text:
\section{Regularized Moment System in 1D}
In 1D space, the structure of the system we derived is significantly
simpler than in multi-dimensional case. In this section, we give the
detailed formation of the moment system in 1D case since the 1D Wigner
equation is already very useful in modelling of RTDs
\cite{Frensley1987}.

The 1D Wigner equation is as
\begin{equation}
\label{eq:Wigner1D}
\pd{f}{t} + p \pd{f}{x}
- \sum_{\lambda=1,3,5,\cdots}
\frac{ (\hbar/2i)^{\lambda-1}}{\lambda !}
\pd{^\lambda V}{x^\lambda} \pd{^\lambda f}{p^\lambda} = \frac{1}{\tau}
(f_{\mathrm{eq}} - f),
\end{equation}
where effective mass is again assumed to be $1$ for convinence.  The
equilibrium distribution $f_{\mathrm{eq}}$ is assumed to be a 1D
Maxwellian distribution
\begin{equation}
f_{\mathrm{eq}}(t,x,p) = \frac{\rho(t,x)}{
	\sqrt{2\pi k_B T} } \exp\left( - \frac{ ( p -u (t,x) )^2 }
		{2 k_B T(t,x) } \right).
\end{equation}
The 1D Wigner function is expanded as
\begin{equation} \label{eq:expansion_1d}
f(t,x,p) = \sum_{n\in\bbN}f_n(t,x,p)
\mathcal{H}_{\mathcal{T},n}\left(  \frac{ p
		-u}{\sqrt{\mathcal{T}}}\right),
\end{equation}
where
\begin{equation}
\mathcal{H}_{\mathcal{T},n}(x) =
\frac{1}{\sqrt{2\pi} } \mathcal{T}^{-(n+1)/2}
\He_{n}(x) \exp(-x^2/2 ),
\end{equation}
and the 1D density $\rho$ and average momentum $u$ are defined by
\begin{equation}
\rho = \int_{-\infty}^{\infty} f(t,x,p) d p, \quad
u = \frac{1}{\rho}\int_{-\infty}^{\infty}p  f(t,x,p) d p,
\end{equation}
and the scaled temperature $\mathcal{T}$ defined by
\begin{equation}
\mathcal{T} = k_B T = \frac{1}{\rho}
\int_{-\infty}^{\infty}(p -u)^2  f(t,x,p) d p.
\end{equation}
The above equation plus the definition for the pressure \eqref{eq:Pij}
for $P$ shows that
\begin{equation}
\label{eq:P1D}
P = \rho \mathcal{T}.
\end{equation}
In one dimension, \eqref{eq:estimate} becomes
\begin{equation}
f_1 = 0, \quad f_2 = 0.
\end{equation}
The relation \eqref{eq:Qi2} between the heat flux $q$ and the
coefficient $f_3$ of the expansion is turned to be
\begin{equation}
\label{eq:Q1D}
q = 3 f_3.
\end{equation}

We present below the details of the regularized moment system
truncated at very low order based on \eqref{eq:regularized_system},
though the closed moment system up to any order can be written in a
general formation. We will demonstrate that with the low order
expansions of the distribution function, the moment system obtained is
able to capture very typical quantum effects. As the simplest case, a
closed regularized moment system truncated up to $M=3$ is
\begin{equation}
\label{eq:Moment0}
\pd{\rho}{t} + u \pd{\rho}{x} + \rho \pd{ u}{x} = 0,
\end{equation}
\begin{equation}
\label{eq:Moment1}
\rho\pd{u}{t}+ \rho u \pd{u}{x}  + \pd{P}{x} = -\pd{V}{x}\rho,
\end{equation}
\begin{equation}
\label{eq:Moment2}
\pd{P/2}{t} + u \pd{P/2}{x} + \dfrac{3}{2} P
\pd{u}{x} +  3\pd{f_3}{x} = 0.
\end{equation}
\begin{equation}
\label{eq:Moment3}
 \pd{f_3}{t}
- \dfrac{P^2}{2\rho^2}\pd{\rho}{x} +\dfrac{P}{2\rho} \pd{P}{x}
+u \pd{f_3}{x}  =
-\dfrac{1}{\tau} f_3
+ \dfrac{\hbar^2}{24}\pd{^3 V}{x^3} \rho,
\end{equation}
where $\dfrac{\hbar^2}{24}\pd{^3 V}{x^3} \rho $ is the quantum
correction term yielded by the nonlocal Wigner potential. We
reformulate \eqref{eq:Moment0}, \eqref{eq:Moment1},
\eqref{eq:Moment2}, \eqref{eq:Moment3} into a matrix form
\begin{equation}
\label{eq:MomentMatrixForm}
\pd{\bw}{t} +{\bf M} (\bw) \pd{\bw}{x} = {\bf G} {\bw} ,
\end{equation}
where $\bw=\left(\rho, u, P/2, f_3 \right)^{T}$ and
\begin{equation}
\label{eq:MatrixM3}
{\bf M}
=\begin{pmatrix}
u & \rho & 0 & 0 \\
0 & u & \dfrac{2}{\rho} & 0 \\
0 & 3P & u & 3 \\
-\dfrac{P^2}{2\rho^2} & 0 & \dfrac{P}{\rho} & u
\end{pmatrix},
\end{equation}
\begin{equation}
\label{eq:MatrixG3}
{\bf G} =
\begin{pmatrix}
0 & 0 & 0 & 0 \\
-\pd{V}{x}\dfrac{1}{\rho} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\dfrac{\hbar^2}{24}\pd{^3 V}{x^3} & 0 & 0 & -\dfrac{1}{\tau}
\end{pmatrix}.
\end{equation}

Here we consider the following example to show the difference between
the quantum moment system and the classical moment system. The
Boltzmann equation in 1D with a potential
\begin{equation}
V(x) =\left\{
\begin{array}[]{ll}
\exp \left( - \frac{1}{1-x^2} \right), & \text{if}\ \ |x|<1 , \\
0, & \text{else},
\end{array}
\right.
\end{equation}
admits a steady solution
\begin{equation}
f(t,x,p) = \frac{1}{\sqrt{2\pi}} \exp\left( -\frac{p^2}{2} - V(x) \right),
\end{equation}
with its moments as
\begin{equation}
\label{eq:RhoEx}
\rho = P = \exp(-V(x)), \quad u = f_3 = 0.
\end{equation}
$\rho$, $u$, $P$ and $f_3$ in \eqref{eq:RhoEx} satisfy the moment
system derived from the Boltzmann equation which can be obtained from
\eqref{eq:MomentMatrixForm} by removing the term $-\dfrac{\hbar^2}{24}
\pd{^3 V}{x^3} \rho$. The solutions given in \eqref{eq:RhoEx} show
that the classical behavior of particles obeys the Newtons law, i.e.,
the particles with low kinetic energy ($p^2/2$ at position $x$ ) can
not move to a new position with energy $V(x')$ which is greater than $
p^2/2 + V(x)$. As a result, $\rho$ in \eqref{eq:RhoEx} will not change
in time.

In the quantum mechanics, the particles behave like a wave that the
particles with lower energy can be tunneled through a potential
barrier. The moment system derived from the Wigner equation will
reflect this point. Let us consider the system
\eqref{eq:MomentMatrixForm} with \eqref{eq:RhoEx} as initial value. We
carry out an asymptotic analysis to study the behavior of the solution
around $t = 0$, using time $t$ as the asymptotic expansion
parameter. The ansatz we adopted is as
\begin{align}
\label{eq:f3Asympotic}
f_3 & = f_{3,0} + f_{3,1} t + \mathcal{O}(t^2), \\
\label{eq:PAsympotic}
P & = P_0 + P_1 t +  P_2 t^2 + \mathcal{O}(t^3), \\
\label{eq:uAsympotic}
u & = u_0 + u_1 t +  u_2 t^2 + u_3 t ^3 +  \mathcal{O}(t^4), \\
\label{eq:rhoAsympotic}
\rho & = \rho_0 + \rho_1 t + \rho_2 t^2 + \rho_3 t ^3 + \rho_4 t ^4
+\mathcal{O}(t^5),
\end{align}
where the leading order terms are as the initial value
\begin{equation}
\label{eq:ZeroOrder}
\rho_0 = P_0 = \exp(-V(x)), \quad u_0 = f_{3,0} = 0.
\end{equation}
Substituting the ansatz into \eqref{eq:Moment3}, we immediately have
\begin{equation}
f_{3,1} = \frac{\hbar^2}{24} V^{(3)}(x)  \rho_0,
\end{equation}
thus $f_3$ is as
\begin{equation}
\label{eq:f3Result}
f_3 =  \frac{\hbar^2}{24} V^{\left( 3 \right)}(x) \rho_0 t +
\mathcal{O}(t^2)
\end{equation}
Plugging \eqref{eq:f3Result} and \eqref{eq:PAsympotic} -
\eqref{eq:rhoAsympotic} into \eqref{eq:Moment2}, we obtain
\begin{equation}
P_1 = 0 , \quad P_2 =  -\frac{ \hbar^2}{24} \opd{}{x} \left(V^{\left( 3
			\right)}(x) \rho_0\right) ,
\end{equation}
which yields
\begin{equation}
\label{eq:PResult}
P = P_0  - \frac{\hbar^2}{8} \opd{}{x}\left(V^{\left( 3 \right)}(x)
		\rho_0 \right) t^2 + \mathcal{O}(t^3).
\end{equation}
Similar procedure gives us
\begin{equation}
\label{eq:uResult}
u = u_0 + \frac{\hbar^2}{24} \opd{^2}{x^2}\left(V^{\left( 3 \right)}(x)
		\rho_0 \right)t ^3 + \mathcal{O}(t^4),
\end{equation}
\begin{equation}
\label{eq:rhoResult}
\rho = \rho_0 - \frac{\hbar^2}{96} \opd{^3 }{x^3 } \left (
		V^{(3)} \rho_0^2  \right) t ^4 +
\mathcal{O}(t^5).
\end{equation}
Noting that $\rho_0 = \exp(-V(x))$, we can rewrite
\eqref{eq:rhoResult} into
\begin{equation}
\label{eq:rhoResultAgain}
\rho = \rho_0 + \frac{\hbar^2 t^4 }{96}
g(x) + \mathcal{O}(t^5),
\end{equation}
where
\begin{equation}
\label{eq:gx}
g(x)=	-\opd{^3 }{x^3 }\left ( V^{(3)} \exp(-2V) \right) ,
\end{equation}
whose figure is plotted in \figref{fig:gx}.
\begin{figure}[htbp]
\centering
\includegraphics[width=4in]{gx.eps}
\caption{The figure of $g(x)$ in \eqref{eq:gx}. The curve  of $g(x)$
is plotted in semi-log scale only in the subintervals where
	$g(x)>0$.  $g(x)=0$ for $x$ is outside of $(-1,1)$
	has not been plotted for it is obvious that $g(x)=0$.    }
\label{fig:gx}
\end{figure}

From \eqref{eq:f3Result}, we see that the heat flux $f_3$ is changed
by the term $\dfrac{\hbar^2}{24} \pd{^3 V}{x^3} \rho_0$, which is the
difference between the Boltzmann equation and the Wigner equation. The
change of heat flux induces the change of the temperature
\eqref{eq:PResult} thus the electrons in part of the domain become
cooler than the electrons in the else part of the domain. This makes
the electrons with higher temperature moving to the domain with lower
temperature \eqref{eq:uResult}, eventually resulting the
redistribution of the density of the electrons
\eqref{eq:rhoResult}. \eqref{eq:rhoResultAgain} shows that how the
density $\rho$ changes at a very small $t$, and whether it will
increase or decrease depends on the sign of $g(x)$, which is shown in
\figref{fig:gx}. So it is observed from \figref{fig:gx} that the
particles will be redistributed because of the quantum
term. Precisely, at a very small time $t$, the electron density in the
subintervals of $(-1, 1)$ where $g(x)>0$, whose curves are plotted in
\figref{fig:gx}, is increasing, and the electron density in the rest
subintervals of $(-1, 1)$ where $g(x)<0$ without its curves plotted,
is decreasing.

In the remain part of this section, we present the 1D system obtained
by truncating the expansion \eqref{eq:expansion_1d} at $M = 4$, $5$
and $6$ for the reader's convenience to carry out numerical
simulations. If we truncate up to $M = 4$, we obtain a closed system
consisting of \eqref{eq:Moment0} - \eqref{eq:Moment2} and
\begin{equation}
\label{eq:Moment3_B}
\pd{f_3}{t}
+ 4 f_3 \pd{u}{x}
+ \dfrac{1}{2} \mathcal{T}\rho   \pd{\mathcal{T}}{x}
+u \pd{f_3}{x} + 4 \pd{f_4}{x}
= -\dfrac{f_3}{\tau}
+ \dfrac{\hbar^2}{24}\pd{^3 V}{x^3} \rho,
\end{equation}
and
\begin{equation}
\label{eq:Moment4}
\pd{f_4}{t}
-\dfrac{f_3}{\rho} \pd{P}{x}
+ \mathcal{T} \pd{f_3}{x} + u \pd{f_4}{x}
= -\dfrac{f_4}{\tau}.
\end{equation}
And the system is collected into the matrix form
\eqref{eq:MomentMatrixForm} with $ \bw=\left(\rho, u, P/2, f_3,
  f_4\right)^{T}$ and
\begin{equation}
\label{eq:MatrixM4}
{\bf M}
=\begin{pmatrix}
u & p & 0 & 0 & 0 \\
0 & u & \dfrac{2}{\rho} & 0 & 0  \\
0 & 3P & u & 3 & 0   \\
-\dfrac{P^2}{2\rho^2} & 4f_3 & \dfrac{P}{\rho} & u & 4 \\
0 & 0 & -\dfrac{f_3}{ \rho} & \dfrac{P}{\rho} & u
\end{pmatrix},
\end{equation}
\begin{equation}
\label{eq:MatrixG4}
{\bf G} =
\begin{pmatrix}
0 & 0 & 0 & 0 & 0  \\
-\pd{V}{x}\dfrac{1}{\rho} & 0 & 0 & 0 & 0  \\
0 & 0 & 0 & 0 & 0  \\
\dfrac{\hbar^2}{24}\pd{^3 V}{x^3} & 0 & 0 & -\dfrac{1}{\tau} & 0  \\
0 & 0 & 0 &  0  & -\dfrac{1}{\tau}
\end{pmatrix}.
\end{equation}

Similarly we can write out the system truncated at $M = 5$ which
consists of \eqref{eq:Moment0} - \eqref{eq:Moment2},
\eqref{eq:Moment3_B} and
\begin{equation}
\label{eq:Moment4_C}
\pd{f_4}{t}
-\dfrac{f_3}{\rho}  \pd{P}{x}
+5 f_4 \pd{u}{x}  + \frac{5f_3}{2}  \pd{\mathcal{T}}{x}
+ \mathcal{T} \pd{f_3}{x} + u \pd{f_4}{x}
+ 5 \pd{f_5}{x}
= -\dfrac{f_4}{\tau} ,
\end{equation}
and
\begin{equation}
\label{eq:Moment5}
\begin{split}
&
\pd{f_5}{t}
-\dfrac{f_4}{\rho} \pd{P}{x}
-\dfrac{3f_3}{\rho}\pd{f_3}{x}
+ \mathcal{T} \pd{f_4}{x} + u \pd{f_5}{x}
 = -\dfrac{f_5}{\tau}
+ \dfrac{\hbar^4}{1920} \pd{^5 V}{x^5} f_0 ,
\end{split}
\end{equation}
which is expressed into the matrix form
\eqref{eq:MomentMatrixForm} with
where $\bw=\left(\rho, u, P/2, f_3, f_4, f_5\right)^{T}$ and
\begin{equation}
\label{eq:MatrixM5}
{\bf M}
=\begin{pmatrix}
u & p & 0 & 0 & 0  & 0 \\
0 & u & \dfrac{2}{\rho} & 0 & 0  & 0  \\
0 & 3P & u & 3 & 0 & 0    \\
-\dfrac{P^2}{2\rho^2} & 4f_3 & \dfrac{P}{\rho} & u & 4 & 0  \\
-\dfrac{5P f_3}{2\rho^2}  & 5 f_4 & \dfrac{3f_3}{\rho} & \dfrac{P}{\rho} & u & 5 \\
0 & 0 & -\dfrac{ f_4}{\rho} & -\dfrac{3f_3}{\rho}
& \dfrac{P}{\rho} & u
\end{pmatrix},
\end{equation}
\begin{equation}
\label{eq:MatrixG5}
{\bf G} =
\begin{pmatrix}
0 & 0 & 0 & 0 & 0  & 0  \\
-\pd{V}{x}\dfrac{1}{\rho} & 0 & 0 & 0 & 0 & 0   \\
0 & 0 & 0 & 0 & 0  & 0  \\
\dfrac{\hbar^2}{24}\pd{^3 V}{x^3} & 0 & 0 & -\dfrac{1}{\tau} & 0  & 0 \\
0 & 0 & 0 &  0  & -\dfrac{1}{\tau} &  0 \\
-\dfrac{\hbar^4}{1920}\pd{^5 V}{x^5} & 0 & 0 & 0 & 0
& -\dfrac{1}{\tau}
\end{pmatrix}.
\end{equation}

In the case of truncating at $M = 6$, the moment system obtained
consists of \eqref{eq:Moment0} - \eqref{eq:Moment2},
\eqref{eq:Moment3_B}, \eqref{eq:Moment4_C} and
\begin{equation}
\label{eq:Moment5_D}
\begin{split}
\pd{f_5}{t}
-\dfrac{f_4}{\rho} \pd{P}{x}
&-\dfrac{3f_3}{\rho}\pd{f_3}{x}
 +6 f_{5} \pd{u}{x}
+ 3f_{4} \pd{\mathcal{T}}{x}  \\
 &+ \mathcal{T} \pd{f_4}{x} + u \pd{f_5}{x}
+ 6 \pd{f_6}{x}
= -\dfrac{f_5}{\tau}
- \dfrac{\hbar^4}{1920} \pd{^5 V}{x^5}f_0 ,
\end{split}
\end{equation}
\begin{equation}
\label{eq:Moment6}
\begin{split}
&
\pd{f_6}{t}
-\dfrac{f_5}{\rho} \pd{P}{x}
-\dfrac{3f_4}{\rho}\pd{f_3}{x}
+\dfrac{1}{2}\pd{\mathcal{T}}{x}  \mathcal{T} f_3
+ \mathcal{T} \pd{f_5}{x} + u \pd{f_6}{x}
 = -\dfrac{f_6}{\tau}
+ \dfrac{\hbar^2}{24}\pd{^3 V}{x^3} f_3 .
\end{split}
\end{equation}
This system can be formulated in the matrix form of
\eqref{eq:MomentMatrixForm} with $\bw=\left(\rho, u, P/2, f_3, f_4,
  f_5, f_6\right)^{T}$ and
\begin{equation}
\label{eq:MatrixM6}
{\bf M}
=\begin{pmatrix}
u & p & 0 & 0 & 0  & 0 & 0 \\
0 & u & \dfrac{2}{\rho} & 0 & 0  & 0 & 0  \\
0 & 3P & u & 3 & 0 & 0  & 0   \\
-\dfrac{P^2}{2\rho^2} & 4f_3 & \dfrac{P}{\rho} & u & 4 & 0 & 0 \\
-\dfrac{5P f_3}{2\rho^2}  & 5 f_4 & \dfrac{3f_3}{2\rho} & \dfrac{P}{\rho} & u & 5 & 0 \\
-\dfrac{3P f_4}{\rho^2}  & 6f_5 &  \dfrac{2f_4}{\rho}
 & -\dfrac{3f_3}{\rho}
& \dfrac{P}{\rho} & u  & 6  \\
-\dfrac{P^2 f_3}{2 \rho^3}  & 0 & \dfrac{Pf_3}{2\rho^2}-\dfrac{f_5}{\rho}
&  -\dfrac{3 f_4}{\rho} & 0
& \dfrac{P}{\rho} & u
\end{pmatrix},
\end{equation}
\begin{equation}
\label{eq:MatrixG6}
{\bf G} =
\begin{pmatrix}
0 & 0 & 0 & 0 & 0  & 0 & 0   \\
-\pd{V}{x}\dfrac{1}{\rho} & 0 & 0 & 0 & 0 & 0 & 0    \\
0 & 0 & 0 & 0 & 0  & 0 & 0  \\
\dfrac{\hbar^2}{24}\pd{^3 V}{x^3} & 0 & 0 & -\dfrac{1}{\tau} & 0  & 0 &
0 \\
0 & 0 & 0 &  0  & -\dfrac{1}{\tau} & 0 &
  0 \\
-\dfrac{\hbar^4}{1920}\pd{^5 V}{x^5} & 0 & 0 & 0 & 0   &
-\dfrac{1}{\tau} &  0 \\
0 & 0 & 0 & \dfrac{\hbar^2}{24}\pd{^3 V}{x^3} & 0 &
0 &
-\dfrac{1}{\tau}
\end{pmatrix}.
\end{equation}

Observing {\bf M}'s in \eqref{eq:MatrixM3}, \eqref{eq:MatrixM4},
\eqref{eq:MatrixM5}, \eqref{eq:MatrixM6} and {\bf G}'s in
\eqref{eq:MatrixG3}, \eqref{eq:MatrixG4}, \eqref{eq:MatrixG5},
\eqref{eq:MatrixG6}, we find that the left hand side of the moment
system of \eqref{eq:MomentMatrixForm} is the exactly the same as the
moment system derived from the classical Boltzmann equation, and the
quantum mechanical terms with a typical characteristic which involves
the high order derivatives of the potential $V$ are only appear in the
matrices $G$ in the right-hand side of \eqref{eq:MomentMatrixForm}. It
is clear that every $G$ is lower triangular. The formation of matrices
$G$ shows that the quantum potential term works in the way by letting
high-order moments get information from the moments of different lower
orders, as low as to order $0$.

\section{Introduction}
The Wigner equation was proposed by Wigner in 1932 as a counterpart in
quantum mechanics of the Boltzmann equation
\cite{Wigner1932}. Different from the distribution function in the
Boltzmann equation, the Wigner function may take negative values, so
it is not a true probability distribution function. But it is called a
``quasi-probability distribution function" in the phase space because
it allows one to express quantum mechanical averages (related to
statistical moments) in a form which is very similar to that for
classical averages. Readers may refer to the review article by Hillery
{\it et al.} \cite{Hillery1984} for more details of the properties of
the Wigner function. It offers a convenient interpretation of the
ensemble in the form of a quasi-probability distribution in the phase
space defined by position $\bx$ and momentum $\bp$ and has been proved
to of great use not only as a calculation tools but can also provide
insights into the connections between classical and quantum mechanical
mechanics. Actually, Lions {\it et al.} have given a rigorous
mathematical proof that in the semiclassical limit, the Wigner
equation gives the Vlasov (or Liouville) equation in the phase space
\cite{Lions1993}.  As one of the most accurate equations in quantum
mechanics, the Wigner equation has many advantages over the
Schr\"odinger equation, the nonequilibrium Green method and the
density matrix method in simulating the carrier transport in
semiconductor devices because the description of boundary conditions
and collision operator for the semi-classical Boltzmann equation can
be extended to the Wigner equation due to their strong similarity
\cite{ferry_transport}. There has been an increasing interest in the
Wigner equation as the size of semiconductor devices goes into the
nano-scale under which the quantum effects becomes not negligible any
more \cite{itrs2009}. In some devices, the quantum effect even plays a
dominating role, e.g., the resonant tunneling diode (RTD) has been
extensively studied recently using the numerical methods based on the
Wigner equation \cite{Frensley1987}.  The numerical method for the
Wigner equations has attracted many researchers from different fields
\cite{ShLu09, GeKo05, dollfus09, KKFR89, JaBo04, KoNe06, zhao_05,
  Biegel1996}. The deterministic numerical methods have been
successfully used in simulating one-dimensional devices, but are not
expected to be directly used for multi-dimensional devices simulation
due to its formidable expense in memory storage and computational
time.  One practical approach to investigate a higher dimensional
devices where quantum effects are relevant is to use quantum
hydrodynamics models which are moment systems derived from the Wigner
equation.  Because the close connection of the Wigner equation and the
Boltzmann equation, many moment methods devised for the Boltzmann
equation have been extended to the Wigner equation, see
\cite{Gardner1994, Degond2003} and references therein. Equations
derived from the Wigner equation are called quantum drift-diffusion
equations, quantum Euler equations and quantum hydrodynamics equation,
and numerical simulations based on such moment equations are
extensively studied \cite{Zhou1992, Degond2005QETDD, Jungel1997,
  Tang2008}. In this paper, we will extended the moment method
recently proposed in \cite{Fan_new} for the Boltzmann equation to the
Wigner equation.

In 1940s, Grad \cite{Grad} proposed a moment method to approximate the
Boltzmann equation, and a 13-moment model is given as an extension of
the classic Euler equations. However, this model was quickly found to
be problematic \cite{Grad1952}. Its major deficiencies include the
appearance of subshocks in the structure of a strong shock wave and
the loss of global hyperbolicity. Later on, a number of
regularizations were raised to solve or alleviate these problems.
Levermore \cite{Levermore} introduced a promising way to achieve
global hyperbolicity, while 14 moments are needed and the explicit
expressions of the equations cannot be written. Jin and Slemrod
\cite{Jin} gave a regularization of the Burnett equations via
relaxation, which resulted in a set of equations containing the same
variables with Grad's 13-moment theory, and no subshocks appear in the
structure of shock waves. Struchtrup and Torrilhon
\cite{Struchtrup2003} regularized Grad's system by integrating the
moment method with Chapman-Enskog expansion, and the resulting system
is called as the R13 equations. And in \cite{Struchtrup2004},
Struchtrup improved the R13 equations by the ``order of magnitude''
method. The R13 system also removes the discontinuities in the shock
wave, and it extends the region of hyperbolicity considerably
\cite{TorrilhonRGD}. Recently, Torrilhon \cite{Torrilhon2010} tackled
the problem of hyperbolicity by introducing the Pearson-Type-IV
distributions, and the resulting system is proven to be able to gain a
much larger hyperbolicity region.

Due to the complexity of the explicit expressions, systems with large
number of moments are not investigated until recently. In
\cite{Torrilhon}, Torrilhon and his coworkers developed a software
named $ET_{XX}$ \cite{ETXX} which is able to generate moment systems
with almost any number of moments in one-dimensional space. And in
\cite{Weiss}, some numerical results for a shock tube were carried out
to show the behavior of characteristic waves in the extended
thermodynamics. A numerical method solving large moment systems was
proposed in \cite{NRxx}, and therein the regularization technique in
\cite{Struchtrup2003} was applied to general moment systems. In
\cite{NRxx_new}, the order of magnitude method was also integrated
into large moment systems. In \cite{Fan}, the authors focused on the
one-dimensional velocity space, and the characteristic polynomial of
the quasi-linear coefficient matrix is found to be very simple; thus a
brand-new regularized model with global hyperbolicity is proposed by
the correction of the characteristic speed. Such regularization is
extended to the multi-dimensional velocity space in \cite{Fan_new}.

It is clear that the moment expansion of the drift term for Boltzmann
equation can be extended to the Wigner equation. The resulting
convective terms in the moment system expanded from the drift term of
the Wigner equation has exactly the same format as that of the
Boltzmann equation. Thus the method of the hyperbolic regularization
in \cite{Fan_new} is applied to the Wigner equation, too, to achieve
the global hyperbolicity. As the difference of the Wigner equation
from the Boltzmann equation, the nonlocal Wigner potential term due to
the electric potential is also expanded using Hermite polynomial. It
is interesting for us to find that this term can be represented in the
moment expansion style with very compact expressions. The overall
formation of the nonlocal Wigner potential term in the moments is
turned into a linear source term, with compact sparse coefficient
matrix. Moreover, the contribution from the lower order moments in
this term is always on the higher order moments. Immediately, the
coefficient matrix in the linear source term is strictly lower
triangular, thus it is a nilponent matrix. Noticing that the moment
expansion of the relaxation scattering term produces a linear source
term providing an exponential decay of the high order moments. As a
result, the growth of the high order moments in time due to the source
term is essentially slower than exponential growth rate. This makes
that the derived moment system is formulated as a quasi-linear system,
plus a linear source term which can induce only very mild growth of
the high order moments. Since the convection term in the system is
guaranteed to be globally hyperbolic by the regularization, the local
well-posedness of the system is out of box.

The rest of this paper is arranged as follows: in Section 2 we present
the elementary formula of Wigner equation. The moment expansion of
Wigner equation is carried out in Section 3 and the system obtained is
closed by truncation of the expansion and regularized using method in
\cite{Fan_new} in Section 4 to achieve the final hyperbolic system. In
Section 5, we discuss the simple case in 1D space for better
understanding of the structure of the derived moment
system. Concluding remarks are in the last section.

\section{Moment Closure with Global Hyperbolicity}
\label{the moment closure}
The moment system derived from the Wigner equation consists of
\eqref{mass_con}, \eqref{eq:mtm}, \eqref{eq:moment_pi} and
\eqref{eq:mnt_system}. It is clear that this is a system with infinite
number of equations taken $\rho$, $u_d$, $p_{ij}$ and $f_\alpha$,
$|\alpha| \geqslant 3$, as unknowns. To obtain a system with finite
unknowns, we will truncate the expansion \eqref{eq:expansion} and
close the system following the method in \cite{Fan_new}.

With a truncation of \eqref{eq:expansion}, \eqref{eq:mnt_system} will
result in a finite moment system. Precisely, we let $M\geqslant 3$ be
a positive integer and only the coefficients in the set $\mathcal{M}
= \{f_{\alpha}\}_{|\alpha|\leqslant M}$ are considered. Let
$F_{M}(\bu, \mathcal{T})$ denotes the linear space spanned by all
$\mathcal{H}_{\mathcal{T}, \alpha}
\left(\frac{\bv -\bu(t,\bx)}{\sqrt{\mathcal{T}(t,\bx)}}\right)$'s
with $|\alpha|\leqslant M$,
and the expansion \eqref{eq:expansion} is truncated as
\begin{equation}
  \label{eq:expansion_1}
  f(t,\bx,\bv) \approx \sum_{|\alpha| \leqslant M} f_{\alpha}(t,\bx) \mathcal{H}_{\mathcal{T},
    \alpha}\left(\frac{\bv -\bu(t,\bx)}{\sqrt{\mathcal{T}(t,\bx)}}\right),
\end{equation}
with $f(t,\bx,\bv) \in F_{M}(\bu, \mathcal{T}) $ and $f_{\alpha} \in
\mathcal{M}$. The moment equations which contain
$\partial{f_{\alpha}}/\partial{t}$ with $|\alpha|>M$ are disregarded
in \eqref{eq:mnt_system}.  Then, \eqref{mass_con}, \eqref{eq:mtm},
\eqref{eq:moment_pi}  and \eqref{eq:mnt_system} with $2\leqslant
|\alpha|\leqslant M$ lead to a system with finite number of equations.

Following \cite{Fan_new}, we let
\[
\cS_M = \{\alpha\in\bbN^3\mid |\alpha| \le M\}.
\]
Then for any $\alpha\in \cS_M$, let
\begin{equation}\label{eq:def_mN}
\mN(\alpha) = \sum_{i=1}^3\binom{\sum_{k=4-i}^3\alpha_k + i-1}{i}+1
\end{equation}
to be the ordinal number of $\alpha$ in $\cS_M$, and the cardinal
number of set $\cS_M$ is \[N = \mN(M e_3) = \binom{M+3}{3},\] which is
total number of moments if a truncation with $|\alpha|\le M$ is
applied.

Let $\bw = (w_1, \cdots, w_N)^T \in\bbR^N$ and for each $i,j\in \{1,
2, 3\}$ and $i\neq j$,
\begin{subequations}\label{eq:basic_moments}
\begin{align}
w_1 &= \rho,    &   w_{\mN(e_i)} &= u_i,\\
w_{\mN(2e_i)} &= \frac{p_{ii}}{2},    &
    w_{\mN(e_i+e_j)} &= p_{ij}, \\
w_{\mN(\alpha)} &= f_{\alpha}, \quad 3\le|\alpha|\le M.
\end{align}
\end{subequations}
The moment system \eqref{mass_con}, \eqref{eq:mtm},
\eqref{eq:moment_pi} and \eqref{eq:mnt_system} is collected in
quasi-linear format as
\begin{equation}\label{eq:grad_system}
  \pd{\bw}{t} + \sum_{j = 1}^3 {\bf M}_j(\bw) \pd{\bw}{x_j} =
  {\bf G} \bw,
\end{equation}
by taking the derivatives of $f_{\alpha + e_j}$, $|\alpha| = M$ to be
zero, where ${\bf M}_j$ and ${\bf G}$ are $N \times N$ matrices.  The
entries of ${\bf M}_j$ are given as the coefficients of the terms in
\eqref{mass_con}, \eqref{eq:mtm}, \eqref{eq:moment_pi} and
\eqref{eq:mnt_system} with derivatives of $\bw$. The entries of ${\bf G}$
arise from the nonlocal Wigner potential term and the scattering term.
From \eqref{eq:moment_pi}, one can observe that
\begin{equation}\label{eq:matrix_pi}
{\bf G}_{\mN(2e_i),\mN(2e_j)} = -\frac{1}{\tau}
(\delta_{ij}-\frac{1}{3}), \quad i,j=1,2,3.
\end{equation}
And \eqref{eq:mnt_system} indicates that the diagonal entries of the
lower right part of ${\bf G}$ are
\begin{equation}\label{eq:matrix_diagonal}
  {\bf G}_{\mN(\alpha),\mN(\alpha)} = -\frac{1}{\tau}
  \mathrm{H}(|\alpha| - 2), \quad \text{ for } |\alpha|\ge 2
  \text{ and } \alpha\neq2e_i,~~i=1,2,3.
\end{equation}
From \eqref{eq:mtm}, it is clear that
\begin{equation}
{\bf G}_{\mN(e_i),1} = -\dfrac{1}{\rho} \pd{V}{x_i}, \quad i = 1, 2, 3.
\end{equation}
The other nonzero entries of ${\bf G}$ from the nonlocal Wigner
potential are as
\begin{equation}\label{eq:G_a}
  {\bf G}_{\mN(\alpha), \mN(\alpha - \blambda)} = -\frac{ (\hbar/2i)
  ^{ |\blambda| -1 }}{ \blambda ! } \frac{ \partial ^{\blambda } V}
{\partial \bx ^ {\blambda} },
\end{equation}
where $|\blambda|$ is odd and $|\blambda|$ is greater than $1$ and
$|\alpha - \blambda| \neq 1$ or $2$. In case of $|\alpha - \blambda| =
1$, we have ${\bf G}_{\mN(\alpha), \mN(\alpha - \blambda)} = 0$ since
$f_{e_i} = 0$, $i = 1, 2, 3$. In case of $|\alpha - \blambda| = 2$,
we have \eqref{eq:G_a} if $\alpha - \blambda = e_i + e_j$, $i \neq j$,
since $w_{\mN(e_i + e_j)} = p_{ij} = f_{e_i + e_j}$. The difference is
in case of $\alpha - \blambda = 2 e_i$, $i = 1, 2, 3$. Precisely by
\eqref{eq:pij_fij}, we have
\begin{equation}
  {\bf G}_{\mN(\alpha),\mN(2 e_i)} = -\sum_{j=1}^3 \left(\delta_{ij} -
    \dfrac{1}{3} \right) \frac{ (\hbar/2i)^{|\alpha| - 3}}{ (\alpha - 2 e_j) ! }
  \frac{\partial^{\alpha - 2 e_j } V} {\partial \bx ^ {\alpha - 2 e_j} }.
\end{equation}

All other entries of ${\bf G}$ vanishes except for the ones specified
above. It is clear that the non-zero entries defined by
\eqref{eq:matrix_pi} and \eqref{eq:matrix_diagonal} by the scattering
term provide us a linear damping of the high order moment. The
contribution of the nonlocal Wigner potential matrix ${\bf G}$ is
strictly lower triangular. Thus these part of the matrix is nilponent.
As a result, the growth of the high order moments in time due to the
nonlocal Wigner potential term is essentially slower than exponential
growth rate.

In \eqref{eq:grad_system}, we following Grad \cite{Grad} take
${\partial f_{\alpha + e_j}}/{\partial x_j}$, $|\alpha| = M$, as zero
to make the system to be closed. It has been pointed out in
\cite{Fan_new} that it is not appropriate to set $\partial f_{\alpha +
  e_j}/\partial x_j = 0$, $|\alpha| = M$, as the closure proposed in
\cite{Grad} since the system is lack of hyperbolicity if the
distribution function is far away from the equilibrium. To obtain a
system with global hyperbolicity, we have to adopt the regularization
given in \cite{Fan_new}. For any $\alpha$ with $|\alpha| = M$, we
define
\begin{equation}\label{defRMj}
  \RM^j(\alpha) =
  (\alpha_j+1)\left[\sum_{d=1}^3 f_{\alpha-e_d+e_j}\pd{u_d}{x_j} +
    \frac{1}{2}\left(\sum_{d=1}^3 f_{\alpha-2e_d+e_j}\right)
  \pd{\mathcal{T}}{x_j}\right].
\end{equation}
and
\begin{equation} \label{eq:Grad_hme}
\hat{\bf M}_j \pd{\bw}{x_j} =
  {\bf M}_j \pd{\bw}{x_j} -
  \sum_{|\alpha|=M} \RM^j(\alpha) I_{\mN(\alpha)},
\quad \text{ for any admissible } \bw,
\end{equation}
where $I_k$ is the $k$-th column of the $N \times N$ identity
matrix. We regularize the system \eqref{eq:grad_system} as
\begin{equation}\label{eq:regularized_system}
  \pd{\bw}{t} + \sum_{j = 1}^3 \hat{\bf M}_j(\bw) \pd{\bw}{x_j} =
  {\bf G} \bw ,
\end{equation}
which is the quantum hydrodynamics model we derived. It has been
proved in \cite{Fan_new} that
\begin{theorem} \label{thm:rot_inv} The regularized moment system
  \eqref{eq:regularized_system} is hyperbolic for any $\bw$ with
  positive temperature. Precisely, for a given unit vector $\bn =
  (n_1, n_2, n_3)$, the matrix
\begin{equation} \label{eq:rot_inv}
\sum_{j=1}^3 n_j \hat{\bf M}_j(\bw)
\end{equation}
is diagonalizable with eigenvalues as
\begin{equation}
\bu \cdot \bn + \rC{n}{m} \sqrt{\mathcal{T}},
  \qquad 1 \leqslant n \leqslant m \leqslant M+1,
\end{equation}
where $\rC{n}{m}$ is a root of $m$-order Hermite polynomial, and
satisfies $\rC{1}{m}<\cdots<\rC{m}{m}$. The structure of the $N$
eigenvectors can be fully clarified.
\end{theorem}
Based on this theorem, the regularized moment system
\eqref{eq:regularized_system} is locally well-posed due to the
hyperbolicity. We would like to mention here that the regularization
here actually does not add any new terms to the system
\eqref{eq:grad_system}. On the contrary, it has erased the terms in
\eqref{eq:mnt_system} with a factor $\alpha_j + 1$ in its coefficient
for the equations of $f_\alpha$ with $|\alpha| = M$ only.

\section{Conclusion}
We extend the moment closure method \cite{Fan_new} for the Boltzmann
equation to its quantum counterpart, the Wigner equation. And we
obtain a hyperbolic moment system, i.e., quantum hydrodynamic system.
The method to derive the quantum hydrodynamic system with the global
hyperbolicity is systematic that systems for arbitrary number of
moments are obtained at once. Numerical simulations are to be carried
out to demonstrate that the quantum effects are able to be captured by
using the quantum hydrodynamic model derived.

\section*{Acknowledgements}
This research of R. Li was supported in part by the
National Basic Research Program of China (2011CB309704) and Fok Ying
Tong Education and NCET in China. T. Lu was supported in part by the
NSFC (11011130029) and by SRF for ROCS, SEM.

\section{Grad Moment System}
In this section, we derive the moment system of the Wigner equation
using the Grad type moment expansion.

\subsection{Hermite expansion of the distribution function}
\label{sec:discretization}
Following the method in \cite{NRxx, NRxx_new}, we expand the
distribution function into Hermite series as
\begin{equation}
  \label{eq:expansion}
  f(t,\bx, \bv) = \sum_{\alpha \in \bbN^3}f_{\alpha}(t,\bx)
	\mH_{\mT,\alpha}
	\left( \frac{\bv-\bu(t,\bx)}{\sqrt{\mT(t,\bx)}} \right),
\end{equation}
where $\alpha=(\alpha_1,\alpha_2,\alpha_3)$ is a three-dimensional
multi-index. The basis functions $\mH_{\mT,\alpha}$
are the 3-dimensional Hermite functions defined by
\begin{equation}
  \label{eq:base}
  \mH_{\mT,\alpha}(\bxi) =\prod\limits_{d=1}^3
 \frac{1}{\sqrt{2\pi}}\mT^{-\frac{\alpha_d+1}{2}}
\He_{\alpha_d}(\xi_d)\exp \left(-\frac{\xi_d^2}{2} \right),
\end{equation}
where $\He_{n}(x)$ is the Hermite polynomial of order $n$
\begin{equation}
  \label{eq:hermit}
  \He_n(x) = (-1)^n\exp \left( \frac{x^2}{2} \right) \frac{\dd^n}{\dd x^n}
  \exp \left(-\frac{x^2}{2} \right).
\end{equation}
For convenience, $\He_{n}(x)$ is taken as zero if $n<0$, thus
$\mH_{\mT,\alpha}(\bxi)$ is zero when any component of
$\alpha$ is negative. The parameter $\mT$ in the expansion is
the scaled local temperature as
\begin{equation}
\label{eq:DefScaledTem}
\mT(t,\bx) = k_B T(t,\bx).
\end{equation}

It is clear that the equilibrium distribution $f_{\mathrm{eq}}$ is
coincidently equal to the first term of expansion, i.e.,
\begin{equation}
  f_{\mathrm{eq}}(t,\bx,\bv) = f_0(t,\bx) \mH_{\mT,0}
  \left( \frac{\bv-\bu(t,\bx)}{\sqrt{\mT(t,\bx)}}\right),
\end{equation}
where $f_0(t,\bx) = \rho(t,\bx)$. So in this case the relaxation time
approximation scattering term \eqref{eq:collision} can be written as
the linear combination of the basis functions of orders greater than
or equal to $1$,
\begin{equation}
  \label{eq:collision}
  \frac{1}{\tau}(f_M-f) = -\frac{1}{\tau}\sum_{|\alpha| \geq 1}f_{\alpha}
  \mH_{\mT,\alpha} \left( \frac{\bv-\bu(t,\bx)}
  {\sqrt{\mT(t,\bx)}}\right).
\end{equation}

The definition of the Hermite function \eqref{eq:base} shows that
each basis function is an exponentially decaying function multiplied
by a multi-dimensional Hermite polynomial shifted by the local
macroscopic momentum $\bu$ and scaled by the square root of the
local temperature $\mT$.

If one uses arbitrary known function $\bu'(t,\bx)$ and
$\mT'(t,\bx)$ in \eqref{eq:expansion} to expand the
distribution function $f(t,\bx,\bv)$ as
\begin{equation}
  \label{eq:expansion-o}
  f(t,\bx, \bv) = \sum_{\alpha \in \bbN^3}f_{\alpha}'(t,\bx)
  \mH_{\mT',\alpha}
  \left( \frac{\bv-\bu'(t,\bx)}{\sqrt{\mT'(t,\bx)}} \right),
\end{equation}
then the following relations between the macroscopic quantities $\bu$,
$\mT$, the coefficients $f_{\alpha}$, and $\bu'$,
$\mT'$, the coefficients $f_{\alpha}'$ can be derived as
follows,
\begin{subequations}
\label{eq:similar_relation}
 \begin{align}
    \rho &=  f_0 = f_0', \\
\rho \bu &=  \rho \bu' + (f_{e_d}')_{d=1,2,3}^T, \\
\rho |\bu - \bu'|^2 + 3\rho \mT &=
\sum\limits_{d=1}^3(\mT' f_0' + 2 f_{2e_d}'),
\end{align}
\end{subequations}
where $e_d$ is the unit vector with its $d$-th entry to be $1$. It is
clear that the coefficients $f_\alpha$ expanded using parameters $\bu$
and $\mT$ satisfy the following conditions:
\begin{equation}
\label{eq:estimate}
  f_{e_i} = 0, \quad
  \sum\limits_{d=1}^3 f_{2e_d} = 0, \quad i =1, 2 , 3.
\end{equation}
Moreover, if we define the heat flux $q_i$ and the pressure tensor
$P = \{p_{ij}\}, \ i,j =1,2,3$ with
\begin{gather}
\label{eq:Qi1}
  q_i  = \frac{1}{2}\int_{\bbR^3}|\bv - \bu|^2(p_i - u_i)f
  \dd \bv,\\
\label{eq:Pij}
p_{ij} = \int_{\bbR^3}(p_i - u_i)(p_j - u_j)f \dd \bv ,
\end{gather}
then direct calculations give us the relations between them and  the
coefficients  $f_{\alpha}$ in \eqref{eq:expansion} as
\begin{gather}
\label{eq:Qi2}
q_{i} = 2f_{3e_i} + \sum_{d=1}^3f_{2e_d+e_i}, \\ \label{eq:q_p}
p_{ij} - \frac{1}{3} \delta_{ij} \sum_{d=1}^3p_{dd} =
(1 + \delta_{ij}) f_{e_i+e_j}.
\end{gather}
By the definition of the temperature \eqref{eq:DefTemperature} and
\eqref{eq:DefScaledTem} and the definition of the tensor pressure
\eqref{eq:q_p}, the scaled temperature $\mT$ is a linear
combination of $p_{ij}$ as
\begin{equation}
\rho \mT = \dfrac{1}{3}\sum_{d=1}^{3} p_{dd}.
\end{equation}
With the relation \eqref{eq:q_p}, we then have
\begin{equation} \label{eq:pij_fij}
p_{ij} = \delta_{ij} \rho \mT +
(1 + \delta_{ij}) f_{e_i+e_j}.
\end{equation}

\subsection{Moment expansion of the Wigner equation}
\label{moment_equations}
Now we are ready to derive the moment system by taking the moments of
the Wigner equation. The general method to get the moment system is to
first multiply the Wigner equation \eqref{eq:vlasov} of by polynomials
of momentum $\bp$ of different order and then integrate both sides
over momentum $\bp$ on $\bbR^3$. One equivalent way is as
follows. First, we substitute the expansion of the Wigner function
\eqref{eq:expansion} into the Wigner equation \eqref{eq:vlasov}, then
we collect the coefficients of basis functions of the same order at
both sides, and finally we equate the coefficients of the basis
functions of the same order on both sides to yield the derived moment
system.  We plug the Hermite series \eqref{eq:expansion} into the
Wigner equation \eqref{eq:vlasov}, and make calculations by noting
that the Hermite function \eqref{eq:base} used in this paper depends
also on the time $t$ and position $\bx$ through $\bu(t,\bx)$ and
$\mT(t,\bx)$, which is different from the general expansion
using the Hermite functions depending only on the momentum $\bv$
\cite{Shan1998}. For convenience, we list some useful relations of
Hermite polynomials as below \cite {Abramowitz}:
\begin{enumerate}
\item Orthogonality: $\displaystyle \int_{\bbR} \He_l(x)
\He_n(x) \exp(-x^2/2) \dd x = l! \sqrt{2\pi} \delta_{l,n}$;
\item Recursion relation: $\He_{n+1}(x) = x \He_n(x) - n \He_{n-1}(x)$;
\item Differential relation: $\He_n'(x) = n \He_{n-1}(x)$.
\end{enumerate}
And the following equality can be derived from the last two relations:
\begin{equation}
[\He_n(x) \exp(-x^2/2)]' = -\He_{n+1}(x) \exp(-x^2/2).
\end{equation}
Especially, we have
\begin{equation}\label{eq:partial_H}
  \pd{}{p_j} {\mH_{\mT,\alpha}\left( \frac{\bv-\bu}{\sqrt{\mT}} \right)}=
  -\mH_{\mT,\alpha+e_j}\left(
    \frac{\bv-\bu}{\sqrt{\mT}} \right).
\end{equation}
With these relations, the part
\[
\pd{f}{t} + \bv \cdot \nabla_{\bx} f
\]
of \eqref{eq:vlasov} is expanded as
\begin{equation}
\label{eq:DriftExpansion}
\begin{split}
&~~\sum_{\alpha \in \bbN^3} \Bigg\{ \left(
    \frac{\partial f_{\alpha}}{\partial t} +
    \sum_{d=1}^3 \frac{\partial u_d}{\partial t} f_{\alpha-e_d} +
    \frac{1}{2} \frac{\partial \mT}{\partial t}
      \sum_{d=1}^3 f_{\alpha-2e_d}
  \right) \\
& \qquad + \sum_{j=1}^3 \Bigg[ \left(
    \mT \frac{\partial f_{\alpha - e_j}}{\partial x_j} +
    u_j \frac{\partial f_{\alpha}}{\partial x_j} +
    (\alpha_j + 1) \frac{\partial f_{\alpha+e_j}}{\partial x_j}
  \right) \\
& \qquad \qquad + \sum_{d=1}^3 \frac{\partial u_d}{\partial x_j}
    \left( \mT f_{\alpha-e_d-e_j} + u_j f_{\alpha-e_d}
      + (\alpha_j + 1) f_{\alpha-e_d+e_j} \right) \\
& \qquad \qquad + \frac{1}{2} \frac{\partial \mT}{\partial x_j}
    \sum_{d=1}^3 \left(
      \mT f_{\alpha-2e_d-e_j} + u_j f_{\alpha-2e_d}
      + (\alpha_j + 1) f_{\alpha-2e_d+e_j}
    \right)
  \Bigg] \Bigg\} \mH_{\mT,\alpha} \left(
    \frac{\bp - \bu}{\sqrt{\mT}}
  \right).
\end{split}
\end{equation}
Then using \eqref{eq:partial_H}, we calculate the pseudo-operator term
$\Theta[V]f$ expressed in \eqref{eq:ThetaVDiff}, and obtain
\begin{equation}
\label{eq:ThetaVDiffExpansion}
(\Theta[V]f)(t,\bx,\bv)=
 \sum_{\alpha,\blambda}
\frac{ (\hbar/2i) ^{ |\blambda| -1 }}{ \blambda ! }
\frac{ \partial ^{\blambda } V}
{\partial \bx ^ {\blambda} }
f_{\alpha - \blambda }
\mH_{\mT,\alpha} \left(\frac{\bp - \bu}
		{\sqrt{\mT}} \right) ,
\end{equation}
where the summation over $\blambda$ means the same as that in
\eqref{eq:ThetaVDiff}, and $f_{\alpha-\blambda}$ is taken as zero when
any component of $\alpha-\blambda$ is negative. Finally, the
scattering term $\left. \pd{f}{t} \right \vert_{\mathrm{Scat}}$
given in \eqref{eq:Maxwellian} becomes
\begin{equation}
\label{eq:CollisionExpansion}
-\frac{1}{\tau}\sum_{|\alpha|\geq 2}f_{\alpha}\mH_{\mT,\alpha} \left(
    \frac{\bp - \bu}{\sqrt{\mT}}
  \right),
\end{equation}
noticing that $f_{e_d} = 0$, $d = 1, 2, 3$.

Collecting the three terms \eqref{eq:DriftExpansion},
\eqref{eq:ThetaVDiffExpansion} and \eqref{eq:CollisionExpansion}
yielded after the substitution of the Hermite expansion
\eqref{eq:expansion} into the Wigner equation \eqref{eq:vlasov}, we
can get the following general moment equations with a slight
rearrangement by matching the coefficients of the same weight
function:
\begin{equation} \label{eq:mnt_eq}
\begin{split}
& \frac{\partial f_{\alpha}}{\partial t} +
  \sum_{d=1}^3 \left(
    \frac{\partial u_d}{\partial t} +
    \sum_{j=1}^3 u_j \frac{\partial u_d}{\partial x_j}
  \right) f_{\alpha-e_d}
    +  \frac{1}{2} \left(
    \frac{\partial \mT}{\partial t} +
    \sum_{j=1}^3 u_j \frac{\partial \mT}{\partial x_j}
  \right) \sum_{d=1}^3 f_{\alpha-2e_d} \\
& \qquad\qquad + \sum_{j,d=1}^3 \Big[
    \frac{\partial u_d}{\partial x_j} \left(
      \mT f_{\alpha-e_d-e_j} + (\alpha_j + 1) f_{\alpha-e_d+e_j}
    \right) \\
&\qquad\qquad+ \frac{1}{2} \frac{\partial \mT}{\partial x_j} \left(
      \mT f_{\alpha-2e_d-e_j} + (\alpha_j + 1) f_{\alpha-2e_d+e_j}
    \right)
  \Big] \\
& \qquad \qquad+ \sum_{j=1}^3 \left(
    \mT \frac{\partial f_{\alpha - e_j}}{\partial x_j} +
    u_j \frac{\partial f_{\alpha}}{\partial x_j} +
    (\alpha_j + 1) \frac{\partial f_{\alpha+e_j}}{\partial x_j}
  \right)\\
&\qquad \qquad= -\frac{1}{\tau}
 \mathrm{H}( |\alpha|-2 )f_{\alpha}
 -\sum_{\blambda}
 \frac{ (\hbar/2i) ^{ |\blambda| -1 }}{ \blambda ! }
 \frac{ \partial ^{\blambda } V}
{\partial \bx ^ {\blambda} }
f_{\alpha - \blambda } ,
\end{split}
\end{equation}
where $\mathrm{H}(x)$ is the Heaviside function defined by
\begin{equation}
\label{eq:delta}
  \mathrm{H}(x) = \left\{
    \begin{array}{ll}
      0, & \text{if} ~~  x < 0, \\
      1, & \text{if}~~  x \geqslant 0.
    \end{array}
\right.
\end{equation}
By setting $\alpha=0$ in \eqref{eq:mnt_eq}, we deduce the mass
conservation
\begin{equation}
\label{mass_con}
\pd{\rho}{t} +
  \sum_{j=1}^3 \left(
    u_j \pd{\rho}{x_j} +
    \rho \pd{u_j}{x_j}
  \right) = 0.
 \end{equation}
By setting $\alpha = e_d$, with $ d = 1,2,3$ and
noting that $f_{e_d} = 0$ in \eqref{eq:mnt_eq}, we obtain
\begin{equation}
\label{eq:alpha = e_d}
\rho \left(
  \pd{u_d}{t} +
  \sum_{j=1}^3 u_j \pd{u_d}{x_j}
\right) + \rho \pd{\mT}{x_d}
  + \mT \pd{\rho}{x_d}
  + \sum_{j=1}^3 (\delta_{jd} + 1)
    \pd{f_{e_d + e_j}}{x_j} = -\rho \pd{V}{x_d},
\end{equation}
which is simplified as
\begin{equation} \label{eq:mtm}
\rho \left(
  \pd{u_d}{t} +
  \sum_{j=1}^3 u_j \pd{u_d}{x_j}
\right) + \sum_{j=1}^3 \pd{p_{jd}}{x_j} = -\rho \pd{V}{x_d}.
\end{equation}
By setting $\alpha = 2e_d$, with $d=1,2,3$ and noting that
$f_{e_d}=0$, we obtain
\begin{equation}
\label{eq:alpha=2e_d}
\begin{split}
\pd{f_{2e_d}}{t}&+\frac{\rho}{2}\left(\pd{\mT}{t}
            +\sum_{j=1}^3u_j\pd{\mT}{x_j}\right)
    +\rho\mT\pd{u_d}{x_d}
    +\sum_{j,l}(1+\alpha_j)f_{2e_d-e_l+e_j}\pd{u_l}{x_j}\\
&\qquad\qquad\qquad\qquad
    +\sum_{j=1}^3u_j\pd{f_{2e_d}}{x_j}+(1+2\delta_{jd})\pd{f_{2e_d+e_j}}{x_j}
    =-\frac{1}{\tau}f_{2e_d}.
\end{split}
\end{equation}
Noting that $\sum_{d=1}^3f_{2e_d}=0$, we sum the upper equations over
$d$ to get
\begin{equation}
\label{eq:moment_eq}
\rho \left( \pd{\mT}{t}
  + \sum_{j=1}^3 u_j \pd{\mT}{x_j} \right)
  + \frac{2}{3} \sum_{j=1}^3 \left(
    \pd{q_j}{x_j} +
    \sum_{d=1}^3 p_{jd} \pd{u_d}{x_j}
  \right) = 0.
\end{equation}
\comment{\begin{remark}
  We notice that $|\bv-\bu|^2$ is a linear sum of the first three
  Hermite polynomials and $f_{e_d}$, $d = 1, 2, 3$ are zero, thus
\begin{equation}
  \int_{\bbR^3} |\bv-\bu|^2 \pd{^{\blambda} f}{p^{\blambda}}\dd \bv
  = 0, \quad |\blambda| \text{ is odd.}
\end{equation}
The contribution of the nonlocal Wigner potential term does not appear
in \eqref{eq:moment_eq}.
\end{remark}}

Since $\rho\mT=\frac{1}{3}\sum_{d=1}^3p_{dd}$, we have
\begin{equation} \label{eq:partial_T2p}
\pd{\mT}{x_j} =
\frac{1}{3\rho}\sum_{d=1}^3\pd{p_{dd}}{x_j}-\frac{\mT}{\rho}\pd{\rho}{x_j},\quad
j=1,2,3.
\end{equation}
Substituting \eqref{eq:mtm}, \eqref{eq:moment_eq} and
\eqref{eq:partial_T2p} into \eqref{eq:mnt_eq}, we eliminate the time
derivatives of $\bu$ and $\mT$ and the spatial derivatives of
$\mT$. Then the quasi-linear form of
the moment system reads:
\begin{equation} \label{eq:mnt_system}
\begin{split}
&\pd{f_{\alpha}}{t}+\sum_{j=1}^3\left(
    \mT\pd{f_{\alpha-e_j}}{x_j}+u_j\pd{f_{\alpha}}{x_j}
    +(\alpha_j+1)\pd{f_{\alpha+e_j}}{x_j}\right) \\
&\qquad +\sum_{j=1}^3\sum_{d=1}^3\pd{u_d}{x_j}\left(
    \mT f_{\alpha-e_d-e_j} + (\alpha_j+1)f_{\alpha-e_d+e_j}
        -\frac{p_{jd}}{3\rho}\sum_{k=1}^3f_{\alpha-2e_k}\right)\\
&\qquad -\sum_{j=1}^3\sum_{d=1}^3
        \frac{f_{\alpha-e_d}}{\rho}\pd{p_{jd}}{x_j}
        -\frac{1}{3\rho}\left(\sum_{k=1}^3f_{\alpha-2e_k}\right)\sum_{j=1}^3\pd{q_j}{x_j}\\
&\qquad+\sum_{j=1}^3\left(\left(-\frac{\mT}{2\rho}\pd{\rho}{x_j}
        +\frac{1}{6\rho}\sum_{d=1}^3
           \pd{p_{dd}}{x_j}\right)\sum_{k=1}^3\left(\mT
                f_{\alpha-2e_k-e_j}+(\alpha_j+1)f_{\alpha-2e_k+e_j}\right)\right)\\
&\qquad=-\frac{1}{\tau}\mathrm{H}(|\alpha|-2)f_{\alpha}
    - \sum_{\blambda}
    \frac{ (\hbar/2i) ^{ |\blambda| -1 }}{ \blambda ! }
    \pd{^{\blambda } V}
    {\bx ^ {\blambda} }
    f_{\alpha - \blambda } , \qquad \forall ~|\alpha| \geq 2.
\end{split}
\end{equation}
We should point out again the summation over $\blambda$ is extended
over all the non-negative integers $\lambda_d$, $d=1, 2, 3$ for which
$\vert \blambda \vert $ is odd and greater than $1$. Especially for
$|\alpha|\leq 2 $, the moment system \eqref{eq:mnt_system} derived
from the Wigner equation is the same as that derived from the
Boltzmann equation \cite{Li}.

With \eqref{eq:pij_fij}, we can have the equations for $p_{ij}$ by
\eqref{eq:mnt_system}. Precisely, we have the equation for $p_{ii}/2$,
$i = 1, 2, 3$, as
\begin{equation}\label{eq:moment_pi}
\begin{split}
 \pd{p_{ii}/2}{t} &+ \sum_{j=1}^3 u_j \pd{p_{ii}/2}{x_j}
  +\sum_{j=1}^3 \left( \frac{1}{2}+\delta_{ij} \right)
  \rho\mT\pd{u_j}{x_j}
 + \sum_{j=1}^3\sum_{d=1}^3
  (2\delta_{ij}+1)f_{2e_i-e_d+e_j}\pd{u_d}{x_j} \\
&\qquad\qquad+\sum_{j=1}^3(2\delta_{ij}+1)\pd{f_{2e_i+e_j}}{x_j} =
  -\frac{1}{2\tau}\left(p_{ii}-\frac{1}{3}\sum_{d=1}^3p_{dd}\right),
  \quad i = 1, 2, 3.
\end{split}
\end{equation}
If $i \neq j$, we have $p_{ij} = f_{e_i + e_j}$, thus its equation is
already in \eqref{eq:mnt_system}.

We collect the equations \eqref{mass_con}, \eqref{eq:mtm},
\eqref{eq:moment_pi} and \eqref{eq:mnt_system} together to obtain a
moment system with infinite number of equations. Noting that the
relation between $u_d$ and $f_{e_d}$ given in
\eqref{eq:similar_relation} and the definition of $q_i$ and of
$p_{ij}$ given in \eqref{eq:q_p}, we can see that what we obtain is a
quasi-linear system for $f_{\alpha}$. We would like to point out that
the only difference between the system derived from the Wigner
equation and that from the Boltzmann equation is the term with
high-order derivatives of the potential $V(t,\bx)$, which is a source
term of the quasi-linear system of $f_{\alpha}$.

\section{Wigner Equation with Smooth Potential}
We consider a one-particle system in a statistical mixture with $N$
states described by the wave functions $\Psi_{k} = \Psi_k (t,\bx)$,
each with a probability $P_k \geqslant 0 $, with
$k=1,\cdots,N_{\text{state}}$, which satisfies
$\sum_{k=1}^{N_{\text{state}}} P_k = 1$. The wave functions
$\Psi_k(t,\bx)$ of state $k$ obeys the Schr\"odinger equation
\begin{equation}
\label{eq:schrodinger}
i \hbar \pd{\Psi_{k}(t,\bx)}{t} = -\frac{\hbar^2}{2m}\nabla_{\bx}^2
		\Psi_{k}(t,\bx)
+ V(t,\bx) \Psi_{k}(t,\bx) ,
\end{equation}
where $i=\sqrt{-1}$, $\hbar$ is the reduced Planck constant, $m$ is the particle
effective mass assumed to a constant in this paper, and $V(t,\bx)$ is
the electric potential energy (which will be called potential for
short hereafter). We can also regard the one-particle system as a
many-particle system with $P_k$ interpreted as the percentage of the
particles occupying state $k$.  This is the general way the Wigner
equation used in modeling carrier transport in semiconductor devices.
We construct the Wigner function $f(t,\bx,\bp)$ which is a
quasi-probability density function in the phase space $(\bx,\bp)$, as
usual, by
\begin{equation}
  f(t,\bx,\bp) =
  \frac{1}{(2\pi \hbar)^3} \sum_{k=1}^{N_{\text{state}}} P_k
  \int_{\bbR^3} \Psi_k^{*}(t, \bx + \by/2) \Psi_k(t,\bx-\by/2)
	e^{i\by\cdot \bp/\hbar} d^3 \by.
\end{equation}
Using the Schr\"odinger equation \eqref{eq:schrodinger} and the
definition of the Wigner function given above, we can derive the
Wigner equation (or the quantum Vlasov equation) as been done in
\cite{Wigner1932, Hillery1984}
\begin{equation}
  \label{eq:vlasovBallistic}
  \pd{f}{t} + \frac{\bv}{m} \cdot \nabla_{\bx} f + \Theta[V] f
  = 0,
  \quad \bx \in \bbR^3, \ \bp \in \bbR^3 ,
\end{equation}
where the nonlocal Wigner potential term $\Theta[V] f$ is a
pseudo-differential operator. It is defined by
\begin{equation}
\label{eq:ThetaV}
(\Theta[V]f)(t,\bx,\bv)=
\int_{\bbR^3}
V_w(t,\bx,\bv')
 f(t,\bx,\bv-\bv') \dd  \bp',
\end{equation}
where the Wigner potential is as
\begin{equation*}
V_{w}(t,\bx,\bv)=
\frac{-i}{(2\pi\hbar)^3 \hbar} \int_{\bbR^3}
\left[ V\left(t, \bx+\frac{\by}{2}\right)
		-V\left(t, \bx-\frac{\by}{2}\right) \right]
e^{i \by\cdot \bv/\hbar
}  \dd \by .
\end{equation*}
For convenience, we simply take $m = 1$ later on. Similar to a
Vlasov-Poisson system, a Wigner-Poisson system could be also
considered to include the self-consistent electric field induced by
the redistribution of the electrons. In this paper, we focus on the
case that $V(t,\bx)$ is a given smooth function.  The pseudo-operator
$\Theta[V] f $ can be written into an equivalent form
\cite{Hillery1984}
\begin{equation}
\label{eq:ThetaVDiff}
(\Theta[V]f)(t,\bx,\bv)=
-\sum_\blambda
\frac{ (\hbar/2i) ^{ |\blambda| -1 }}{ \blambda ! }
\frac{ \partial ^{\blambda } V}
{\partial \bx ^ {\blambda} }
\frac{ \partial ^{\blambda } f}
{\partial \bv ^ {\blambda} } ,
\end{equation}
where $\blambda=(\lambda_1,\lambda_2,\lambda_3)$,
$|\blambda|=\sum_{i=1}^3 \lambda_i $, $\bx
^{\blambda} = \prod_{i=1}^3 x_i^{\lambda_i}$,
\[
\dfrac{\partial^\blambda}{\partial \bx^\blambda} = \prod_{i=1}^3
\dfrac{\partial^{\lambda_i}}{\partial x_i^{\lambda_i}}, \qquad
\dfrac{\partial^\blambda}{\partial \bp^\blambda} = \prod_{i=1}^3
\dfrac{\partial^{\lambda_i}}{\partial p_i^{\lambda_i}},
\]
and the summation over $\blambda$ has to be extended over all
non-negative integer values of $\lambda_1,\lambda_2,\lambda_3$ for
which $\vert \blambda \vert$ is odd. In the semi-classical limit
$\hbar\rightarrow 0$, $\Theta[V]f$ converges to the usual operator
$-\nabla_{\bx}V \cdot \nabla_{\bp} f$. The equation
\eqref{eq:vlasovBallistic} describes particles movement without
scattering corresponding to ballistic transport. The scattering effect
may be considered by adding a scattering term to the right hand side of
\eqref{eq:vlasovBallistic}. So one obtains the following Wigner
equation with a scattering term:
\begin{equation}
  \label{eq:vlasov}
  \pd{f}{t} + \bv \cdot \nabla_{\bx} f + \Theta[V] f
  = \left. \pd{f}{t} \right \vert_{\mathrm{Scat}},
  \quad \bx \in \bbR^3, \ \bp \in \bbR^3.
\end{equation}
The time-relaxation approximation of the scattering term
is often used for the Wigner equation \cite{Frensley1987}, and it has
the same form of the BGK scattering term used for the collision of
gas \cite{BGK}. It is easy to be study analytically and takes
of the following simple form
\begin{equation}
\left. \pd{f}{t} \right \vert_{\mathrm{Scat}}
= \frac{ f_{\mathrm{eq}} - f } {\tau} ,
\end{equation}
where $\tau$ is the relaxation time and $f_{\mathrm{eq}}$ is the
equilibrium distribution. For example, when the density of electrons
are not extremely high, we can assume that the equilibrium
distribution is a Maxwellian distribution,
\begin{equation}
\label{eq:Maxwellian}
f_{\mathrm{eq}}(t,\bx,\bp) =
\frac{\rho(t,\bx)}{ \left( 2\pi k_B T  \right)^{3/2}}
\exp\left( - \frac{(\bp - \bu(t,x))^2}{2 k_B T(t,\bx)} \right)
\end{equation}
where $\rho(t,\bx)$ is the number density of particles at position
$\bx$, $k_B $ is the Boltzmann constant, $T(t,\bx)$ is the particle
temperature, and $\bu(t,\bx)$ is the average momentum of particles.
These macroscopic variables are related with the distribution function
as below:
\begin{equation}
\rho(t,\bx) = \int_{\bbR^3} f(t,\bx,\bv) \dd \bv ,
\end{equation}
\begin{equation}
\rho(t,\bx)\bu(t,\bx) = \int_{\bbR^3} \bv f(t,\bx,\bv) \dd \bv ,
\end{equation}
\begin{equation}
\label{eq:DefTemperature}
\rho(t, \bx) k_B T(t,\bx) = \int_{\bbR^3} |\bv - \bu|^2 f(t,\bx,\bv)
	\dd \bv .
\end{equation}

The conservation of mass, momentum and total energy are all valid for
the Wigner equation. Multiplying the equation \eqref{eq:vlasov} by
$1$ and $\bv$, direct integration with $\bv$ and $\bx$ gives us
\begin{gather}
  \label{eq:mass_conservtion}
  \frac{\dd }{\dd t}\int_{\bbR^3\times
    \bbR^3}f(t,\bx,\bv)\dd \bx \dd \bv = 0,
  \quad  t
  \in \bbR^+, \\
\label{eq:momentun_conservation}
\frac{\dd }{\dd t}\int_{\bbR^3\times \bbR^3}\bv f(t,\bx,\bv)\dd \bx
\dd \bv = - \int_{\bbR^3} \rho(t,\bx) \pd{V(t,\bx)}{\bx} \dd \bx, \quad t \in
\bbR^+.
\end{gather}
Multiplying the equation \eqref{eq:vlasov} by $|\bv|^2$ and
integrating by parts, we get the conservation of the total energy for
the system \eqref{eq:vlasov} and \eqref{eq:ThetaVDiff}:
\begin{equation}
  \label{eq:energy_conversvation}
  \frac{\dd }{\dd t}\int_{\bbR^3\times \bbR^3}f(t,\bx,\bv)|\bv|^2\dd
  \bx\dd \bv
= -2\int_{\bbR^3}\rho(t,\bx) \bu(t,\bx)\cdot\nabla_{\bx}
  V(t, \bx)\dd \bx, \quad  t \in \bbR^+ .
\end{equation}

redpajama_set_name: RedPajamaArXiv


text:
The life of Swathi Thirunal was an offering in music to God. A rare renaissance personality, Swathi Thirunal was a king among musicians and musician among kings. His musical compositions and his life were a mutual translation of his devotion to LordPadmanabha, the presiding deity of the Travancore Royal House....
More>>>>
Swathi Thirunal Rama Varma
was a progressive ruler who lived ahead of his times and was committed to the welfare of the people. Being an able administrator his far sighted reforms were responsible for introducing modern medicine, English education and a humane judicial system in Kerala....
Composers and Musicians
M.S. Subbalakshmi
Bharatha Rathna M.S. Subbalakshmi, popular called MS, was born in the temple town of Madurai on September 16th, 1916. Her Grandmother Akkammal was a violinist. Her father, a lawyer by profession, was a music lover. Her mother was a veena exponent.
Her first guru Madurai Srinivasa Iyengar passed away rather too soon. But she kept practicing on her own and having a musician mother helped a lot. Her first recording was at the age of ten, when she recorded a couple of songs for HMV in Madras.
She started giving concerts at a very young age, first accompanying her mother and then as a solo vocalist. In 1940 she married Sadasivam, a well known figure in the Madras Congress circle, and a protege of Rajaji. They had met four years earlier and with his wide connections in the journalistic and political world, he became instrumental in the continued success of her already flourishing career. She started acting in Films too, in 1938. Her movies were quite successful and her final movie "Meera" released both in Hindi and Tamil was a mega hit. After that she quit movies to concentrate solely on music. Her rendering of Meera Bhajans are very popular even today.The money from movies went into the magazine Kalki.
Semmangudi, Musiri, Brinda, composer Papanasam Sivan, 'bhajan' singer Sidheswari Devi of Banares taught MS new compositions and styles of singing. Kalki magazine played a big role in projecting M.S. as a saintly musician,that has endured to this day.
She has got every award a musician in India can get ... from "Isai Vani" in 1940 to Bharath Rathna in 1998. At 80, one finds her still learning, rehearsing new pieces with note books balanced on Sruthi box. That perhaps explains her phenomenal success. Pandit Jawaharlal Nehru, the first Prime Minister of India commented after listening to her singing "what am I ? A mere Prime Minister". He was a great fan of MS and called her the nightingale of India.
MS is considered as the face and voice of the Carnatic tradition. She is perhaps the most loved respected woman in India. Many admire her for her grace, beauty and dedication to her chosen field. India wakes up to a new day listening to her rendering of the Venkitesha Suprabhatam for the last five decades or so. Lalgudi is also a good vocalist.The most significant contribution of Lalgudi to Carnatic Music is his Tillanas and Varnams.

redpajama_set_name: RedPajamaCommonCrawl


Macau's Chief Executive, Chui Sai On, is due to lead a delegation from Macau to Hong Kong on Thursday to attend the launch of the General Development Plan for the Guangdong-Hong Kong-Macau Greater Bay.
The Vice-Chairman of China's National Development and Reform Commission (NDRC) Lin Nianxiu, Guangdong Province Governor Ma Xingrui and Hong Kong Chief Executive Carrie Lam Cheng-Yuet-ngor are also due to attend the meeting.
Guo Lanfeng, managing director of the NDRC Regional Department of Economics will announce the details of the plan that have yet to be disclosed.
During the meeting, Macau Chief Executive Chui Sai On will speak on the measures that the special administrative region has already taken in the framework of the Development Plan for the Guangdong-Hong Kong-Macau Greater Bay.
In his speech he will address the holding of the first edition of the Youth Forum on the Greater Bay, the preparation of an annex to the Five-Year Development Plan of the MSAR, the content of which was focused on adapting to the development of the Greater Bay and the creation in November 2018 of the Commission for the construction of the Guangdong-Hong Kong-Macau Greater Bay which will coordinate the general plans and short, medium and long-term preparatory work for Macau's participation in the Greater Bay area.
The Macau delegation includes representatives of Macau, including members of the Government Executive Council and members of the National People's Congress and the Chinese People's Political Consultative Conference.
The Secretary for Economy and Finance, Leong Vai Tac and the Director of the Policy and Regional Development Study Services, Mi Jian are part of the delegation.


Decide when to spill the beans: Couples are generally advised to wait 12 weeks to announce a pregnancy. We kept our little secret as long as possible but couldn't help sharing the news with our families before this milestone. Announcing a potential switch in ERP systems can send waves of panic throughout your organization. Prioritize those who will be involved in the decision-making process early on. Also consider how you'll make the news public to the rest of your employees-- it is important to develop a communications plan to address the change formally, through emails, all-hands meetings, or one-on-ones.
Create a short list: Having attended my fair share of baby showers, I was aware of how many gadgets exist to clothe, feed, and comfort a child. Building a registry has been overwhelming but with the help of my mom friends, I have been able to focus on a few must have items. Similarly, there are a variety of modern cloud ERPs in the marketplace that may potentially meet your business needs. When developing a technology strategy, educate yourself on the benefits and risks associated with a range of applications including on-premise, hosted and cloud. By hiring the right consultants, you can quickly arrive at a short list of vendors that are worth evaluating.
Document your journey: My friends and family can hardly see a difference in my physical appearance to date but my husband and I are amazed at just how much our child has grown. I have been keeping a journal to track my pregnancy, where I store photos of my baby bump and keep advices I've received from friends and family handy. Make time to document your current and future state processes as part of your ERP selection and implementation. Process flows which define end-to-end scenarios, users, and data inputs and outputs will enable you to better articulate how procedures, responsibilities, and systems are going to change.
Define your goals: Soon after our wedding, my husband and I wrote down our personal and professional goals for the year. This has allowed us to direct our time and resources toward starting a family and purchasing a new home. Provide ERP vendors a list of goals as part of your RFP. Whether it be increasing revenues, reducing manual processes or satisfying new regulatory requirements, using goal-setting language will enable vendors to focus their responses on the strengths of their solution and how it can support your strategic objectives.
Understand the costs: While we may not have a real sense of what the cost of raising a child is, we have done our best to estimate future expenses. There are one-time costs that we are anticipating like a stroller and bedroom furniture as well as recurring ones such as food, daycare and clothing. Ask for a high-level estimate for potential vendors early to eliminate ERPs that are outside of your budget. As you get closer to selecting a solution, request that the vendors recast their initial quotes to account for more informed implementation costs and their best pricing for annual subscription fees.
The path to starting a family or selecting cloud-based ERPs is paved with both excitement and uncertainty. When companies take time to plan for and carefully consider the changes in store, they are better set up to successfully adopt a new system into their lives.
As a leader of the Technology Strategy & Selection service offering, Nicole executes client initiatives to define enterprise system strategies and evaluate alternatives to existing applications. She leverages over 10 years of experience in accounting operations and business analysis to enable organizations to meet their technology objectives.


text:
English » » Shared for en » Gideon the fifth judge of Israel – Judges 6:1-8:35
Gideon the fifth judge of Israel – Judges 6:1-8:35
^ BACK ^
6:1 Again the Israelites did evil in the eyes of the LORD, and for seven years he gave them into the hands of the Midianites. 6:2 Because the power of Midian was so oppressive, the Israelites prepared shelters for themselves in mountain clefts, caves and strongholds. 6:3 Whenever the Israelites planted their crops, the Midianites, Amalekites and other eastern peoples invaded the country. 6:4 They camped on the land and ruined the crops all the way to Gaza and did not spare a living thing for Israel, neither sheep nor cattle nor donkeys. 6:5 They came up with their livestock and their tents like swarms of locusts. It was impossible to count the men and their camels; they invaded the land to ravage it. 6:6 Midian so impoverished the Israelites that they cried out to the LORD for help.
6:7 When the Israelites cried to the LORD because of Midian, 6:8 he sent them a prophet, who said, "This is what the LORD, the God of Israel, says: I brought you up out of Egypt, out of the land of slavery. 6:9 I snatched you from the power of Egypt and from the hand of all your oppressors. I drove them from before you and gave you their land. 6:10 I said to you, 'I am the LORD your God; do not worship the gods of the Amorites, in whose land you live.' But you have not listened to me."
11 After this, the angel of the LORD arrived and sat down in the shade of the oak tree in Ophrah that belonged to Joash, a descendant of Abiezer, while his son Gideon was threshing wheat in a wine press in order to safeguard it from the Midianites. 12 The angel of the LORD appeared to him and told him, "The LORD is with you, you valiant warrior!"
13 But Gideon replied, "Right… Sir, if the LORD is with us, then why has all of this happened to us? And where are all of his miraculous works that our ancestors recounted to us when they said, 'The LORD brought us up from Egypt, didn't he?' But now the LORD has abandoned us and given us over to Midian!"
14 The LORD looked straight at him and replied, "Go with this determination of yours and deliver Israel from Midian's domination. I've directed you, haven't I?"
15 "Right…," Gideon responded. "Sir, how will I deliver Israel? Look—my family is the weakest in Manasseh, and I'm the youngest in my father's household."
16 The LORD told him, "Because I'll be with you, and you'll defeat Midian—every single one of them!"
17 So Gideon asked him, "Please, if I have received favor from you, then do a miracle for me that shows that you're making this promise to me. 18 And please don't leave here until I've come back to you, brought my offering, and set it down in front of you."
And the LORD said, "I will wait until you return."
6:19 Gideon went in, prepared a young goat, and from an ephah of flour he made bread without yeast. Putting the meat in a basket and its broth in a pot, he brought them out and offered them to him under the oak.
6:20 The angel of God said to him, "Take the meat and the unleavened bread, place them on this rock, and pour out the broth." And Gideon did so. 6:21 With the tip of the staff that was in his hand, the angel of the LORD touched the meat and the unleavened bread. Fire flared from the rock, consuming the meat and the bread. And the angel of the LORD disappeared. 6:22 When Gideon realized that it was the angel of the LORD, he exclaimed, "Ah, Sovereign LORD! I have seen the angel of the LORD face to face!"
6:23 But the LORD said to him, "Peace! Do not be afraid. You are not going to die."
6:24 So Gideon built an altar to the LORD there and called it The LORD is Peace. To this day it stands in Ophrah of the Abiezrites.
6:25 That same night the LORD said to him, "Take the second bull from your father's herd, the one seven years old. Tear down your father's altar to Baal and cut down the Asherah pole beside it. 8:26 Then build a proper kind of altar to the LORD your God on the top of this height. Using the wood of the Asherah pole that you cut down, offer the second bull as a burnt offering."
6:27 So Gideon took ten of his servants and did as the LORD told him. But because he was afraid of his family and the men of the town, he did it at night rather than in the daytime.
6:28 In the morning when the men of the town got up, there was Baal's altar, demolished, with the Asherah pole beside it cut down and the second bull sacrificed on the newly built altar!
6:29 They asked each other, "Who did this?"
When they carefully investigated, they were told, "Gideon son of Joash did it."
6:30 The men of the town demanded of Joash, "Bring out your son. He must die, because he has broken down Baal's altar and cut down the Asherah pole beside it."
6:31 But Joash replied to the hostile crowd around him, "Are you going to plead Baal's cause? Are you trying to save him? Whoever fights for him shall be put to death by morning! If Baal really is a god, he can defend himself when someone breaks down his altar." 6:32 So that day they called Gideon "Jerub—Baal," saying, "Let Baal contend with him," because he broke down Baal's altar.
6:33 Now all the Midianites, Amalekites and other eastern peoples joined forces and crossed over the Jordan and camped in the Valley of Jezreel. 6:34 Then the Spirit of the LORD came upon Gideon, and he blew a trumpet, summoning the Abiezrites to follow him. 6:35 He sent messengers throughout Manasseh, calling them to arms, and also into Asher, Zebulun and Naphtali, so that they too went up to meet them.
6:36 Gideon said to God, "If you will save Israel by my hand as you have promised— 6:37 look, I will place a wool fleece on the threshing floor. If there is dew only on the fleece and all the ground is dry, then I will know that you will save Israel by my hand, as you said." 6:38 And that is what happened. Gideon rose early the next day; he squeezed the fleece and wrung out the dew—a bowlful of water.
6:39 Then Gideon said to God, "Do not be angry with me. Let me make just one more request. Allow me one more test with the fleece. This time make the fleece dry and the ground covered with dew." 6:40 That night God did so. Only the fleece was dry; all the ground was covered with dew.
Gideon Defeats the Midianites
7:1 Early in the morning, Jerub—Baal (that is, Gideon) and all his men camped at the spring of Harod. The camp of Midian was north of them in the valley near the hill of Moreh. 7:2 The LORD said to Gideon, "You have too many men for me to deliver Midian into their hands. In order that Israel may not boast against me that her own strength has saved her, 7:3 announce now to the people, 'Anyone who trembles with fear may turn back and leave Mount Gilead.'" So twenty—two thousand men left, while ten thousand remained.
7:4 But the LORD said to Gideon, "There are still too many men. Take them down to the water, and I will sift them for you there. If I say, 'This one shall go with you,' he shall go; but if I say, 'This one shall not go with you,' he shall not go."
7:5 So Gideon took the men down to the water. There the LORD told him, "Separate those who lap the water with their tongues like a dog from those who kneel down to drink." 7:6 Three hundred men lapped with their hands to their mouths. All the rest got down on their knees to drink.
7:7 The LORD said to Gideon, "With the three hundred men that lapped I will save you and give the Midianites into your hands. Let all the other men go, each to his own place." 7:8 So Gideon sent the rest of the Israelites to their tents but kept the three hundred, who took over the provisions and trumpets of the others.
Now the camp of Midian lay below him in the valley. 7:9 During that night the LORD said to Gideon, "Get up, go down against the camp, because I am going to give it into your hands. 7:10 If you are afraid to attack, go down to the camp with your servant Purah 7:11 and listen to what they are saying. Afterward, you will be encouraged to attack the camp." So he and Purah his servant went down to the outposts of the camp. 7:12 The Midianites, the Amalekites and all the other eastern peoples had settled in the valley, thick as locusts. Their camels could no more be counted than the sand on the seashore.
7:13 Gideon arrived just as a man was telling a friend his dream. "I had a dream," he was saying. "A round loaf of barley bread came tumbling into the Midianite camp. It struck the tent with such force that the tent overturned and collapsed."
7:14 His friend responded, "This can be nothing other than the sword of Gideon son of Joash, the Israelite. God has given the Midianites and the whole camp into his hands."
7:15 When Gideon heard the dream and its interpretation, he worshiped God. He returned to the camp of Israel and called out, "Get up! The LORD has given the Midianite camp into your hands." 7:16 Dividing the three hundred men into three companies, he placed trumpets and empty jars in the hands of all of them, with torches inside.
7:17 "Watch me," he told them. "Follow my lead. When I get to the edge of the camp, do exactly as I do. 7:18 When I and all who are with me blow our trumpets, then from all around the camp blow yours and shout, ' For the LORD and for Gideon.'"
7:19 Gideon and the hundred men with him reached the edge of the camp at the beginning of the middle watch, just after they had changed the guard. They blew their trumpets and broke the jars that were in their hands. 7:20 The three companies blew the trumpets and smashed the jars. Grasping the torches in their left hands and holding in their right hands the trumpets they were to blow, they shouted, "A sword for the LORD and for Gideon!" 7:21 While each man held his position around the camp, all the Midianites ran, crying out as they fled.
7:22 When the three hundred trumpets sounded, the LORD caused the men throughout the camp to turn on each other with their swords. The army fled to Beth Shittah toward Zererah as far as the border of Abel Meholah near Tabbath. 7:23 Israelites from Naphtali, Asher and all Manasseh were called out, and they pursued the Midianites. 7:24 Gideon sent messengers throughout the hill country of Ephraim, saying, "Come down against the Midianites and seize the waters of the Jordan ahead of them as far as Beth Barah."
So all the men of Ephraim were called out and they took the waters of the Jordan as far as Beth Barah. 7:25 They also captured two of the Midianite leaders, Oreb and Zeeb. They killed Oreb at the rock of Oreb, and Zeeb at the winepress of Zeeb. They pursued the Midianites and brought the heads of Oreb and Zeeb to Gideon, who was by the Jordan.
Zebah and Zalmunna
8:1 Now the Ephraimites asked Gideon, "Why have you treated us like this? Why didn't you call us when you went to fight Midian?" And they criticized him sharply.
8:2 But he answered them, "What have I accomplished compared to you? Aren't the gleanings of Ephraim's grapes better than the full grape harvest of Abiezer? 8:3 God gave Oreb and Zeeb, the Midianite leaders, into your hands. What was I able to do compared to you?" At this, their resentment against him subsided.
8:4 Gideon and his three hundred men, exhausted yet keeping up the pursuit, came to the Jordan and crossed it. 8:5 He said to the men of Succoth, "Give my troops some bread; they are worn out, and I am still pursuing Zebah and Zalmunna, the kings of Midian."
8:6 But the officials of Succoth said, "Do you already have the hands of Zebah and Zalmunna in your possession? Why should we give bread to your troops?"
8:7 Then Gideon replied, "Just for that, when the LORD has given Zebah and Zalmunna into my hand, I will tear your flesh with desert thorns and briers."
8:8 From there he went up to Peniel and made the same request of them, but they answered as the men of Succoth had. 8:9 So he said to the men of Peniel, "When I return in triumph, I will tear down this tower."
8:10 Now Zebah and Zalmunna were in Karkor with a force of about fifteen thousand men, all that were left of the armies of the eastern peoples; a hundred and twenty thousand swordsmen had fallen. 8:11 Gideon went up by the route of the nomads east of Nobah and Jogbehah and fell upon the unsuspecting army. 8:12 Zebah and Zalmunna, the two kings of Midian, fled, but he pursued them and captured them, routing their entire army.
8:13 Gideon son of Joash then returned from the battle by the Pass of Heres. 8:14 He caught a young man of Succoth and questioned him, and the young man wrote down for him the names of the seventy-seven officials of Succoth, the elders of the town. 8:15 Then Gideon came and said to the men of Succoth, "Here are Zebah and Zalmunna, about whom you taunted me by saying, 'Do you already have the hands of Zebah and Zalmunna in your possession? Why should we give bread to your exhausted men?'" 8:16 He took the elders of the town and taught the men of Succoth a lesson by punishing them with desert thorns and briers. 8:17 He also pulled down the tower of Peniel and killed the men of the town.
8:18 Then he asked Zebah and Zalmunna, "What kind of men did you kill at Tabor?"
"Men like you," they answered, "each one with the bearing of a prince."
8:19 Gideon replied, "Those were my brothers, the sons of my own mother. As surely as the LORD lives, if you had spared their lives, I would not kill you." 8:20 Turning to Jether, his oldest son, he said, "Kill them!" But Jether did not draw his sword, because he was only a boy and was afraid.
8:21 Zebah and Zalmunna said, "Come, do it yourself. 'As is the man, so is his strength.'" So Gideon stepped forward and killed them, and took the ornaments off their camels' necks.
Gideon's Ephod
8:22 The Israelites said to Gideon, "Rule over us—you, your son and your grandson—because you have saved us out of the hand of Midian."
8:23 But Gideon told them, "I will not rule over you, nor will my son rule over you. The LORD will rule over you." 8:24 And he said, "I do have one request, that each of you give me an earring from your share of the plunder." (It was the custom of the Ishmaelites to wear gold earrings.)
8:25 They answered, "We'll be glad to give them." So they spread out a garment, and each man threw a ring from his plunder onto it. 8:26 The weight of the gold rings he asked for came to seventeen hundred shekels, not counting the ornaments, the pendants and the purple garments worn by the kings of Midian or the chains that were on their camels' necks. 8:27 Gideon made the gold into an ephod, which he placed in Ophrah, his town. All Israel prostituted themselves by worshiping it there, and it became a snare to Gideon and his family.
Gideon's Death
8:28 Thus Midian was subdued before the Israelites and did not raise its head again. During Gideon's lifetime, the land enjoyed peace forty years.
8:29 Jerub—Baal son of Joash went back home to live. 8:30 He had seventy sons of his own, for he had many wives. 8:31 His concubine, who lived in Shechem, also bore him a son, whom he named Abimelech. 8:32 Gideon son of Joash died at a good old age and was buried in the tomb of his father Joash in Ophrah of the Abiezrites.
8:29 No sooner had Gideon died than the Israelites again prostituted themselves to the Baals. They set up Baal—Berith as their god and 8:34 did not remember the LORD their God, who had rescued them from the hands of all their enemies on every side. 8:35 They also failed to show kindness to the family of Jerub—Baal (that is, Gideon) for all the good things he had done for them.
The 7 days of Creation, Genesis 2:4-3:24
Day One – Night and Day
Day Two – sky and sea
Day Three – Land and Vegetation
Day Four – Stars, Sun and Moon
Day Five – Sea creatures including fish and Birds
Day Six – Animals and Mankind
Day Seven – Rest
Ark, Genesis 8:1- 10:20
Hagar and Ishmael – Genesis 16:11–17:27
Rebekah the wife of Isaac the son of Abraham, Genesis 24:1-67
Laban Meets Jacob, Genesis 29:1-30:43
Laban pursues Jacob, Genesis 31:1-55
Esau Meets Jacob, Genesis 32:1-33:20 and Genesis 36:1-43
Jacob Moves to Egypt in a cart, Genesis 46:8 – 47:12
2nd 40 Years, Exodus 2:26-7:6
Moses – 3rd set of 40 years, the book of Exodus 7:8 – 40:38
Moses – 10 Plagues, Exodus 7:14-12:36
Passsover
Feast of Tabernacle – Succouth or Sykkot
Moses – Yom Kippur, means "Day of Atonement"
Moses – 37 years, Numbers 14:1-36:13
Balaam, Numbers 22:1-24:25
Three Sermons, the whole book of Deuteronomy 1:1-34:12
Deborah the 4th Judge of Israel sends for Barak her army commander, Judges 4:9 to 5:31
Eli the Priest and the ninth Judge of Israel, 1 Samuel 1:1-4:22
Ruth the Moabitess, Ruth 1:16-4:22
Jephthah the Gileadite the tenth Judge of Israel, Judges 11:16-40 & 12:1-7
Samson the fourteenth Judge of Israel, Judges 14:1-16:31
House hold Idols, Judges 18:1-31
David's soldiers, 1 Samuel 21:1-30:31
King Rehoboam the first King of Judah, 1 Kings 11:41-12:24
The 20 Kings of Judah who ruled between – 931BC-586BC = 345 years
King Jeroboam, 1 Kings 11:34-14:20
The 19 Kings of Israel – 931BC-722BC = 209 years
King Ahab 7th King of Israel, 1 Kings 17:1-22:40
Elijah fire, 1 Kings 18:20-46
Hezekiah King of Judah – Continued from 2 Kings 18:1-12
Down Well, Jeremiah 38:7-40:6
Nehemiah Inspects Jerusalem's Walls, Nehemiah 2:11-3:32
The Gospel of Luke
The Gospel of Matthew
Dates that Jesus might have been born
Herodian Dynasty
The Sermon in a level place, Luke 6:17-49
Lazarus raised from the dead, John 11:1-45
Possible Dates of Jesus' Death
Stephen Killed
OUTLINE HISTORY OF THE APOSTLES


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how get gold ore crushing . how get gold ore crushing. Gold mining in Alaska ... There is often some very small sized gold there and it can be a lot of work to get all the gold out.
Gold ore, when discovered in nature, is usually found combined with stones and other metals, such as silver and lead. This natural state does not display gold's aesthetic virtues, nor does it easily permit workers in gold to fashion jewelry.
The best way to begin prospecting is to get a book that describes the gold mines and placers and visit these as I have found there are always many deposits near old gold mines that have been overlooked.
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get gold out of ore . Gold Minecraft Wiki Guide IGN. Mar 22, 2013Gold Minecraft This page covers all Gold blocks, including Gold Ore and Gold Ingot. Gold is a rare block that is commonly found buried in the bottom.


text:
By a.ross
What is a Negro Spiritual and where did they come from?
Negro spirituals are the religious songs and music that were sung and created during slavery. The songs were mainly strutted in a call-and-response structure and they represented Christian Ideals and values. Many of the enslaved people were converted to Christianity when they were brought over, known as the Great Awakening, and that is why a large portion of negro spirituals are religious. When these spirituals were originally created, there was no real structure to follow and they just improvised as they sang. These songs not only represented Christianity, but also the little bit of freedom that the enslaved did have, they were able to be free through song and dance that also helped them stay connected to their roots and culture.
Where were Negro Spirituals heard?
Negro spirituals were used as a from of expression and communication between those brought over from Africa. So, naturally the only place to really hear spirituals were on a plantation. However, because spirituals were religious it was very common to also hear them in church services held by the enslsved. During this time since the gathering of black people was illegal, these religious gatherings were very risky. Black churches began to thrive where they were able to express themselves in whatever they wanted. They became known as invisible church where slaves would worship in secret. Many of these songs had double meanings to them and that is how they communicated without the knowledge of any white person.
These singing of these songs was not the only element to Negro Spirituals. Performing practices began to develop and soon became a normal part of negro spirituals. Ring shouts are a form of folk spiritual characterized by leader-chorus antiphonal singing, hand clapping, and other percussion, which incorporated stylized religious dance as participants move in a counterclockwise circle (pg. 94). Ring shouts were a form of worship and helped influence the genre of music that we now call Folk Spiritual.
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PubMed Central is a free digital archive of full-text articles from important life sciences journals. Growing exponentially each month, PMC has become a true digital counterpart to NLM's extensive collection of print journals.
From the cellular level up, the Genetics Home Reference site provides information on genetic conditions in straightforward, easy-to-understand language.
The Oncolink site provides information on a wide range of cancer topics. It includes recent news about cancer, coping tips, clinical trial matching and referral service and links to other resources.
WebMD offers searchable information on diseases and conditions, genetics, drugs and herbs, pregnancy and healthy living.
The mission of MayoClinic is to help people find answers and the support they need to manage illness and stay healthy. Articles on illness and diseases, drugs, first aid and more.
HON provides useful and reliable health information. It is a non-government site of medical resources.


text:
LDF conquers 2 sitting seat of UDF; Aroor moves to photo finish
Oct 24, 2019, 01:10 PM IST
CPM workers celebrating victory in front of AKG centre in Thiruvananthapuram. Photo: Biju Varghese
Thiruvananthapuram: The counting of votes in the by-election to five assembly constituencies is entering the final phase. According to the latest reports, LDF conquered the sitting seats of UDF in Vattiyoorkavu and Konni whereas UDF retained its seats in Ernakulam and Manjeshwaram.
The counting in Aroor constituency, the sitting seat of LDF, is likely to end in a photo finish as UDF candidate Shanimol Usman is leading by a slight margin. At present, Shanimol is leading by 2029 votes when the last round of counting process in underway.
UDF candidate T. J. Vinod managed to retain the seat in Ernakulam despite unfavourable political atmosphere. His victory has been declared with a margin of 3673 votes. Hibi Eden of UDF had won the seat for a margin of 21,949 votes in the previous election. Compared to the two previous elections, there is a considerable decline in the votes earned by the UDF.
LDF candidate in Vattiyoorkavu V K Prasanth has won by more than 14251 votes. Prasanth was leading right from the initial stage itself over UDF candidate K Mohan Kumar and NDA candidate S Suresh. An LDF victory in Vattiyoorkavu could turn to be a major boost for Left cadres in the constituency and a heavy blow to the UDF faction as it was UDF's sitting seat.
K. U. Janeesh Kumar of the LDF has secured victory in the Konni bypolls by a margin of 9953 votes. Janeesh Kumar's victory marks the grand comeback of LDF to the constituency after a gap of 23 years. He emerged victorious after securing a comfortable lead over UDF's P Mohan Raj and NDA's K Surendran.
In Manjeshwaram, UDF candidate M. C. Kamarudeen won by 7923 votes. BJP candidate Raveesha Tantri Kundar is in the second place and LDF candidate Sankar Rai is trailing behind in the third spot.
Konni debacle: No lapses on district leadership's part, says DCC president
National leadership to decide on BJP chief, have no personal opinion, says Kummanam Rajasekharan
Adoor Prakash blames Pathanamthitta DCC for bypoll debacle in Konni
RSS crossvoting helped LDF victory in Vattiyoorkavu: K Muraleedharan


text:
Q: Can't turn screen on after I turned off once I turn the screen off when the right command appear.
TURN OFF SCREEN:
    WindowManager.LayoutParams params = getWindow().getAttributes();
    params.flags |= LayoutParams.FLAG_KEEP_SCREEN_ON;
    params.screenBrightness = 0.0f;
    getWindow().setAttributes(params);

This is working fine. At least it seems like that.
But when the command comes to turn the screen on, it doesn't work. This is the code for turn on:
TURN ON SCREEN:
    WindowManager.LayoutParams on = getWindow().getAttributes();
    on.screenBrightness = 0.7f;
    getWindow().setAttributes(on);

The brightness doesn't get up. And I don't know why.
But when I change the screenBrightness at TURN OFF SCREEN to params.screenBrightness = 0.1f; then the brightness of the screen is not quite black but its very low. And if it's 0.1f I can call the BrightnessUp function and the brightness of the screen go on right level.
All in all I can't set the brightness higher if it's 0.0f before, only if it's 0.1f or higher.
I also tried this solution to turn of the screen:
    PowerManager pm = (PowerManager) getSystemService(Context.POWER_SERVICE);
    screenWakeLock = pm.newWakeLock(PowerManager.PARTIAL_WAKE_LOCK,   "screenWakeLock");
    screenWakeLock.acquire();

Manifest:
<uses-permission android:name="android.permission.WAKE_LOCK" />
<uses-permission android:name="android.permission.DEVICE_POWER"/>

But this doesn't work anyway.
So I hope there is anybody out there to help me and give me a solution for this problem.

A: Ok I solved my problem.
I just set the screenBrightness in TURN OFF SCREEN params.screenBrightness=0.01f; and then all work fine.
The screen is black if it should turn off and it get the right brightness if it should on again.

redpajama_set_name: RedPajamaStackExchange


VirtualBox is an open source and multi-platform tool, available for Windows, Linux and Mac OS X, as well as other operating systems, which lets you create virtual disk units where you can install guest operating system on your own computer, and it is actually Use as was installed.
Virtual disk where running the system is completely customizable, and it lets you modify the virtual hardware that you need to processor, RAM memory or storage capacity. You need to consider taking part in the actual physical assets of your computer, so you both need a tool powerful enough to run both guest and host operating system.
The program supports almost all versions of the latest operating systems, such as Windows 10, Mac OS X, and the latest updates to Ubuntu or any other Linux distribution. The benefits of creating virtual units are many programs, such as an operating system or in conjunction with others, such as accepting virtual modular units with you and making copies of their security. In fact, an interesting feature of the virtualbox can save sessions on the operating system in the virtual unit, so you can turn it off and return to wherever you are.
The best way to 'test' your PC is to virtualize them. VirtualBox is probably the best option for creating a hard drive to start another operating system without affecting the home machine. But many do not think the useful chances of creating snapshots are to remember the different positions of the same machine that can be run in a few seconds. I have to explain a bit more.
The Android-X86 project has been effective for several months now. The Android operating system is a free and open enterprise to control the quality of smartphones and tablets instead of limiting ARM architecture to computers with AMD and Intel processors. When you use a Linux kernel, it is possible to run and run Android on a desktop computer natively. We are going to virtualize this tutorial using the Android 4.4 KitKat virtual box, though you can extend the actual installation of the operating system on a computer by burning it to a hard disk or running it via USB.


We now believe that the US and China will reach a trade deal, but it is likely to be a shallow one that only temporarily resolves tensions in that relationship. China's government is applying moderate policy stimulus to slow the rate at which economic growth is decelerating, and we expect this to be sufficient to deliver real GDP expansion of 6.3% in 2019. Growth is forecast to weaken to 6.1% in 2020.
How reliable are China�s GDP data?
Overall China's official GDP data remain problematic in several areas, but we judge it to be highly unlikely that they are falsified on a wide scale.
Ties between Italy and China are likely to strengthen, which will help China to position the Belt and Road Initiative more strategically as an attractive option for the rest of Europe.
The law will go into effect on January 1st 2020. However, questions over enforcement will be complicated by the preservation of aggressive industrial policies.
The authorities have made labour market stability the priority for 2019, and outlined an "employment first" plan, elevating it to the same level as monetary and fiscal policy.
Following the stockmarket surge, the China Securities Regulatory Commission called for stricter monitoring of any signs of abnormal trading and unregulated margin financing.
What next for bitcoin in China?
Is China really planning to open its financial sector?
This was the first attack against Chinese interests in the city for many years. It will keep the spotlight on political turbulence in Pakistan, but will not impede joint infrastructure plans.
Apr 18th 2019 – China/North Korea politics: Security ministry agents defected while in China?