain point,
$c=|v_0|$ occurs there and the $(t,t)$-component of the metric (\ref{g_mn}) vanishes similar to the event horizon of  a black hole.
Correspondingly, we can show that no phonon can propagate from the supersonic region $(c<|v_0|) $ to the subsonic region $(c>|v_0|)$ across the point $c=|v_0|$.
For this reason, the point $c=|v_0|$ is called acoustic event horizon.
Then we expect that the phonon $\psi$ in the subsonic region ($c<|v_0| $) may be thermally excited similar to the Hawking radiation.
We will confirm it later.

\subsubsection{Acoustic black holes and white holes}
\label{sec-BH-WH}

\begin{figure}
\begin{center}
\includegraphics[scale=1]{phase-space.eps}
\caption{The sketch of the fermions in the phase space.
They compose the droplet(s) in the phase space.
The doted lines describe the separatrices $p=\pm \sqrt{\alpha/m}x$.
(LEFT)
$P_+(x,t)$ and $P_-(x,t)$ (\ref{P_+}) are the upper and lower boundaries of the droplet, respectively.
(RIGHT)
If the boundaries of the droplet have the multiple values as functions of $x$,
$P_+(x,t)$ and $P_-(x,t)$ are not enough to define the boundaries.
Such a multiple value configuration is called ``fold" \cite{Ginsparg:1993is, Polchinski:1991uq} and
the hydrodynamic equation (\ref{euler}) does not work there \cite{KMM}.
We do not consider this case in this article.
}
\label{Fig-phase}
\end{center}
\end{figure}

When does $c=|v_0|$ occur in our matrix model?
In order to see it,
it is convenient to define $P_{\pm}$ as
\begin{align}
P_\pm(x,t):= m v(x,t) \pm \hbar \pi \rho(x,t).
\label{P_+}
\end{align}
It means that
\begin{align}
\rho(x,t) & =\frac{1}{2\pi \hbar} (P_+(x,t) - P_-(x,t)), \qquad
v(x,t)  = \frac1{2m} (P_+(x,t) + P_-(x,t)).
\label{quad-densities}
\end{align}
Here $P_{\pm}$ describe the boundaries of the droplet of the fermions in the phase space as shown in figure \ref{Fig-phase}. (We have implicitly excluded the case in which the folds \cite{Ginsparg:1993is, Polchinski:1991uq} appear on the boundaries of the droplet.)
Then from (\ref{sound speed}) and (\ref{quad-densities}), we can rewrite $c$ and $v_0$ by using $P_{0\pm}$,
\begin{align}
c\pm v_0= \pm \frac{1}{m} P_{0\pm}(x,t),
\label{c-v-P}
\end{align}
where we have expanded
\begin{align}
P_{\pm}=P_{0\pm}+ \epsilon P_{1\pm}+ \epsilon P_{2\pm}+ \cdots,
\label{expand-P}
\end{align}
corresponding to the expansion (\ref{expansion}).
Therefore $c= |v_0|$ occurs when $P_{0+}$ or $P_{0-}$ vanishes, and
an acoustic event horizon may appear there.

Now we solve  the fluid equation (\ref{euler}), and find the acoustic metric involving an acoustic event horizon.
By substituting (\ref{quad-densities}) to (\ref{euler}), we obtain equations for $P_\pm$,
\begin{align}
\partial_t P_+ + \partial_x\left(  \frac{1}{2m} P_+^2 + V(x) \right) =0,\; \qquad
\partial_t P_- + \partial_x\left( \frac{1}{2m} P_-^2 + V(x) \right)=0.
\label{eom-P}
\end{align}
By using the expansion (\ref{expand-P}) in these equations and considering $O(\epsilon^0)$ terms, we obtain equations for $P_{0\pm}$ which are just $P_{\pm} \to P_{0\pm}$ in (\ref{eom-P}).
We assume that the background is stationary $\partial_t P_{0\pm}=0$.
Then the solution of these equations are given by
\begin{align}
 P_{0\pm}  = \sigma \sqrt{ 2 m ( E_\pm - V(x))},
\end{align}
where $E_{\pm}$ are constants and $\sigma$ is $+1$ or $-1$.

\begin{figure}
\begin{center}
\includegraphics[scale=1]{BH-WH.eps}
\\
\hspace{3pt}
\caption{Four possible acoustic event horizons ($c=|v_0|$).
(TOP) Droplets in the phase space.
(MIDDLE) The particle motion in the potential.
The arrows indicate the possible directions of the phonon propagation.
(BOTTOM) The plots of the fluid velocity $v_0$ and the speed of sound $c$ (dashed line).
(a) Black hole (right): The phonons in the supersonic region ($x>0$) cannot propagate into the subsonic region ($x<0$).
(b) Black hole (left): $x \to -x$ of the black hole (right).
(c) White hole (right): The phonons in the subsonic region ($x<0$) cannot propagate to the supersonic region ($x>0$).
(d) White hole (left): $x \to -x$ of the white hole (right).
}
\label{Fig-BH-WH}
\end{center}
\end{figure}

To realize the acoustic event horizon, $ P_{0+} $  or  $ P_{0-} $ has to cross 0.
We can easily show that it occurs only when
\begin{align}
 &P_{0+}  = \sqrt{ 2 m  E_+ + m \alpha x^2}, \qquad  P_{0-}  = \sigma \sqrt{ m \alpha }x, \nonumber \\
  \text{or} \qquad &P_{0+}  = \sigma \sqrt{ m \alpha }x, \qquad  P_{0-}  = - \sqrt{ 2 m  E_- + m \alpha x^2},
\end{align}
with positive $E_+$ and $E_-$.
Here the acoustic event horizon appears at $x=0$ in both cases.
Then depending on the choice of $\sigma=\pm 1$, there are four possibilities.
We call them as
 black hole (right), black hole (left), white hole (right) and  white hole (left).
See figure \ref{Fig-BH-WH}.
These names refer to the ways of the classical propagation of the phonons, and (left) and (right) are related by parity $x \to -x$.
To see the phonon propagation, we consider the black hole (right) case as an example.
As shown in figure \ref{Fig-BH-WH}, $P_{0\pm}$ in this case are given by
\begin{align}
 P_{0+}  =  \sqrt{ 2 m  E_+ + m \alpha x^2}, \qquad  P_{0-}  =  \sqrt{m \alpha }x.
 \label{BH-right}
\end{align}
From (\ref{c-v-P}) (and the plots of $v_0$ and $c$ in figure \ref{Fig-BH-WH}), $x<0$ is the subsonic region ($c>|v_0| $) while $x>0$ is the supersonic region ($c<|v_0| $).
Then we expect that no phonon can propagate into $x<0$ from $x>0$. Let us confirm it.
The propagation of the phonons can be read off from the wave equation (\ref{wave-phonon}).
By using new coordinates (called time-of-flight coordinates)
\begin{align}
z_{\pm}:= t - m \int^x \frac{dx'}{P_{0 \pm}(x')},
\label{TOF-coordinate}
\end{align}
the wave equation (\ref{wave-phonon}) becomes $\partial_{z_+} \partial_{z_-} \psi =0$.
Thus the solution is given by
\begin{align}
\psi=\psi_+(z_+)+\psi_-(z_-).
\label{sol-psi}
\end{align}
Then $P_{1\pm}$ become
\begin{align}
P_{1\pm} =m v_1 \pm \hbar \pi \rho_1 = \frac{2m^2}{P_{0\pm}} \psi'_\pm (z_\pm) .
\label{sol-P}
\end{align}
This relation shows (the well-known result for experts) that  the fluctuation of the boundaries of the droplet $P_{\pm}$ describes
 the excitations of the phonons and they propagate depending on $z_\pm$.

 In the black hole (right) case (\ref{BH-right}),  $P_{1+}(z_+)$ always propagates toward the right direction, since $P_{0+}$ is always positive in (\ref{TOF-coordinate}).
 On the other hand, we need to take care $P_{1-}(z_-)$. Since $P_{0-}$ becomes zero at $x=0$ and $z_-$ diverges there, $z_-$ covers only the left region ($x<0$) or the right region ($x>0$) as
\begin{align}
z_{-}=&t - \sqrt{\frac{m}{\alpha}} \log x + \text{const.} \qquad (x>0) \nonumber \\
=& t - \sqrt{\frac{m}{\alpha}} \log (-x) + \text{const.} \qquad (x<0).
\label{tortoise}
\end{align}
Thus $P_{1-}(z_-)$ propagates to left in $x<0$ and to right in $x>0$.
Therefore both $P_{1\pm}$ cannot propagate into $x<0$ from $x>0$ as we expected.
This is similar to the black hole geometry in which no mode can propagate to the outside from the inside.
For this reason we call this case as ``black hole (right)", by identifying the supersonic region ($x>0$) as the inside of the black hole\footnote{Note that this result is valid only for sufficiently small fluctuations.
 The ways of the propagation of the finite fluctuations of $P_{\pm}$ can  be read off from the particle trajectories as shown in figure \ref{Fig-fluct}.
Particularly if the fluctuations are large and they cross the separatrix $p=-\sqrt{m\alpha}x$, even the fluctuations in the supersonic region can propagate into the subsonic region.
Besides, it is known that arbitrary non-stationary fluctuations develop to folds at late times. }
.

\begin{figure}
\begin{center}
\includegraphics[scale=1]{fluct.eps}
\caption{The sketch of the propagation of finite fluctuations of $P_{\pm}$ on the black hole (right) and the white hole (right) in the phase space.
We can read off the propagation from the classical particle trajectories  $E=p^2/2m - \alpha x^2/2$ in the phase space. (Recall that the droplet is the collection of the fermionic particles, and each point in the droplet moves according to (\ref{particle-chaos}).)
In the black hole case, small fluctuations in the supersonic region ($x>0$) cannot propagate into the subsonic region ($x<0$).
However, large fluctuations which cross the separatrix $p=-\sqrt{m\alpha}x$ can propagate.
In the white hole case, even small fluctuations on $P_+$ in the subsonic region ($x<0$) can propagate into the supersonic region ($x>0$).
It means that the ``causality" of the acoustic white hole is unstable against perturbations on $P_+$.
}
\label{Fig-fluct}
\end{center}
\end{figure}

Lastly, we take a detour and discuss the white hole (right) case.
In figure \ref{Fig-BH-WH} (c),  $P_{0\pm}$ is given by
\begin{align}
P_{0+}  = - \sqrt{ m \alpha }x, \qquad  P_{0-}  = - \sqrt{ 2 m  E_- + m \alpha x^2}.
\end{align}
In this case, from (\ref{c-v-P}), $v_0$ is always negative and the background current flows from the right to left.
Importantly the fluid velocity $|v_0|$ decelerates along the flow, and the supersonic flow in $x>0$ becomes subsonic at $x=0$.
As a result, $\psi$ in the subsonic region ($x<0$) cannot propagate into the supersonic region.
This is an analogue of a white hole in fluid mechanics.

However, this conclusion is valid only for $\epsilon \rightarrow 0$ limit.
 $P_{0+}$ is unstable against the perturbations and even tiny fluctuations in the subsonic region can propagate into the supersonic region as shown in figure \ref{Fig-fluct}.

\subsubsection{Acoustic Hawking radiation}
\label{sec-HR}

Now we are ready to derive the acoustic Hawking radiation.
We consider the black hole (right) case.
In this case, the classical solution of $\psi$ is given by (\ref{sol-psi}), and importantly the coordinate $z_-$ (\ref{tortoise}) covers only either $x<0$ or $x>0$.
This is an analogue of the tortoise coordinates in the black hole geometries, and $z_-$ is not suitable to define the vacuum of the quantized phonon.

A coordinate which is well defined on  $-\infty < x < \infty$ is a Kruskal-like coordinate $U:=-e^{-\kappa z_-}=xe^{-\sqrt{\frac{\alpha}{m}}t} $, where we have defined ``the surface gravity" \footnote{
Up to an overall factor, the line element of the acoustic metric (\ref{g_mn}) can be written as
$ds^2 = c^2 f dt^2+ 2v_0 dt dx -dx^2$, where $f:=1-v_0^2/c^2= - P_{0+}P_{0-}/m^2c^2$.
Then the surface gravity $\kappa$ can be written as the familiar formula $\kappa= \frac{c}{2} \partial_x f |_{x=0}$.
Also $\kappa$ is written as $\kappa=\frac{1}{m} \partial_x P_{0-}|_{x=0}$.
The equations in this footnote work even for a general potential $V(x)$ if the local maximum is at $x=0$ and the acoustic event horizon appears there.
},
\begin{align}
\kappa:=\sqrt{\frac{\alpha}{m}}.
\label{kappa}
\end{align}
The vacuum of the phonon $\psi_-(z_-)$ should be defined by using this coordinate.
It implies that $\psi_-$ in the asymptotic region (large negative $x$)  will be thermally excited at temperature
\begin{align}
T=\frac{\hbar \kappa}{2\pi} = \frac{\hbar }{2\pi} \sqrt{\frac{\alpha}{m}}
\label{Hawking-temp}
\end{align}
through the Bogoliubov transformation.
Here the obtained temperature precisely agrees with the lowest temperature $T_L$ (\ref{T-mini}).

For example, we can calculate the expectation value of the energy momentum tensor of the phonon by using the two dimensional conformal field theory technique \cite{Birrell:1982ix, Thorlacius:1994ip,Iso:2007kt},
\begin{align}
\langle  T_{z_- z_-}  \rangle =-\frac{c_{\psi}}{24\pi} \{ U, z_- \}=
 \frac{\hbar}{48\pi} \kappa^2, \qquad
\langle  T_{z_+ z_+}  \rangle =
 0,
  \label{EM-tensor}
\end{align}
where $\{ U, z_- \}:=U'''/U' - \frac{3}{2}(U''/U')^2  $ is the Schwarzian derivative and $c_{\psi}$ is the central charge which we have taken $c_{\psi}=1$ since phonon is a boson.
Note that $\langle  T_{z_+ z_+}  \rangle = 0$, since $z_+$ is well defined on $-\infty < x < \infty$.

\subsection{Acoustic Hawking radiation from quantum mechanics}
\label{sec-HR-QM}

So far we have seen that the acoustic Hawking radiation occurs in the $c=1$ matrix model, since this model can be regarded as a two dimensional fluid.
Interestingly the Hawking temperature (\ref{Hawking-temp}) is given by the temperature $T_L$ which is also coincident with the temperature (\ref{T-mini-2}) derived from the quantum mechanics of the fermions in section \ref{sec-c=1}.
Although the configuration of the fermions for the bosonic non-critical string case sketched in figure \ref{Fig-string} and the acoustic Hawking radiation case sketched in figure \ref{Fig-BH-WH} are different\footnote{In terms of the fluid variables, the configuration of the bosonic non-critical string is given by $P_{0\pm}= \pm \sqrt{-2m\mu +m \alpha x^2} $, ($x<0$). See figure \ref{Fig-string} (RIGHT).}, the same temperature is obtained.
We argue why these two temperatures agree.

\begin{figure}
\begin{center}
\includegraphics[scale=1]{HR.eps}
\caption{The mechanism of the appearance of the Hawking radiation in the acoustic black hole (right) in the phase space.
The holes and the reflected fermions which are not allowed in the classical motions cause the acoustic Hawking radiation when we consider the quantum corrections.
}
\label{Fig-HR}
\end{center}
\end{figure}

In the classical particle picture, both the bosonic non-critical string and the acoustic black hole are described by the flow of  the fermions from the left.
The difference of these two configurations are just the maximum energies.
The energy of the fermions are filled up to the fermi energy $-\mu <0$ in the bosonic non-critical string case while up to $E_+ >0$ in the acoustic black hole case.
Correspondingly all the fermions are reflected by the potential in the classical limit $\hbar \to 0$ in the bosonic non-critical string case, and, in the acoustic black hole case, they are reflected if $E<0$ and go through the potential if  $0<E \le E_+$.
As we have seen in section \ref{sec-c=1}, for the fermions with $E<0$, the holes appear in the fermi sea through the quantum tunneling, and they cause the thermal radiation.
In the acoustic black hole case, in addition to these holes, even the fermions with $0<E \le E_+$ are reflected quantum mechanically, and they may also contribute to the thermal radiation.
Indeed, the probability of the reflectance of this process for the fermion with energy $E$ can be calculated as \cite{Moore:1991zv,Giovanazzi:2004zv}
\begin{align}
 P_R(E)= \frac{1}{\exp\left( \frac{2\pi}{\hbar} \sqrt{\frac{m}{\alpha}} E \right)+1}
\label{PR}
\end{align}
The derivation is shown in appendix \ref{app-QM}.
Again this probability can be interpreted as the Fermi-Dirac distribution at the temperature $T_L$.
Thus the Hawking radiation in the acoustic black hole may be explained as the net effect of the holes and reflected fermions\footnote{Thus the acoustic Hawking radiation is related to the quantum tunneling of the fermions. This result may remind us the derivation of the Hawking radiation through the quantum tunneling proposed by Parikh and Wilczek \cite{Parikh:1999mf}. However they are not directly connected, since the former is the tunneling of the fermions which compose the fluid and the latter is the tunneling of the phonons in the acoustic geometry.}. See figure \ref{Fig-HR}.
Since both the holes and reflected fermions obey the same Fermi-Dirac distribution, the temperature of the bosonic non-critical string case and the acoustic black hole case are the same.

Now it is natural to interpret the thermal radiation in the bosonic non-critical string case as a variety of the acoustic Hawking radiation.
Although there is no supersonic region in the bosonic non-critical string case, the mechanism of the appearance of the radiation is related to the Hawking radiation in the acoustic black hole.

Finally, in order to confirm that the Hawking radiation in the acoustic black hole can be explained by the quantum mechanics of the fermions, we evaluate the density of the observable $O(\hat{x},\hat{p})$ in the two different ways: the quantum mechanics of the fermions and the second quantized phonon.
In appendix \ref{app-QM-phonon}, we show that these two derivations provide the same result,
\begin{align*}
 \frac{m}{2\pi \hbar}
 \int_{0}^{\infty} \frac{d E}{e^{\beta_L E}+1}
 \left(
 \frac{O(x,p(E,x))}{|p(E,x)|}
-  \frac{O(x,p(-E,x))}{|p(-E,x)|}
\right).
\end{align*}
The first term is the contribution of the reflected fermions and the second one is that of the holes which is equivalent to (\ref{VEV-O-semi}).
(One important point to derive this result  in the phonon calculation is that we need to evaluate the $O(\epsilon^2)$ correction $\rho_2$ and $v_2$ in the $\epsilon$ expansion (\ref{expansion}), since they are the same order to $\langle \psi  \psi  \rangle$ which is the leading contribution of the Hawking effect.)
In this way, we can show that the acoustic Hawking radiation of the phonon is equivalent to the quantum tunneling and reflection of the fermions in the quantum mechanics.

\section{Discussions}
\label{sec-discussion}

We have studied the thermal radiation in the $c=1$ matrix model which is related to the acoustic Hawking radiation in the supersonic fluid.
Remarkably, although the system is not chaotic, the temperature  $T_L$ (\ref{T-mini}) of this radiation saturates the chaos bound (\ref{L-bound}) \cite{Maldacena:2015waa}.
The mechanism of the appearance of the bound in our model is very simple.
The classical motion of the particle in the inverse harmonic potential (\ref{particle-chaos}) receives the quantum correction (\ref{PT}) or (\ref{PR}), and it causes the thermal radiation.
This mechanism would be universal since only the dynamics around the tip of the potential is relevant for the radiation and it does not depend on the details of the potential in the far region.
Thus our result may be applicable to various other systems.
Particularly, applications to the classical chaotic systems would be interesting.
It would be valuable to investigate whether any thermal radiation associated with the classical Lyapunov exponent arises in these systems in the semi-classical regime.
If the mode which classically shows the exponential sensitivity of the initial condition (\ref{chaos}) effectively feels the inverse harmonic potential approximately, thermal radiation might be induced through the quantum effect.

Besides the acoustic black hole may provide a chance to explore the nature of the chaos bound in experiment.
Several interesting phenomena related to chaos may occur near the event horizon of black holes (e.g. \cite{Maldacena:2015waa, Sekino:2008he, Hashimoto:2016dfz,Qaemmaqami:2017jxz}), and some of them may occur even in the acoustic event horizon.
Since acoustic black holes may be realized in laboratories \cite{Lahav:2009wx,Steinhauer:2014dra,Steinhauer:2015saa}, these predictions of the chaotic behaviors might be tested by these experiments.

\paragraph{Acknowledgements}
The author would like to thank Koji Hashimoto, Pei-Ming Ho, Shoichi Kawamoto, Manas Kulkarni, Gautam Mandal, Yoshinori Matsuo, Joseph Samuel, Tadashi Takayanagi and Asato Tsuchiya for valuable discussions and comments.
The work of T.~M. is supported in part by Grant-in-Aid for Scientific Research (No. 15K17643) from JSPS.

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text:
Apostóli, en  est un village du dème de Minóa Pediáda, en Crète, en Grèce. Selon le recensement de 2011, la population d'Apostóli compte . Il est situé à une altitude de ,  à  de Héraklion et à  de Kastélli.

Notes et références

Notes

Références

Localité dans le dème de Minóa Pediáda

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text:
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text:
Matakana Island homicide victim's tangi under way
4 Jan, 2019 3:54pm 2 minutes to read
Man discharged from Whanganui Hospital, died hours later
Local Focus: Marching girls become marching grannies
4 Jan, 2019 4:00pm
Marching Grannies keep the drill alive in Hawke's Bay. Made with funding from NZ On Air.
By: Patrick O'Sullivan
Video Journalist Hawke's Bay/Wairarapa, NZH Local Focus
patrick@localfocus.nz @padyoo7
Marching girls are as 'kiwiana' as pavlova, and they still are in Hawke's Bay - although the marchers are not always girls.
When Kellie Fulton was growing up, competitive marching was "really big". There was even a New Zealand television series based on a marching team.
"We must have had at least maybe 12 teams here in Hawke's Bay," she said. "There would have been probably about five clubs and it is only two clubs here in Hawke's Bay now. Six teams here in the Bay and that's it."
The coach of the Burlington Marching Under-16 Team, she said the emphasis on major sporting codes is to blame.
"It's not advertised the same as netball and rugby and soccer. It is not part of the school curriculum - I think it just comes down to that. It is very sad to see.
"Children at school are hassled for marching because it's not cool."
She said the marchers soon become a family.
"You spend lots of time together, you make new friends, you travel."
Her mum, Jennie MacDonald helps run the three Burlington Marching teams.
"It is a very disciplined sport and not all children will handle that," MacDonald said.
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"It is also a lot of fun and they like meeting new people and they like travelling and the medals and awards that go along with it.
"They get to dance and jump around and make up the moves. And in the display part the girls actually made a lot of those moves themselves."
Recreational marching for girls was introduced to New Zealand and Australia in the 1930s to keep them fit and active.
In Australia it has morphed into drill dancing, which has some of the traditional moves, along with Vegas-style razzle dazzle.
But in New Zealand, despite generations of competitive marchers still active, competitive marching is in serious decline.
Marching Grannies
But it's not all bad news. Non-competitive leisure marching, aimed at people over-50 is growing.
The Taradale Super Grans are one of the biggest teams in the country, fielding 24 marchers.
Val Mitchell is one of several in their 80s at the Napier club. She said a large part of their success was having the same coach for 26 years, Adele Wakely, also in her 80s.
"She is very strict - you've got to be there on time. She keeps us on our toes," Mitchell said.
"We do very well really, we are always pleased with ourselves. We had a couple of years where we boo-booed it up, but most of the time we are pleased with ourselves."
The Super Grans took part in the Dannevirke Christmas Parade last month, before taking part in a North Island Display Day with 17 teams from the Lower North Island, hosted by the Ruahine Ramblerz.
Ramblerz president Pam Bassett said increased awareness of fitness was partly the reason for the growth of Leisure Marching, which has as its motto: Fun, friendship and fitness.
Tararua mayor Tracey Collis said the Ramblerz had "put marching back on the map".
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Joseph Miller earns Outstanding Major in Writing Award
Joseph Miller, son of David and Sandra Scallion of Aurora, was awarded the Outstanding Major in Writing Award during Honors Day May 13 at North Central College in Naperville.
A senior, Joseph is majoring in English: writing at North Central College. He is a graduate of Wakefield Junior-Senior High School.
Joseph presented his project, titled "The Living Dead as Individuals: A Candid Look at the Evolution of the Zombie," at the 26th Annual Meeting of the Society for Literature, Science and the Arts in 2012 held in Milwaukee. An active member of the creative writing club, Joseph writes primarily fiction works based on his work experience and his time in the military. He is a staff member at the North Central Review, the College's literary magazine, and The Kindling, the College's humor magazine.
His name has regularly appeared on the Dean's List.
North Central Review