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redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

The study of the initial condition of the system produced by high energy collisions and of its evolution
to a quark-gluon plasma (QGP) is one of the most difficult but interesting problems related to the physics of
relativistic heavy ion collisions (RHICs), as well as to that of high energy proton-proton (pp) and
proton-nucleus (pA) collisions. If the energy of the collision is very large then the two colliding nuclei
in the backward light cone can be described within the color-glass-condensate (CGC) effective theory
\cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd},
see~\cite{Gelis:2010nm,Iancu:2003xm,McLerran:2008es,Gelis:2012ri} for reviews:
because of Lorentz contraction along the flight direction the nuclei appear as thin sheets of a colored glass,
in which the fast partons dynamics is frozen by time dilatation and these degrees of freedom act as static sources
for low momentum gluons: their large occupation number allows to treat them as classical fields, in particular as
transverse color-electric and color-magnetic fields.

Immediately after the collision the situation changes quite drastically. As a matter of fact,
the solution of the classical Yang-Mills (CYM) equations in the forward light cone shows that
new fields are
formed \cite{Kovner:1995ja,Kovner:1995ts,Gyulassy:1997vt,Lappi:2006fp,Fukushima:2006ax,
Fries:2006pv,Chen:2015wia,Fujii:2008km,Krasnitz:2000gz,Krasnitz:2003jw,Krasnitz:2001qu},
because of the non abelian interaction of the two CGC
sheets: these fields are called the
Glasma \cite{Lappi:2006fp} and are characterized by
having, at initial time, nonvanishing longitudinal components
while the transverse components are zero.
The Glasma fields connect the effective color charges
that are formed on the transverse plane of the
two colliding nuclei after the collision,
which are opposite on the two light cones $x^0 = \pm x^3$ as it should be
in order to generate longitudinal fields  \cite{Lappi:2006fp}.
These fields are classical since they are characterized by a large gluon occupation number,
typically $A_\mu^a \simeq 1/g$ with $g$ the QCD coupling.
The classical theory that describes the evolution of these fields is named the Classical Yang-Mills  (CYM) theory.

On the top of the Glasma it is possible to add
quantum fluctuations \cite{Romatschke:2005pm,Romatschke:2006nk,Fukushima:2011nq,
Fukushima:2013dma,Iida:2014wea,Gelis:2013rba,Epelbaum:2013waa,Tanji:2011di,Ryblewski:2013eja,Ruggieri:2015yea,
Berges:2012cj,Berges:2013fga,Berges:2013lsa,Berges:2013eia}
which are known to trigger plasma instabilities and are helpful to produce entropy during the early stage
of high energy nuclear collisions.
Quantum fluctuations appear when one considers the finite coupling corrections to the Glasma solution which is obtained
in the small coupling limit;  the spectrum of these fluctuations has been computed within a perturbative calculation in~\cite{Gelis:2013rba}
and it has been shown that they affect both the gauge potential and the color electric field.
Although quantum in nature, it is interesting to study what are the predictions of the CYM theory
on the evolution of the system made of a Glasma plus the fluctuations, although the application of the classical theory
is justified only if the amount of fluctuations is small in comparison with the background classical field.
In this article we will follow \cite{Romatschke:2005pm,Romatschke:2006nk,Iida:2014wea,Berges:2012cj}
and we will consider for the sake of simplicity
the fluctuations of the color electric field only, neglecting those of the gauge potentials
that will be considered in future works.

We present here some result on the role of classical fluctuations on the isotropization of Glasma.
The quantity that we will consider for this problem is the ratio of the longitudinal over the
transverse pressures, $P_L/P_T$, which has been the subject of studies on the onset of the
hydrodynamical regime in the early stage of relativistic nuclear collisions
\cite{Romatschke:2005pm,Romatschke:2006nk,Fukushima:2011nq,Gelis:2013rba,
Ryblewski:2013eja,Ruggieri:2015yea,Li:2016eqr,Kurkela:2015qoa,Bellantuono:2015hxa}.
We consider here the Glasma in a three-dimensional static box with periodic boundary conditions, neglecting for simplicity the
longitudinal expansion that will be considered in a next work: the longitudinal expansion is necessary
to describe the early stage of the system created in realistic collisions but focusing on the static box
allows for a clear intepretation of the physical results without the complication of the dilution of the fields.

We will follow closely the work of \cite{Iida:2014wea} in which gaussian fluctuations have been studied
in relation to entropy production, but almost no emphasis has been put on isotropization.
Moreover, our work has some link to \cite{Fukushima:2013dma} in which a similar problem has been considered
but within a simplified model of fluctuations.
Instead of insisting to study the evolution of the system up to very late times as in
\cite{Fukushima:2013dma,Iida:2014wea} we will focus on the very early stage,
which is the only time range in which the description based on CYM
has some phenomenological interest for high energy nuclear collisions.

We can anticipate one of our results, namely that
as long as fluctuations carry a not so large part of the initial energy of the Glasma, their impact on the
evolution of the longitudinal over transverse pressure is marginal in the early stage.
This result will hardly be modified by the longitudinal expansion because it is known that the latter
can only lower the amount of isotropy as well as delay the isotropization \cite{Ruggieri:2015yea}.
We also find that increasing the amount of energy carried by the fluctuations it is possible to obtain
a fairly isotropic system within a short time range: most likely this is due to the saturation of the instabilities
as mentioned in \cite{Fukushima:2013dma,
Romatschke:2005pm,Romatschke:2006nk,Fukushima:2011nq}.
Due to our limitated computing power we do not perform long time running simulations here,
therefore we are not able to put a firm statement on the fact that instabilities rather than a collisional dynamics lead to
the quick isotropization in cases with substantial fluctuations: in this article we limit ourselves to present few results
about the isotropization (or missed isotropization) in the early stage and how the fluctuation size affects the evolution
of $P_L/P_T$, leaving more specific studies to a near future work.

Besides the pressures there are other quantities that are interesting to understand
the evolution of the color fields produced in high energy nuclear collisions: correlation functions of the classical color fields
represent an example of these . The correlators have been studied in several
articles before \cite{Fujii:2008km,Dumitru:2013wca,Dumitru:2014nka} and it has been shown that
at $\tau=0^+$ the Glasma presents a very short correlation magnetic length, $\lambda_M Q_s \ll 1$: this means that a description
of the initial stage in terms
of uncorrelated color strings is appropriate.
Moreover, the investigation of the gauge invariant
magnetic correlators shows that for $Q_s x_T \gtrsim 1$ the Glasma fields present anticorrelation, which
roughly speaking implies
that in domains of transverse area $A_T\approx Q_s^2$ the magnetic field flips its sign \cite{Dumitru:2014nka}.
On the other hand,  for $\tau = O(1/Q_s)$ the anticorrelation disappears and the correlation length increases:
the interaction tends to align the magnetic field and to correlate the fields in domains with $A_T\approx Q_s^2$.

Another problem that we study in this article
is the impact of fluctuations on the evolution of the field correlators.
Once again we consider only the static box case: the longitudinal expansion
can dilute the correlators but cannot affect drastically their structure in the transverse plane, that is the one
we consider here. Also in this case we can anticipate our main results, namely that
fluctuations do not change drastically the picture described above.
Another novelty that we bring here is the study of the longitudinal correlation functions:
these are important to understand if and how the color strings experience some loss of correlation
along the longitudinal direction during the CYM evolution. We anticipate our main result on this, namely that
even in absence of fluctuations the evolving Glasma experiences a partial loss of correlation
on the long distance within a very short time. We call this as the gauge invariant partial string breaking.
We dub this as the partial string breaking since we find some residual correlation also at large distance
for $g^2\mu t\simeq 10$, although this correlation is smaller than the one we have at a very small distance;
moreover, we specify that this is gauge invariant since it is related to the calculation of the gauge invariant
correlation function, while the naive gauge dependent correlator would be $z-$independent at any time.
We also study the effect of fluctuations on the longitudinal correlations: we find that string breaking
is accelerated by the fluctuations and waiting for a sufficiently long time fluctuations lead to
the complete breaking of the color strings.

The plan of the article is as follows: in section II, we review the initial condition that we implement in
the calculations, as well as the spectrum of the fluctuations and the CYM equations that we solve to study the
evolution of the Glasma. In section III, we present results about pressures evolution with and without
fluctuations. In section IV, we discuss the correlation functions of the Glasma fields in the transverse plane.
Finally, in section V we draw our conclusions.

\section{Glasma and classical Yang-Mills equations}
In this section we briefly review how the Glasma is built up within the McLerran-Venugopalan (MV)
model \cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd,Kovchegov:1996ty} and gaussian fluctuations
are added on the top of it,
then how this initial condition is evolved by means of the CYM equations.

\subsection{The Glasma\label{sec:glasma}}
In the MV model, the color charge densities $\rho_a$ that act like the static sources of the CGC fields
in the two colliding nuclei are assumed to be random variables that are
normally distributed with zero mean
and variance specified by the equation
\begin{equation}
\langle \rho^a(\bm x_T)\rho^b(\bm y_T)\rangle = (g^2\mu)^2 \delta^{ab}\delta^{(2)}(\bm x_T-\bm y_T);
\label{eq:dfg}
\end{equation}
$a$ and $b$ denote the adjoint color index;
in this work we limit ourselves for simplicity to the case of the $SU(2)$ color group therefore
$a,b=1,2,3$.
In Eq.~(\ref{eq:dfg}) $g^2\mu$ is the only energy scale of the model that is related to the saturation momentum $Q_s$:
lattice calculations of the Wilson line correlations for deep inelastic scattering show that
$Q_s / g^2\mu$ in the MV model lies approximately in the range (0.6,1.2) \cite{Lappi:2007ku},
while for realistic high energy nuclear collisions the relation between the two quantities is less clear
because one has to sum up over gluons produced with different transverse momenta, hence related
to partons with different values of $x$ in the nuclear wave function; in this case
$Q_s/g^2\mu\approx 0.6$ for nuclear collisions at the RHIC energy \cite{Lappi:2007ku}.
Because of the $\delta^{(2)}(\bm x_T-\bm y_T)$ the fluctuations of the color charge
density in the two CGC sheets are uncorrelated in the transverse plane.

On the lattice Eq.~(\ref{eq:dfg}) is implemented by distributing the fluctuating color charges with
variance given by
\begin{equation}
\langle \rho^a(\bm x_T)\rho^b(\bm y_T)\rangle =
(g^2\mu)^2 \delta^{ab}
\frac{1}{a^2}\delta_{\bm x_T,\bm y_T},
\label{eq:poi1}
\end{equation}
where $a\equiv L_x/N_x=L_y/N_y$ is the lattice spacing, with
$L_x$, $L_y$ corresponding to the physical length of the lattice in the $x$ and $y$ directions respectively,
while $N_x$, $N_y$ denote the number of cells in the $x$ and $y$ directions;
in this work we assume that $L_x = L_y$  and $N_x = N_y$. In the lattice implementation we remove the zero mode
from the color charge density: physically this amounts to require that the net color charge carried by the
distribution is vanishing; this is achieved by Fourier transforming
$\rho_a(\bm x_T)$ for each $a$, then constructing a new density $\tilde\rho_a(\bm x_T)$
summing over all the Fourier modes but the zero mode.
For the sake of notation we denote in the following by $\rho_a(\bm x_T)$ the density obtained in this way,
keeping in mind that it corresponds to a charge distribution that has been color neutralized.

The static color sources $\{\rho\}$ generate CGC fields that can be computed as follows.
Firstly we  solve the Poisson equations for the gauge potentials
generated by the color charge distributions of the nuclei $A$ and $B$, namely
\begin{equation}
-\partial_\perp^2 \Lambda^{(A)}(\bm x_T) = \rho^{(A)}(\bm x_T)
\end{equation}
(a similar equation holds for the distribution belonging to $B$). The Wilson lines are then computed as
\begin{equation}
V^\dagger(\bm x_T) = e^{i \Lambda^{(A)}(\bm x_T)},~~~
W^\dagger(\bm x_T) = e^{i \Lambda^{(B)}(\bm x_T)},
\end{equation}
and the pure gauge fields of the two colliding nuclei are given by
\begin{equation}
\alpha_i^{(A)} = i V \partial_i V^\dagger,~~~\alpha_i^{(B)} = i W \partial_i W^\dagger.
\label{eq:pgp}
\end{equation}
In terms of these fields the solution of the CYM in the forward light cone
at initial time, namely the Glasma gauge potential,
can be written as \cite{Kovner:1995ja,Kovner:1995ts}
\begin{eqnarray}
&&A_i = \alpha_i^{(A)} + \alpha_i^{(B)}~,~~i=x,y, \\
&&A_z = 0,
\end{eqnarray}
and the initial longitudinal Glasma fields are
\begin{eqnarray}
&& E^z = i\sum_{i=x,y}\left[\alpha_i^{(B)},\alpha_i^{(A)}\right], \label{eq:f1}\\
&& B^z = i\left(
\left[\alpha_x^{(B)},\alpha_y^{(A)}\right]  + \left[\alpha_x^{(A)},\alpha_y^{(B)}\right]
\right),\label{eq:f2}
\end{eqnarray}
while the transverse fields are vanishing\footnote{
It is worth mentioning that in the more correct implementation of the MV model the gauge potentials are
computed as path ordered
exponentials of multiple layers of color charges, describing the propagation
of a colored probe through a thick nucleus
\cite{Lappi:2007ku}, while in the present work we limit ourselves to consider a single layer of charges.
We will implement the sources with a finite thickness in our future works.}.

On the top of the Glasma specified by Eqs.~(\ref{eq:f1}) and~(\ref{eq:f2}) it is possible to add gaussian
fluctuations \cite{Romatschke:2005pm,Romatschke:2006nk,Fukushima:2011nq,Iida:2014wea}
which are known to trigger instabilities and might be helpful to achieve isotropization
as well as to produce entropy during the early stage of high energy nuclear collisions;
in this article we study the Glasma in a static box therefore we use
the definitions of \cite{Iida:2014wea}. Below we report the definitions of the longitudinal and transverse
fluctuations in the continuum limit: the ones on the lattice can be obtained by extracting the trivial scaling
of the physical quantities with the proper power of the lattice spacing $a$ and the derivatives with
central difference operators.
Firstly, we define the auxiliary random fields $\{\xi_i\}$, $i=x,y$, that are assumed to be white noise with variance
\begin{equation}
\langle\xi_i^a(\bm x_T)\xi_j^b(\bm y_T)\rangle = \delta_{ab}\delta_{ij}\delta^{(2)}(\bm x_T - \bm y_T);
\label{eq:xi1}
\end{equation}
in addition to these we consider a random fluctuation in the longitudinal direction, again normally distributed with
variance
\begin{equation}
g^2\mu \langle F(z)F(z^\prime)\rangle = \Delta^{2}\delta(z - z^\prime).
\label{eq:xi2}
\end{equation}
We notice that the $g^2\mu$ on the left hand side of the above equation has been introduced
in \cite{Iida:2014wea} to balance the inverse length dimension carried by the $\delta(z - z^\prime)$ on the
right hand side: this is necessary in the case of the static three dimensional box but it does not
appear the case of the expanding geometry \cite{Romatschke:2006nk,Fukushima:2011nq}
where $\delta(\eta - \eta^\prime)$, with $\eta$ corresponding to the space-time rapidity,
appears instead of $\delta(z - z^\prime)$.
The fluctuations of the electric field are then defined in the continuum limit as
\begin{eqnarray}
\delta E_i^a (\bm x_T,z) &=& \partial_z F(z) \xi_i^a(\bm x_T), \label{eq:f11}\\
\delta E_z^a (\bm x_T,z) &=& -F(z) D_i \xi_i^a(\bm x_T).\label{eq:f12}
\end{eqnarray}
These fluctuations allow to store part of the initial energy in the transverse
color electric field, as well as to break the longitudinal invariance (the boost invariance
in the case of the expanding geometry) that characterizes the Glasma fields in Eq.~(\ref{eq:f1}).

\subsection{Classical Yang-Mills equations}
In this subsection we describe how we study
the evolution of the initial conditions specified above.
The dynamical evolution that we study here is given by the
classical Yang-Mills (CYM) equations.
In this study we closely follow the work of \cite{Iida:2014wea} therefore we refer
to that reference for more details;
we write below the relevant equations in lattice units therefore the
physical quantities do not carry any mass dimension.

The CYM hamiltonian density is given by
\begin{equation}
H = \frac{1}{2}\sum_{x,a,i}E_i^a(x)^2 + \frac{1}{4}\sum_{x,a,i,j}F_{ij}^a(x)^2,
\label{eq:H}
\end{equation}
where the magnetic part of the field strength tensor is
\begin{equation}
F_{ij}^a(x) = \partial_i A_j^a(x) - \partial_j A_i^a(x)  - \sum_{b,c}f^{abc} A_i^b(x) A_j^c(x);
\label{eq:Fij}
\end{equation}
here $\partial_i A_j^a(x) = (A_j(x+i)-A_j(x-i))/2$ corresponds to the central difference
operator in the $i^{\mathrm{th}}$-direction, $f^{abc} = \varepsilon^{abc}$ with $\varepsilon^{123} = +1$.
The equations of motion for the fields and conjugate momenta, namely the CYM equations, are
\begin{eqnarray}
\frac{dA_i^a(x)}{dt} &=& E_i^a(x),\\
\frac{dE_i^a(x)}{dt} &=& \sum_j \partial_j F_{ji}^a(x) -
\sum_{b,c,j} f^{abc} A_j^b(x)  F_{ji}^c(x).
\end{eqnarray}
We solve the above equations on a static box in three spatial dimension, using the common
fourth order Runge-Kutta method as in \cite{Iida:2014wea}.
Although the method used in this work sounds a bit primitive in comparison with
recent works that formulate the problem in terms of gauge links,
see for example \cite{Fukushima:2011nq,Gelis:2013rba},
the approach used here gives results that are in good agreement with these methods and
we leave the more rigorous implementation of the numerical problem based on gauge links
to a future work.

\section{Results: fields and pressures\label{sec:ruv}}
In this section we show our results obtained for the evolution of the Glasma in the case of a static box
with a square cross section of side $L$.
Our main goal is to study the color fields produced in high energy nuclear collisions,
although within a simplified static geometry:
we set the physical length of the box in the transverse plane  to $L = 2$ fm; moreover,
we consider $g^2\mu=1$ GeV.

We notice that although it would be possible to extend the numerical calculations in the static box up to
$g^2\mu t \gg 1$, in the real collisions the longitudinal expansion would dilute the fields thus invalidating the
classical approximation used here: previous studies with the expanding geometry show that dilution of the fields
becomes important for  $g^2\mu t =O(1)$, therefore for times
$g^2\mu t\gtrsim O(1)$ particle quanta dynamically evolving in interaction with  the classical fields should be
introduced, see for example \cite{Gelis:2013rba,Ryblewski:2013eja,Ruggieri:2015yea}.
For this reason in this study we will focus to times ranging up to
few units of $1/g^2\mu$, that in physical units correspond to $t\approx 2$ fm/c.

\subsection{Fields}

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{FIELDS.png}\\
\includegraphics[scale=0.3]{FIELDS_fluct.png}
\end{center}
\caption{\label{Fig:1}Glasma fields, averaged over the whole box, as a function of time measured in units of $g^2\mu$.
All the fields are measured in units of $(g^2\mu)^2$. Indigo lines correspond to the magnetic color fields and green lines
to the electric color fields. Moreover, solid lines denote longitudinal fields while dashed lines correspond to the
averaged transverse fields.
Upper panel stands for the case without fluctuations, while the lower panel collects the results obtained
adding gaussian fluctuations with the seed $\Delta^2=2.2\times 10^{-4} g^2\mu a$.
The results correspond to a grid with $N=71^3$ lattice sites with lattice spacing $a$ equal to
$g^2\mu a = 0.14$. }
\end{figure}

In Fig.~\ref{Fig:1} we plot the Glasma fields, averaged over the whole box, as a function of time measured in units
of $g^2\mu$.
All the fields are measured in units of $(g^2\mu)^2$. Indigo lines correspond to the magnetic color fields and green lines
to the electric color fields. Moreover, solid lines denote longitudinal fields while dashed lines correspond to the
averaged transverse fields.
The upper panel stands for the case without fluctuations, while the lower panel collects the results obtained
adding gaussian fluctuations with the seed $\Delta^2=2.2\times 10^{-4} g^2\mu a$.
We have averaged each component summing over color indices building up gauge invariant
quantities: for example we have defined
\begin{equation}
B_L^2 = \frac{1}{N}\sum_{i,j=-N_x}^{N_x}\sum_{k=-N_z}^{N_z}
\sum_{a=1}^3 \left[B_z^a(x_i,y_j,z_k)\right]^2,
\end{equation}
and
\begin{eqnarray}
B_T^2 &=&\frac{1}{2}\left( \frac{1}{N}\sum_{i,j=-N_x}^{N_x}\sum_{k=-N_z}^{N_z}
\sum_{a=1}^3
\left[B_x^a(x_i,y_j,z_k)\right]^2
\right.\nonumber\\
&& +\left.\frac{1}{N}\sum_{i,j=-N_x}^{N_x}\sum_{k=-N_z}^{N_z}
\sum_{a=1}^3  \left[B_y^a(x_i,y_j,z_k)\right]^2\right);
\end{eqnarray}
similar definitions hold for the color electric fields. In the above equations $N$ corresponds to the number
of lattice sites $N=(2N_x+1)\times (2N_y+1) \times (2N_z+1)$ with $N_x = N_y = N_z = 35$.

The results collected in the upper panel of Fig.~\ref{Fig:1} show that at the initial time
the system is made of purely longitudinal fields; however, in a time range $g^2\mu t \approx 0.5$ the bulk structure
of the fields is formed, with both longitudinal and transverse fields having reached their asymptotic value:
in fact going on with time the evolution does not change drastically the value of the fields.

In the lower panel of Fig.~\ref{Fig:1}  we show the evolution of the classical fields with gaussian fluctuations in the
initial condition. We notice the main effect of the fluctuations on the averaged fields is to have nonvanishing
transverse electric fields at the initial time. Besides this the evolution of the fields follows that measured in case
of the pure Glasma, with the bulk structure of the fields that is formed within $g^2\mu t \approx 0.5$.

\subsection{Pressures evolution}

For a system made of only classical color fields the longitudinal and transverse pressures are given by
\begin{eqnarray}
P_T &=&\frac{E_z^a E_z^a + B_z^a B_z^a}{2},\label{eq:uyt1}\\
P_L &=&-P_T+
  \frac{\bm{E}_T^a \bm{E}_T^a +\bm{B}_T^a \bm{B}_T^a}{2},
  \label{eq:uyt2}
\end{eqnarray}
with $\bm{E}_T^a \bm{E}_T^a = E_x^a E_x^a + E_y^a E_y^a$,
$\bm{B}_T^a \bm{B}_T^a = B_x^a B_x^a + B_y^a B_y^a$; isotropy requires $P_L = P_T$.

At initial time $P_L/P_T=-1$ in the pure Glasma, see also below, and a hydro description
of the fireball created in relativistic collisions would prefer that a nearly isotropic system is formed within a short time
range $\approx 1$ fm/c:
\footnote{A perfect isotropy is however not necessary, in fact it has been realized that
anisotropic hydro can succesfully describe the evolution of the quark-gluon plasma
even if at the initialization time the isotropy is not perfect, see
\cite{Martinez:2012tu,Martinez:2010sd,Strickland:2016ezq,Alqahtani:2017tnq,
Florkowski:2010cf,Florkowski:2013lza,Florkowski:2013lya,Ryblewski:2012rr} and
references therein.}
it is therefore interesting to address the question whether the classical evolution of the Glasma can
produce a fairly isotropic system within this short amount of time.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{pressures_D0.png}\\
\includegraphics[scale=0.3]{pressures_D30.png}
\end{center}
\caption{\label{Fig:3before}Pressures over energy density as a function of $g^2\mu t$.
Maroon lines in panel {\bf{(a)}} correspond to $P_L/\varepsilon$ (solid) and $P_T/\varepsilon$ (dashed) for the case without fluctuations;
orange lines  in panel {\bf{(b)}} correspond to $P_L/\varepsilon$ (dot-dashed) and $P_T/\varepsilon$ (dotted)
for the case with gaussian fluctuations with $\Delta^2 = 2.2\times 10^{-4}g^2\mu a$.
In both panels the lattice specifications as well as the value of $g^2\mu$ are the ones used for the
data shown in Fig.~\ref{Fig:1}.}
\end{figure}

In the upper panel of Fig.~\ref{Fig:3} we plot the
pressures as a function of $g^2\mu t$ for the cases of the pure longitudinal initial fields,
while in the lower panel we show the results for the case of gaussian fluctuations with $\Delta^2/g^2\mu a = 2.2\times 10^{-4}$
analogously to the data shown in  Fig.~\ref{Fig:1}.
In the figure, solid lines correspond to $P_L/\varepsilon$ and dashed lines to $P_T/\varepsilon$.

We firstly comment on the case $\Delta=0$.
At initial time the Glasma consists of purely longitudinal fields therefore
$P_L/P_T = -1$, see Eq.~(\ref{eq:uyt2}), and the system presents
a strong anisotropy. However, the Yang-Mills evolution tends to remove part of this initial anisotropy by making
the longitudinal pressure positive for $g^2\mu t \approx 0.15$. On the other hand, the interactions are not enough to
isotropize the system, indeed at later times a strong anisotropy remains with $P_L\ll P_T$
in agreement with the static box case studied in \cite{Fukushima:2013dma}
and similarly to what happens in case of a longitudinally expanding
geometry \cite{Fukushima:2011nq,Gelis:2013rba,Li:2016eqr}.

Turning on the gaussian fluctuations results in the increase of the
initial value of $P_L$, thanks to the presence of transverse electric fields in the initial state.
For the case $\Delta^2=2.2\times 10^{-4} g^2\mu a$ shown in Fig.~\ref{Fig:3before}
we find that the disturbances approximately
add a $30\%$ to the total energy.
We notice that fluctuations
are not enough to gain a considerable amount of isotropization within a short time range.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{press_ratio_revi.png}
\end{center}
\caption{\label{Fig:3}{\em Upper panel.} Pressures as a function of $g^2\mu t$.
Maroon lines correspond to $P_L/\varepsilon$ (solid) and $P_T/\varepsilon$ (dashed) for the case without fluctuations;
orange lines correspond to $P_L/\varepsilon$ (dot-dashed) and $P_T/\varepsilon$ (dotted)
for the case with gaussian fluctuations with $\Delta^2 = 2.2\times 10^{-4}g^2\mu a$.
{\em Lower panel.} Ratio $P_L/P_T$ for several values of $\Delta^2$.
In both panels the lattice specifications as well as the value of $g^2\mu$ are the ones used for the
data shown in Fig.~\ref{Fig:1}.}
\end{figure}

In Fig.~\ref{Fig:3} in which we plot $P_L/P_T$ for
several values of $\Delta$: in particular, the maroon solid line corresponds to $\Delta=0$;
the brown dashed line to $\Delta^2/g^2\mu a = 4.7\times 10^{-5}$ for which the energy carried by the fluctuations
is $\approx 10\%$ of the total energy; the dot-dashed orange line corresponds to $\Delta^2/g^2\mu a = 2.2\times 10^{-4}$
for which the energy carried by the fluctuations
is $\approx 30\%$ of the total energy; finally, the green dot-dot-dashed line denotes the case
$\Delta^2/g^2\mu a = 5\times 10^{-4}$
for which the energy carried by the fluctuations
is $\approx 50\%$ of the total energy.
We notice that
one of the effects of the  fluctuations is to increase the value of $P_L/P_T$
that is reached for $g^2\mu t \gtrsim 1$: in fact, for $\Delta=0$ the ratio of pressures
remains positive but below $\approx 0.3$ for almost all the range of time studied here, namely
$0.5 \lesssim g^2\mu t \lesssim 10$ that corresponds to the physical time range
$0.1~\mathrm{fm/c} \lesssim g^2\mu t \lesssim 2~\mathrm{fm/c}$.
Small fluctuations do not change significantly this result, see the brown dashed line in Fig.~\ref{Fig:3}.
For a more substantial amount of fluctuations the ratio of pressures in the aforementioned time range
is larger: for example, $0.3 \lesssim P_L/P_T \lesssim 0.5$ for $\Delta^2/g^2\mu a = 2.2\times 10^{-4}$
and $0.5 \lesssim P_L/P_T \lesssim 0.73$ for $\Delta^2/g^2\mu a = 5\times 10^{-4}$.
We thus obtain that in order to have a fairly isotropic system within a short time range
it is important to have a substantial amount of fluctuations in the initial stage.

Before going ahead we would like to mention that the results collected in Fig.~\ref{Fig:3}
lead to a picture that is consistent with the one discussed in \cite{Gelis:2013rba}. As a matter of fact,
quantum fluctuations have been considered in \cite{Gelis:2013rba} for two values of the QCD coupling, $g$, and it has been found that
for the relatively small value $g=0.1$ the fluctuations are not enough to gain a substantial amount of isotropization in the early stage,
while for the larger value $g=0.5$ the instabilities affect in a more substantial way the early evolution of $P_L/P_T$
and the final amount of isotropy grows up thanks to these instabilities. The results presented here confirm the scenario already anticipated
in \cite{Gelis:2013rba} and suggest that the early isotropization in the Glasma can be obtained via fluctuations-induced induced instabilities
if fluctuations are large enough.
Our numerical estimates suggest that if fluctuations carry about the $30\%$ of the initial total energy then at later times
$P_L/P_T$ stays between $0.3$ and $0.5$, while pushing the initial energy fraction to $50\%$ brings the amount of isotropy
between $0.5$ and $0.73$. Calculations with different values of $g^2\mu$ and $a$ will be performed in the near future in order to establish
the dependence of the results on these quantities.
It will be interesting in the near future to use long time simulations of the classical Yang-Mills dynamics
to explore the dependence of the isotropization times on the fluctuation seed: in the weak coupling scenario it is known that
the time scale for the excitation of the secondary instabilities, which are responsible for the rise of $P_L/P_T$ at very late
time, depends logarithmically on the seed \cite{Berges:2012cj}: it will be interesting to check whether this dependence remains also for
larger values of the seed.

\section{Correlation functions of the longitudinal fields}
In this section we compute the correlation functions of the longitudinal fields
in the evolving Glasma: these quantities are
important to understand the structure of the Glasma color strings and how this changes because of the CYM
evolution.
In particular, at the initial time the Glasma consists of longitudinal color strings that are (almost)
uncorrelated in the transverse plane; on the other hand, the CYM evolution is capable to increase the correlation
length in the transverse plane in a time range $\tau\approx 1/Q_s$.
The correlation functions of the magnetic field have been already studied in \cite{Dumitru:2013wca,Dumitru:2014nka}
for the case of the pure Glasma.
We present here the results for the magnetic as well as the electric correlations
in the transverse plane for the cases without and with fluctuations, although we have found that the fluctuations
do not have a substantial impact on the evolution of these correlators.
Then we present our results for the case of the correlations of the magnetic field along the longitudinal direction,
which have not been computed in \cite{Dumitru:2013wca,Dumitru:2014nka} and offer a way to discuss the string breaking
in the Glasma: again, we present both the results with and without fluctuations.
It is interesting to mention since now that this loss of correlation of the magnetic field at later times
comes purely from the gluon dynamics, since in the model that we implement in the initial condition
the fluctuations are added only on the top of the electric field, giving an infinite magnetic correlation length
at the initial time both with and without fluctuations.

\subsection{Correlation functions in the transverse plane}
In the continuum limit the correlator of the longitudinal magnetic field reads
\begin{eqnarray}
C_{M}(x_\perp)&=&\mathrm{Tr}\left\langle B_z(0)
U_{0\rightarrow x}B_z(x_\perp)U^{-1}_{0\rightarrow x} \right\rangle,
\end{eqnarray}
where $x_\perp$ corresponds to distance measured in the transverse plane and
we have defined $B_z = F_{12} = F_{12}^a T_a$. In the above equation
$U_{0\rightarrow x} =P e^{-i \int A_i dx}$ stands for the parallel transporter along a straight line that connects
the point $x$ with the point $x=0$, where $P$ denotes the path ordering.
In the continuum limit the path ordering would require to divide the path between two points
into infinitesimally small paths, then compute the matrix exponential of the integral along each of these
small paths and multiply these in order to build up the full path.
In the numerical calculations the smallest path is of course that connecting two lattice neighbours points
so the parallel transporter is implemented as follows:
suppose we aim to compute $U_{X_1\rightarrow X_{M}}$ along the $x$ direction between
two lattice points.
We then have to compute the $SU(2)$ matrix
\begin{equation}
U_{X_1\rightarrow X_{M}} = \prod_{j=1}^{M-1} e^{-i A_x(X_j)},
\label{eq:PathOrdering}
\end{equation}
where we remind $A_x$ is understood in lattice units and the argument of the exponential is a discretized
version of the integral $\int_{X_j}^{X_{j+1}} A_x dx$. Then we
use the generalized Euler identity for an $SU(2)$ matrix to compute each of the exponentials
in the path ordered expression~(\ref{eq:PathOrdering}).
For the electric field we compute a similar correlation function
replacing $B_z$ with $E_z$.
On the lattice we define the average of the quantity $O$ as
\begin{equation}
\langle O\rangle = \frac{1}{N}\sum_{x_j} O(x_j),
\end{equation}
where $x_j$ denotes the $j^\mathrm{th}$ lattice site
and $N$ is the number of lattice sites involved in the summation.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{correl_B_nofluct.png}\\
\includegraphics[scale=0.3]{correl_E_nofluct.png}
\end{center}
\caption{\label{Fig:3a}Color-magnetic (upper panel) and color-electric (lower panel) correlators
as a function of the transverse plane coordinate, for several values of time.
Maroon solid lines corresponds to the correlators at initial time; green dashed lines denote that
at $g^2\mu t = 2$ and blue dot-dashed lines correspond to $g^2\mu t = 10$.
Lattice specifications and physical parameters are the same used for the data in Fig.~\ref{Fig:1}.}
\end{figure}

In Fig.~\ref{Fig:3a} we plot the  magnetic (upper panel) and electric (lower panel) correlation functions
versus the transverse plane coordinate, for several values of time.
Maroon solid lines corresponds to the correlators at initial time; green dashed lines denote that
at $g^2\mu t = 2$ and blue dot-dashed lines correspond to $g^2\mu t = 10$.
Lattice specifications and physical parameters are the same used for the data in Fig.~\ref{Fig:1}.
We have checked that changing $g^2\mu$ does not change qualitatively
the results.

We begin by discussing the result at the initial time: in fact the data
for this correlator show a couple of interesting features. First of all we notice that the correlator
decays at short distances, becoming negative for $g^2\mu x\approx 0.4$.
Following \cite{Dumitru:2013wca} we fit this first portion
of the correlator by the standard two-dimensional screened propagator,
\begin{equation}
C_M(r) = \frac{A}{(m r)^{1/2}}e^{-m r},~~r=g^2\mu x,
\label{eq:2Dp}
\end{equation}
in order to estimate the value of the screening mass $m$ in units of $g^2\mu$: we find $m\approx 8.3$ that implies
a screening length for the magnetic modes, $\lambda_M$, equal to $g^2\mu\lambda_M\approx 0.11$.
This result shows the existence of very tiny correlation domains in the transverse plane for the Glasma,
with a transverse size much smaller than the characteristic length scale of the system given by
$1/g^2\mu$. The existence of this screening of the magnetic modes
is in agreement with the one found in \cite{Dumitru:2013wca}
where it has been related to the existence of sources and sinks of the magnetic field lines in the transverse plane.

The magnetic correlator at initial time becomes negative on a length scale,
$\lambda_A$, approximately given by $\lambda_A\approx  0.4/g^2\mu$: this result, in qualitative agreement with
the one found in \cite{Dumitru:2014nka}, shows an anticorrelation in the initial Glasma
that developes at $\lambda_A$, meaning that walking on the transverse plane over distances
of the order of  $\lambda_A$ it is possible to cross domains in which the magnetic field flips its sign.

On the other hand at later times the situation changes both qualitatively and quantitatively.
As a matter of fact, already for $g^2\mu t=2$ the anticorrelation disappears:
within a very short time range the CYM interaction has been capable to align the fields within domains with transverse area
$(g^2\mu)^2 A_T \approx 1$. Moreover, the fit with the two-dimensional propagator in Eq.~(\ref{eq:2Dp})
gives $g^2\mu\lambda_M\approx 0.22$ showing that the CYM interactions have enlarged
(in fact, doubled) the correlation domains
of the magnetic fields, albeit
these correlations domains are still microscopic since their spatial extension in the transverse plane
is quite smaller than the typical size of a nucleon.

For the color-electric correlator shown in the lower panel of Fig.~\ref{Fig:3a}  we can perform a similar analysis:
in this case we find that the initial correlation length is $g^2\mu\lambda_E\approx 0.82$;
the qualitative behavior of the electric correlator agrees with that of the magnetic one,
in fact we find that anticorrelation develops for $g^2\mu\lambda_A\approx  1$ at the initial time,
and this anticorrelation disappears already for $g^2\mu t=2$. The electric correlation length does not seem to be
very affected by the CYM evolution, indeed at $g^2\mu t=3$ we find $g^2\mu\lambda_E\approx 0.76$ which,
within numerical uncertainties, agrees with the value of the correlation length at $t=0$.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{correl_B_fluct_30.png}\\
\includegraphics[scale=0.3]{correl_E_fluct_30.png}
\end{center}
\caption{\label{Fig:3aa}Color-magnetic (upper panel) and color-electric (lower panel) correlators
as a function of the transverse plane coordinate, for several values of time and for a fluctuation seed
$\Delta^2 = 2.2\times 10^{-4} g^2\mu a$ that corresponds to that used in Fig.~\ref{Fig:1}.
Maroon solid lines corresponds to the correlators at initial time; green dashed lines denote that
at $g^2\mu t = 2$ and blue dot-dashed lines correspond to $g^2\mu t = 10$.
Lattice specifications and physical parameters are the same used for the data in Fig.~\ref{Fig:1}.}
\end{figure}

For completeness, in Fig.~\ref{Fig:3aa} we plot the color-magnetic (upper panel) and color-electric (lower panel) correlators
as a function of the transverse plane coordinate, for several values of time,
in the case of the Glasma plus gaussian fluctuations with the seed
$\Delta^2 = 2.2\times 10^{-4} g^2\mu a$ that corresponds to that used in Fig.~\ref{Fig:1}.
Maroon solid lines corresponds to the correlators at initial time; green dashed lines denote that
at $g^2\mu t = 2$ and blue dot-dashed lines correspond to $g^2\mu t = 10$.
Lattice specifications and physical parameters are the same used for the data in Fig.~\ref{Fig:1}.
We remind that we study a three dimensional box and that the correlators are computed on the transverse
plane, averaging over the volume of the box, therefore the results shown in Fig.~\ref{Fig:3aa} are obtained
averaging over the contribution of all the transverse planes corresponding to different values of
the longitudinal coordinate.
We find that both the magnetic and the electric correlation functions are not affected
in a substantial way by the gaussian fluctuations; we measure a small bump of the average of the electric correlator
at small $g^2\mu x$, but taking into account the dispersion of the data the behavior of the correlator with fluctuations
is in agreement with that without fluctuations.

\subsection{Correlation functions along the longitudinal direction}

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{corr_long_late.png}\\
\includegraphics[scale=0.3]{corr_long_late_ratio.png}
\end{center}
\caption{\label{Fig:3aaab}Gauge invariant color-magnetic  correlators
as a function of the longitudinal coordinate.
Panel {\bf (a)} shows the correlation function for several selected values of time: maroon line corresponds to
$t=0$, green dashed line to $g^2\mu t=0.25$, indigo dot-dashed line to $g^2\mu t=0.5$,
and blue solid line to $g^2\mu t=10$.
Panel {\bf (b)} shows the ratio $C_M(z)/C_M(0)$ for a larger set of times.
Data shown for $g^2\mu = 1$ GeV, $g^2\mu a = 0.14$ as in Fig.~\ref{Fig:1} and
for a lattice $71\times71\times151$.}
\end{figure}

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{corr_long_late_ratio_TIME.png}
\end{center}
\caption{\label{Fig:3aaaT}Gauge invariant color-magnetic  correlator along the longitudinal direction,
normalized to its value at $z=0$,
as a function of $g^2\mu t$ for several values of the longitudinal separation $z$.
Data have been obtained by those of panel  {\bf (b)} of Fig.~\ref{Fig:3aaab}.
Data shown for $g^2\mu = 1$ GeV, $g^2\mu a = 0.14$  and
for a lattice $71\times71\times151$.}
\end{figure}

In Fig.~\ref{Fig:3aaab} we plot the magnetic correlator as a function of the longitudinal separation $z$.
The results are shown for the case without fluctuations.
In panel {\bf (a)} we plot the correlation function for selected values of time: maroon solid line corresponds to
$t=0$, green dashed line to $g^2\mu t=0.25$, indigo dot-dashed line to $g^2\mu t=0.5$ and blue solid line to $g^2\mu t=10$.
On panel {\bf (b)} we plot the ratio $C_M(z)/C_M(0)$ for a larger set of times.
Data shown for $g^2\mu = 1$ GeV, $g^2\mu a = 0.14$ as in Fig.~\ref{Fig:1} and
for a lattice $71\times71\times151$. WIth respect to the lattice used in the previous sections we have built up
a lattice that is longer along the longitudinal direction in order to study better these correlation functions.
In Fig.~\ref{Fig:3aaaT} we arrange data shown in panel {\bf (b)} of  Fig.~\ref{Fig:3aaab}
in the form of a normalized correlator as a function of time for several values of the longitudinal separation.
We show the normalized ratio $C_M(z)/C_M(0)$ because this quantity allows us to describe more
clearly the behavior of the correlations along the longitudinal direction.
As a matter of fact, as it is evident from the data in panel {\bf (a)} of Fig.~\ref{Fig:3aaab},
at each time the value of $C_M(z)$ at the origin fluctuates in particular in the very early stage,
because that corresponds to twice the average of the magnetic longitudinal energy which changes with time,
see Fig.~\ref{Fig:1}: therefore, what is really important is the $z-$variation of the correlation function with respect
to the value at $z=0$ rather than its magnitude.

The data shown in Figg.~\ref{Fig:3aaab} and~\ref{Fig:3aaaT} are interesting for several reasons.
Firstly, we notice that although the magnetic field is $z-$independent due to the lack of longitudinal fluctuations,
the correlation function at $t>0$ is $z-$dependent. This might sound counterintuitive:
in fact, $B_z^a$ is $z-$independent due to the lack of fluctuations so one naively would expect
the correlation function to be $z-$independent as well.
However,
we should remind that we compute the gauge-invariant correlator, $C_M$, which differs from the naive one,
$C_\mathrm{naive}=\langle B_z^a(0) B_z^a(z)\rangle$, that would be $z-$independent; the two would
coincide only if $A_z = 0$ which in turn happens at $t=0$ and explains why at the initial time
$C_M$ does not depend on the longitudinal coordinate.

The difference among the two correlators is clearly given by the presence of the parallel transporter along the $z$ direction,
$U=P e^{i\int_0^z dz^\prime A_z(z^\prime)}$.
As a matter of fact, in the case without fluctuations both $B_z$ and $A_z$ are $z-$independent so
$[A_z(z_1),A_z(z_2)]=0$ and path ordering gives the standard
exponential matrix of the integral over the full path so we can write,
for a straight path from $0$ to $z$ along the $z$ direction
\begin{equation}
U = Pe^{i\int_0^z dz^\prime A_z(z^\prime)} = e^{i\int_0^z dz^\prime A_z^a(z^\prime)\tau_a}
=e^{i z A_z^a \tau_a},
\end{equation}
with $\tau_a$ denoting the standard Pauli matrices in color space; then for each point in the transverse plane
we are left with
\begin{equation}
C_M (z)= B_a(0) B_b(z) G^{ab}(z)
\end{equation}
where
\begin{equation}
G^{ab}(z) = \mathrm{Tr}\left[\tau^a e^{i\int_0^z dz^\prime A_z^c(z^\prime)\tau_c} \tau^b
e^{-i\int_0^z dz^\prime A_z^d(z^\prime)\tau_d}\right];
\end{equation}
noticing that $B_z$ does not depend on $z$ the term $B_a(0) B_b(z)$ is symmetric for the exchange of $a$ and $b$
and it suffices to compute the symmetric part of $G^{ab}(z)$;
this is a straightforward calculation that can be performed by using the generalized Euler idendity for $SU(2)$ matrices and it leads to
\begin{eqnarray}
\mathrm{Sym}\left[G^{ab}(z)\right] &=& \frac{A_a^2 +(A_c^2 + A_d^2)\cos(2|Az|)}{|A|^2},~~a=b,\nonumber\\
&&
\end{eqnarray}
where $a\neq c \neq d$, and
\begin{eqnarray}
\mathrm{Sym}\left[G^{ab}(z)\right] &=& \frac{2 A_a A_b\sin(2|Az|)^2}{|A|^2},~~a \neq b.
\end{eqnarray}
We notice the explicit $z-$dependence of the matrix elements that can not disappear even after
assuming a uniform random distribution of $A_z$ over the transverse plane.
Also notice that in the limit $A_z\rightarrow 0$ then $\mathrm{Sym}\left[G^{ab}(z)\right]  \rightarrow 1$
and the $z-$dependence disappears: the correlator in this limiting case coincides with the naive one.
As a further check of the numerical code we have verified that if we replace the
parallel transporter with the unit matrix we obtain a $z-$independent correlation function at any time.

This result can be summarized in other words by saying that the statement that a $z-$indpedentent color-magnetic field gives a $z-$independent
correlator is not gauge invariant unless $A_z = 0$; if this was the case then the naive correlator would be gauge invariant
and independent on the longitudinal separation.
From the physical point of view, we interpret this result as the effect of the interaction of two gluon fields
separated by $z=Z$ with the background with $A_z \neq 0$:  this interaction screens the zero mode
and causes the lowering of the correlator for a large value of $g^2\mu Z$.

We notice that for small values of $g^2\mu t$ the variation of the correlator at small $z$
is slower than the one we measure for larger separation: for example, compare the maroon dots
with the brown triangles up to $g^2\mu t = 0.3$ in  Fig.~\ref{Fig:3aaaT} (we easily get the same
message from panel {\bf{(b)}} in Figg.~\ref{Fig:3aaab}). On the other hand, for
$g^2\mu t\approx 0.3$ the correlations at large separation slow down their variation and approach an approximantely
constant value for a long time range, while correlations at small separation still evolve and decrease with time.
This evolution seems to continue up to $g^2\mu t=10$.

The picture that we read from the above results is that the classical gluon dynamics is capable to affect the long range
gauge invariant correlations of the evolving Glasma, but only within a very short time range,
while it affects the short range dynamics for a somehow larger
time range. During the very early stage the Yang-Mills evolution lowers the long range correlations
but it is not able to cancel them completely, leaving a residual long range order: most likely this will affect the
particle production from the classical gluon field and it might be the reason of the ridge measured in proton-proton collisions.

We call the process described above as the partial gauge-invariant string breaking in the Glasma.
In fact, at the initial time the Glasma is made of $z-$independent color-magnetic fields that develop
along the longitudinal direction, almost uncorrelated in the transverse plane, so it looks like an ensemble
of color strings that extend among the two nuclei: this system is characterized by a perfect
long range order along the longitudinal direction. As the time goes on $A_z$ is formed dynamically
and this results in a quick and partial loss of the gauge-invariant correlation of the longitudinal magnetic fields with $z$;
this loss of correlation happens quickly for large separations while it is slower for smaller separations.
Eventually this results in a residual long range order, with a correlation that is almost constant for large $z$
but being approximately only the $30\%$ of the correlation measured for small $z$.
For this reason we say that the original Glasma strings have been partly broken in a gauge invariant way by the dynamics.
We have verified that this evolution of the longitudinal correlator happens also for the electric correlation function,
and that changing the longitudinal size of the box does not change qualitatively the results.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{FT_vari_COMBI_2.png}
\end{center}
\caption{\label{Fig:3png}Longitudinal Fourier transform of the correlation function at different times.
Lines and colors conventions agree with those used in panel {\bf (b)} of Fig.~\ref{Fig:3aaab}.
Data shown for $g^2\mu = 1$ GeV, $g^2\mu a = 0.14$  and
for a lattice $71\times71\times151$.
In the inset we show $|C_0|^2$ and $|C_1|^2$ versus time.}
\end{figure}

The gauge invariant partial loss of correlation can be described also in the longitudinal momentum space.
In Fig.~\ref{Fig:3png} we plot the squared magnitude of the Fourier coefficients of the correlation function,
the latter being defined as
\begin{equation}
C_{k} = \sum_{{n}=0}^{N_z} \frac{C_M(n)}{C_M(0)}e^{i\frac{2\pi k n}{N_z -1}},
\end{equation}
where $N_z = 150$ corresponds to the longitudinal lattice size and $n$ is
the discrete index representing a lattice site in the longitudinal direction. In Fig.~\ref{Fig:3png}
the main panel represents $|C_k|^2$ for $k\geq 2$ at different times, while for the sake of keeping the picture clear
we have collected
the results for $|C_0|^2$ and $|C_1|^2$ in the inset of the same figure.
From these results we notice that at the initial time the only contribution to the correlator comes from the zero mode,
which is obvious because at this time the field has no $z-$dependence.
On the other hand, the dynamics lowers the zero mode coefficient very quickly; in fact,
for $g^2\mu t \leq 0.5$ we find that $|C_0|$ changes approximately of a factor of $2$
then it stays almost constant and nonzero.
At the same time modes with $k\neq 0$ are populated and these are responsible of the acquired $z-$dependence of the
correlation function.
We notice that while the zero-mode
as well as the low momentum modes evolve mainly for  $g^2\mu t \leqslant 0.3 $,
the higher modes have a relative substantial variation also for larger times.
In the future we will perform simulations with larger lattices in order to have a better resolution of the Fourier
transform of this correlation function.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{corr_long_30p_ratio.png}
\end{center}
\caption{\label{Fig:3aaa}Color-magnetic  correlators
as a function of the longitudinal coordinate, for several values of time and for a fluctuation seed
$\Delta^2 = 2.2\times 10^{-4} g^2\mu a$.  Colors and line conventions agree with panel (b) of Fig.~\ref{Fig:3aaab}.
Data shown for $g^2\mu = 1$ GeV, $g^2\mu a = 0.14$  and
for a lattice $71\times71\times151$.}
\end{figure}

In Fig.~\ref{Fig:3aaa} we plot the correlation function along the longitudinal direction, normalized to the value at $z=0$,
in the case fluctuations are added on the top of the Glasma with seed
$\Delta^2 = 2.2\times 10^{-4} g^2\mu a$. We find that the correlations of the magnetic field
are affected by the fluctuations. In particular, in the very early stage the main effect of the fluctuations is the damping of the oscillations
that we have found for the case without fluctuations, compare Fig.~\ref{Fig:3aaab} with Fig.~\ref{Fig:3aaa}.
For larger $z$  we do not observe the freezing of the correlations that we have
found without fluctuations; in fact comparing the data at $g^2\mu t=10$ in the two cases we notice that
in the case of fluctuations the correlator at large $z$ decreases to $\approx 15\%$ of the value at $z=0$,
while in the case without fluctuations the correlator decreases to $\approx 30\%$ of the value at $z=0$.
This suggests that fluctuations enhance the breaking of the color strings.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.3]{comparison_correlatori_50.png}
\end{center}
\caption{\label{Fig:3aaaBBB}Correlation function of the color-magnetic field along the longitudinal direction at $g^2\mu t = 50$
for three values of the fluctuation seed: dot data stand for the case $\Delta=0$, dot-dashed green line corresponds to
$\Delta^2 = 2.2\times 10^{-7} g^2\mu a$, dashed indigo line stands for $\Delta^2 = 2.2\times 10^{-6} g^2\mu a$,
finally solid orange line denotes the case $\Delta^2 = 2.2\times 10^{-4} g^2\mu a$. }
\end{figure}

In order to put a firm statement on the enhancement of string breaking by fluctuations
we have followed the evolution of the system up to a larger time for different values of the fluctuation seed.
The results are collected in Fig.~\ref{Fig:3aaaBBB} where we plot the correlation of the
color-magnetic field along the longitudinal direction at $g^2\mu t = 50$
for three values of the fluctuation seed: dotted maroon data stand for the case without fluctuations,
dot-dashed green line corresponds to
$\Delta^2 = 2.2\times 10^{-7} g^2\mu a$, dashed indigo line stands for $\Delta^2 = 2.2\times 10^{-6} g^2\mu a$,
finally solid orange line denotes the case $\Delta^2 = 2.2\times 10^{-4} g^2\mu a$.
The results in Fig.~\ref{Fig:3aaaBBB}  show that
at a given large time fluctuation tend to lower the magnitude of the correlator;
in fact in the case $\Delta^2 = 2.2\times 10^{-4} g^2\mu a$ the correlation seems to vanish already for
$g^2\mu z \approx 1$ whole being clearly nonvanishing for smaller values of $\Delta$ as well as for $\Delta=0$.
We can then conclude that fluctuations accelerate the  breaking of the Glasma strings.
We remark however that $g^2\mu t =50$ corresponds to the physical time $t \approx 10$ fm/c which is quite
beyond the expected range of validity of the CYM approach that instead should describe the system
up to $t\approx 1$ fm/c; therefore, in realistic  collisions
the results in Figg.~\ref{Fig:3aaab}  and~\ref{Fig:3aaa} suffice and we expect only a partial breaking of the Glasma strings
to occur in these processes.

Concluding this section, we have found that the CYM dynamics leads to a partial loss of the gauge invariant correlation along the longitudinal
direction (the naive correlator would  be $z-$independent) even without fluctuations, and that fluctuations
enhance the string breaking since they accelerate the degradation of the gauge invariant correlation function.
We have shown the results for the color-magnetic strings: for the electric fields we have obtained similar results.
It is useful to remark that these results have been obtained in simulations of the color fields in a static box:
the effect of the longitudinal expansion on the correlations will be the subject of a forthcoming article.

\section{Conclusions and outlook}

We have studied the evolution of the color fields produced in high energy nuclear
collisions, namely the Glasma with gaussian fluctuations,
by means of classical Yang-Mills equations that we have solved numerically on a lattice.
Neglecting in this first study the longitudinal expansion, that we will consider in a near future work,
we have followed the evolution of the pressures of the system
and computed the effect of the fluctuations in the early stage, up to
$t\approx 2$ fm/c:
this early stage roughly corresponds to the time range in which the classical Yang-Mills dynamics is
relevant for potential applications to high energy collisions.

We have considered one representative value of $g^2\mu$, namely $g^2\mu=1$ GeV\footnote{We have checked that
changing the value of $g^2\mu$ does not affect the results qualitatively.}
 and following the evolution up to
$g^2\mu t=10$, that  corresponds to
a maximum physical time of $t\approx 2$ fm/c: this time range is
already beyond that in which the classical field theory description can have
an interest for realistic collisions.
We have set the
transverse area to $A_T = 4$ fm$^2$.  Among other things, this choice allows for a small lattice spacing
even if the lattice size is not large. Since we solve the equations of motion
applying periodic boundary conditions, the solutions found here should represent quite
well also the evolution of the color fields in the central region of nucleus-nucleus collisions.

We have measured the ratio of the longitudinal over the
transverse pressure, $P_L/P_T$, as a function of time; this quantity is usually taken as the reference
for studying isotropization of the system, a perfect isotropy being achieved if $P_L=P_T$.
We have found that unless the fluctuations carry a substantial amount of the Glasma energy
density at the initial time, they do not change drastically the evolution of $P_L/P_T$ in the early stage.
For example, if the if the initial disturbances carry about the $65\%$ of the energy to the Glasma then
the system is capable to obtain a fair amount of isotropy, $P_L/P_T\approx0.8$ within $g^2\mu t = 1$.

Our limited computing power does not allow us to perform long time runs, but it will be important to do these in the near
future: in fact, it is well known that
secondary instabilities trigger a rise of $P_L/P_T$ at very large times in the case of small fluctuations size \cite{Berges:2012cj},
and that the time at which these instabilities become important depends logarithmically on the fluctuation size.
Computing this in the future will be necessary to understand if the instabilities are the leading mechanism for the more efficient
isotropization in case of larger fluctuation seeds, or if a collisional-like dynamics sets in. We will address this problem in a
forthcoming article. For the time being we take our results as preparatory of a more complete work,
emphasizing that what we have found suggests that it is possible that the classical Yang-Mills dynamics leads to a fairly isotropic state within
a short time range, regardless of the actual mechanism used to produce the isotropy, but that in order to achieve this
the fluctuations have to be large enough.
This result might suggest that the fair amount of isotropy invoked in hydrodynamics simulations
can be the consequence of an initial state in which fluctuations are substantial;
on the other hand, we are forced to take this conclusion with a grain of salt, since
it might put doubts on the use of the classical dynamics for the description
of the early stage of high energy nuclear collisions: indeed, if fluctuations are substantial,
and being them of a quantum nature, it might be not safe to apply a classical theory to describe the evolution
of the system.
The discussion of this complicated topic is however too far from the purpose
of our simple work, therefore we limit ourselves to summarize our results and we leave the discussion
to future works.

Finally, we have measured the
color-magnetic and color-electric correlation functions. We have found a qualitative agreement  with previous
studies \cite{Dumitru:2013wca,Dumitru:2014nka};
moreover, we have found that the gaussian fluctuations hardly affect the correlation functions.
We support the idea that at the initial time the Glasma fields have anticorrelation
in the transverse plane, and that the CYM evolution aligns the fields in a time range
$g^2\mu t = O(1)$ on a transverse area $(g^2\mu)^2 A_T = O(1)$.

A novelty that we have brought with our study has been that of the gauge invariant correlators along the longitudinal direction;
these are important in order to address the question of the evolution of the color strings.
We have found that also in absence of fluctuations in the initial state, the gauge invariant magnetic correlator
decreases at large $z$; this is different from what happens to the naive, gauge dependent correlation function
which instead remains $z-$indepentent. We call this phenomenon as the gauge invariant partial string breaking.
The string breaking is related to the fact that
at the initial time the correlation function along the longitudinal direction is $z-$independent so the field looks like
a string that connects the two transverse planes, while this
correlation function becomes $z-$dependent for $t>0$ and in particular the correlation for large $z$
is smaller than the one we measure near the origin.
It is gauge invariant since it is obtained studying a gauge invariant correlation function; it is partial
since we still find some correlation at large distance.
We remark that this has been obtained for simulations
of the classical dynamics of color fields in a static box: most likely these correlations will experience a more
substantial decay in the expanding case that we will study in the near future.
The difference between the naive and the gauge invariant correlation functions is given formally by the
presence of the parallel transporter which in turn depends on $A_z^a$: this is zero in the initial state
but it becomes nonzero because of the Yang-Mills evolution, therefore the propagation of the gluon field
along the longitudinal direction is affected by the interaction with this $A_z$ background.
We have found that the dynamics at large $z$ is a bit faster than the one at smaller $z$:
indeed, the correlations at large $z$ evolve within $g^2\mu t \leqslant 0.3 $ then correlations seem to freeze while those at small $z$
still evolve in the full time range studied. We have confirmed this by the computation
of the Fourier transform along the longitudinal direction, finding that while the zero-mode
as well as the low momentum modes evolve drastically for  $g^2\mu t \leqslant 0.3 $,
the higher modes have a relative substantial but slow variation also for larger times.
In the future we will perform simulations with larger lattices in order to have a better resolution of the Fourier
transform of this correlation function.

Finally, we have computed the effect of the fluctuations
on the correlations along the longitudinal direction. We have found that
fluctuations enhance this gauge invarant string breaking since they accelerate the decay of the longitudinal
correlation function, see Figg.~\ref{Fig:3aaa} and~\ref{Fig:3aaaBBB}.
However, we have found a substantial breaking of the color strings only for times
that are well beyond the alleged applicability of the CYM approach to realistic collisions;
it is likely that for these processes the results in Figg.~\ref{Fig:3aaab} and~\ref{Fig:3aaa} suffice to describe the correlations
of the gluon fields in the early stage, thus that substantial correlation remains for large longitudinal separation
and the Glasma strings are only partially broken even when fluctuations are included.

We think that the most important improvement of the work presented here is the
introduction of the longitudinal expansion
in order to describe more closely the early stage of the system created by the collision.
In addition to this, we also would like to improve our simulation code by means of a parallelizzation
that will allow us to perform the long time runs like those in \cite{Romatschke:2005pm,Fukushima:2013dma,Berges:2012cj}
and study the interplay of the collisional dynamics with the Glasma instabilities.
We plan to report on these topics in future works.

\begin{acknowledgements}
M. R. is indebted to F. Gelis and Y. Nara for the many suggestions given
during the preparation of this article.
The work of M. R. has been supported by the National Natural Science Foundation of China
(11575190 and 11475110) and the Chinese Academy of Sciences President International Fellowship
Initiative (2015PM008).
G.X.P and V.G would like to thank the National Natural Science Foundation of China (11575190 and 11475110) and the
President International Fellowship Initiative (2016VMA063).
Numerical calculations have been performed on the QGPDyn Cluster of Catania University and on the
Cluster of School of Nuclear Science and Technology in Lanzhou University.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


The day before the Canadian men's national team faced Scotland in Edinburgh, Octavio Zambrano addressed the team.
So, well, Fraser Aird sure looked like he was pretty pumped to play for Canada in Scotland, didn't he?
With Ledgerwood off to play for Canada, coach Colin Miller said that fullback Netan Sansara will get the captain's armband for the Stoke City match. Dustin Corea, who scored and set up a goal against Newcastle, will be out with a groin strain. And goalkeeper Nathan Ingham is nursing a strain in his kicking leg.
So, did Shiels feel the weight of expectation on his shoulders when he took the field with his new Eddies teammates for the first time Monday?
"The guys are in a good place, but we need to get out of here," Miller said after Tuesday's training session. "It's getting claustrophobic. Guys are on top of each other, there's a bit of chirping going on at training, as you would imagine. We're now into our fourth or fifth week of preseason at this little place and it's understandable that we need to get out and get some fresh air."
A preseason trip to Europe in no way suggests that Fath is doing anything more but increasing his investment in a team he and his brother, Dave Fath, have lost money on since day one. But, with the dollar slumping to less than 73 cents US on Friday, and it likely to drop even further as the oil price continues to sag, the fact remains that a trip to America has become increasingly expensive, and will likely be even more pricey in the spring of 2016.
We're now at the stage of the poker game where it's time to call. We've gone through the bets, the bluffs and the folds.
And one name jumps right out at the Canadian soccer supporter. Fraser Aird.


text:
Tunyawaj Kamolwongwat
New marriage bills to go before parliament soon, says committee
Prachatai
The Drafting Committee for the Civil Partnership bill and the Marriage Equality bill has said that it will push for both bills to go before parliament before the end of the current session, while concerns have been raised about homophobia among committee members.
Marriage equality bill delayed, forwarded to Cabinet
The Thai parliament voted yesterday (9 February 2022) to forward the marriage equality bill, proposed by Move Forward Party (MFP), to the Cabinet for consideration, delaying the bill's first reading.
Parliament votes no to forming an LGBT rights committee
On 22 August, parliament voted not to support a proposal to set up a separate Standing Committee on LGBT rights. Instead, LGBT rights will be subsumed under the Standing Committee on children, young people, women, the elderly, persons with disabilities, and ethnic groups.
A small victory for Future Forward's transgender MPs
Prachatai
Amidst the storm of legal prosecutions against its party leaders, the Future Forward Party (FFP) has gained a small victory after its transgender MPs were allowed to dress according to their gender identity to the opening of parliament.


edwardmurphy wrote: I never said there was collusion. How would I possibly know? I'm waiting to hear what the Special Prosecutor has to say on the matter. But Abe, the fact that Trump's AG was forced to recuse himself and the acting AG appointed a Special Prosecutor does sort of hint at the possibility of impropriety.
Wrong! Because it did not stop the investigation because Comey was fired,it goes on.However he is not a special prosecutor - Mueller,as only the DOJ can prosecute,Mueller does not have the power to prosecute.You were defending this Russia conspiracy theory before and even thinking and hoping it would lead to impeachment because you listen to NPR,you may have changed your mind,but you were propagandized.Trump wants this investigation because he knows he is innocent.
Yep,this is what I was getting at Mueller is a special counsel and is not a special prosecutor,there is a difference that is overlooked.


Recruiter description: The AviationPower UK Ltd. is part of the AviationPower Group, a Joint Venture between Manpower Group and Lufthansa Technical Training, is a cooperation partner in all areas of aviation. Our clients benefit from our tailor-made project and high qualified staff solutions and the experience of our employees. Experienced professionals, as well as career entrants with an affinity for aviation, find their dream job at AviationPower. We set a high value on training, further qualifications and personal support of our employees. In cooperation with educational providers like Lufthansa Technical Training and others.


Sheep placenta, unbeknownst to many, is a very useful component used in many health products in the market today. This is because the benefits of sheep placenta are numerous. Developed correctly, sheep placenta presents many uses in the fields of cosmetics, hair care, pharmaceuticals and even food products.
There are many benefits of sheep placenta. To take advantage of the many benefits of sheep placenta, NS-mart effectively uses it as the core component for its oral supplement products. The very popular MFIII Sheep Placenta Capsules and the more recently developed MFIII PE Softgels Advanced Formula both take advantage of the anti-aging benefits of sheep placenta. Studies have shown that sheep placenta offers anti-aging benefits because it is a rich source of nutrients and vitamins, the same nutrients and vitamins the placenta used to nourish the sheep that grew inside of it.
The nutrients and vitamins of sheep placenta include nucleic acids, antioxidants, essential amino acids, hyaluronic acid, immune co-factors, epidermal growth factors, melatonin and quite a few others. All these bring about the many anti-aging benefits of sheep placenta. Many of these benefits are easily obtainable by taking either MFIII Sheep Placenta Capsules or MFIII PE Softgels Advanced Formula.
One of the most important benefits of sheep placenta has to do with its healthy effect of the skin. Sheep placenta promotes beautiful, young-looking skin, hydrating it and thereby reducing wrinkles, sagging, blemishes and freckles. Aside from this, another one of the benefits of sheep placenta it is ability to improve the skin's renewal process when it comes to wounds and scars.
The body's immune system also benefits from the intake of sheep placenta products. One of the benefits of sheep placenta is its ability to regulate hormones and promote immunity and general health. Additionally, it increases your energy, stamina and overall vitality.
Sheep placenta also offers a number of benefits to women's reproductive health. The benefits of sheep placenta in this area include the effective regulation of menstrual cycle, the reduction of pre-menstrual tension and finally, the alleviation of menopausal symptoms.
There are indeed several benefits of sheep placenta to health and well-being. Make sure to consider MFIII Sheep Placenta Capsules or MFIII PE Softgels Advanced Formula as your oral supplements.


text:
Q: It needs to create Predicate with date in Criteria API, but still gets an exception. How to build such Predicate?     private CriteriaBuilder criteriaBuilder;
    private CriteriaQuery<Points> criteriaQuery;
    private Root<Points> root;

    @PostConstruct
    private void init() {
        criteriaBuilder = entityManager.getCriteriaBuilder();
        criteriaQuery = criteriaBuilder.createQuery(Points.class);
        root = criteriaQuery.from(entityManager.getMetamodel().entity(Points.class));
    }
    public void t(Date lowerLimit, Date upperLimit) {
      Predicate startPredicate = criteriaBuilder.greaterThanOrEqualTo(root.<Date>get(Points_.createdOn), lowerLimit);
    }

Exception:
java.lang.NullPointerException: null
    at org.hibernate.query.criteria.internal.path.AbstractPathImpl.get(AbstractPathImpl.java:123)
I use hibernate 5.2.10.Final version.
Root is not null:

Metamodel and entity are in the same package:

Points_ class:
package ...domain.points.model;
@StaticMetamodel(Points.class)
public class Points_ {
    public static volatile SingularAttribute<Points, Date> createdOn;
    ....
}

Points class:
package ...domain.points.model;
@Entity
@Table(name = "POINTS")
public class Points extends AuditingEntity {
...
}

AuditingEntity class:
@Audited
@MappedSuperclass
@EntityListeners(AuditingEntityListener.class)
public class AuditingEntity implements Serializable {
    @CreatedDate
    @Column(name = "CREATED_ON")
    protected Date createdOn;
    ......
}

A: https://stackoverflow.com/a/20097800/4587961
Is metamodel and entity in the same package? Points_ and Points
You say yes.
Now, let's try to replace metamodel with String field name.
 Predicate startPredicate = criteriaBuilder.greaterThanOrEqualTo(root.<Date>get("createdOn"), lowerLimit);

I just want to see if it works, and then continue to search a solution.
Check this
https://stackoverflow.com/a/4256642/4587961

I am afraid there is a bug in Hibernate
https://hibernate.atlassian.net/browse/HHH-9259
I used hardcoded Strings. I would create a class PointTable with all fields from Point. These classes can be in different packages.
Point
package entities;

public class Point {

    private Long id;
    //Other fields.

    @ID
    @Column(PointTable.ID)
    public Long getId() {
        return id;
    }

    //Other getters and setters.
}

PointTable
package entities.table;

public class PointTable {

    public static final String ID = "id";
    public static final String CREATED_ON = "createdOn";
    //Other fields.
}

redpajama_set_name: RedPajamaStackExchange


Two weeks ago, for the official release of GATK version 4.0, we held a live online event that was both a launch party and a comprehensive if condensed overview of everything that's new in GATK4. Over the course of two hours, members of the GATK development team and a great lineup of external guests gave presentations about the new capabilities, discussed their implications in small panels and answered questions from the online audience.
I had the privilege of serving as host -- and unintentional comic relief, between forgetting panelists and bumping into the set furniture -- so I'm probably biased, but I'd say it was the most fun-yet-informative event we've done so far on GATK. Not that we do a lot of events -- and it's mostly just workshops -- but this felt pretty special. We had a great time doing it, and lots of people showed up to watch and ask questions. So we're now considering doing others in a similar vein, though they would each be focused on a specific topic and have more time for answering questions from the online audience. If that sounds like something you'd be interested in, let us know in the comments!
Anyway, the recording of the full event is now available on YouTube, and you can always find it again on our Presentations page.


Julie Pike is a talented fashion photographer based in Oslo, Norway, who has done editorials for magazines such as Bon, Elle, Mirage, Nylon, S Publication, and many more. Her commercial clients include American Express Gold, EMI Music, Sony Music, Universal Music, etc.
5 comments to ''Photography by Julie Pike"
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text:
192 UIC medical students graduate early
New doctors ready to help during COVID-19 pandemic
10-Apr-2020 6:15 AM EDT, by University of Illinois at Chicago
UIC/Brad Cavanaugh)
Aerial view of the University of Illinois College of Medicine buildings in Chicago
Newswise — Today, 192 medical students from the University of Illinois College of Medicine, one of the largest and most diverse medical schools in the country, will graduate early.
Among the early grads, one-third have been matched to residency programs in Illinois.
"This will help meet the growing demand for health care workers during the COVID-19 pandemic, and the college is joining other medical schools across the country in sending our graduating students into the world early," said Dr. Mark Rosenblatt, executive dean of the college, which is part of the University of Illinois at Chicago.
"Today, our new medical school graduates take the Hippocratic oath and, during this pandemic, they do so at a time that physicians and other health care professionals represent the best that humanity has to offer," UIC Chancellor Michael Amiridis said. "Now, more than ever before in our lifetimes, we need their contributions 'for the cure of their patients and the prevention of disease,' as the oath says. I know our graduates are ready and determined to carry out this mission. All of us at UIC are very proud of them!"
"This is a very dynamic time for these graduates to be entering into the medical profession," said Dr. Robert Barish, vice chancellor for health affairs at UIC. "We are proud of the enthusiasm that these young doctors have shown to make a difference in our communities already. Now, they are sure to directly apply what they have learned in their studies and clinical experiences to this historic time in medicine."
In order to qualify for early graduation, the students needed to meet all components of the college's graduation competencies, which includes completion of all required course and clinical rotation objectives.
The early grads represent about two-thirds of the college's 2020 graduating class.
"Many of our students are ready and eager to do their part. We couldn't be prouder of them and wish them well," said Rosenblatt, who oversees the college's four campuses, located in Chicago, Peoria, Rockford and Urbana. "Our students want to help our world, and they are finding ways and seeking opportunities to use their precious skills where it is needed most.
"The college is committed to doing what it can to support these efforts. We are proud to offer early graduation and proud of the extraordinary accomplishments of our students," Rosenblatt said.
Students who have graduated early will still need to work directly with their individual residency programs to determine if they will be able to begin work early.
The early graduates will join their classmates who will graduate on schedule in a virtual commencement ceremony in May.
Register for reporter access to contact details
COMMENTS | COMMENTING POLICY


text:
Home | acebook
History | Calendar | Membership - Join the Mailing List | Links | Seating Plan | Brochure 2018 A+B (1Mb.docx each) | Contact us: 011 728 5492 or 083 228 2917
2020 is an auspicious year; musicians and music-lovers all over the world will be celebrating the 250th year since the birth of Ludwig van Beethoven - a universal genius who transformed the musical universe. Beethoven's great spirit sought to encompass the ideals of the French Revolution and the new spirit of humanism that was sweeping the world in which he lived. His music, passionate and personal, reveals the power that music has to convey the deepest emotions as well as a philosophy of life.
This year, the Johannesburg Musical Society joins with the global music community in featuring much of Beethoven's music. However, our listeners will still enjoy programmes that are varied and which feature the music of other composers. As always, the Johannesburg Musical Society offers up a veritable feast of music, performed by superb artists from all over the world.
Join music-lovers in the Linder Auditorium, St Andrews Road, Parktown - Johannesburg's most popular hall - to hear the superb classical concerts presented by the Johannesburg Musical Society.
Here, in the comfort of the Linder, you can sit back and listen to masterful performances by artists from all over the world. Secure parking is yet another advantage offered by this venue, and you can also enjoy a glass of pre-concert wine or an excellent cappuccino and a slice of cake in the adjoining 27 Degrees Café, where you can also partake of an excellent pre-concert dinner.
All Johannesburg Musical Society concerts are open to the public, who may buy tickets at the box office or pre-book them through Computicket. However, if you become a subscriber you'll enjoy the privilege of a permanent and pre-allocated seat at each concert as well as a free programme. A further benefit is that, as a member of the JMS, you'll be paying up to R45 less per ticket.
Simply fill in the feed-back form on this website and send it to us. We will contact you immediately to assist you in becoming a member.
Alternatively, telephone 011 728 5492 for help.
Charl du Plessis Trio
Sunday 7 November: 16:00
THE PERCY BANESHIK MEMORIAL CONCERT
The concert will feature the charismatic Charl du Plessis Trio, playing a specially designed programme that moves from Beethoven, through Albeniz, Piazzolla, Rodrigues and more, showcasing Charl's usual blend of fine musicality and good humour.
The concert will be approximately one hour in length, with no interval. Since the Linder is a vast space, there is plenty of room for social distancing, and all audience members will be masked. There will be no reserved seats, but the JMS asks people to make sure that there are two empty seats on either side of them.
Seats are R200 each. A deposit into the bank account of the Johannesburg Musical Society and a proof of payment advice will serve to secure your seats. Payments will be acknowledged.
Details as follows:
Bank : Standard Bank
Account Name : Johannesburg Musical Society
Account Number : 001 805 908
Proof of Payment : info@jms.org.za
For more info: 011 728 5492 or Herman on 083 228 2917.
Calendar 2021:
Please browse through our 2021 season at your leisure!
1.	ANMARI VAN DER WESTHUIZEN cello and ANNEKE LAMONT piano
Saturday 18 April : 16:00
6.	Date and Artist to be announced.
9. Date and Artist to be announced.
10. Charl du Plessis Trio
Sunday 7 November : 16:00
All Johannesburg Musical Society concerts are open to the public, who can book at Computicket. But remember, that if you're a member, you pay far less for tickets and you also have the privilege of preferential booking and a free programme.
For more info telephone 011 728 5492 or e-mail us.
Percy Baneshik Memorial Trust
Nabarro Chamber Concert Trust
History | Calendar | Membership - Join the Mailing List | Links | Seating Plan | Brochure 2016 (3Mb.pdf) | Contact us: 011 728 5492 or 083 228 2917


There is constant demand to refine this powerful research tool. This conference will present the latest information regarding how to improve the accuracy of this technique - including optimising cellular conditions, developing quantitative technologies and enhancement of biomarkers. Speakers will focus on the ways in which this technique facilitates research in numerous therapeutic areas, notably stem cells and oncology.
Other conference tracks at this event include Cancer & Stem Cells, Cell Culture, Circulating Tumour Cells and Flow Cytometry. Registered delegates will have access to all five meetings ensuring a very cost-effective trip.


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