text:
\section{Introduction} \label{s:intro}
The color screening of heavy quarks, as it was proposed in \cite{Matsui}, should lead to dissociation of  $J/\psi$ bound state and may indicate a formation of the quark-gluon plasma (QGP) arising in heavy-ion collision experiments \cite{Laine}.
The idea of a screening mass and running coupling constant allow us to understand the interaction of a particles, inserted into a medium, intuitively.
The simplest way is to consider a static quark-antiquark probe in color plasma, wherein all medium effects are taken into account by two body interaction potential $U(r)$.
In medium both the Coulomb and string-like parts of interaction between the heavy quark and anti-quark are modified \cite{Koma,Kaczmarek,Burnier}, and the potential $U(r)$ is generally a complex quantity \cite{Beraudo,Brambilla1,Brambilla2}.
An effective description have to capture the effects of screening and Landau damping, related to the real and imaginary parts of the potential correspondingly.
Several approaches were proposed to treat this problem using e.g. effective theories
\cite{Kajantie,Hart,Philipsen}, spatial correlation functions
or the behavior of the color singlet free energies \cite{Nadkarni1,Nadkarni2,Maezawa,Digal}.
However the perturbative expansion in terms of the QCD coupling constant $g$ fails \cite{Linde}.
Therefore it is necessary to use a non-perturbative approaches to make prediction of QGP properties.
Currently, the lattice QCD simulation is
the only systematic method allowing to extract various properties
of QGP and to study the inter-quark interactions and screening properties of QGP.
However interpretation of these very complicated numerical computations requires
application of various QCD motivated, albeit schematic, models simulating
various aspects of the full theory and allowing for a deeper physical
understanding.
Moreover, such models are needed in cases when the lattice QCD
fails, e.g. at large quark chemical potentials and out of thermodynamic equilibrium.
For temperatures higher than the QCD transition temperature  ( of order $175$~MeV) this issue can be addressed
by adopting the quasiparticle approaches
\cite{Chandra1,Chandra2,Chandra3,Peshier1,Peshier2,Peshier3,Peshier4,Dumitru,Fukushima,Ghosh,Abuki,Tsai,
Ruggieri,DElia1,DElia2,Castorina1,Castorina2,Plumari}.
quasiparticle models have to account such high temperatures for the hot QCD medium effects and to describe the influence
of strong inter-particle interaction.
To consider here the microphysics insights into the inter-quark interactions and screening mechanism, we investigate QGP in framework of quasiparticle QGP model, which has been proved in reproducing various aspects of QGP thermodynamics and kinetic properties \cite{QGP1,QGP2,EbelForFil}.
The approach used in \cite{QGP1,QGP2,EbelForFil} is based on the quasiparticle pattern and is motivated by expectation that the main features of non-Abelian plasmas can be understood in simple semi-classical terms without difficulties inherent to a full quantum field-theoretical analysis.
This also sets the motivation for the investigations, which are using the continuous classical color variable interacting with the chromodynamic field \cite{LM105,Mr,reveiw,shuryak1}.
Quantum Monte Carlo simulations presented in \cite{EbelForFil}
were able to reproduce the lattice equation of state even near the critical temperature and
at nonzero baryon chemical potential.

The quasiparticle model of QGP is seen to be consistent with the lattice simulations  \cite{Boyd1,Boyd2,Panero,Karsch,Cheng1,Cheng2,Bazavov,Borsanyi1,Borsanyi2,Aoki1,Aoki2}.
As a consequence, the very definition and numerical determination of screening mass are obscured by the complications of the non-abelian nature of QCD and the strong coupling.
Nevertheless, let us note that parametrization of the real part of effective potential $U(r)$ can reproduce the lattice data quite well \cite{Dixit}.
In this paper, to identify a running coupling constant and a Debye screening mass of quasiparticles, we are fitting the spatial pair distribution functions with a form, corresponding to a screened Coulomb potential.
The pair distribution functions also allow us to identify the gluon bound states in the form
of glueballs, but only at densities related to the average inter-particle distances more than $r_s\ge 0.4$~fm. ($r_s=\sqrt[3]{3/4\pi n}$, n is the density of all quasiparticles).

For detailed studies of the quark, antiquark and gluon color screenings we have also developed the new spatial color pair correlation function with respect to the inter-particle distance and introduced related new color screening masses as function of temperature.

The quasiparticle model, considered here,  also allows us to investigate the kinetic properties of QGP.
For this purpose the quantum color dynamics in the color phase space has been developed in \cite{EbelForFil}, moreover  diffusion coefficient and shear viscosity of QGP, calculated there, are in a quite good agreement with available data.
In the present paper in the framework of color phase space Wigner approach to quasiparticle model we propose Monte Carlo calculations of the quark and gluon momentum distribution functions for strongly coupled QGP in thermal equilibrium at zero baryon chemical potential.
To do this we rewrite the Wigner function of QGP in the form of color path integrals.
To integrate over color variables, we develop a procedure of sampling the color quasiparticle variables in accordance with the Haar measure of the SU(3) group with the quadratic and cubic Casimir conditions.
The developed approach self-consistently
takes into account Fermi (Bose) statistics of quarks (gluons).

The paper is organized as follows.
Section II deals with assumptions of the quasiparticle description of hot QCD.
In section III and IV we discuss the path integral representation of the matrix elements of the density operator and the Wigner functions for canonical ensemble correspondingly.
Section V is devoted to the brief description  of the Monte Carlo simulations.
Results of simulations of the pair distribution functions are considered in section VI.
In section VII the spatial pair distribution functions, the Debye screening mass and the running coupling constant are discussed.
In sections VIII the new color pair correlation functions are presented and the related color screening mass is discussed.
Section IX deals with the quantum ``tails'' of QGP momentum distribution functions.

\section{Assumptions of the model}
The basic assumptions of the considered quasiparticle model are similar to those in \cite{shuryak1,shuryak2017,EbelForFil}.\\
1) Masses of quasiparticles $m$ are of order or higher than the mean kinetic energy per particle; this assumption is based on the analysis of QCD lattice data \cite{Lattice02,LiaoShuryak,Karsch:2009tp}. \\
2) We consider the model with quarks of three flavors ; for  simplicity, we assume the masses of $u$, $d$ and $s$  quarks to be equal; as for gluon quasi- particles, we allow their masses to differ from that of quarks (heavier). \\
3)The interparticle interaction is dominated by color-electric Coulomb potential; color-magnetic effects are neglected as subleading ones. \\
4) Since the color charges are large, the color operators are replaced by their average values, i.e. by Wong's classical color vectors [eight-dimensional (8D) in SU(3)] with the quadratic and cubic Casimir conditions \cite{Wong}.

Applicability of this approach has been discussed in \cite{LM105,shuryak1} in details.
Our approach differs from \cite{LM105,shuryak1} by a quantum treatment to quasiparticles instead of the classical one.
This model requires the following quantities as functions of temperature $T$ and quark chemical potential $\mu_q$: \\
1) quasiparticle masses $m_q$ for quarks and $m_g$ for gluons; \\
2) the coupling constant $g^2$, or $\alpha_s = g^2/4\pi$. \\
It would be ideal if the input quantities are deduced from lattice QCD data or from other appropriate model.
However, this task is still quite ambiguous presently, so in the present simulations we take
only a possible set of parameters ($m_g$ and $m_q$ ) given by the HTL perturbative approach \cite{Lebel96}:
\begin{widetext}
	\begin{eqnarray}
		\label{m_g-qg-T}
		m_g^2 (\{\mu_{q}\},T) =
		\frac{1}{12}\left((2N_c+N_f)T^2 +
		\frac{3}{\pi^2}\sum_{q=u,d,s}
		\mu_{q}^2\right)g^2(\{\mu_{q}\},T \gg T_C) \,
		\\
		\label{m_q-qg-T}
		m_{q}^2(\{\mu_{q}\},T) =
		\frac{N_g}{16N_c}\left(T^2+\frac{\mu_{q}^2}{\pi^2}\right)g^2(\{\mu_{q}\},T \gg T_C),
	\end{eqnarray}
\end{widetext}
where $N_f$ is the number of quark flavors which can be excited, $N_c = 3$ for SU(3) group, and $g^2$ is square of 		the QCD running coupling constant, generally depending on $T$ and all $\mu_q$.
All masses depend on combinations of
$ z_{g}=\left(T^2 +
\frac{3}{\pi^2 (2N_c+N_f)}\sum_{q=u,d,s}
\mu_{q}^2\right)^{1/2} $
and $ z_{q}=\left(T^2+\frac{\mu_{q}^2}{\pi^2}\right)^{1/2} $
rather than on two independent variables $T$ and $\mu_{q}$.
It is also reasonable to assume that $g^2$ is a function of this single variable $z_{g}$ , because $g^2$ is related to the whole system rather than one specific quark flavor.
Then we can use ``one-loop analytic coupling constant'' \cite{Shirkov1,Shirkov2,Prosperi}:
\begin{eqnarray}
	\alpha_s (Q^2) = \frac{4\pi}{11-(2/3)N_f} \left[
	\frac{1}{\ln (Q^2/\Lambda_{\rm QCD}^2)} + \frac{\Lambda_{\rm QCD}^2}{\Lambda_{\rm QCD}^2-Q^2}\right],
	\nonumber

\end{eqnarray}
where $Q$ is the momentum transfer, $ \Lambda_{\rm QCD} = 206$ MeV is the QCD scale, $N_f = 3$ is the number of flavors and $Q$ is replaced by $2\pi z_g$.

We consider a multi-component QGP consisting of $\tilde{N}$ color quasiparticles: $N_g$ gluons, $N_q$ quarks
and $N_{\overline q}$ antiquarks.
The Hamiltonian of this system is $\hat{H}=\hat{K}+\hat{U}^C$ with the kinetic and color Coulomb interaction parts:
\begin{eqnarray}
	\label{Coulomb}
&&		\hat{K}=\sum_i \sqrt{\hat{\bf p}^2_i+m^2_i(T,\mu_i)},
\nonumber\\&&
	\hat{U}^C=\frac{1}{2}\sum_{i\neq j}

	\frac{g^2(T,\mu_i)
	(Q_i \cdot Q_j)}
	{4\pi| {\bf x}_i-{\bf x}_j|}.
\end{eqnarray}
Here $i$ and $j$ run over all quark and gluon quasiparticles,
$\mu_i$ are their chemical potentials,
$i,j=1,\ldots,\tilde{N}$, $\tilde{N}=N_q+N_{\overline q }+N_g$,
$N_q=N_u+N_d+N_s$ and $N_{\overline q }=N_{\underline u}+N_{\underline d}+N_{\underline s}$ are total numbers of quarks and antiquarks of all flavors ($u$, $d$, $s$),
3D vectors ${\bf x}_i$ are quasiparticle dimensionless spatial coordinates,  $g^2(T,\mu_i)/4\pi$ is  coupling constant,
the $Q_i$ denotes the Wong's quasiparticle color variable (8D-vector in the group $SU(3)$),
$(Q_i \cdot Q_j)$ denotes the scalar product of color vectors.
Non-relativistic approximation for potential energy is used, while for kinetic energy we still keep the relativistic form,  since the temperature is not negligible in comparison with quasiparticle masses.
The equation of eigenvalues of this Hamiltonian is usually called  "spinless Salpeter equation".
The grand canonical ensemble with given temperature, net-quark-number ($\mu_q$ ), strange ($\mu_s$ ) chemical potentials, and fixed volume $V$ is completely described by grand partition function:

	%

	%
\begin{widetext}
	\begin{eqnarray}
		Z\left(\mu_q,\mu_s,\beta,V\right)=

		\sum_{\{N\}}
		\frac{\exp\{\mu_q(N_q-N_{\overline q })/T\}\;\exp\{\mu_s(N_s-N_{\underline s})/T\}}%
		{N_u!\;N_d!\;N_s!\;N_{\overline q }!\;N_{\underline d }!\;N_{\underline s }!\;N_g!}
		Z\left(\{N\},V,\beta\right),
		\label{Gq-def}
	\end{eqnarray}
\end{widetext}
where $\{N\}=\{N_u,N_d,N_s,N_{\underline u },N_{\underline d }N_{\underline s } ,N_g\}$.
In equation (\ref{Gq-def}) we explicitly wrote sum over different quark flavors (u,d,s).
The sum over quark degrees of freedom is understood in the same way below.
Usual choice of the strange chemical potential is $\mu_s=-\mu_q$ (nonstrange matter),
such that the total factor in front of $(N_s-N_{\underline s })$ is zero,	$\beta=1/T$ is the reciprocal temperature.
Therefore, we omit $\mu_s$ from the list of variables below.

The partition function in canonical ensemble $Z\left(\{N\},V,\beta\right)$ and related thermodynamic properties of many particle system are defined by  diagonal matrix elements of the density operator ${\hat \rho} = \exp (-\beta{\hat H})$:
\begin{widetext}
	\begin{eqnarray}
		Z\left(\{N\},V,\beta\right) &=&
		\sum_{\sigma,\acute{\sigma}} \int \rm  dx\; \rm d\acute{x}\; \rm d\mu Q \; \rm d\mu \acute{Q} \;

		\delta_{\sigma,\acute{\sigma}}\delta(x-\acute{x})\delta(Q-\acute{Q})
		\langle x,Q,\sigma|e^{-\beta\hat H(Q)}|\acute{x},\acute{Q},\acute{\sigma}\rangle \,
		\nonumber\\
		&=&\sum_{\sigma} \int	\rm  dx\; \rm d\mu Q \;\rho(x,Q, \sigma),
		\label{Z-def}
	\end{eqnarray}
\end{widetext}
where $x$, $\sigma$, $Q$ denote the multi-dimensional vectors, related to spatial, spin and color degrees of freedom of $N$ quasiparticles with related flavor indexes respectively.
The summation over $\sigma$, spatial ($\rm dx \equiv \rm d^{3} x_1 \dots \rm d^{3} x_N $)
and color ($\rm d\mu Q\equiv \rm d\mu Q_1 \dots \rm d\mu Q_N $) integrations
run over all individual degrees of freedom of the quasiparticles,
while $\rm d\mu Q_i$ denotes integration over SU(3) group Haar measure \cite{LM105,QGP1}.

\section{Path integral representation of the density matrix}
The exact matrix elements of density operator $\rho=\rm e^{-\beta {\hat H}}$ of interacting quantum system can be constructed using a path integral approach \cite{Feynm,zamalin}, based on operator identity
$\rm e^{-\beta \hat{H}}=\rm e^{-\varepsilon {\hat H}} \cdot \rm e^{-\varepsilon {\hat H}}\dots \rm e^{-\varepsilon {\hat H}}$,
where the r.h.s. contains $M$ identical factors with $\varepsilon = \beta/M$, allowing us to rewrite the integral in equation (\ref{Z-def}) as follows:
\begin{widetext}
	\begin{eqnarray}
	&&\sum_{\sigma } \int\limits \rm  dx \rm d\mu Q\,
		\rho(x,{\rm Q},\sigma) =
		\sum_{\sigma} \int\limits \rm  dx \rm d\mu Q \int\limits \rm  dx^{(1)}\rm d\mu Q^{(1)}\dots
		\rm dx^{(M-1)}\rm d\mu Q^{(M-1)} \, \rho^{(1)}\cdot\rho^{(2)} \, \dots \rho^{(M-1)}
		\nonumber\\&&
		\times \sum_{{\rm P}_q} \sum_{{\rm P}_{ \overline{q}}}\sum_{{\rm P}_g}(-1)^{\kappa_{{\rm P}_q}+ \kappa_{{\rm P}_{\overline{q}}}}
		\sum_{\sigma'} {\cal S}(\sigma, {\rm P}_{q\overline{q}g} \sigma^\prime)\delta_{\sigma',\sigma} \,
		\nonumber\\&&
		\times \int\limits {\rm d}x^{(M)} {\rm d}\mu {\rm Q}^{(M)} \delta(x-{\rm P}_{q\overline{q} g}x^{(M)})
		\delta({\rm Q}-{\rm P}_{q\overline{q}g}{\rm Q}^{(M)})  \rho^{(M)},

		\label{Grho-pimc0}
	\end{eqnarray}
\end{widetext}
where $x \equiv x^{(0)}$, ${\rm Q} \equiv {\rm Q}^{(0)}$, spin gives rise to the spin part of the density matrix
(${\cal S}$) with exchange effects accounted for by the permutation
operators ${\rm P}_q$, ${\rm P}_{ \overline{q}}$ and ${\rm P}_g$ acting on the quasiparticle indexes of quarks,
antiquarks and gluons in $x^{(M)}$, ${\rm Q}^{(M)}$ and the spin projections $\sigma'$,  $ {\rm P}_{q\overline{q}g}={{\rm P}_q}{{\rm P}_{ \overline{q}}}{{\rm P}_g}$.

The sum runs over all permutations with parity $\kappa_{P_q}$ and $\kappa_{P_{\overline{q}}}$, while
\begin{eqnarray}
&&\rho^{(m)} \equiv
		\rho\left(x^{(m-1)},{\rm Q}^{(m-1)};x^{(m)},{\rm Q}^{(m)};\{N\};\varepsilon \right)
		\nonumber\\&&
=\langle x^{(m-1)}|e^{-\varepsilon {\hat H}}|x^{(m)}\rangle\delta({\rm Q}^{(m-1)}-{\rm Q}^{(m)})

		\label{rho(m)}
\end{eqnarray}
is the off-diagonal element of the density matrix.
Since the color charge is treated classically, we keep only diagonal terms
($\delta({\rm Q}^{(m-1)}-{\rm Q}^{(m)})$) in color degrees of freedom.
Here each quasiparticle is presented by … set of coordinates 	$\{x_i^{(0)}, \dots, x_i^{(M-1)}\}$, called ``beads'', in units of $\lambda_a=\sqrt{2\pi\varepsilon /m_a}$, $a=q,\overline{q},g$, $\hbar=k_B=c=1$)
and a 8-dimensional color vector $Q_i^{(0)}$ in the $SU(3)$ group.
Thus, all ``beads'' of each quasiparticle are characterized by the same spin projection, flavor and color charge.
Notice that masses and coupling constant $g^2(T,\mu_i)/4\pi$ in each $\rho^{(m)}$ are the same as those
for the original quasiparticles, i.e. these are still defined by the actual temperature $T$.
Details of analytical calculation of matrix elements $\rho^{(m)}$ are presented in \cite{EbelForFil,QGP1,QGP2}.

The main advantage of this approach is that it allows us to use perturbation theory to obtain approximation for density matrices $\rho^{(m)}$, which is applicable due to smallness of artificially introduced factor $1/(M)$.
Each factor $\rho^{(m)}$ should be calculated with the accuracy of order of $1/M^{\theta}$ with $\theta > 1 $, because in this case the error of the whole product in the limit $M \to \infty$ tends to zero.

\section{The Wigner function for canonical ensemble}
Now we are going to obtain a new path integral representation of Wigner functions in the color phase space, which allows us numerical simulations of strongly coupled quantum systems of particles in canonical ensemble \cite{Feynm,LarkinFilinovCPP,JAMP,Wiener,NormanZamalin,zamalin}.
The Wigner function of many-particle system in canonical ensemble can be defined as a Fourier transform \cite{Wgnr,Tatr} of the matrix element of the density operator \cite{QGP1,QGP2} in the coordinate representation:
\begin{widetext}
\begin{eqnarray}\label{pathint_wignerfunctionint2}
&&W(p,x,{\rm Q})=\sum_{\sigma,\acute{\sigma}}
\int \rm  d\xi\;  \rm d\mu \acute{Q} \; \;
\delta_{\sigma,\acute{\sigma}}\delta(Q-\acute{Q})\exp (i\langle \xi |p\rangle)
\langle x+\xi/2,Q,\sigma|e^{-\beta\hat H(Q)}|x-\xi/2,\acute{Q},\acute{\sigma}\rangle \,
\nonumber \\&&
=\frac{C(M)}{Z\left(\{N\},V,\beta\right)}
\sum_{\sigma}\sum_{{ {\rm P}_q}{{\rm P}_{ \overline{q}}}{{\rm P}_g}} (-1)^{\kappa_{{\rm P}_q}+ \kappa_{{\rm P}_{ \overline{q}}}}
{\cal S}(\sigma, {{\rm P}_{q\overline{q}g}} \sigma^\prime)\big|_{\sigma'=\sigma}\,
\int\limits {\rm d}\mu {\rm Q}^{(M)} \delta({\rm Q}-{\rm P}_{q\overline{q}g}Q^{(M)})
\nonumber \\&&
\times \int {\rm d} \xi \,
\int {\rm d}q^{(1)} \dots {\rm d}q^{(M-1)}\, {\rm d}\mu {\rm Q}
\exp\Biggl\{-\pi \frac{\langle \xi|{\rm P}_{q\overline{q}g}+E |\xi \rangle}
{2M } + i\langle \xi |p\rangle
-\pi \frac{|{\rm P}_{q\overline{q}g} x-x|^2}{M}
\nonumber\\&&
-\sum\limits_{m = 0}^{M-1}
\biggl[\pi | q^{(m)}-q^{(m+1)}|^2 +
\varepsilon U\biggl(({\rm P}_{q\overline{q}g} x-x)\frac{m}{M}+x + q^{(m)}
-\frac{ (M-m) \xi}{2M}+\frac{ m {\rm P}_{q\overline{q}g}\xi}{2M} \biggr)
\biggr]
\Biggr\} ,
\nonumber \\
\end{eqnarray}
\end{widetext}
where $C(M)=M^{6\tilde{N}(M-1)/2}$ is constant and $q^{(M)}=q^{(0)}$.
Details of analytical calculation of the matrix elements of density operator are presented in \cite{EbelForFil,QGP1,QGP2}.
Here the interaction energy $U$ is the sum of the
two-particle color quantum Kelbg potentials.
The antisymmetrization for quarks and symmetrization for gluons takes into account quantum statistics.
Here we have replaced variables of integration $x^{(m)}$
for any given permutation ${{\rm P}_q}{{\rm P}_{ \overline{q}}}{{\rm P}_g}$ by relation
\begin{widetext}
\begin{eqnarray}
x^{(m)} = ({\rm P}_{q \overline{q} g }x-x)\frac{m}{M}+x + q^{(m)}
-\frac{ (M-m) \xi}{2M}+\frac{ m {\rm P}_{q\overline{q}g}\xi}{2M} \,.
\label{pathint_variableschange}
\end{eqnarray}
\end{widetext}
In equation~(\ref{pathint_wignerfunctionint2}) $E$ is the unit matrix, while the matrix presenting permutation
${{\rm P}_q}{{\rm P}_{ \overline{q}}}{{\rm P}_g}$ is equal to unit
matrix with appropriately transposed columns.
In equation (\ref{pathint_wignerfunctionint2}) $E$ is the unit matrix, while the permutation matrix ${{\rm P}_q}{{\rm P}_{ \overline{q}}}{{\rm P}_g}$  is obtained from $E$ by appropriate transposition of columns.

To avoid the problems with definition of the relativistic Wigner function, discussed in \cite{reveiw,Zav1,Zav2,Lar1}, we use here the non-relativistic limit for kinetic energy operator (\ref{Coulomb}).

The expression for the Wigner function (\ref{pathint_wignerfunctionint2}) is inconvenient for Monte Carlo simulations since it does not contain explicit result of integration over $\xi$, even for free particles with $U(x) \equiv 0$.
In general, this integral can not be calculated
analytically.
Exceptions are linear and quadratic functions $U (x)$, known as the linear and harmonic potentials correspondingly.
To perform the integration over $\xi$ analytically and obtain an explicit expression for Wigner function , let us take the approximation for potential $U(x)$, given by the Taylor expansion up to the first or second order in $\xi$, strongly restricted by exponentially decaying factor with quadratic form of  $\xi$ in (\ref{pathint_wignerfunctionint2}).
This approximations were tested by calculations of thermodynamic values and ground state wave functions for quantum particle in 1D and 3D potential field \cite{LarkinFilinovCPP, JAMP}; they gives practically exact results even for potentials, which differ from linear or harmonic ones significantly.

As it was shown in \cite{JPA2017} for electromagnetic plasma, the main contribution to Fermi repulsion in (\ref{pathint_wignerfunctionint2}) comes from the pair permutations at moderate plasma degeneracy, when temperature is about Fermi energy.
This is the physical reason to take into account only pair permutations and neglect the others.
In this approximation the Wigner function can be presented in the next form:

\begin{widetext}
	\begin{eqnarray}
		\label{pathint_wignerfunctionint4}
	&&W(p,x,{\rm Q}) \approx\,
		\frac{C(M)}{Z\left(\{N\},V,\beta\right)}

		\int \rm dq^{(1)} \dots \rm dq^{(M-1)}\,
		\nonumber\\&&
		\times	\exp\Biggl\{
		-\sum\limits_{m = 0}^{M-1}
		\biggl[\pi | q^{(m)}-q^{(m+1)}|^2 +
		\varepsilon U\biggl(x + q^{(m)}

		) \biggr] \Biggr\}
\exp\Biggl\{\frac{M}{4 \pi}
		\Biggr|i p + \frac{\varepsilon}{2}\sum\limits_{m = 0}^{M-1} \frac{ (M-2m) }{M}
		\frac{\partial U(x+q^{(m)})}{\partial x}

		\Biggr|^2\Biggr\}
		\nonumber\\&&
		\times\sum_{\sigma_q}\Biggl\{1-\sum_{l<t}^{N_q}\delta_{\sigma_{l,q}\sigma_{t,q}}
		\delta_{f_{l,q}f_{t,q}} \delta(({\rm Q}_{l,q}-{\rm Q}_{t,q}))
		\exp(-2\pi\frac{|x_{l,q}-x_{t,q}|^2}{M})
		\delta \biggl(\frac{(\tilde{p}_{l,q}-\tilde{p}_{t,q})\sqrt{M}}{2\pi}
		\biggr)\Biggr\}
		\nonumber\\&&
		\times\sum_{\sigma_{\overline{q}}}\Biggl\{1-\sum_{l<t}^{\bar{N}_q}\delta_{\sigma_{l,{\overline{q}}}\sigma_{t,{\overline{q}}}}
		\delta_{f_{l,\overline{q}}f_{t,\overline{q}}} \delta(({\rm Q}_{l,\overline{q}}-{\rm Q}_{t,\overline{q}}))
		\exp(-2\pi\frac{|x_{l,{\overline{q}}}-x_{t,{\overline{q}}}|^2}{M})
		\delta \biggl(\frac{(\tilde{p}_{l,{\overline{q}}}-\tilde{p}_{t,{\overline{q}}})\sqrt{M}}{2\pi}
		\biggr)\Biggr\}
		\nonumber\\&&
		\times\sum_{\sigma_g}\Biggl\{1+\sum_{l<t}^{N_g
		\delta(({\rm Q}_{l,g}-{\rm Q}_{t,g}))
		\exp(-2\pi\frac{|x_{l,g}-x_{t,g}|^2}{M})
		\delta \biggl(\frac{(\tilde{p}_{l,g}-\tilde{p}_{t,g})\sqrt{M}}{2\pi}
		\biggr)\Biggr\},
	\end{eqnarray}
\end{widetext}

	where
	\begin{eqnarray}
		\tilde{p}_{t,a}=p_{t,a} + \frac{\varepsilon }{2 }\sum\limits_{m = 0}^{M-1}
		\frac{\partial U(x+q^{(m)})}{\partial x_{t,a}}.
		\nonumber
	\end{eqnarray}
Here $\delta_{f_i,f_j}$ are the Kronecker symbols, depending on flavor indexes $f_i$ of quasiparticles and taking values $u$, $d$ and $s$.
For simplicity, we write here the expression, corresponding only to the linear terms in the Taylor expansion of color potentials.

To avoid difficulties, arising at Monte Carlo simulations due to presence of the delta-function in expression (\ref{pathint_wignerfunctionint4}), and to regularize integration over momenta, the positive Husimi distributions, being a coarse-grained Wigner function, can be used  with a Gaussian smoothing for small phase space cells of parameters $\Delta^2_x$ and $\Delta^2_p$  \cite{Tatr}.
	The final expression for Wigner function can be written in the form:
\begin{widetext}
	\begin{eqnarray}
		\label{pathint_wignerfunctionint5}
	&&W^H(p,x,{\rm Q}) \approx\,
		\frac{C(M)}{Z\left(\{N\},V,\beta\right) }
		\int \rm dq^{(1)} \dots \rm dq^{(M-1)}\,
\exp\Biggl\{
		-\sum\limits_{m = 0}^{M-1}
		\biggl[\pi | q^{(m)}-q^{(m+1)}|^2 +
		\varepsilon U(x + q^{(m)}

		) \biggr] \Biggr\}
		\nonumber\\&&
		\times \exp\Biggl\{\frac{M}{4 \pi}
		\Biggr| i p + \frac{\varepsilon}{2}\sum\limits_{m = 0}^{M-1} \frac{ (M-2m) }{M}
		\frac{\partial U(x+q^{(m)})}{\partial x}

		\Biggr|^2\Biggr\}
\sum_{\sigma}\exp (-\beta \sum_{l<t}^{N_q} v^{q}_{ lt})\exp (-\beta \sum_{l<t}^{\bar{N}_q}
		v^{{ \overline{q}}}_{ lt})\exp (-\beta \sum_{l<t}^{N_g}v^{{g}}_{ lt}),
		\nonumber\\
	\end{eqnarray}
\end{widetext}

where the final expression for the phase space pair pseudopotentials, accounting for quantum statistical effects, looks like:

\begin{widetext}
	\begin{eqnarray}
	&&v^{a}_{ lt}=-kT \ln\Biggl\{
		1-\delta_{\sigma_{l,a}\sigma_{t,a}}\delta_{f_{l,a}f_{t,a}}\exp \biggl(-\frac{|{\rm Q}_{l,a}-{\rm Q}_{t,a})|^2}{2\tilde{\Delta}^2_Q}\biggr)
		\nonumber\\&&
		\times\exp \biggl(-\frac{2\pi|x_{l,a}-x_{t,a}|^2 (1-\frac{\tilde{\Delta}^2_{a,x}/\lambda^2_a}{1+\tilde{\Delta}^2_{a,x}/\lambda^2_a})}{\lambda^2_a}\biggr)
		\exp \biggl(-\frac{|(\tilde{p}_{l,a}-\tilde{p}_{t,a})|^2\lambda^2_a}{(2\pi\hbar)^2(\tilde{\Delta}^2_p/\lambda^2_a)}\biggr)
		\Biggr\},
		\nonumber\\&&
		v^{g}_{ lt}=-kT \ln\Biggl\{
		1 + \exp \biggl(-\frac{ |{\rm Q}_{l,g}-{\rm Q}_{t,g})|^2}{2 \tilde{\Delta}^2_Q}\biggr)
		\nonumber\\&&
		\times\exp \biggl(-\frac{2\pi|x_{l,g}-x_{t,g}|^2 (1-\frac{\tilde{\Delta}^2_{g,x}/\lambda^2_g}{1+\tilde{\Delta}^2_{g,x}/\lambda^2_g})}
		{\lambda^2_g}\biggr) \exp \biggl(
		\frac{|(\tilde{p}_{l,g}-\tilde{g}_{t,g})|^2\lambda^2_g}{(2\pi\hbar)^2\tilde{(\Delta}^2_p/\lambda^2_g)}\biggr)
		\Biggr\}.
	\end{eqnarray}
\end{widetext}
Here $a=q,\overline{q}$, $\Delta^2_{a,x}=\frac{2}{(\pi^2-2)}$ and $z=1/\sqrt{2}$.
To extent the region of applicability of obtained phase space pair pseudopotential, $\tilde{\Delta}^2_p$ and $\tilde{\Delta}^2_Q $ can be considered as fit functions with values much smaller than unity.
Our test calculations \cite{JPA2017} have shown that the best fit for $\tilde{\Delta}^2_p$ can be written in the form
$\tilde{\Delta}^2_p/\lambda^2_a=0.00505+0.056n\lambda_a^3$, while $\tilde{\Delta}^2_{a,x}$ and $\tilde{\Delta}^2_Q$ were of order $0.1$.

The pseudopotentials $v^{q}_{ lt}$ in the phase space allow us to avoid the famous 'fermionic sign problem' and to realize the Pauli blocking for quarks/antiquarks with the same spin, flavor and color.
The pseudopotentials $v^{g}_{ lt}$  provides Bose statistics for gluons.
Note also that the expression (\ref{pathint_wignerfunctionint5}) explicitly contains the term, related to the classical Maxwell distribution,  modified by terms accounting for influence of interaction on the momentum distribution function.

An average value of arbitrary quantum operator $\hat A$ can be written as Weyl symbol $A(p,x,Q)$, averaged over the color phase space with the 	Wigner function $W(p,x,Q)$ or $W^H(p,x,Q)$:
\begin{eqnarray}\label{wigfunc_average0}
\langle \hat A\rangle = \int{\rm d}\mu {\rm Q} \;\frac{{\rm d}p {\rm d}x } {(2\pi)^{6\tilde{N}}}
A(p,x,{\rm Q}) W^H(p,x,\rm Q),
\end{eqnarray}
where the Weyl symbol of operator $\hat A$ is:
\begin{eqnarray}\label{wigfunc_weylsymbol}
A(p,x,{\rm Q}) = \int \rm d\xi \rm e^{-\rm i\langle
	\xi|p\rangle /\hbar} \langle x-\xi/2|\hat A(Q)|x + \xi/2\rangle.
\end{eqnarray}
Weyl symbols of common operators like $\hat p$, $\hat x$, $\hat p^2$, $\hat x^2$, $\hat H$, $\hat H^2$ etc. can be easily calculated directly from the definition (\ref{wigfunc_weylsymbol}).

\section{Monte Carlo simulations of QGP distribution functions}
\label{s:model}
The basic idea of the Monte Carlo simulations of QGP is to construct a Markovian chain of different quasiparticle states in the color configuration space or in the color phase space \cite{EbelForFil}.
The computational procedure comprises two stages.
At the first stage a dominant, i.e., maximal, $\{N\}$-term in the sum of Eq.~(\ref{Gq-def})
is determined by calculations in the grand canonical ensemble.
This term is indeed the dominant one in thermodynamic limit of the box volume ($V \to \infty$).
In grand canonical ensemble, the quasiparticle numbers in the simulation box are varied, i.e., the sequential states of the Markovian chain can differ from each other by numbers of quarks, antiquarks or gluons.
Transitions between these states are the first type of Markovian elementary step.
In the second type of elementary step, the coordinates of randomly chosen quasiparticle are changed.
The color variables are changed according to the SU(3) group Haar measure \cite{LM105,LM106,LM107,LM108,QGP1,EbelForFil} in the third type of the Markovian elementary step.
For simplicity, numbers of spins "up' and "down" are fixed and equal to each other for quarks (antiquarks).
The Markovian chain is generated until complete convergence of calculated values is achieved.
This allows one to determine the average numbers of quarks, antiquarks, and gluons in the box volume at fixed temperature.
Here, only the densities of each type of particles, i.e., average number of particles to box volume ratio, have physical sense.
Usually, after several million elementary steps, the average numbers of quasiparticles of each type become stable, and, for example, at zero baryon chemical potential, the average number of quarks is practically equal to the average number of antiquarks.
This equality can be considered as an inherent test of self-consistency of the calculations.
In this way, the dependence of quasiparticle density on baryon chemical potential is calculated in grand canonical ensemble according to (\ref{Gq-def}).

At the second stage the fixed numbers of quarks, antiquarks, and gluons have to be chosen  equal to the obtained average values at the first stage, and calculations were carried out in the canonical ensemble.
Wherein the second and the third types of the elementary Markovian step described above are used.
To generate the Markovian chain for integration in the color phase space (see (\ref{wigfunc_average0})), it is necessary to sample the momentum of some quasiparticle as the new Markovian elementary step \cite{JPA2017}.
\begin{figure}[htb]
  \begin{minipage}[ht]{1.0\linewidth}
  		\includegraphics[width=7.5cm,clip=true]{Dens.eps}  \\(a)
  \end{minipage} \\
 \begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=7.5cm,clip=true]{GGMUPT4.eps} \\(b)
 \end{minipage}	\\
 \begin{minipage}[ht]{1.0\linewidth}
   \includegraphics[width=7.5cm,clip=true]{DellPMU.eps} \\(c)
 \end{minipage}

	\caption{(Color online) (Plot (a)) The quark and gluon densities versus temperature at baryon chemical potential equal to zero.
(Plot (b)) Equation of state of QGP for different baryon chemical potentials.
(Plot (c)) Pressure for $\mu >0 $ in exceess of the pressure at $\mu =0$.
The Monte Carlo (PIMC) results (symbols) are compared  to lattice data (lines) of Refs.~\cite{Borsanyi1,Borsanyi2}.
}
	\label{density}
\end{figure}

For reader convenience, Fig.~\ref{density} shows the results of the Monte Carlo calculations (PIMC) for averaged over spin and flavor variables, obtained in \cite{QGP1,QGP2} within quasiparticle approach, discussed above.
The top plot shows dependences of quark and gluon densities on temperature at zero baryon chemical potential ($ n_q=n_{\overline{q}}$).
For different chemical potentials the central and bottom plots present the equations of state and the pressure difference compared to the limit of zero chemical potential.
As  it follows from Fig.~\ref{density}, for chemical potentials $\mu/T_c=133/175<1$ our approach agrees quite
well with lattice QCD data based on a Taylor expansion around $\mu=0$.

In this paper we calculate the spatial and momentum distribution functions of QGP quasiparticles in canonical ensemble, using the quark, antiquark and gluon densities discuissed above.

\section{Pair distribution functions}
To undestand physical properties of QGP, let us start from consideration of spatial arrangement of  quasiparticles by discussing the pair distribution functions (PDF) $\rm g_{ab}(r)$, which are obtained by integration of the Wigner functions over all variables except coordinates of two quasiparticles of the types 'a' and 'b':
\begin{eqnarray}\label{g-def}
&&\rm g_{ab}(|{\bf R}_1-{\bf R}_2|) \frac{N_aN_b}{V^2} =
\frac{1}{Z}
\sum_{\sigma}
\sum_{i,j,i\neq j}
\delta_{a_i,a}\, \delta_{b_j,b}  \int \rm dp
\nonumber\\&&
\times  \rm dr d\mu  Q \delta({\bf R}_1-{\bf r}_i)\,\delta({\bf R}_2-{\bf r}_j)\;W^H(p,x,Q),
\end{eqnarray}
where $a(a_i)$ and $b(b_j)$ can take values $q,\overline{q}$ or $g$.
Functions $\rm g_{ab}(r)$ give probability density to find a pair of quasiparticles of types a and b at the distance $r$ between them as the PDFs depending only on the difference between coordinates due to translational invariance of the system.
\begin{figure}[htb]
\begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=6.5cm,clip=true]{crt4r04-6g.eps}  \\(a)
\end{minipage} \\
\begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=6.5cm,clip=true]{r2crt2-4sum.eps}  \\(b)
\end{minipage} \\

	\caption{(Color online) (Plot (a)) The pair distribution functions $\rm g_{ab}(r)$ of the quark and gluon quasiparticles at temperatures $T=4T_c$ at baryon chemical potential equal to zero (a,b=quark, antiquark or gluon, $a_0=1.16$~fm). Lines: 1-6 ~-~ $r_s=0.6$~fm; 7 - $r_s=0.4$~fm.
	(Plot (b))  Product $r^2 \rm g_{ab}(r)$. Lines for $T=4T_c$:
	1,2 - $r_s=0.6$~fm; 3 - $r_s=0.5$~fm; 4 - $r_s=0.4$~fm.
	Line 5 - $T=2T_c$, $r_s=0.6$~fm.

	}
	\label{crt4}
\end{figure}
In a non-interacting classical system $\rm g_{ab}(r)\equiv 1$, whereas interactions and  quantum effects result in
a re-distribution of the particles.
Results for PDFs at temperature $T/T_c=2$ and  $T/T_c=4$ and different average interparticle distances $r_s$
($r_s=\sqrt[3]{3/(4\pi n)}$, n is the density of all quasiparticles)
are shown on Fig.~\ref{crt4}.
The PDFs of identical particles are presented by lines 1, 2, 4 and 7 on the top plot of Fig.~\ref{crt4}, while other lines show PDFs of the quasiparticles of different types.
All PDFs reveal a similar behavior.
At the distances $r/a_0 \ge 0.5$ all PDFs tends to unity (i.e. the ideal gas limit), while near zero
PDFs are monotonously growing.
This behavior of the PDFs at small distances is manifestation of effective pair attraction of quarks, antiquarks and gluons.
This attraction suggests that the color vectors of nearest neighbor quasiparticles of any type are anti-parallel.
QGP lowers its total energy by minimizing the color Coulomb interaction energy via a spontaneous ``anti-ferromagnetic or colorless'' ordering of color vectors.
This may result in clustering of quarks, antiquarks and gluons and, possible, appearance of  bound states.
Such low-distance behavior is also observed in a non- ideal dense astrophysical electron-ion plasma and in a nonideal electron-hole plasmas in semiconductors \cite{EbelForFil}.

We can also notice fundamental difference between quark and gluon PDFs (antiquark PDFs are identical to the quark PDFs).
The short-distance attraction is stronger for gluon-gluon and gluon-(anti)quark pairs	than for (anti)quark-(anti)quark ones because of the corresponding difference in values of quadratic Casimir invariants $\breve{q}_2$ \cite{EbelForFil},
which determine the maximal values of the effective color charge scalar products
$(\rm Q_i \cdot \rm Q_j)$ in color Kelbg (Coulomb) potentials:
for gluon-gluon pairs $\left|(\rm Q_g \cdot \rm Q_g)\right|_{max} = 24$,
for gluon-(anti)quark pairs $\left|(\rm Q_g \cdot \rm Q_q) \right|_{max}=
\left| (\rm Q_g \cdot \rm Q_{\overline{q}} ) \right|_{max}\approx$ 10,
and for (anti)quark-(anti)quark pairs
$\left| (\rm Q_q \cdot \rm Q_q ) \right|_{max}=
\left| (\rm Q_{\overline{q}} \cdot \rm Q_{\overline{q}} ) \right|_{max}=\left| ( \rm Q_q \cdot \rm Q_{\overline{q}} ) 			\right|_{max}=$ 4.

The short-distance correlation implies formation of the gluon-gluon
clusters, which are uniformly distributed in space (see ${\rm g}_{gg}(r)$ at large distance).
In case of the gluon-gluon clusters we can even talk about $gg$-bound states (i.e. glueballs) due to the following well known statements of quantum mechanics for two particles.
The gluon-gluon PDFs can be formed either by correlated scattering states or by bound states of quasiparticles, depending on the relative fractions of these states.
However, strictly speaking, there is no clear subdivision into bound and free "components" in plasma media due to mutual overlap of the quasiparticle clouds.
In addition, there is not any rigorous criterion for a bound state at high densities due to the strong affection of the surrounding plasma.

Nevertheless, a rough estimate of existance and even fraction of quasiparticle bound states  \cite{ElHol}  can 	be obtained	by  the following reasonings.
The product $r^2 \rm g_{ab}(r)$ has sense of probability to find a pair of quasiparticles at the distance $r$ between them.
On the other hand, the corresponding quantum mechanical probability is the product of $r^2$ and two-particle Slater sum
\begin{equation}
	\label{slsm}
	\Sigma_{ab}=8\pi
	^{3/2}\lambda_{ab}^{3}\sum_{\alpha}|\Psi_{\alpha}(r)|^{2}\exp(-\beta
	E_{\alpha})
	= \Sigma_{ab}^{d}+\Sigma_{ab}^{c},
\end{equation}
where $E_{\alpha}$ and $\Psi_{\alpha}(r)$ are the energy (without center of mass energy) and the wave function of a quasiparticle pair correspondingly.
$\Sigma_{ab}$ is, in essence, the diagonal part of the corresponding density matrix.
In Eq.~(\ref{slsm}) the summation runs over all possible states $\alpha$
with contributions from the discrete ($\Sigma_{ab}^{d}$) and continuous ($\Sigma_{ab}^{c}$) parts of the spectrum.

At the temperatures smaller than the binding energy and at distances smaller than or of the order of several bound state radii the main contribution to the Slater sum comes from bound states and the product $r^2\Sigma_{ab}^d$ is sharply peaked at distances around the Bohr radius \cite{ElHol}.
Similarly, in QGP the product $r^2\rm g_{gg}(r)$ forms distinct maximum, which can be interpreted as  evidence of bound states of $gg$ pairs or clusters.
Sharp peak on the right plot of Fig.~\ref{crt4} at small distances indicates existence of the bound states; it demonstrates the fast decreasing fraction of the glueballs with increasing density at temperatures $T=4T_c$.
One can notice that at density related to $r_s=0.6$~fm fraction of bound states almost is not changed while temperature decreases from $T=4T_c$ to $T=2T_c$.

\section{Debye screening mass and running coupling constant}
When studying the system by numerical methods, it is interesting to know how the free energy  depends on the interparticle distance.
The free energy surface along the selected coordinate is called the potential of mean force (PMF) \cite{Reith,Kirkwood1,Kirkwood2}.
PMF can be obtained through Monte Carlo or Molecular Dynamics simulations, which examine how the energy of the system varies with a specific parameter.
For example, it can examine how the energy of the system changes as function of the distance between two given particles.
This energy change $w^{(2)}(r)$ is an average work required to bring two particles from infinite distance to the distance $r$.
The potential of mean force $w^{(2)}(r)$ is usually applied in the Boltzmann inversion method as a first guess for 		the effective pair interaction potential that ought to reproduce the correct pair distribution function \cite{Reith}.  	For low density of particles  the virial expansion in terms of bare potential $U_{ab}(r,T)$ gives  $w^{(2)}(r)=U_{ab}		(r,T)= -T \ln \rm g_{ab} (r,T)$ \cite{EbelForFil}.
In general case the PMF of a system with $\tilde{N}$ particles is the potential that
gives the average force over all the configurations of all the $n+1,\dots,\tilde{N}$ particles acting on a particle j 	at any fixed configuration keeping fixed a set of particles $1, \dots, n$ (see Eq.~(\ref{Z-def})) \cite{Kirkwood1,Kirkwood2} :
\begin{widetext}
\begin{eqnarray}\label{avf}
-\nabla_j w^{(n)}(x^{(n)})=\frac{\sum_{\sigma} \int	\rm  dp \rm  dx^{(\tilde{N}-n)}\; \rm d\mu Q \;\nabla_jW^H(p,x,Q)}
{\sum_{\sigma} \int	\rm  dp \rm  dx^{(\tilde{N})}\; \rm d\mu Q \;W^H(p,x,Q)}, j=1, \dots,n
\end{eqnarray}
\end{widetext}
Above expression $-\nabla_j w^{(n)}$ is the averaged force, i.e. "mean force" on particle j and $w^{(n)}$ is the so-called potential of mean force.
According to the Ref.~\cite{Chandler} the pair distribution functions can be expressed through the potential of mean force $w^{(2)}$:
\begin{eqnarray}
\label{crg}
\rm g(r)=\exp (-w^{(2)}(r)/T).
\end{eqnarray}

As it follows from the left plot of Fig.~\ref{crt4}, the logarithm of PDF, describing the PMF, can be approximated by a 	linear functions at distances smaller than the half of average inter quasiparticle distance with good accuracy.
Thus the PMFs are almost linear functions of the interparticle distances, which are formed by the contributions of the color Coulomb potentials (strictly speaking color Kelbg pseudopotentials).
Below we will see that at large distances all PDFs (except ${\rm g}_{gg}(r)$) can be described by exponentially screening Yukawa-type effective potential, similarly with electromagnetic plasma.

In lattice QCD calcualtions \cite{Maezawa} the effective Debye mass and running coupling constant at temperatures above $T_c$ have been estimated from the appropriate fit range for free energies $w^{(2)}_M(r,T)$ by the screened Coulomb potential:
\begin{eqnarray}
\label{dbm}
&&w^{(2)}_M(r,T)/T=V_M(r,T)
\nonumber\\&=&
C(M)\frac{\alpha_{eff}(T,M)}{r}\exp(-m_D(T,M)r),
\end{eqnarray}
where $\alpha_{eff}(T,M)$, $m_D(T,M)$ and $C(M)$
are the running coupling constant, the Debye screening mass and the Casimir factors
for related color channel M respectively
($C(1)=-\frac{4}{3},C(8)=\frac{1}{6},C(6)=\frac{1}{3},C(3)=-\frac{2}{3}$).

The PIMC simulations allow also to estimate the Debye screening masses $m^{ab}_D(T)$ and
running coupling constant $\alpha_{eff}(T,M)$ for the quasiparticle pairs free from
forming bound states (see above).
According to the equations (\ref{crg}, \ref{dbm}) we have:
\begin{eqnarray}
\label{dbmm}
&&-\ln (\rm g_{ab}(r))=w^{(2)}_{ab}(r,T)/T=V_{ab}(r,T)
\nonumber\\&=&
\frac{\tilde{\alpha}^{ab}_{eff}(T)}{r}\exp(-m^{ab}_D(T)r),
\end{eqnarray}
where
$\tilde{\alpha}^{ab}_{eff}(T)=<C(M)>\alpha^{ab}_{eff}(T)$ with the averaged Casimir factors $<C(M)> \approx (C(1)+C(8)+C(6)+C(3))/4$.
The Debye screening masses $m^{ab}_D(T)$ and running coupling constant $\tilde{\alpha}^{ab}_{eff}(T)$ can be estimated from the long distance behavior of the $\rm g_{ab}(r)$, approximated accordingly to (\ref{dbmm}) with the Debye potential $V_{ab}(r,T)$ (see top plots of Fig.~\ref{crt4} and Fig.~\ref{crt31}).
Disagreement with Debye approximation has place only at short interparticle distances, where the main contribution to $\rm \rm g_{ab}	(r)$ is given by the interaction with the nearest neighbor quasiparticle.
\begin{figure}[htb]
  \begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=5.5cm,clip=true]{debqaq.eps}   \\(a)
  \end{minipage} \\
  \begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=5.5cm,clip=true]{crrs04.eps}	\\(b)
  \end{minipage} \\
  \begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=5.5cm,clip=true]{alfrs04.eps}  \\(c)
  \end{minipage} \\

	\caption{(Color online)
(Plot (a)) The pair quasiparticle quark-antiquark distribution function
at temperature $T=3T_c$
 and $r_s=0.4$~fm (baryon chemical potential equal to zero, $a_0=1.16$~fm ).
Lines: 1~-~ PIMC results; 2~-~ Debye approximation.
(Plot (b)) The PIMC and lattice QCD \cite{Maezawa} Debye screening masses. PIMC scatters: 1 ~-~ quark--quark;  2 ~-~ antiquark--antiquark;
3 ~-~ quark--antiquark; 4 ~-~ quark--gluon; 5 ~-~ antiquark--gluon.
Lattice QCD scatters: 6 ~-~ Wilson quark action; 7 ~-~ staggered quark action.
(Plot (c)) The running coupling constant. The same notation with the central plot. The PIMC statistical errors
are of order sizs of the of the related scatters.
}
	\label{crt31}
\end{figure}

The Debye screening masses and running coupling constant, obtained by PIMC and lattice QCD calculations \cite{Maezawa}  correspondingly, are presented on the central and bottom plots of Fig.~\ref{crt31}.
Vertical line shows the scatter region of the Debye masses for color channel 	$M=1, 8, 6, 3$ at $T/T_c=1$, while at other temperatures these regions are approximately of the twice size of the scatters presenting data \cite{Maezawa}.
Comparison of these results shows quite good agreement between PIMC and lattice QCD results.
Note that the Debye screening masses, obtained from PIMC,  are closer to ones obtained by Wilson quark action than ones,  obtained from improved staggered quark action.
Of course, existing disagreements should be further investigated by more detailed PIMC simulations and as well as at smaller lattice spacing.

\section{Pair color  correlation functions}
Independent estimations of screening parameters can be carried out from pair color correlation functions.
To introduce the pair color correlation functions, let us consider the probability of an elementary configuration for $\tilde{N}$ particles  given by the Eq.~(\ref{Z-def}):

\begin{eqnarray}
	\label{crg1}
&&\rm P^{[\tilde{N}]}(x,Q) =
\frac{\sum_{\sigma} \int \rm  dp  W^H(p,x,Q)}{\sum_{\sigma} \int	\rm  dp \rm  dx\; \rm d\mu Q \;W^H(p,x,Q)}.
\nonumber\\
\end{eqnarray}
Total number of particles is huge, so that $\rm P^{[\tilde{N}]}$ in itself is not very useful.
However, one can also obtain the probability of reduced configuration, where the degrees of freedom of $n$ ($n < \tilde{N}$ ) particles $x_1Q_1,\dots,x_ nQ_n$ are fixed, with no constraints on the remaining $\tilde{N}-n$ indices.
To do this, one has to integrate (\ref{crg1}) over the remaining degrees of freedom.
Let us consider the probability for quasiparticles to have positions in the color configuration space $x_1,Q_1, \dots,  x_{n_{q}}, Q_{n_q}$ 	by  expression:
\begin{eqnarray}
	\label{crg2}
&&\rm P^{[n]}(x^{[n]},Q^{[n_q]})
\nonumber\\&=&
\frac{\sum_{\sigma} \int  \rm  dp \rm  dx_{(n+1)}\; \dots \rm d x_{\tilde{N}}

\int\rm  dx\; \rm d\mu Q \;W^H(p, \sigma)}{\sum_{\sigma}
\int \rm  dp \rm  dx\; \rm d\mu Q \;W^H(p,x,Q)},
\end{eqnarray}

Generally speaking, the thermodynamic functions $M_n$ in canonical ensemble can be presented as integrals of the next form:
\begin{widetext}
\begin{eqnarray}
	\label{therm}
&&<M_n>=
\frac{\sum_{\sigma}  \int \rm  dp \rm  dx\; \rm d\mu Q \; M_n(x,Q) \;W^H(p, x, \sigma)}
	{\sum_{\sigma} \int \rm  dp \rm  dx\; \rm d\mu Q \;W^H(p,x,Q)},
	\nonumber\\&
	\mbox{where }\qquad
	&M_n(x,{\rm Q})=
	\sum_{1 \le i_1 < i_2 < \dots < i_n \le N} f(x_{i_1},{\rm Q}_{i_1}; \dots ;x_{i_n},{\rm Q}_{i_n}).
\end{eqnarray}
\end{widetext}
\begin{figure}[htb]
\begin{minipage}[ht]{0.3\linewidth}
	\includegraphics[width=5.5cm,clip=true]{clcrt3r04qq.eps} \\(a)
\end{minipage}
\begin{minipage}[ht]{0.3\linewidth}
	\includegraphics[width=5.5cm,clip=true]{clct3r04gg.eps}	\\(b)
\end{minipage}
\begin{minipage}[ht]{0.3\linewidth}
	\includegraphics[width=5.5cm,clip=true]{clrs04.eps}  \\(c)
\end{minipage}

	\caption{(Color online)
The PIMC quark-quark  (Plot (a)) and quark/gluon-gluon (Plot (b))  color  pair correlation functions
for strongly coupled QGP.
Lines: 1 ~-~ quark--quark;  2 ~-~ antiquark--antiquark; 3 ~-~ quark--antiquark;
4 ~-~ quark--gluon; 5 ~-~ antiquark--gluon; 6 ~-~ gluon--gluon.
Temperature $T=3 T_c$,
$r_s=0.4$~fm and baryon chemical potential equal to zero ($a_0=1.16$~fm).
(Plot (c)) Comparison color screening parameter $m_{Cl}/T$ from the color pair correlation functions
and Debye screening mass \cite{Maezawa}. Scatteres: 1 ~ quark - quark;  2 ~ antiquark - antiquark; 3 ~ quark - antiquark;
4 ~ quark - gluon; 5 ~ antiquark - gluon. Lattice QCD  \cite{Maezawa}:
6 ~-~ Wilson quark action; 7 ~-~ staggered quark action.
	}
	\label{clms}
\end{figure}

Thus, for example, for $n_q$ quasiparticles the number of terms in Eq.~(\ref{therm}) is equal to $N_q!/(N_q-n_q)!n_q!$:
\begin{eqnarray}
	\label{therm1}
&&<M_n> =\frac{N_q!}{(N_q-n_q)!n_q!}
\nonumber\\&\times&
\int  \rm  dx_1\; \rm d\mu Q_1 \dots \rm  dx_{n_q}\; \rm d\mu Q_{n_q}
\nonumber\\&\times&
	f(x_1,{\rm Q}_1; \dots ;x_{n_q},{\rm Q}_{n_q} )  P^{[n]}(x^{[n_q]},{\rm Q}^{[n_q]}).
\end{eqnarray}

Let us introduce the dimensionless color correlation functions by the next equations:
\begin{eqnarray}
	\label{crg5}
	&&\frac{N^{n_q}}{V^{n_q}} \rm g^{[n_q]}(x^{[n_q]},Q^{[n_q]})
\nonumber\\&=&
	\frac{N_q!}{(N_q-n_q)!n_q!}
	P^{[n_q]}(x^{[n_q]},Q^{[n_q]}).
\end{eqnarray}
For example, pair quark-quark color correlation function  can be defined through scalar product of 8D color Wong vectors:
\begin{eqnarray}
\label{clf}
&&\tilde{M}_2(r,x,{\rm Q})
\nonumber\\&=&
\sum_{1 \le i_1 < i_2 \le N} \delta(r-|x_{1_{i_1}}-x_{i_2}|) ({\rm Q}_{i_1} \cdot {\rm Q}_{i_1}).
\end{eqnarray}
Thus, the averaged over canonical ensemble pair color correlation function looks like:
\begin{widetext}
\begin{eqnarray}
	\label{crg7}
({\rm Q}_{a} \cdot {\rm Q}_{b})_{\rm av}(r)=\frac{N_q^2}{V^2}\int \rm  dx_{1,a} \rm  dx_{2,b}  \rm  d\mu {\rm Q}_{1,a} \rm  d\mu {\rm Q}_{2,b}
	\delta(r-|x_{1,a}-x_{2,b}|)({\rm Q}_{1,a} \cdot {\rm Q}_{2,b})\rm \rm g_{ab}(x_{1,a},Q_{1,a}; x_{2,b},Q_{2,b}),
\end{eqnarray}
\end{widetext}
where a, b  correspond to quarks and $\rm g^{[2]}(x^{[2]},Q^{[2]})$ is the pair color distribution function of the system.
Definition of color pair distribution functions for quasipaticles of different types (a and b correspond to quark, antiquark or gluon) is the similar.
These functions are negative due to contribution of quasiparticle attraction and can be approximated by decaying negative exponents:
\begin{eqnarray}
\label{clmm}
({\rm Q}_{a} \cdot {\rm Q}_{b})_{\rm av}(r) \approx -\exp(-m^{ab}_{Cl}(T)r).
\end{eqnarray}
The logarithms of module of the pair color distribution function are shown on the left and central plots of Fig.~\ref{clms}.
Thus, according to its definition, the color screening mass $m^{ab}_{Cl}(T)$ can be estimated from the slope of this logarithm, approximated by straight line at distances less than the average interparticle distance $r_s=0.4$~fm.

Analysis of Fig.~\ref{clms} confirms exponential decay of color pair correlation functions and allows us to calculate the color screening masses in strongly coupled QGP, which turn out to be of order the Debye screening mass, obtained in PIMC and lattice QCD \cite{Maezawa} calculations.
Scatter of the values $m^{ab}_{Cl}(T)$ for different types of quasiparticles is also connected with different values of the related Casimir factors.

\section{Quantum ``tails'' in momentum distribution functions}
The momentum distribution function $w_a(|p|)$ can be obtained by integration of Wigner function over all quasiparticle coordinates and momenta except the momentum of some quasiparticle of type a:
\begin{eqnarray}
W_a(p_a) =
\int	\rm  dx\; \rm  dp\; \rm d\mu Q \; \delta(p_a-p) W^H(p,x,Q)
\label{wdp}
\end{eqnarray}
where $a=q,\overline{q},g$.
From physical point of view the momentum distribution function $w_a(|p|)$ gives probability density for quasiparticle to have the momentum $p$.

Non-ideal classical systems of particles are described by Maxwell distribution (MD) (proportional to $\exp(-(p\lambda_a)^2/4\pi\hbar^2)$ )) even when coupling is strong, due to the commutativity of the kinetic and potential energies.
Ideal quantum systems of particles, due to the quantum statistics , are described by Fermi or Bose momentum distribution functions.
Interparticle interaction may be able to cause formation of the bound states of two particle or many particle clusters and, hence, influence on the momentum distribution function.
Moreover, interaction of not-bounded quantum particle with its surroundings may restrict the available volume of configuration space for the particle and, due to the uncertainty principle, can also influence on the momentum distribution function \cite{Galitskii,Kimball,StarPhys,Eletskii,Emelianov,Kochetov,StMax}.
Thus, all these physical factors can modify the momentum distribution function, making it non-Maxwellial at high momenta.

One of the main objectives of this paper is to study the influence of strong interaction between quasiparticles on the color quasiparticle momentum distribution functions in QGP.

It has been shown in \cite{Galitskii,Kimball,StarPhys,Eletskii,Emelianov,Kochetov,StMax}, that in fully ionized electromagnetic plasma the momentum distribution function at high momenta can be described by the sum of the MD and the product of $const/p^{8}$ and the Maxwell distributions with effective temperature that exceeds the temperature of medium (short notation – P8)\cite{StMax}.

The results of PIMC calculations of quark and gluon momentum distribution functions are shown in Fig.~\ref{wpt4}.
Fig.~\ref{wpt4} shows also two related dependencies: Maxwell distributions (lines 1 and 4)
and analytical high momentum asymptotic (P8) (lines 4 and 6).
Here the constant and the effective temperature in P8 have been considered as adjustable parameters, set to fit PIMC momentum distribution functions at high momenta.
As it follows from the analysis of Fig.~\ref{wpt4}, the dependences P8 can reliably fit the PIMC distributions at high momenta and confirm appearance of quantum ``tails''.

Trends in behavior of momentum distribution functions, when temperature increases, can be understood from comparison of results for temperatures $T/T_c=1$ and $T/T_c=3$, while  $r_s=0.4$ is fixed (see plots of Fig.~\ref{wpt4}).

The basic physical reason of difference between behaviors of momentum distribution functions of quarks and gluons is that the quadratic Casimir value for qluons, responsible for interparticle interaction, is significantly larger than the one for quarks.

\begin{figure}[htb]

 \begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=5.5cm,clip=true]{lpt1rs04.eps}  \\(a)
 \end{minipage}	\\
 \begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=5.5cm,clip=true]{lpt2rs04.eps}   \\(b)
 \end{minipage}	\\
 \begin{minipage}[ht]{1.0\linewidth}
	\includegraphics[width=5.5cm,clip=true]{lpt3r04.eps}	 \\(c)
 \end{minipage}	\\

	\caption{(Color online)
		The averaged over color, flavor and spin variables momentum distribution functions
		$w_a(|p|) \, (a=q, g)$ for quarks and gluons of the strongly coupled QGP plasma at $r_s=0.4$~fm.

		(Plot (a)) Temperature $T/T_c=1$. Lines: 1(4) ~-~ the Maxwell distribution for quarks(gluons);
		2(5) ~-~ the PIMC results for for quarks(gluons); 3(6) ~-~ the high momentum asymptotics (P8) \cite{StMax}.
		(Plot (b)) Temperature $T/T_c=2$. The same notation with left plot.
		(Plot (c))  Temperature $T/T_c=3$. The same notation with central plot.
		The momentum distribution functions for quarks and antiquarks practically coincide with each other,
		here $\lambda$ is the thermal wave length and $\hbar$ is the Plank's constant.
		All momentum distribution functions are normalized to unity.

		Oscillations of  the PIMC distribution functions designate statistical errors.

	}
	\label{wpt4}
\end{figure}

\section{Conclusion}
Thermodynamic and kinetic properties are important for theoretical description of equilibrium states of quark-gluon plasma and need to be considered within unified appropriate model.
In the framework of considered constituent quasiparticle model of quark-gluon plasma (QGP) the matrix elements of density operator and the Wigner function in the color phase space are presented in form of color path integrals with Wiener and SU(3) group Haar measures.
The obtained explicit expression of the Wigner function
resembles the Maxwell--Boltzmann distribution on momentum variables, 	but with quantum corrections.
This approximation contains also oscillatory multiplier describing quantum interference between coordinates and momenta.

Monte Carlo calculations of quark and gluon densities, momentum and spatial pair distribution functions for strongly coupled QGP in thermal equilibrium at zero baryon chemical potential have been carried out.
The Debye screening mass and running coupling constant have been obtained from the spatial pair distribution function; the results are in agreement with available lattice data.
Gluon bound states in form of glueballs has been found at temperatures of order $T=3T_c$ and densities corresponding to the average interparticle distance $r_s\ge 0.4$~fm.

Comparison with classical Maxwell–Boltzmann distribution shows the significant influence of interparticle interaction on high energy asymptotics of the momentum distribution functions, resulting in appearance of quantum ``tails''.
New pair color correlation function, color distribution functions and color screening mass have been developed and discussed.
Quantum effects have proven to be of primary importance in these simulations, showing a valuable understanding of the internal structure of QGP.

Our analysis is still too simplified and incomplete.
It is still confined only to the case of zero baryon chemical potential.
The input data of the model also requires refinement.
Work on these problems is in progress.

\section{Acknowledgements}

We acknowledge stimulating discussions with
Profs.	A.N.~Starostin, D.~Blaschke and Yu.B.~Ivanov.
Authors acknowledge the program of fundamental research of the Presidium
of the Russian Academy of Sciences ``Condensed matter and plasma at high energy densitie'' for financial support.

\bibliographystyle {apsrev}

redpajama_set_name: RedPajamaArXiv


Little red lizards, like all other vices, must die. For neither flesh and blood nor anything natural may enter Heaven without dying first. The all-important thing: "Die before you die. There's no chance after." Till We Have Faces, C.S. Lewis. Made in the image of God, we are born in a predicament. Born broken and bent on self, without fail we go about making matters worse for ourselves and our neighbors.
In The Great Divorce, C.S. Lewis gives us a fantastic, admittedly speculative account of a field-trip from Hell to the plains of Heaven. Hell is a massive, gray city and ever-expanding, because the self-absorbed, ghostly residents always seek to get away from each other. But the bus to Heaven leaves at regularly-scheduled intervals. Whoever wishes may go. When Ghosts arrive in the solid, green fields surrounding the Mount of God, they meet Solid People and Angels. Solid People, who have chosen Heaven as their home, along with the angels, are there to help the unsubstantial visitors who insanely cling to one vice or another. Each Ghost may stay and proceed towards the eternal mountain. Or, each may get back on the bus and return to the infernal place.
Lust is highlighted in an encounter between one of the Ghosts, a dark, oily and unsubstantial creature, and one of God's flaming ministers. Lounging on the shoulder of the Ghost, and whispering incessantly in his ear, is a little red lizard. Of course, the Ghost is embarrassed about his red, tyrannical master gibbering away in his ear. The Ghost, however, argues that he's got the lizard under control. The unsubstantial creature is content to stay and proceed toward the eternal city, so long as the reptile can come along. The flaming spirit, however, repeatedly seeks the Ghost's permission to kill the beast. As the Ghost rationalizes and contends, the ministering Flame of fire has one unrelenting question: "May I kill it?" Permission is imperative. The will of the Ghost will not be violated. The dialogue is pitiful and sad because it's so true.
The Master says to our master, Come up. Share my rest and splendour till all natures that were your enemies become slaves to dance before you and backs for you to ride, and firmness for your feet to rest on.
From beyond all place and time, out of the very Place, authority will be given you: the strengths that once opposed your will shall be obedient fire in your blood and heavenly thunder in your voice.
Overcome us that, so overcome, we may be ourselves: we desire the beginning of your reign as we desire dawn and dew, wetness at the birth of light.
Everything may be raised if it submits to death. What is sown a natural body is raised a spiritual body. When lust, that "poor, weak, whimpering, whispering thing" is killed, "that richness and energy of desire" arises; and "if the risen body even of appetite is as grand a horse as ye saw, what would the risen body of maternal love or friendship be?" (1166-74).
I recommend The Great Divorce. Learn from those who relinquish and from those who cling to their sins, the latter Ghosts getting back on the bus for the return trip to Hell. Lust is not the capital sin that wars against every soul. If not lust, then surely pride, avarice, envy, wrath, gluttony, sloth or acedia crouch at the door seeking to master us. Recalling that neither eye has seen, nor ear heard, nor has it entered the heart of man what God has prepared for those who love Him (1 Cor. 2:9); Come and die before you die; after is too late.


Malaysia runs two parallel legal systems. The civil courts cater to its non-Muslim citizens while the Islamic system decides issues affecting the fate of the country's Muslim majority.
The civil judiciary covers a range of major cases while the Islamic legal system focuses on family law, frequently tackling issues such as divorce, polygamy and custody battles.
An increasing number of cases heard in the Islamic courts involve both Muslims and non-Muslims.
A Christian lawyer in Malaysia has failed in her attempt to be allowed to practise in the Muslim Shariah courts.
Victoria Jayaseele Martin said she wanted to appear for non-Muslim clients fighting in such courts, to provide them with fairer representation.
However, a judge in Kuala Lumpur dismissed her challenge. The judge uphold the decision of a religious council that all lawyers in Islamic courts must be Muslim.
Ms Jayaseele Martin, who says the bar is unconstitutional, plans to appeal against the judge's decision.
Ms Martin's lawyer, Ranjit Singh, said it was difficult for non-Muslims to find legal counsel, who may not want to defend cases that conflict with their own faith.
One of the lawyers for the Federal Territories Islamic Religious Council, Mr Abdul Rahim Sinwan, said it was "not a problem" for non-Muslims to find Muslim lawyers to defend them.
"It goes one step further because it's a question of faith. Because when the Muslim holds the opinion held by the judge, it's a question of faith, in which non-Muslims doesn't (have it)," Mr. Sinwan added.
The Shariah courts are governed by Shariah laws that are based on Islamic traditions. A practicing lawyer must be a Muslim and needs to have a formal training to have an adept understanding of how Shariah laws work relative to Islamic culture in general.


Spacious 3 Bed Semi-Detached house presented to the market in impeccable condition.
The Mount Prospect Estate was constructed some 20/25 years ago in the very popular parish of Clonard.
The location is excellent with the house being close to all major schools, shopping centres and convenient to both the N11 & N25 & serviced by local link bus.
No. 48 is very well situated in a cul-de-sac within the estate itself and has a private garden surrounded by mature hedging with driveway & gate to rear garden.
This is a particularly good house for either First Time Buyers, Owner Occupiers or Investor with rental yields in the area being extremely good.
Open Fire with Cast iron insert surrounded by timber mantle, slate hearth, Laminate Timber Flooring.
Fitted Kitchen with floor & eye Level units, Tiled Floor & splashback.
White goods can be purchased at separate price.
W.C., W.H.B. & Shave Unit.
Shower Unit tiled & fitted with Electric T90 Shower.
With fitted wardrobes & timber flooring.


text:
The Question: Can Liverpool win the Champions League and a sixth European Cup this season?
Liverpool supporters in the Spion Kop are starting to believe that they may win a sixth European Cup Credit: Getty Images
Chris Bascombe, Anfield
6 March 2018 • 10:02pm
The average is three. Sometimes you might wait until number five. This is not just the number of goals in each Liverpool game in 2018 – a feat there was no compulsion to match in this pedestrian second leg – but the number of questions in an Anfield Champions League press conference before somebody referenced Istanbul.
"Can Liverpool do what they did in 2005?"
Players and manager will be rehearsed in their response after this exercise in game control against Porto.
Liverpool's history is a stalker, and now they are back in the quarter-finals the reminders of what's gone before become more frequent.
In fairness, chanting about Istanbul was the only way to keep entertained in a monotonous second leg, the tie already won three weeks ago.
But there are occasions you suspect Klopp would willingly impose a restraining order on the club's past as he seeks to inspire the next generation without sensing the heavy breath of those who tread a glorious path before.
In Istanbul, Liverpool won the European Cup for a fifth time. As fans on Merseyside occasionally remind people Credit: Getty Images
Reducing European nights at Anfield to testimonial pace is certainly a break with tradition – the lack of tension here inevitable with Liverpool protecting a 5-0 lead. So too is the luxury of resting star players such as Mohamed Salah. It is another example how this Liverpool side has far more quality and depth than the side that actually won the competition under Rafael Benítez.
Anfield's honours board is more assistance than hindrance, of course. The romantic tales of yore are fundamental to Liverpool's ability to lure. Without it you could argue coaches of Klopp's ilk would not be at the club.
That does not also mean it is not – just occasionally – a pain in the backside to those seeking to create their own European memories.
Too often the Istanbul question is worthy of a roll of the eyes, in danger of being misappropriated in the services of cliché. By becoming the first Liverpool team to reach the quarter-final since 2009, this team is already in a position to create new memories without the compulsion to summon the spirit of others.
Are managers of other English clubs expected to undergo the same journeys through time when considering what comes next? Is a mention of Barcelona in 1999 a pre-requisite of any Jose Mourinho press conference? Does persistent questioning as to whether Chelsea players can recreate the spirit of Munich in 2012 shadow Antonio Conte?
Jurgen Klopp at Anfield on Tuesday night Credit: Getty Images
Whenever Klopp is asked about replicating Benítez's team, one is put in mind of how the Spanish coach felt when asked if he could repeat Gérard Houllier's treble winning season. Or how Houllier responded when quizzed on the difficulties of recreating the 60s, 70s and 80s. Both were uninterested and sometimes irked by it, but were able to make the step from history chaser to history maker. Both are now guaranteed the VIP seat on the nostalgia trips, but they will empathise with Klopp. He too is obsessed with doing it his way and we have already witnessed how different that is.
Istanbul has its place in the club museum and each May 25th, celebrated for what it meant then and what it will always mean to those who revelled in it, but it has no bearing on whether this Liverpool team will win the club's 6th European Cup.
Liverpudlians are masters of storytelling and, as Houllier and Benítez demonstrated in 2001 and 2005, visionaries are not interested in repeats. They created something fresh and dynamic, standing alone from what had gone.
The attention now turns to the quarter-final, Klopp and Liverpool asked to consider which side they would most like to avoid. A raucous Anfield will be bottom of the wanted list of every other side. That was not the case with a weaker time that achieved the improbable 13 years ago.
Klopp, inevitably, will dutifully and deferentially acknowledge the reminders of Istanbul every time he walks onto that Uefa stage.
But he will be maintaining the finest traditions of Liverpool's heritage by glancing over his shoulder with little more than respectful, ceremonial curiosity.
He is far more interested in ensuring the Liverpool flag flies at another Champions League final venue, giving his players and fans another topic of conversation – Kiev in 2018.
Moment which changed the match
Only a Liverpool calamity was going to change the direction of this tie. It never looked likely. Only Salah's late introduction brought excitement.
Most influential player
Emre Can: It was an evening for disciplined midfielders, Liverpool's high pressing game applied rarely. He controlled the tempo – or what there was of one.
Crowd rating
There were a few attempts at making this seem a 'normal' tie, but with 5-0 lead rallying cries were unnecessary. 5/10
Referee rating
Felix Zwayer (Ger) had nothing to do. 7/10
Match rating
This was a Liverpool European night in name only, lacking the intensity and noise. It will be different in the next round. 3/10

redpajama_set_name: RedPajamaCommonCrawl


SANTIAGO, CHILE—Hilton's largest hotel in South America will open following the signing of the Hampton by Hilton Santiago Las Condes here.
Slated to open in 2021 and owned by Metro Hotels development group, the 221-room hotel is the second Hampton property inked as part of a strategic alliance with the developer.
The 16-story hotel will be part of a mixed-use complex with residential and retail components. It will have 221 guestrooms with complimentary WiFi, coffeemaker and HDTV. There will also be meeting space with high-tech audio/visual capabilities and a 24-hour business center as well as a fitness center.
Situated at Manquehue Norte 245, the hotel will be located in Nueva Las Condes, the city's business district.
Hilton currently has three properties open in Chile: DoubleTree by Hilton Hotel Santiago-Vitacura, Hilton Garden Inn Santiago Airport and Hilton Garden Inn Iquique, and two additional projects in Chile: Hampton by Hilton Antofagasta and Hilton Santiago Las Condes.


text:
Commissioner Wayne Lewis Holds Press Conference, KEA Responds
FRANKFORT, Ky. (LEX 18)– Kentucky Education Commissioner Wayne Lewis said he wanted to clarify his recent demands for the names of teachers who called in sick.
Commissioner Lewis referenced the high number of absences during teacher sick outs in the last few weeks during the legislative session.
An email was sent to 10 school districts Thursday: Bath, Boyd, Bullitt, Carter, Fayette, Jefferson, Letcher, Madison, Marion, and Oldham.
The email requested the following:
1.)The names of all teachers that called in sick for February 28, March 5-7, and/or March 12-14, 2019 and the day(s) for which each teacher called in sick;
2.) Any and all affidavits or certificates of a reputable physician stating that the employee was ill or caring for an ill family member on the days the employee called in sick, as required by KRS 161.155 for the granting of any sick leave; and
3.) Documentation of the district's policies, procedures, and/or protocols for collecting sick leave affidavits or certifications and verifying qualification for the granting of sick leave.
Lewis said he did not ask for names of teachers who went to Frankfort, only the names of those who called in sick to school.
"Teachers have the right to protest at the capitol, but teachers do not have the constitutional right to call out sick when they are not sick," said Lewis.
The Kentucky Education Association released the following statement, "At his press conference earlier today, Commissioner Lewis said that his demand for the names of educators who called in sick on specific days of the legislative session is not a scare tactic. But all educators, including the school administrators who received the Commissioner's "requests" through his general counsel, know better. By requesting this information through his attorney and citing his broad investigatory and subpoena powers under KRS 156.210, the Commissioner is using the power of his office to compile an "enemies list" of educators for the Bevin Administration. This tactic serves only one purpose — to intimidate educators, public employees, school board members and superintendents across the Commonwealth and keep them from speaking out. Each school district determines their sick leave policies and practices in compliance with the applicable statute. Commissioner Lewis claims that he wants it to remain a local issue, but his heavy-handed exercise of state oversight authority speaks otherwise. We also want to believe the Commissioner when he says that it's not his goal to discipline individual teachers who called in sick. But if that's not his goal, then what is he trying to achieve? Who is he targeting? And why now?"
Kentucky Education Commissioner Wayne Lewis is expected to hold a press conference.
Posted by LEX 18 on Friday, March 15, 2019


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Lotus is celebrating their Formula 1 heritage with a special edition Lotus Exige S car known as the Lotus Exige LF1.
According to a press release issued today by Lotus, there will be 81 of these cars created, each with a special livery commemorating the number of grand prix victories that Lotus has achieved.
The LF1's exterior is highlighted by unique 'LF1' badging and a black and gold paint scheme with red accents, paying homage to the racing liveries of the 1970s and 1980s. The most striking exterior modification comes in the form of 17″ front and 19″ rear cast alloy wheels, each finished in gold and wrapped in Pirelli P-Zero Trofeo tires.
The black and gold color scheme is also found in the car's cabin thanks to the bespoke fabric, leather and contrast stitching and piping. Embroidered 'LF1' logos can be found on the headrests and floormats, while the Lotus F1 Team's logo is embroidered on the seat backs.
"Lotus' DNA is borne of the Formula One® arena and with 65 years of motorsport racing experience, 40 of those in F1®, it's fitting we celebrate our sporting origins and our race successes with the new Exige LF1," explained Lotus CEO Jean-Marc Gales in the press release. 81 limited edition cars, each one an acknowledgment to a different Lotus' 81 Grand Prix victory, this car is intended for the truest of Lotus' racing fans.
The right-hand drive Lotus Exige LF1 will make its debut at the Canary Wharf Motorexpo in London, England, taking place June 9-15, while the left-hand drive version will be on display at the Le Mans 24 from June 11-15.


text:
Tile Bluetooth trackers are up to 35 percent off at Amazon
The holiday sale brings the 2022 Tile Sticker down to only $20.
Valentina Palladino|@valentinalucia|December 6, 2022 9:05 AM
Whether you or someone you love is prone to losing things, or you just want the peace of mind of having a digital helper keep track of your stuff, Tile trackers are solid, inexpensive options that work with both iOS and Android devices. The company is having a holiday sale on Amazon at the moment that knocks 35 percent off a number of its 2022 trackers and bundles. You'll find the biggest discount on the Tile Sticker, which is down to just under $20, and you can get a pack of three Tile Mates for $53 or a pack of one Tile Mate and one Tile Slim for $40.
Tile Sticker (2022)
Tile's holiday sale knocks 35 percent off the Sticker tracker, bringing it down to just under $20.
All of the Tile devices included in this sale work similarly, but they have different designs. With the Tile Sticker, you're getting the company's Bluetooth tracking technology in a tiny disk with adhesive on one side that allows you to stick it to things like your TV's remote, your eyeglass case and other important items. You can check the Tile's last known location using its companion mobile app, and if you're within its 250-foot Bluetooth range, you can use the app to force the device to ring so you can more easily find your stuff. That works in the other direction as well — if you have your Tile on hand, you can double-press its button to ring your phone in the event you misplace that important piece of tech.
Since most features are common throughout the entire Tile lineup, you'll just have to decide which form-factor is right for you. The Sticker has the benefit of adhesive, while the standard Tile Mate has a keyring hole that makes it easy to attach to your keys. The Tile Slim is roughly the size of a credit card, making it ideal for keeping track of your wallet.
At this point, there are a number of other Bluetooth trackers on the market. Apple's AirTags are arguably the most popular, and Samsung makes its own trackers for its smartphones, too. But Tile devices are convenient because they'll work on both iOS and Android devices, so if you're thinking of gifting the trackers for the holidays this year, you don't have to know the type of phone your giftee has ahead of time.
Arguably the biggest downside to Tile trackers (if you're not already put off by privacy concerns) is that most of them have three-year, non-replaceable batteries. Tile does make one model, the Tile Pro, that runs on a one-year, replaceable battery, extending the life of the device as a whole. While it doesn't show up in Tile's Amazon sale, it too has been discounted to $25 for the holiday shopping season. If you're willing to shop elsewhere, you can even get a three-pack of Tile Pros with gift sleeves for $75 at QVC.
Buy Tile Pro at Amazon - $25 Buy Tile Pro (3 pack) at QVC - $75
Follow @EngadgetDeals on Twitter and subscribe to the Engadget Deals newsletter for the latest tech deals and buying advice.
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text:
<?php

declare(strict_types=1);

namespace OpenCFP\Test\Integration\Domain\Model;

use OpenCFP\Domain\EntityNotFoundException;
use OpenCFP\Domain\Model\Airport;
use OpenCFP\Test\Integration\WebTestCase;

final class AirportTest extends WebTestCase
{
    /**
     * @var Airport
     */
    private $airports;

    protected function setUp()
    {
        parent::setUp();

        $this->airports = new Airport();
    }

    /** @test */
    public function it_queries_airports_table_by_code()
    {
        $airport = $this->airports->withCode('AAC');

        $this->assertSame('AAC', $airport->code);
        $this->assertSame('Al Arish', $airport->name);
        $this->assertSame('Egypt', $airport->country);
    }

    /**
     * @test
     */
    public function it_squawks_when_airport_is_not_found()
    {
        $this->expectException(\Exception::class);
        $this->expectExceptionMessage('not found');

        $this->airports->withCode('foobarbaz');
    }

    public function testItIsNotCaseSensitive()
    {
        $airport = $this->airports->withCode('aac');

        $this->assertSame('AAC', $airport->code);
        $this->assertSame('Al Arish', $airport->name);
        $this->assertSame('Egypt', $airport->country);
    }

    public function testItThrowsTheCorrectError()
    {
        $this->expectException(EntityNotFoundException::class);
        $this->expectExceptionMessage('not found');

        $this->airports->withCode('foobarbaz');
    }
}

redpajama_set_name: RedPajamaGithub


Executive Inn Kilgore hotel is 3 miles from Kilgore Oil Museum, Kilgore downtown and Kilgore College. Popular attractions such as our annual Great Texas Balloon Race and Lone Star Speedway are just 5 miles away. Executive Inn is surrounded by many restaurants including; Kilgore Café, Kobee Japanese Steak House, Chili's, and Johnny Cace's Seafood and Steak House.
The Inn features 51 rooms, offering both smoking and non-smoking rooms of which 31 are King Bed & 20 two double beds. All guest rooms are generously equipped with Wireless Internet access, coffee pot, hair dryers and dish network TV. The Inn is a great place to stay for just overnight rest or long term place to seek home like atmosphere.


Is Revoice LGBTQ+ Thriving Through Cover Up in PCA Missouri Presbytery?
Evangelical Deep State Harnessing the Resources of the Church to Create "The World Globalists Want"
ERLC Family Conference Redefines "Family" as the "LGBT Inclusive Church"
The Stain of Revoice: Is Heritage Foundation Also Revoiced?
Beyond Revoice: Recruiting Youth Pastors, Targeting Your Teens!
Collapsing Conservative America: Will Evangelical Leaders Succeed in Selling out the American Church?
Make the Gospel Great Again: Reject the False Gospel of "LGBT+ Queer Christianity"
Will the Next President of the SBC Condemn or Condone Revoice "LGBT+ Queer Christianity"?
Where is the Conservative Evangelical Voice in Today's Cultural Tsunami?
Rev. Thomas Littleton is a minister and evangelist with over 35 years experience on university campuses and in urban centers across America, Europe and Latin America. He has engaged in outreach among teens and youth gatherings during Spring/ Summer Break and among the arts culture. Rev. Littleton worked for over a decade during the height of the unchecked AIDS epidemic in New York City from the mid 80s to 90s. He has witnessed firsthand both the destructive impact of the homosexual lifestyle on individuals and the overpowering influence of activists on New York City government. His time in New York City also provided a window on the global push for what has become LGBTQ "Human Rights" and the sexualization of education for K4-12.


The fixed costs include a normal $4,500 allocation for in-house design costs, although no in-house design will be done. Instead, the job will require the use of external designers costing $9,250. What is the total amount to be included in the calculation to determine the minimum acceptable price for the job?
(TCO 1) Who are the users of managerial accounting information? How does their use of accounting information differ from the users of financial accounting information?
that its employees will work 400,000 direct labor hours during the coming year. Total overhead costs are estimated to be $9,600,000 and direct labor costs are estimated to be $12,500,000. Direct Labor hours are actually 450,000.
If Wolf Co. allocates overhead based on direct labor HOURS, what is the predetermined overhead rate?
(TCO 3) The Mixing Department is the third department in the MZS Inc. factory. During January, there were 4,000 units of beginning inventory in the Mixing Department, and 80,000 units were transferred in from the prior process. There were 8,000 units in ending inventory. The transferred-in cost in the beginning inventory was $170,000 and there was $600,000 in transferred-in cost during the month.
What is the cost per equivalent unit for transferred-in cost?
(TCO 4) Assume that we are manufacturing a product and assume that the sales price per unit is $80, the variable cost is $20 per unit, and the fixed cost is $90,000; a) how many units would we need to sell to break even? b) How many units would we need to sell to earn a profit of $120,000? c) How many units do we need to sell to double that profit to $240,000? D) Why didn't the number of units double from Part B to Part C?
(TCO 5) Sivan Co. manufactures and sells one product. For the year, they started with no opening inventory; produced 100,000 units, but only sold 70,000 units. The selling price per each unit is $60.
(a) Prepare the Income Statement using Absorption Costing.
(b) Prepare the Income Statement using Variable Costing.
What is the a) estimated variable cost per machine hour and what is the b) estimated TOTAL fixed cost?
An outside supplier has offered to supply parts to the North Company for only $21.25 per part.(it appears to the President of the company that he could save $2.75 per unit.
100 percent of the traceable or avoidable fixed manufacturing cost is supervisor salaries and other costs that can be ELIMINATED if the parts are purchased. The decision to buy the parts from the outside supplier would have no effect on the common fixed costs of the company, and the space being used to produce the parts would otherwise be idle. Ignore the impact of income taxes in your calculation.
outside supplier rather than making them inside the company?
(TCO 9) Harry Corp buys equipment for $194,000 that will last for 9 years. The equipment will generate cash flows of $36,000 per year and will have no salvage value at the end of its life. Ignore taxes. Use 10% required rate of return.
(a) What is the Present Value (PV) of this investment (at 10%)?
(b) What is the NET Present Value (NPV) of this investment Should you buy the equipment if you need 10%?
(c) What is the Internal Rate of Return (IRR) of this investment?
(d) What is the payback period?
(TCO 10) Tanya Corp sells its products on both credit and cash basis. Monthly sales are sold 20% for cash, 80% for credit. Credit sales are collected 65% in the month of sale and 35% the following month. Sales for the first quarter are BUDGETED as follows: January $300,000; February $200,000; March $300,000.
Compute cash collections budgeted for February. How much cash was collected in the month?


Prices offered in a number of major wine regions this year were not enough to cover the cost of growing the crop and vines have since been removed. The Riverina Winegrapes Marketing Board in (RWGMB) southern New South Wales represents more than 300 growers in the Murrumbidgee Irrigation Area and surrounding districts. RWMG CEO Brian Simpson said many of its members were stretched financially from years of low prices. However, he is encouraged by early price signals. "The domestic focused wineries, once one moves, they're all going to jump and they're all going to push to increase," he said.


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SA can ill-afford to overlook the non-profit sector in its response to Covid-19
Shelagh Gastrow
The networks provided by non-profit organisations reach deeply into communities. In the 'new world' in which we will find ourselves, we will need the NPO sector even more to ensure that vulnerable people and all communities receive the support they need.
The UK government has announced that the charity sector in that country will receive a £750-million bailout to ensure that they can continue to operate during the Covid-19 pandemic. Of that, £360-million will be allocated to organisations that provide "key services", including victims' services, citizens' advisory offices and hospices, while the balance will support smaller organisations, including funds from their lotteries. In addition, the UK government will match donations from the public towards the independent National Emergencies Trust that is fundraising and distributing funds to charities during this crisis.
All the above is in recognition of the fact that charities play a "crucial role in the national fight against Covid-19, supporting those who are most in need". Rishi Sunak, Chancellor of the Exchequer, said: "For them, shutting up shop at this moment would be to contravene their very purpose, their entire reason to exist. Those charities have never been more needed than they are now, and they've never faced such a sudden fall in their funding."
In South Africa, we do not have reliable statistics on the size and scope of the non-profit sector. There are just under 250,000 registered with the Non-Profit Directorate in the Department of Social Development, but as this is voluntary, there are probably many unregistered organisations.
Our non-profits function in almost every sector imaginable, and in this current crisis, they have innovated and adapted to ensure services for the most vulnerable, while monitoring human rights. In fact, the country cannot do without them. While funds for welfare and food have been made available by the government, business and the donor community, it is the non-profit sector that is at the front line. Community-based organisations know their communities, they know who needs help, where the child-headed households are and they usually have a level of credibility and trust.
For example, the HCI Foundation recently raised over R4-million through the eMedia Covid-19 Relief Fund, which was publicised and campaigned for on eNews. They will be partnering with their network of non-profits to ensure distribution of food. The Solidarity Fund is partnering with the Social Change Assistance Trust to distribute food in rural areas in the Eastern Cape. It is far more difficult, for example, for business to arrive in informal settlements to provide food. This is not just a short-term function during lockdown.
The networks provided by non-profit organisations reach deeply into communities and in the "new world" in which we will find ourselves, we will need the NPO sector even more to ensure that vulnerable people and all communities receive the support they need. This can relate to food and health, but also to gender-based violence, education, child care and issues pertaining to community needs such as water, sanitation, shelter and of course, access to legal aid clinics and legal support if required.
Importantly, the civil society sector should be part of engagement with the government around new developmental and economic policies. We should actually be quantifying what non-profits save the state and the taxpayer by providing the services that they do.
Yet, despite the critical role this sector is playing, the substantial funds that have been established to see SMMEs through this crisis do not include the non-profit sector. There is somewhat of a misplaced idea that charities (a sad word to describe what our non-profits do), don't deserve to be supported, that they are as needy as the people they help and should operate on a shoestring.
How many people or companies providing funds for food have considered the cost for organisations to do the distribution? The personnel, the transport, the protective gear, the communications costs, the financial accountability involved are not necessarily catered for. Somehow, they must just find these resources. Who is providing support to non-profit staff who are at home, who may be retrenched?
Nobody seems to be thinking about what it would mean to this country if thousands of non-profits floundered and closed. We don't know how many people they employ, but it is substantial, and yet it is a sector that is totally overlooked.
Currently, most non-profits do not have reserves on which to rely. They cannot withhold dividends as this is not part of their financial model. Many donors do not like the idea of reserves and occasionally don't fund organisations that have them. This crisis has been a lesson in terms of non-profit financial modelling that reserves are essential, especially as it is this sector that is at the coalface of crises and needs to be flexible if necessary.
While the philanthropy sector has been extremely accommodating with grants already made to non-profits by agreeing that their restricted programmatic funds can be used for general purposes, this is not enough to sustain them. It is extremely difficult to fundraise in the current climate and donors are inundated with requests for support. Corporates are tightening their social investment budgets as the stock market has fallen dramatically and many companies are under lockdown. There is therefore no fat in their budgets going forward. Small businesses are collapsing, and many individuals no longer have extra money to donate.
There is no doubt that some key organisations will be closing as the economic downturn will mean a decline in funding. According to Richard Hebditch, Director of External Affairs for the Association of Charitable Foundations in the UK, "a financial and operational risk burden" is falling "disproportionately on NGOs and other implementing partners".
When it comes to government funding for non-profits providing services such as those dealing with welfare, the government should be taking care to ensure that funds are paid on time to avoid inefficiency or delays. There is no indication on the lotteries website that it is open for business, but this should be a key player in sustaining organisations and by now should have put out an urgent call to fund organisations assisting in Covid-19 service delivery. The requirements of our communities are no doubt going to escalate and at this time, we simply cannot afford to lose our active non-profit sector which is front and centre of the humanitarian response.
This then appears to be placing the burden on everyone to keep the sector open. When exploring the support for SMMEs, there seems to be a lack of understanding that non-profits are SMMEs, with a different financial model. This model means that any profits are not paid out to shareholders, but go back into the organisation for social good. Many NPOs are highly professional and are run as well as any business. They are registered, they have governing boards, they are audited, they employ people, they are accountable to their stakeholders and increasingly generate their own income through services, reducing reliance on the donor community.
They are entrepreneurial and founders take the same risks as anyone establishing a company. They comply with labour legislation and other government requirements. Why their lack of profit-seeking precludes these entities from financial support is anathema to me. We know business cannot save the world and care should be taken not to sideline organisations that play a social role as we now see how indispensable they are in times of crisis. Without the Treatment Action Campaign, the HIV crisis would have resulted in many more deaths. And so it is now that our non-profits continue to play a key role.
The only fund that has taken note of the quandary that nonprofits find themselves in is the emergency fund established by the Charities Aid Foundation Southern Africa (CAFSA). This has been set up to support local organisations and has been supported by Oppenheimer Generations. A call for proposals from organisations that are distributing food has been made. Yet we need to go further and understand how important the viability of this sector is going forward and how it can communicate with government and the public at large about how it plays a critical role in our survival as a people and a country.
Shelagh Gastrow provides advisory services to the philanthropy sector, higher education advancement and non-profit sustainability. She works with individuals and families on how to integrate their wealth and their values into meaningful and effective philanthropy. From 2002-2015 she was founder and executive director of Inyathelo and focused her efforts on strengthening civil society and universities through programmes to develop their financial sustainability whilst promoting philanthropy in SA. Her work has gained public recognition locally and internationally. http://www.gastrow.co.za
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