text:
Q: Define a new Java template in Eclipse IDE Im doing the same thing everytime im creating a JSF ManagedBean class :

*

*From eclipse IDE, click on menu File -> New -> Class

*Mannually adding and importing @ManagedBean, @ViewScope, @PostConstruct, etc

Can i automate this by defining a new template, so it'll be as easy as File -> New -> My ManagedBean ? I dont want to break the existing default Class template.
I know i can do this with new xhtml, i can choose which template to use when creating it. But i still cant find this feature for creating a Java class.

A: You can use my plugin to create these kind of classes easily. Please go to http://fast-code.sourceforge.net/ and install. Let me know if any problem.


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text:
Trusted - Free - Independent - Unbiased - Free
Navigation Home Recent Articles Archive About
The Salvator is receiving reports that a number of the current Sabbatical candidates have been visited by the ghost of Pat Mathewson, of elections past, present and future. Best known by students for his winning smile and commitment to never leaving St Andrews, sightings of his ghost have spooked some of this year's candidates. His ghost has been sighted loitering in the renovated Union bar, taunting folk on the Night Bus and wreaking havoc outside the library, in a bid to warn potential candidates that being a SABB is not all it seems.
One of this year's presidential candidates, who will remain unnamed, believes he is responsible for awakening his spirit. He said:
"I was just looking for some advice on how Pat managed it, especially for two terms! So I googled how to perform a spell that would awaken his presidential spirit and look what happened - oops!"
The candidate is believed to have performed a traditional spell, in which you sit in front of a mirror in a dark room, lit only by the light of a single candle. It is said if you peel an apple in front of the mirror on a night when the moon is full, a vision or apparition of your one true sabbatical president will appear behind you at the stroke of midnight.
It is rumoured Mathewson's ghost will wander St Andrews' streets until his unfinished business is dealt with - in his case, the ongoing St Andrews housing crisis which he pledged to solve completely during his two terms serving as President. One witness claimed the ghost wailed "CURSE YOU SHITTY PRIVATE LANDLORDS AND HMO BANS I WILL SEEK VENGEANCE BEFORE I REST," before vanishing into the dark night in a puff of smoke. Alas, whilst it appears the housing crisis is far from solved, Mathewson's ghost is doomed to wander in purgatory indefinitely.
More follows.
© Salvator News 2016

redpajama_set_name: RedPajamaCommonCrawl


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text:
Marquette sells 11 acres of land to developers, soccer and lacrosse moving home fields
John Leuzzi , Executive Sports Editor|May 20, 2022
Photo courtesy of Marquette Today.
Marquette University announced Friday morning that it has sold 11-acres of land, at the northeast side of the Marquette Interchange, to Bear Development and Kacmarcik Enterprises for a new sports and entertainment district.
"This new development reimagines and promises to bring new life into a highly visible strategic corridor, presenting an exceptional opportunity to connect the Marquette University campus and Westown neighborhood with the Downtown Milwaukee renaissance," Marquette President Michael Lovell said in a statement.
The new district will include a building of a sports stadium, an indoor concert venue, a full-service hotel, multi-family residential housing, retail and food and beverage elements. Along with a professional soccer club playing in the 8,000-seat-stadium, both Marquette soccer and lacrosse teams will play their home games there.
All four Marquette teams will continue to practice at Valley Fields, which has served as the home to the Golden Eagles since 1993.
A professional soccer club will house the 8,000-seat-stadium along with Marquette soccer and lacrosse
"The economic revitalization and community pride this multi-purpose development will catalyze is something I am proud Marquette can help facilitate through the strategic sale of this important parcel of land," Lovell said in the statement. "I look forward to cheering on our men's and women's lacrosse and soccer teams in their new competition venue."
Currently, the land consist of vacant parking lots, a hotel building, an office space and Marquette's Behavior Analysis Program Building. Per the release, all building were included in the sale.
As for the Behavior Analysis Program, the university is working with those facility members for a new location.
The university mentioned that it will have no financial or controlling interests in the sports and entertainment district development, as it will served as "a lessee in the athletic stadium, and a cooperative and supportive neighbor of this new premier sports and entertainment and residential destination."
There is no target date as for when the project will be finished.
This story is developing.
This article was written by John Leuzzi. He can be reached at [email protected] or on Twitter @JohnLeuzziMU.
Michigan Street
John Leuzzi, Executive Sports Editor
John Leuzzi is the Executive Sports Editor for the Marquette Wire. He is a senior from Mount Laurel, New Jersey, majoring in journalism and minoring in...
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JOURNAL: Behind the Painted White Lines
Sunesson, Salmon leave their mark on program
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text:
\section{Introduction}
At zero temperature, fluctuations of the quantum electromagnetic field
in vacuum manifest themselves via an attraction of two parallel
ideal-conductor plates.
This Casimir effect (for an introduction see \cite{Duplantier1})
has a universal character in the sense that it does not depend on
the material constitution of the metallic plates.
Casimir's result was extended to arbitrary temperatures and
general dielectric plates \cite{Lifshitz,Schwinger1}, and to
ideal-conductor walls of arbitrary smooth shapes \cite{Balian1}.
For a recent book and review see \cite{Mostepanko,Bordag}.
The studied models can be divided according to the geometry of
fluctuating walls into two basic sets: the semi-infinite systems,
in which at least one of the spatial coordinates is unconstrained
by the walls (e.g. two parallel plates), and the fully-finite systems
(e.g. a sphere).
The applied methods and observed Casimir phenomena usually depend
on this classification.

As concerns semi-infinite geometries, in the high-temperature limit
defined by the validity of the equipartitioning energy law, the
Casimir force becomes purely entropic \cite{Feinberg}; this force is
usually called classical since it does not depend on Planck's constant.
In the purely electrostatic models which do not incorporate the magnetic
part of the Lorentz force due to charge currents, like the system
of scalar photons \cite{Schwinger2}, the Casimir force is divided by
a factor 2.

As concerns fully-finite three-dimensional (3D) conductor systems
\cite{Balian1,Balian2}, in the high-temperature limit, the Casimir
free energy does depend on Planck's constant.
Furthermore, the presence of both electric and magnetic degrees of
freedom is necessary for obtaining a Casimir effect.
This is no longer true for classical 2D Coulomb fluids defined
in fully-finite domains.
There, the consideration of the pure Coulomb potential, defined
as the solution of the 2D Poisson equation, leads to a universal
Casimir term analogous to the one appearing in finite-size expansions
of thermodynamic quantities for 2D critical systems with
{\em short-range} interactions among constituents.
To be more precise, it is known that, according to the principle
of conformal invariance, for a finite system of characteristic size $R$,
at a critical point, the (dimensionless) free energy has a
large-$R$ expansion of the form \cite{Blote, Affleck,Duplantier2,Cardy}
\begin{equation} \label{1.1}
\beta F = A R^2 + B R - \frac{c \chi}{6} {\rm ln}\, R
+ {\rm const} + \cdots ,
\end{equation}
where $\beta$ denotes the inverse temperature.
The coefficients $A$ and $B$ of the bulk and surface parts are
non-universal.
The coefficient of the logarithmic Casimir term is universal,
dependent only on the conformal anomaly number $c$ of the critical
theory and on the Euler number $\chi$ of the manifold on which
the system is confined.
In general, $\chi = 2-2h-b$, where $h$ is the number of handles
and $b$ the number of boundaries of the manifold
($\chi=2$ for the surface of a sphere, $\chi=1$ for a disc,
$\chi=0$ for an annulus or a torus).
A simple example is the Gaussian model \cite{Houches} which is
critical at all temperatures, with the conformal anomaly number
$c=1$.
At any temperature of the conducting regime, the grand potential
of a classical 2D Coulomb system of characteristic size $R$ is supposed
to exhibit a large-$R$ expansion of type (\ref{1.1}), with however a +
sign in front of the logarithmic term:
\begin{equation} \label{1.2}
\beta \Omega_{\rm Coulomb} = A R^2 + B R + \frac{\chi}{6} {\rm ln}\, R
+ {\rm const} + \cdots .
\end{equation}
Plausible arguments for a critical-like behaviour were first given
for Coulomb gases with periodic boundary conditions \cite{Forrester},
then for Coulomb systems confined to a domain by (vacuum) plain hard
walls \cite{Jancovici1}, by inert ideal-conductor walls
(i.e. when the electric potential obeys Dirichlet boundary conditions)
\cite{Jancovici2} and finally by ideal-dielectric boundaries
(i.e. when the electric potential obeys Neumann boundary conditions)
\cite{Tellez}.
The explicit checks were done in the Debye-H\"uckel limit
\cite{Torres}, at the free-fermion point of the Thirring
representation of the symmetric two-component plasma, based
on the formalism developed in \cite{Cornu}, and for the one-component
plasma at $\beta=2$ in a disc \cite{Jancovici1,Jancovici2}.
Only recently, a direct derivation of the universal finite-size
correction term was done, in the whole stability range of temperatures,
for the specific cases of the symmetric two-component plasma
living on the surface of a sphere \cite{Jancovici3} and in a disc
surrounded by vacuum \cite{Samaj1}.
In both cases, the universal prefactor to the ${\rm ln}\, R$
correction term in (\ref{1.2}) was related to the bulk second moment
of the density structure factor which is known \cite{Janco}.
This is very different from the semi-infinite geometries where
the universality of the Casimir force is related to the second moment
sum rule for the {\em charge} structure factor \cite{Jancovici2,Buenzli}.

It is generally believed that, for semi-infinite systems, the presence
of an electrolyte between fluctuating conductor walls screens the
long-ranged Casimir force to some residual short-ranged force
\cite{Attard,Dean}.
The study of Casimir phenomena via fully microscopic Coulomb models
has the advantage of a coherent description of electrostatic fluctuations
inside conducting walls and the image forces acting on the electrolyte
particles, without any ad hoc separation Ansatz used in usual
macroscopic treatments \cite{Attard}.
We used this strategy to show the screening effect of Casimir forces for
semi-infinite geometries in paper \cite{Jancovici4}, in what follows
referred to as I.

The aim of the present work is to extend the study of the screening
(cancellation) phenomena to universal Casimir terms appearing
in the large-size expansions of the grand potential (\ref{1.2})
for the Coulomb systems confined in fully-finite 2D geometries,
in particular the disc geometry.
The confining disc walls as well as the electrolyte inside the disc
are modelled by two different microscopic two-component plasmas of
point-like particles with $+/-$ unit charges in thermal equilibrium.
Two cases are solved exactly: the high-temperature (Debye-H\"uckel)
limit $\beta\to 0$ and the Thirring free-fermion point $\beta=2$
corresponding to the collapse of positive and negative pairs
of point-like charges.
From the technical point of view, the circular symmetry of the problem
leads to infinite summations over specific products of modified
Bessel functions which have to be evaluated by using the asymptotic
Debye expansion; we apply a few technicalities which help us to
simplify a relatively complicated algebra.
Similarities and differences with respect to screening phenomena
in semi-infinite geometries, described in paper I, are pointed out.

\begin{figure}[h]
\begin{center}
\includegraphics{fig1.eps}
\mbox{}
\mbox{Figure 1. Disc geometry.}
\end{center}
\end{figure}

The model is defined as follows.
We shall consider Coulomb-gas systems of point-like particles with
symmetric $+/-$ unit charges.
Thermal equilibrium is treated in the grand canonical ensemble
characterized by the inverse temperature $\beta$ and by the couple
of equivalent (there is no external electrostatic potential),
possibly position-dependent, particle fugacities
$z_+({\bf r}) = z_-({\bf r}) = z({\bf r})$.
In the disc geometry presented in figure 1, there are two domains:
the disc of radius $R$, $\Lambda_R = \{ {\bf r}, \vert {\bf r}\vert < R \}$,
and its complement ${\bar \Lambda}_R = \{ {\bf r},\vert{\bf r}\vert\ge R \}$.
The electrolyte in $\Lambda_R$ is modelled by a two-component
plasma with the particle fugacity denoted by $z$.
The wall ${\bar \Lambda}_R$ is modelled by another two-component plasma
with the particle fugacity $z_0$; the choice $z_0> 0$ corresponds to
a conducting wall $(\epsilon\to\infty)$ while $z_0=0$ corresponds to
vacuum (plain hard wall with $\epsilon=1$).
In 2D, the Coulomb potential $v$ at a spatial position ${\bf r}$,
induced by a unit charge at the origin ${\bf 0}$, is the solution
of the 2D Poisson equation
\begin{equation} \label{1.3}
\Delta v({\bf r}) = - 2 \pi \delta({\bf r}) .
\end{equation}
Explicitly,
\begin{equation} \label{1.4}
v({\bf r}) = -{\rm ln}\,( \vert {\bf r}\vert /a)
\end{equation}
where $a$ is a free length scale, which only determines the zero of
the potential and should not enter statistical mean values.
The interaction energy of charged particles $\{ i, q_i \}$,
immersed in a homogeneous medium of dielectric constant = 1, is
$\sum_{i<j} q_i q_j v(\vert {\bf r}_i-{\bf r}_j \vert)$.

The paper is organized as follows.

In general section 2, the Coulomb gas confined to some 2D regions
is shown to be equivalent, in the ideal-conductor limit, to the
massless Gaussian model defined in the complementary empty regions.
Since in the derivation \cite{Houches} of the large-size expansion
of the free energy (\ref{1.1}) for the critical Gaussian model with
$c=1$ only the curvature of the boundary is used, this explains
the difference in sign of the logarithmic term between
(\ref{1.1}) and (\ref{1.2}).

In section 3, the underlying 2D fully-finite Coulomb system is solved
in the Debye-H\"uckel limit.
All possible realizations of the model are considered, and the
cancellation of the universal Casimir terms is documented when
both the disc domain and its complement are occupied by a Coulomb gas.
Fundamental differences between fully-finite and semi-infinite
geometries are pointed out.

The exact solution of the model system at the Thirring free-fermion
point $\beta=2$ is presented in section 4.
It is shown that basic features of Casimir phenomena, predicted by
the mean-field theory, persist also at this specific temperature.

A brief recapitulation is given in section 5.

\section{The Gaussian model approach}
The universal term $(\chi/6)\ln R$ in the grand potential
(\ref{1.2}) of a finite 2D Coulomb system closely resembles
the logarithmic term in (\ref{1.1}) valid for critical systems.
At first sight, it is surprising that a Coulomb system,
with short-range particle correlations, exhibits a critical-like behaviour.
It has been argued \cite{Jancovici1,Jancovici2} that this is related to
the existence of long-ranged critical-like correlations of
the electric potential; reasons for the difference in sign of the
logarithmic term between (\ref{1.1}) and (\ref{1.2}) have been given.
A slightly different, and we believe more convincing, argument
will now be presented.

\subsection{General formalism}
In the present subsection, we consider a 2D manifold on which some
regions $C$ are classical ideal conductors and some regions $E$ are empty.
The ideal conductors are obtained as the infinite fugacity limit
of conductors having some microscopic structure, which amounts to take
the limit in which the microscopic length scale goes to zero.
Thus, the results which will be obtained in this limit are expected
to be valid when the length scales under consideration are much larger
than the microscopic lengths.
For simplicity, we restrict ourselves to an infinite-plane manifold,
$C\cup E = R^2$, and start with conductors which are symmetric Coulomb
gases of the kind described in the Introduction, but some generalizations
are straightforward.

The interaction energy $E_N$ of $\pm 1$ charged particles
$\{ q_j, {\bf r}_j \}_{j=1}^N$ is expressible in terms of the
microscopic charge density
${\hat \rho}_N({\bf r}) = \sum_{j=1}^N q_j \delta({\bf r}-{\bf r}_j)$
as follows
\begin{equation} \label{2.1}
E_N(\{ q_j,{\bf r}_j \}) = \frac{1}{2} \int_{R^2} {\rm d}^2 r
\int_{R^2} {\rm d}^2 r' {\hat\rho}_N({\bf r}) v(\vert {\bf r}-{\bf r}'\vert)
{\hat\rho}_N({\bf r}') - \frac{1}{2} N v(0) ,
\end{equation}
where $v(0)$ is the (diverging) self-energy.
The thermodynamic properties of the Coulomb system are determined
by the grand partition function $\Xi$ defined as follows
\numparts
\begin{equation} \label{2.2a}
\Xi = \sum_{N_+=0}^{\infty} \sum_{N_-=0}^{\infty}
\frac{z_+^{N_+}}{N_+!} \frac{z_-^{N_-}}{N_-!} Q(N_+,N_-) ,
\end{equation}
where
\begin{equation} \label{2.2b}
Q(N_+,N_-) = \int_C \prod_{j=1}^N {\rm d}^2 r_j
\exp\left[ - \beta E_N(\{ q_j, {\bf r}_j \}) \right]
\end{equation}
\endnumparts
is the configuration integral of $N_+$ positive and $N_-$ negative
charges, $N=N_+ + N_-$ and $z_+ = z_- = z$ are the equivalent
fugacities of charged particles.
Let us insert the energy representation (\ref{2.1}) into (\ref{2.2b}).
The self-energy simply renormalizes the particle fugacities.
With regard to the fact that, according to (\ref{1.3}),
$-\Delta/(2\pi)$ is the inverse operator of the Coulomb potential
$v({\bf r})$, the standard Hubbard-Stratonovich transformation
(see e.g. ref. \cite{Samuel}) can be used to express the integral
bilinear term as
\begin{eqnarray}
\fl \exp\left[ - \frac{\beta}{2} \int_{R^2} {\rm d}^2 r \int_{R^2}
{\rm d}^2 r' {\hat\rho}_N({\bf r}) v(\vert {\bf r}-{\bf r}'\vert)
{\hat\rho}_N({\bf r}') \right]
\nonumber \\
= \frac{\int {\cal D}\phi \exp\left[ \int_{R^2} {\rm d}^2 r
\left( \frac{\beta}{4\pi} \phi \Delta \phi + {\rm i} \beta \phi {\hat\rho}_N
\right) \right]}{\int {\cal D}\phi \exp\left( \int_{R^2} {\rm d}^2 r
\frac{\beta}{4\pi} \phi \Delta \phi \right)} , \label{2.3}
\end{eqnarray}
where $\phi({\bf r})$ is a real scalar field and $\int {\cal D} \phi$
denotes the functional integration over this field.
The consequent factorization of the contributions from $(N_+,N_-)$
particle states in (\ref{2.2a}) then allows to express
the grand partition function of the system in the form \cite{Minnhagen}
\numparts
\begin{equation} \label{2.4a}
\Xi = \frac{\int{\mathcal D}\phi\exp(-S[z({\bf r})])}
{\int{\mathcal D}\phi\exp(-S[0])},
\end{equation}
with
\begin{equation} \label{2.4b}
S[z({\bf r})]=\int_{C\cup E} {\rm d}^2 r\left[\frac{\beta}{4\pi}\phi({\bf r})
(-\Delta)\phi({\bf r})
-2z({\bf r})\cos(\beta\phi({\bf r}))\right].
\end{equation}
\endnumparts
being the 2D Euclidean action of the sine-Gordon theory and
\begin{equation} \label{2.5}
z({\bf r}) = \left\{
\begin{array}{ll}
z & \mbox{for ${\bf r} \in C$,} \cr
0 & \mbox{for ${\bf r} \in E$.}
\end{array}  \right.
\end{equation}
The normalization of $z$ is fixed by the short-distance expansion
of the two-point correlation function
\begin{equation} \label{2.6}
\langle {\rm e}^{{\rm i}\beta \phi({\bf r})}
{\rm e}^{-{\rm i}\beta \phi({\bf r}')} \rangle \sim
\vert {\bf r}-{\bf r}' \vert^{-\beta} \qquad
\mbox{as $\vert {\bf r}-{\bf r}' \vert \to 0$}
\end{equation}
under which the self-energy factor disappears from statistical
relations; for more details see ref. \cite{Samaj2} and the references
cited therein.

Since the scalar $\phi$-field has to be regular at infinity,
the term $\phi (-\Delta) \phi$ in (\ref{2.4b}) can be transformed
via the integration by parts into $\vert \nabla \phi \vert^2$.
The sine-Gordon action (\ref{2.4b}) thus takes its minimum
at a $\phi({\bf r})$ constant in space.
Due to a discrete symmetry $\phi\to \phi + 2\pi n/\beta$
with any integer $n$, the action has infinitely many ground states
$\vert 0_n \rangle$ characterized by the associate expectation
values of the field $\langle \phi \rangle_n = 2 \pi n / \beta$.
These ground states become all degenerate when the size of the Coulomb
domain $\vert C\vert$ is large \cite{Zamolodchikov}.
It is therefore sufficient to develop the sine-Gordon action
(\ref{2.4b}), on the classical level as well as on higher
quantum-correction levels, around any one of these ground states
\cite{Zamolodchikov},
say $\vert 0_0 \rangle$ with $\langle \phi \rangle_0 = 0$.
Now, in the ideal-conductor limit $z\to\infty$,
the minimum of $-2z\cos(\beta\phi)$ around $\phi=0$ becomes infinitely
sharp and $\phi({\bf r})$ identically vanishes in the regions $C$.
Up to an irrelevant multiplicative constant,
the numerator of (\ref{2.4a}) becomes
\begin{equation} \label{2.7}
Z_{\rm G}=\int{\mathcal D}\phi\exp\left[-\frac{\beta}{4\pi}
\int_E{\rm d}^2 r\phi({\bf r})(-\Delta)\phi({\bf r})\right],
\end{equation}
i.e. the partition function of the massless Gaussian model in the empty
region(s).
The boundary condition for $\phi$ is that it vanishes at the interface(s)
of $E$ and $C$, since it has to be continuous and vanishes in $C$.
The denominator of (\ref{2.4a}) is just an (infinite) constant,
independent of the geometry of the regions $E$ and $C$.

Although $\phi$ has some resemblance with the electric potential, it
also has some drastically different properties.
For instance, in a bulk Coulomb gas, the $\phi$ of the sine-Gordon
representation has short-range two-point correlations while
the electric potential has long-range ones \cite{Caillol}.
Here, the electric potential has fluctuations in the conductor(s)
(these fluctuations survive in the ideal-conductor limit) and
by continuity the electric potential has fluctuations at the interface(s)
of $E$ and $C$, while $\phi$ is identically zero at the interface(s).

\subsection{Parallel conducting plates in $\nu$ dimensions}
In the case of two parallel conducting plates, at a distance $d$ of each
other, separated by vacuum, the Gaussian model (\ref{2.7}), generalized
to $\nu$ dimensions, has been shown \cite{Blote,Jancovici2} to have in its
free energy per unit area of one plate a universal $d$-dependent term $F$
such that the corresponding electrostatic Casimir force per unit area
$-\partial F/\partial d$ is attractive and given by
\begin{equation}  \label{2.8}
-\beta\frac{\partial F}{\partial d}=
-\frac{s_{\nu-1}\Gamma(\nu)\zeta(\nu)}{(2\pi)^{\nu-1}(2d)^{\nu}},
\end{equation}
where $s_{\nu}$ is the area of the unit sphere in $\nu$ dimensions and
where $\Gamma$ and $\zeta$ are the gamma function and the Riemann zeta
function, respectively.
In the Appendix, we give another derivation of (\ref{2.8}) for
the Gaussian model.
In subsection 2.2. of I, we have obtained the same Casimir force (\ref{2.8})
by using the fluctuations of the electric potential
(which do not vanish at the surface of the plates).
This is a check about the validity of the present method using
the Gaussian model with simple Dirichlet boundary conditions
for the $\phi$ field.

\subsection{A hole in a 2D ideal conductor}
We now come back to two dimensions.
We consider an ideal conductor filling the whole plane except for
an empty hole of characteristic size $R$, with a smooth boundary.
It has been shown \cite{Houches} that the Gaussian model has
a free energy of the form (\ref{1.1}) with $c=1$ and $\chi=1$.
Thus, $\beta$ times the grand potential of the system has the
universal term $-(1/6)\ln R$.
If the hole is a disc in a plane, $R$ may be chosen as its radius.

\subsection{A 2D Coulomb gas surrounded by an ideal conductor}
We now consider the opposite geometry of an ideal conductor filling some
connected region $C$ of the plane, with a smooth boundary, surrounded by
vacuum. Now the relevant Gaussian model fills the exterior of $C$.
In the derivation of ref. \cite{Houches}, the curvature of the boundary
is used, and it is easy to see that the present geometry differs from
the one of Section 2.3 by a change of sign of the curvature, leading to
the universal term $+(1/6)\ln R$ in (\ref{1.2}).

\section{Debye-H\"uckel theory}

\subsection{General result}
The high-temperature limit of the model system in figure 1 is described
by the Debye-H\"uckel theory.
For the general formalism, see e.g. section 3.1. of paper I.
The formalism applied to the 2D symmetric Coulomb gas
can be briefly summarized as follows.

In the bulk (homogeneous) regime, the total particle number density $n$
as a function of the fugacity $z$ is given by
\begin{equation} \label{3.1}
\frac{n^{1-(\beta/4)}}{z a^{\beta/2}} =
2 \left( \frac{\pi\beta}{2} \right)^{\beta/4}
\exp \left( \frac{\beta C}{2} \right) ,
\end{equation}
where $C$ is the Euler's constant.
The inverse Debye length is defined by $\kappa^2 = 2\pi \beta n$.

In the inhomogeneous regime, the whole system domain $\Lambda$
can be separated into disjunct physically non-equivalent subdomains,
$\Lambda = \cup_{\alpha} \Lambda^{(\alpha)}$.
Within the grand canonical formalism, each subdomain is characterized
by a constant fugacity, $z({\bf r}) = z^{(\alpha)}$  for
${\bf r}\in \Lambda^{(\alpha)}$; the choice $z^{(\alpha)}=0$ corresponds
to a vacuum subdomain with no particles allowed to occupy the space.
The corresponding ``bulk'' particle density $n^{(\alpha)}$ is related to
the particle fugacity $z^{(\alpha)}$ via (\ref{3.1}), and the corresponding
inverse Debye length is $\kappa_{\alpha} = (2\pi\beta n^{(\alpha)})^{1/2}$.
One introduces the screened Coulomb potential $G$ which obeys
within each domain $\Lambda^{(\alpha)}$ the differential equation
\begin{equation} \label{3.2}
\left[ \Delta_1 - \kappa_{\alpha}^2 \right] G({\bf r}_1,{\bf r}_2)
= - 2\pi \delta( {\bf r}_1 - {\bf r}_2 ), \qquad
{\bf r}_1 \in \Lambda^{(\alpha)} .
\end{equation}
Here, the spatial position of the source point ${\bf r}_2$ is arbitrary.
These equations are supplemented by the usual electrostatic conditions
at each subdomain boundary $\partial \Lambda^{(\alpha)}$: $G$ and its
normal derivative with respect to the boundary surface,
$\partial_{\perp} G$, are continuous at $\partial\Lambda^{(\alpha)}$.
The leading $\beta$-correction to the constant particle density
$n^{(\alpha)}$ in the subdomain $\Lambda^{(\alpha)}$ is then determined
by linearizing the exponential in the expression
\begin{equation} \label{3.3}
n^{(\alpha)}({\bf r}) = 2 z^{(\alpha)} \exp \left\{ \frac{\beta}{2}
\lim_{{\bf r}'\to {\bf r}}
\left[ - G({\bf r},{\bf r}') + v(\vert {\bf r} - {\bf r}' \vert) \right]
\right\}
\end{equation}
where the 2D Coulomb potential $v$ is defined in equation (\ref{1.4}).

We now apply the above formalism to the geometry of interest
presented in figure 1.
The inverse Debye length will be denoted by $\kappa$ for the
disc domain $\Lambda_R = \{ {\bf r}, r<R \}$ and by $\kappa_0$
for the complement wall domain ${\bar\Lambda}_R = \{ {\bf r}, r>R \}$.
We first consider the case when both $\kappa$ and $\kappa_0$
are nonzero.
Let the source point ${\bf r}_2$ be first in the disc domain,
i.e. $r_2<R$.
Equations (\ref{3.2}) then take the form
\numparts
\begin{eqnarray}
\left[ \Delta_1 - \kappa^2 \right] G({\bf r}_1,{\bf r}_2)
= - 2\pi \delta({\bf r}_1-{\bf r}_2) , \qquad & (r_1 < R) , \label{3.4a} \\
\left[ \Delta_1 - \kappa_0^2 \right] G({\bf r}_1,{\bf r}_2)
= 0 , & (r_1 > R). \label{3.4b}
\end{eqnarray}
\endnumparts
In terms of polar coordinates, the solution of these equations can be
written as an expansion of the form
\numparts
\begin{eqnarray}
\fl G({\bf r}_1,{\bf r}_2) = \sum_{l=-\infty}^{\infty}
\left[ I_l(\kappa r_<) K_l(\kappa r_>) + a_l I_l(\kappa r_1) I_l(\kappa r_2)
\right] \exp[{\rm i}l(\varphi_1-\varphi_2)], \quad
& ( r_1 < R), \label{3.5a} \\
\fl G({\bf r}_1,{\bf r}_2) = \sum_{l=-\infty}^{\infty}
b_l K_l(\kappa_0 r_1) I_l(\kappa r_2) \exp[{\rm i}l(\varphi_1-\varphi_2)],
& ( r_1 > R). \label{3.5b}
\end{eqnarray}
\endnumparts
Here, $r_< = {\rm min}\{ r_1,r_2 \}$ and $r_> = {\rm max}\{ r_1,r_2 \}$.
$I_l$ and $K_l$ are modified Bessel functions possessing the symmetry
$I_l = I_{-l}$, $K_l = K_{-l}$.
They satisfy the same differential equation
\begin{equation} \label{3.6}
f'' + \frac{1}{x} f' - \left( 1 + \frac{l^2}{x^2} \right) f = 0 ,
\qquad \mbox{$f=I_l(x)$ or $K_l(x)$,}
\end{equation}
but exhibit different asymptotic behaviours \cite{Gradshteyn}:
\begin{equation} \label{3.7}
I_l(x) { \atop \stackrel{\sim}{\scriptstyle{x\to\infty}} }
\frac{\rme^x}{\sqrt{2\pi x}} , \qquad
K_l(x) { \atop \stackrel{\sim}{\scriptstyle{x\to\infty}} }
\left( \frac{\pi}{2 x} \right)^{1/2} \rme^{-x}
\end{equation}
and
\begin{equation} \label{3.8}
I_l(x) { \atop \stackrel{\sim}{\scriptstyle{x\to 0}} }
\frac{1}{\vert l\vert !} \left( \frac{x}{2} \right)^{\vert l \vert},
\qquad K_l(x) { \atop \stackrel{\sim}{\scriptstyle{x\to 0}} }
\frac{\vert l\vert !}{2\vert l\vert}
\left( \frac{x}{2} \right)^{- \vert l \vert},
\end{equation}
except for the special $l=0$ case of $K_0(x) \sim - \ln(x/2)-C$
in the limit $x\to 0$ where $C$ is Euler's constant.
We see that, in representations (\ref{3.5a}) and (\ref{3.5b}) of $G$,
these asymptotic behaviours ensure the regularity of $G$ at
the origin and at infinity.
The coefficients $a_l$ and $b_l$ are determined by the above defined
boundary conditions for $G({\bf r}_1,{\bf r}_2)$ at $r_1 = R$:
\numparts
\begin{eqnarray}
K_l(\kappa R) + a_l I_l(\kappa R) = b_l K_l(\kappa_0 R) , \label{3.9a} \\
\kappa \left[ K_l'(\kappa R) + a_l I_l'(\kappa R) \right]
= \kappa_0 b_l K_l'(\kappa_0 R) . \label{3.9b}
\end{eqnarray}
\endnumparts
Using the recursion formulas for the modified Bessel functions
\numparts
\begin{eqnarray}
x I_l'(x) = x I_{l\pm 1}(x) \pm l I_l(x) , \label{3.10a} \\
x K_l'(x) = - x K_{l\pm 1}(x) \pm l K_l(x) , \label{3.10b}
\end{eqnarray}
\endnumparts
and the Wronskian relation \cite{Gradshteyn}
\begin{equation} \label{3.11}
I_l(x) K_{l+1}(x) + I_{l+1}(x) K_l(x) = \frac{1}{x} ,
\end{equation}
equations (\ref{3.9a}) and (\ref{3.9b}) give
\numparts
\begin{eqnarray}
a_l = - \frac{K_l(\kappa R)}{I_l(\kappa R)} + \frac{1}{R W_l}
\frac{K_l(\kappa_0 R)}{I_l(\kappa R)} , \label{3.12a} \\
b_l = \frac{1}{R W_l} , \label{3.12b}
\end{eqnarray}
\endnumparts
where the auxiliary quantity $W_l$ is given by
\begin{eqnarray}
\fl W_l = \kappa I_l'(\kappa R) K_l(\kappa_0 R) -
I_l(\kappa R) \kappa_0 K_l'(\kappa_0 R) \nonumber \\
\lo= \kappa I_{l+1}(\kappa R) K_l(\kappa_0 R) +
I_l(\kappa R) \kappa_0 K_{l+1}(\kappa_0 R) \label{3.13} \\
\lo= \kappa I_{l-1}(\kappa R) K_l(\kappa_0 R) +
I_l(\kappa R) \kappa_0 K_{l-1}(\kappa_0 R) . \nonumber
\end{eqnarray}
Note the symmetries $a_l = a_{-l}$, $b_l = b_{-l}$ and $W_l = W_{-l}$.
With regard to the differential equation (\ref{3.6}) obeyed by
the modified Bessel functions, it is easy to show that
$W_l$ fulfills the equality
\begin{equation} \label{3.14}
\frac{\partial}{\partial R} {\rm ln}\, ( R W_l ) = \frac{1}{W_l}
(\kappa^2 - \kappa_0^2) I_l(\kappa R) K_l(\kappa_0 R) .
\end{equation}
The same procedure can be applied when the source point ${\bf r}_2$
lies outside of the disc, i.e. $r_2>R$.
For $r_1>R$, one obtains the solution of the form
\begin{equation} \label{3.15}
\fl G({\bf r}_1,{\bf r}_2) = \sum_{l=-\infty}^{\infty}
\left[ I_l(\kappa_0 r_<) K_l(\kappa_0 r_>)
+ c_l K_l(\kappa_0 r_1) K_l(\kappa_0 r_2) \right]
\exp[{\rm i}l(\varphi_1-\varphi_2)] .
\end{equation}
The coefficients $c_l$ are determined by the boundary conditions for
$G$ at $r_1=R$ as follows
\begin{equation} \label{3.16}
c_l = - \frac{I_l(\kappa_0 R)}{K_l(\kappa_0 R)} +
\frac{1}{R W_l} \frac{I_l(\kappa R)}{K_l(\kappa_0 R)} .
\end{equation}
They fulfill the symmetry $c_l = c_{-l}$.

To obtain the density profile, note that in relations (\ref{3.5a}) and
(\ref{3.15}) the first terms in the sums over $l$ correspond to
the expansion of the modified Bessel functions
$K_0(\kappa\vert {\bf r}_1-{\bf r}_2\vert)$ and
$K_0(\kappa_0\vert {\bf r}_1-{\bf r}_2\vert)$, respectively.
In formula (\ref{3.3}) for the particle density, these bulk
contributions, minus the pure Coulomb potential $v$, imply the
density-fugacity relationship (\ref{3.1}).
The second terms in the sums over $l$ in equations (\ref{3.5a}) and
(\ref{3.15}) are ``reflected'' contributions due to the boundary
at $r=R$.
After the linearization of (\ref{3.3}) in $\beta$, they lead to
\numparts
\begin{eqnarray}
n(r) = n - \frac{\beta n}{2} \sum_{l=0}^{\infty} \mu_l a_l
I_l^2(\kappa r) , & (r < R) , \label{3.17a} \\
n(r) = n_0 - \frac{\beta n_0}{2} \sum_{l=0}^{\infty} \mu_l c_l
K_l^2(\kappa_0 r) , \qquad & (r > R) . \label{3.17b}
\end{eqnarray}
\endnumparts
Here, $\mu_l$ is the Neumann factor: $\mu_0=1$ and $\mu_l=2$ for $l\ge 1$.

Having at one's disposal the density profile, the grand canonical
partition function $\Xi_R(\kappa,\kappa_0)$ can be deduced in
the following way.
For the present geometry, $\Xi_R$ is defined by
\begin{equation} \label{3.18}
\fl \Xi_R = \sum_{N_+,N_-=0}^{\infty} \frac{1}{N_+! N_-!} \prod_{i=1}^N
\left( \int_0^R \rmd^2 r_i \, z + \int_R^{\infty} \rmd^2 r_i \, z_0 \right)
\exp\left[ - \beta \sum_{i<j} q_i q_j
v(\vert {\bf r}_i - {\bf r}_j \vert) \right] ,
\end{equation}
where $N_+$ $(N_-)$ is the number of positively (negatively)
charged particles and $N = N_+ + N_-$.
The averaged particle density at position ${\bf r}$ is given by
$n({\bf r}) = \langle \sum_i \delta({\bf r}-{\bf r}_i)\rangle$.
Combining this with equation (\ref{3.18}), one finds that
\begin{equation} \label{3.19}
\frac{\partial}{\partial R} {\rm ln}\, \Xi_R =
2 \pi R \left[ n(R^-) - n(R^+) \right] ,
\end{equation}
where $R^-$ $(R^+)$ means the left (right) limit to $R$.
With respect to the density profile [equations (\ref{3.17a}) and
(\ref{3.17b}) with coefficients $a_l$ and $c_l$ given by (\ref{3.12a})
and (\ref{3.16}), respectively], this relation takes the explicit form
\begin{eqnarray}
\fl \frac{\partial}{\partial R} {\rm ln}\, \Xi_R = 2 \pi R ( n - n_0 )
- \frac{1}{2} (\kappa^2 - \kappa_0^2) \sum_{l=0}^{\infty}
\mu_l \frac{1}{W_l} I_l(\kappa R) K_l(\kappa_0 R) \nonumber \\
+ \frac{R}{2} \sum_{l=0}^{\infty} \mu_l
\left[ \kappa^2 I_l(\kappa R) K_l(\kappa R) -
\kappa_0^2 I_l(\kappa_0 R) K_l(\kappa_0 R) \right] . \label{3.20}
\end{eqnarray}
The integration with respect to $R$ of the second term on the rhs
of (\ref{3.20}) can be done by using (\ref{3.14}) and the integration
of the last term follows from the indefinite integration formula
for Bessel functions \cite{Erdelyi}
\begin{eqnarray}
\fl \int \rmd x \, x I_l(x) K_l(x) = \frac{x^2}{4} \left[
2 I_l(x) K_l(x) + I_{l+1}(x) K_{l-1}(x) + I_{l-1}(x) K_{l+1}(x) \right]
\nonumber \\
\lo= \frac{1}{2}
\left[ (x^2+l^2) I_l(x) K_l(x) - x^2 I_l'(x) K_l'(x) \right] . \label{3.21}
\end{eqnarray}
Thence, from (\ref{3.20}) one gets
\begin{eqnarray}
\fl {\rm ln}\, \Xi_R(n,n_0) = {\rm const} + \pi R^2 ( n - n_0 )
- \frac{1}{2} \sum_{l=0}^{\infty} \mu_l {\rm ln}\, ( R W_l ) \nonumber \\
+ \frac{1}{4} \sum_{l=0}^{\infty} \mu_l \left\{
\left[ (\kappa R)^2 +l^2 \right] I_l(\kappa R) K_l(\kappa R)
-(\kappa R)^2 I_l'(\kappa R) K_l'(\kappa R) \right\}
\label{3.22} \\
- \frac{1}{4} \sum_{l=0}^{\infty} \mu_l \left\{
\left[ (\kappa_0 R)^2 +l^2 \right] I_l(\kappa_0 R) K_l(\kappa_0 R)
-(\kappa_0 R)^2 I_l'(\kappa_0 R) K_l'(\kappa_0 R) \right\} .
\nonumber
\end{eqnarray}
The integration constant is fixed by considering the $R\to 0$ limit.
With the aid of the asymptotic formulas (\ref{3.8}) one gets,
for instance, $\lim_{R\to 0} ( R W_l ) = (\kappa/\kappa_0)^{\vert l\vert}$.
After simple algebra, the final result reads
\begin{eqnarray}
\fl {\rm ln}\, \Xi_R(n,n_0) = {\rm ln}\, \Xi_{R=0} + \pi R^2 ( n - n_0 )
- \frac{1}{2} \sum_{l=0}^{\infty} \mu_l {\rm ln}\, \left[ R W_l \left(
\frac{\kappa_0}{\kappa} \right)^l \right] \nonumber \\
+ \frac{1}{4} \sum_{l=0}^{\infty} \mu_l \left\{
\left[ (\kappa R)^2 +l^2 \right] I_l(\kappa R) K_l(\kappa R)
-(\kappa R)^2 I_l'(\kappa R) K_l'(\kappa R) - l \right\}
\label{3.23} \\
- \frac{1}{4} \sum_{l=0}^{\infty} \mu_l \left\{
\left[ (\kappa_0 R)^2 +l^2 \right] I_l(\kappa_0 R) K_l(\kappa_0 R)
-(\kappa_0 R)^2 I_l'(\kappa_0 R) K_l'(\kappa_0 R) - l \right\} .
\nonumber
\end{eqnarray}
Here, $\Xi_{R=0}$ is the grand canonical partition function of the system
with zero disc radius, given by the obvious relation
$\lim_{\Lambda\to\infty} {\rm ln}\, \Xi_{R=0}/\vert\Lambda\vert =
\beta p(n_0) = n_0 [1 - (\beta/4)]$.
Formula (\ref{3.23}) was derived under the assumption that both
particle densities $n$ and $n_0$ are nonzero.

The limit $n\to 0$ corresponds to no particles present inside the disc,
i.e. the vacuum disc hole surrounded by the fluctuating conductor wall.
Taking the $n\to 0$ limit in (\ref{3.23}), one gets
\begin{eqnarray}
\fl {\rm ln}\, \Xi_R(n=0,n_0) = {\rm ln}\, \Xi_{R=0} -\pi R^2 n_0
- \frac{1}{2} \sum_{l=0}^{\infty} \mu_l
{\rm ln} \left[ 2 \left( \frac{\kappa_0 R}{2} \right)^{l+1}
\frac{1}{l!} K_{l+1}(\kappa_0 R) \right] \label{3.24} \\
- \frac{1}{4} \sum_{l=0}^{\infty} \mu_l \left\{
\left[ (\kappa_0 R)^2 +l^2 \right] I_l(\kappa_0 R) K_l(\kappa_0 R)
-(\kappa_0 R)^2 I_l'(\kappa_0 R) K_l'(\kappa_0 R) - l \right\} .
\nonumber
\end{eqnarray}

The limit $n_0\to 0$ corresponds to no particles present outside
of the disc, i.e. the Coulomb system in the disc surrounded
by vacuum (the plain hard wall).
One can take the $n_0\to 0$ limit in every term of equation (\ref{3.23}),
except for the $l=0$ term in the first summation on the rhs of
that equation.
This $l=0$ term makes a problem due to the logarithmic divergence of
$K_0(\kappa_0 R)$ in $W_0$ in the limit $\kappa_0\to 0$.
The problem with the $l=0$ term, caused by the fact that the effective
Coulomb potential $G(r)$ is not screened at asymptotically large distances
$r$ in the present limit $\kappa_0\to 0$, was discussed e.g.
in \cite{Jancovici5}).
According to this reference, when calculating the particle density
(\ref{3.17a}) one finds that the coefficient $a_0$ depends on the
free length scale $a$ of the 2D Coulomb potential (\ref{1.4}).
Since a statistical mean value should not depend on $a$,
the limit $a\to\infty$, which puts the zero of the Coulomb potential
to infinity, has to be considered.
Our formalism in the limit $\kappa_0\to 0$ is equivalent to that
outlined in \cite{Jancovici5} if one sets $\kappa_0 = 1/a$.
Thus, returning to the particle density (\ref{3.17a}), the auxiliary
quantity $W_l$ (\ref{3.13}), considered in the limit $\kappa_0\to 0$
and for $R>0$, behaves like
\begin{equation} \label{3.25}
W_l \sim \kappa I_{l-1}(\kappa R) K_l(\kappa_0 R)
\qquad \mbox{for $l\ge 0$,}
\end{equation}
and the coefficients $a_l$, given by (\ref{3.12a}), take the form
\begin{equation} \label{3.26}
a_l = \frac{K_{l-1}(\kappa R)}{I_{l-1}(\kappa R)} ,
\qquad l=0,1, \ldots .
\end{equation}
On the basis of (\ref{3.19}) taken with $n(R^+)\equiv 0$, it holds
\begin{equation} \label{3.27}
\frac{\partial}{\partial R} {\rm ln}\, \Xi_R(n,n_0=0) = 2\pi R n
- \frac{\kappa^2 R}{2} \sum_{l=0}^{\infty} \mu_l
\frac{K_{l-1}(\kappa R)}{I_{l-1}(\kappa R)} I_l^2(\kappa R) .
\end{equation}
The integration of this equation finally implies
\begin{eqnarray}
\fl {\rm ln}\, \Xi_R(n,n_0=0) = \pi R^2 n - \frac{1}{2}
{\rm ln}\left[ (\kappa R) I_1(\kappa R) \right] + {\rm const}
\nonumber \\ - \sum_{l=1}^{\infty} {\rm ln}
\left[ \left( \frac{2}{\kappa R} \right)^{l-1}
(l-1)! I_{l-1}(\kappa R) \right] \label{3.28} \\
+ \frac{1}{4} \sum_{l=0}^{\infty} \mu_l \left\{
\left[ (\kappa R)^2 +l^2 \right] I_l(\kappa R) K_l(\kappa R)
-(\kappa R)^2 I_l'(\kappa R) K_l'(\kappa R) - l \right\} . \nonumber
\end{eqnarray}
Here, since the grand partition function depends on the length
scale $a=1/\kappa_0$, the integration constant is in fact infinite
in the considered limit $a\to\infty$.
It is clear from the derivation procedure that formula (\ref{3.28})
is valid only for $R>0$, and it cannot serve as a basis for an
expansion around the $R=0$ point.

\subsection{Large-$R$ analysis}
Each of the above derived grand potentials
$\Omega = - (1/\beta) {\rm ln}\, \Xi$ is given in terms of infinite
sums which cannot be summed up explicitly.
What can be done is the evaluation of first few terms of the asymptotic
expansion of the sums for large disc radius $R\to\infty$.
Denoting by $\alpha$ either of the dimensionless combinations
$\kappa R$ or $\kappa_0 R$, one has to use the Debye expansion
\cite{Abramowitz} of the modified Bessel functions $I_l(\alpha), K_l(\alpha)$
and of their derivatives, since this expansion is valid for large $l$
{\em uniformly} with respect to $\alpha/l$.
In particular, one has
\numparts
\begin{eqnarray}
I_l(\alpha) = \frac{1}{\sqrt{2\pi}}
\frac{1}{(\alpha^2+l^2)^{1/4}}\, \rme^{\eta} \left[ 1+\frac{3t-5t^3}{24 l}
+ O\left( \frac{1}{\alpha^2+l^2} \right) \right] , \label{3.29a} \\
K_l(\alpha) = \sqrt{\frac{\pi}{2}}
\frac{1}{(\alpha^2+l^2)^{1/4}}\, \rme^{-\eta} \left[ 1-\frac{3t-5t^3}{24 l}
+ O\left( \frac{1}{\alpha^2+l^2} \right) \right] , \label{3.29b}
\end{eqnarray}
\endnumparts
and
\numparts
\begin{eqnarray}
I'_l(\alpha) = \frac{1}{\sqrt{2\pi}}
\frac{(\alpha^2+l^2)^{1/4}}{\alpha}\, \rme^{\eta}
\left[ 1+\frac{-9t+7t^3}{24 l}
+ O\left( \frac{1}{\alpha^2+l^2} \right) \right] , \label{3.30a} \\
K'_l(\alpha) = - \sqrt{\frac{\pi}{2}}
\frac{(\alpha^2+l^2)^{1/4}}{\alpha}\, \rme^{-\eta}
\left[ 1-\frac{-9t+7t^3}{24 l}
+ O\left( \frac{1}{\alpha^2+l^2} \right) \right] , \label{3.30b}
\end{eqnarray}
\endnumparts
where
$$
\eta(l,\alpha) = \sqrt{\alpha^2+l^2} - l\, {\rm sinh}^{-1}(l/\alpha),
\qquad t = l/\sqrt{\alpha^2+l^2} .
$$
The consequent sums over $l$ can be performed by applying the
(generalized) Euler-MacLaurin summation formula \cite{Abramowitz}:
\begin{equation} \label{3.31}
\fl \sum_{l=m}^{n} f(l) = \int_m^n f(l) \rmd l
+ \frac{1}{2}[ f(n)+f(m) ] + \frac{B_2}{2!} [ f'(n)-f'(m) ]
+ \frac{B_4}{4!}[ f'''(n)-f'''(m)] + \cdots ,
\end{equation}
where $B_{\nu}$ are Bernoulli numbers: $B_2=1/6$, $B_4=-1/30$, etc.

In the case of the Coulomb gas of particle density $n_0$, localized outside
of the disc of radius $R$ with vacuum in the disc hole, formula (\ref{3.24})
evaluated in the $R\to\infty$ limit implies
\begin{equation} \label{3.32}
\fl \beta \Omega_R(n=0,n_0) = - \beta p(n_0)(\vert \Lambda\vert -\pi R^2)
+ \beta \gamma(n_0)\, 2\pi R - \frac{1}{6} {\rm ln}(\kappa_0 R) + O(1) .
\end{equation}
The first term on the rhs of (\ref{3.32}) is the bulk contribution
with the pressure $p$ given by
\begin{equation} \label{3.33}
\beta p(n_0) = \left( 1-\frac{\beta}{4} \right) n_0 ,
\end{equation}
the second term is the surface contribution with the surface tension $\gamma$
given, in the Debye-H\"uckel limit, by \cite{Jancovici4}
\begin{equation} \label{3.34}
\beta \gamma(n_0) = \int_0^{\infty} \frac{\rmd l}{4\pi}
{\rm ln}\left[ \frac{(l+\sqrt{\kappa_0^2+l^2})^2}{4 l \sqrt{\kappa_0^2+l^2}}
\right] = \frac{\kappa_0}{8\pi} (4-\pi) .
\end{equation}
Finally, the third logarithmic term has the universal coefficient $-1/6$.
This Casimir term, caused by electrostatic fluctuations inside the wall,
tends to {\em dilate} the empty disc domain.
This is the fundamental difference in comparison with semi-infinite
geometries where fluctuating walls {\em attract} one another.

In the case of the Coulomb gas of particle density $n$, localized
inside the disc of radius $R$ and surrounded by vacuum,
formula (\ref{3.28}) evaluated in the $R\to\infty$ limit gives
\begin{equation} \label{3.35}
\fl \beta \Omega_R(n,n_0=0) = - \beta p(n)\, \pi R^2 +
\beta \gamma(n)\, 2\pi R + \frac{1}{6} {\rm ln}(\kappa R) + O(1) .
\end{equation}
As before, the first and second terms on the rhs  of (\ref{3.35})
correspond to the volume and surface parts of $\beta\Omega$, respectively.
The third logarithmic term has the coefficient $1/6$, in agreement
with the general relation (\ref{1.2}) taken at the disc value of $\chi=1$.
This coefficient is opposite to the one in (\ref{3.32}) which
confirms the relation of the universal term to the curvature
of the constraining surface.
It is important to note that in semi-infinite geometries (see e.g. paper I)
the grand potential of a Coulomb system constrained by vacuum plain hard
walls does not exhibit the universal Casimir term.
This is another fundamental difference between fully-finite and
semi-infinite geometries.

From a technical point of view, it is useful to sum the two expressions
for the grand potential (\ref{3.24}) and (\ref{3.28}), taken at
the same particle density denoted say by ${\bar n}$:
\begin{eqnarray}
\fl \beta \Omega_R(n={\bar n},n_0=0) + \beta \Omega_R(n=0,n_0={\bar n})
= \beta \Omega_{R=0} + {\rm const} \nonumber \\ + \frac{1}{2} {\rm ln}
\left[ ({\bar\kappa}R)^2 K_1({\bar\kappa}R) I_1({\bar\kappa}R) \right]
+ \sum_{l=1}^{\infty} {\rm ln} \left[ \frac{1}{2l}
({\bar\kappa} R)^2 I_{l-1}({\bar\kappa}R) K_{l+1}({\bar\kappa}R) \right] .
\label{3.36}
\end{eqnarray}
Using first the recursion formulas (3.10) and subsequently the
asymptotic expansions (3.29) and (3.30) for the modified Bessel functions,
the application of the Euler-MacLaurin summation formula (\ref{3.31})
implies after simple algebra that
\begin{equation} \label{3.37}
\fl \beta \Omega_R(n={\bar n},n_0=0) + \beta \Omega_R(n=0,n_0={\bar n})
= \beta \Omega_{R=0} + 2 \beta \gamma({\bar n}) 2 \pi R + O(1).
\end{equation}
This relation proves the consistency of asymptotic expansions
(\ref{3.32}) and (\ref{3.35}), and enables us to write down one
knowing the explicit form of the other.
Simplifying technicalities of this kind will be used in what follows.

We are now ready for studying the large-$R$ asymptotic behaviour of the
grand potential when both particle densities $n$ and $n_0$ are
nonzero, see formula (\ref{3.23}).
One has for the specific combination of grand potentials
\begin{eqnarray}
\fl \beta \Omega_R(n,n_0) -
\left[ \beta \Omega_R(n,n_0=0) + \beta \Omega_R(n=0,n_0) \right]
= {\rm const} + \frac{1}{2} {\rm ln} \left[
\frac{W_0}{\kappa \kappa_0 R I_1(\kappa R) K_1(\kappa_0 R)} \right]
\nonumber \\
- \sum_{l=1}^{\infty} {\rm ln}\left\{
\frac{[\kappa R I_{l-1}(\kappa R)] [\kappa_0 R K_{l+1}(\kappa_0 R)]}{2
l R W_l} \right\} . \label{3.38}
\end{eqnarray}
As above, using the recursion formulas and the asymptotic expansions
for the modified Bessel functions, the Euler-MacLaurin summation
formula leads to
\begin{eqnarray}
\fl \beta \Omega_R(n,n_0) -
\left[ \beta \Omega_R(n,n_0=0) + \beta \Omega_R(n=0,n_0) \right] \nonumber \\
\lo= R \int_0^{\infty} {\rm d}l\,
{\rm ln}\left[ \frac{2 l \left( \sqrt{\kappa^2+l^2}+\sqrt{\kappa_0^2+l^2}
\right)}{\left( l+\sqrt{\kappa^2+l^2}\right)\left(l+\sqrt{\kappa_0^2+l^2}
\right)} \right] + O(1) . \label{3.39}
\end{eqnarray}
With regard to the asymptotic expansions (\ref{3.32}) and (\ref{3.35}),
one finally arrives at
\begin{equation} \label{3.40}
\fl \beta \Omega_R(n,n_0) = -\beta p(n) \pi R^2 -
\beta p(n_0) ( \vert \Lambda\vert - \pi R^2) +
\beta \gamma(n,n_0) 2\pi R + O(1) ,
\end{equation}
where
\begin{equation} \label{3.41}
\beta \gamma(n,n_0) = \int_0^{\infty} \frac{\rmd l}{4\pi}
{\rm ln}\left[ \frac{\left( \sqrt{\kappa^2+l^2} + \sqrt{\kappa_0^2+l^2}
\right)^2}{4 \sqrt{\kappa^2+l^2} \sqrt{\kappa_0^2+l^2}} \right]
\end{equation}
is the (dimensionless) contact surface tension of the two 2D plasmas
in the Debye-H\"uckel limit, see equation (3.26) of paper I.
The universal logarithmic Casimir term does not appear in (\ref{3.40}),
it is ``screened''.
This phenomenon is intuitively expected: since there are Coulomb
systems on both sides of the disc boundary, the curvature contributions
with opposite signs cancel with one another.
The same cancellation of long-ranged Casimir forces takes place in
semi-infinite geometries, see paper I.

\section{The free-fermion point}

\subsection{General result}
The 2D Coulomb gas of symmetric unit charges is exactly solvable at
the collapse point $\beta = 2$; for general formalism see e.g.
section 4.1. of paper I.
In the grand canonical formalism, at $\beta=2$, the bulk system
is characterized by the rescaled particle fugacity $m=2\pi a z$
[$a$ is a free length scale introduced in (\ref{1.2})]
which has the dimension of an inverse length.
The many-particle densities can be expressed in terms of
specific Green functions $G_{qq'}({\bf r},{\bf r}')$ $(q,q'=\pm)$;
because of the symmetry between positive and negative particles
one only needs $G_{++}$ and $G_{-+}$.
These Green functions are determined by the equations
\begin{equation} \label{4.1}
( \Delta_1 - m^2 ) G_{++}({\bf r}_1,{\bf r}_2) =
- m \delta({\bf r}_1-{\bf r}_2)
\end{equation}
and
\begin{equation} \label{4.2}
G_{-+}({\bf r}_1,{\bf r}_2) = - \frac{1}{m}
\left( \frac{\partial}{\partial x_1} + \rmi \frac{\partial}{\partial y_1}
\right) G_{++}({\bf r}_1,{\bf r}_2) ,
\end{equation}
supplemented by the vanishing boundary conditions when
$\vert {\bf r}_1-{\bf r}_2 \vert \to \infty$.
In infinite space, the solution of (\ref{4.1}) reads
\begin{equation} \label{4.3}
G_{++}({\bf r}_1,{\bf r}_2) = \frac{m}{2\pi}
K_0(m\vert {\bf r}_1-{\bf r}_2 \vert) .
\end{equation}
The one-particle densities $n_+ = n_- = n/2$ ($n$ is the total
particle density), given by
\begin{equation} \label{4.4}
n_q({\bf r}) = m G_{qq}({\bf r},{\bf r}) ,
\end{equation}
are infinite since $K_0(m r)$ diverges logarithmically as $r\to 0$.
Regularization of the Coulomb interaction by a short-distance
cutoff $L$ implies for the particle density
\begin{equation} \label{4.5}
n = \frac{m^2}{\pi} K_0(m L)
{ \atop \stackrel{\sim}{\scriptstyle{m L \to 0}} }
\frac{m^2}{\pi} \left[ {\rm ln}\, \left( \frac{2}{m L} \right) - C \right] .
\end{equation}
In the inhomogeneous case when the system domain
$\Lambda = \cup_{\alpha} \Lambda^{(\alpha)}$, each subdomain
$\Lambda^{(\alpha)}$ is characterized by a constant rescaled fugacity,
$m({\bf r}) = m_{\alpha}$ for ${\bf r}\in \Lambda^{(\alpha)}$,
and the corresponding bulk density $n_{\alpha}$ defined as a function
of $m_{\alpha}$ by (\ref{4.5}).
Within each domain, the Green function $G_{++}$ obeys the
differential equation
\begin{equation} \label{4.6}
( \Delta_1 - m_{\alpha}^2 ) G_{++}({\bf r}_1,{\bf r}_2)
= - m_{\alpha} \delta({\bf r}_1-{\bf r}_2) , \qquad
{\bf r}_1 \in \Lambda^{(\alpha)} ,
\end{equation}
where the spatial position of the source point ${\bf r}_2$ is
arbitrary.
$G_{-+}$ is determined by relation (\ref{4.2}) with $m$ substituted
by the subdomain-dependent $m({\bf r}_1)$.
The boundary conditions are that both $G_{++}$ and $G_{-+}$ must be
continuous at each subdomain boundary $\partial \Lambda^{(\alpha)}$.
The one-particle densities are again given by (\ref{4.4})
with $m\to m({\bf r})$.

For the disc geometry of figure 1, the rescaled particle fugacity
is equal to $m$ in the disc domain $r<R$ and to $m_0$ in the
complementary wall $r>R$.
Let the source point ${\bf r}_2$ be first in the disc domain,
i.e. $r_2<R$.
Equations (\ref{4.1}) then take the form
\numparts
\begin{eqnarray}
( \Delta_1 - m^2 ) G_{++}({\bf r}_1,{\bf r}_2) =
- m \delta({\bf r}_1-{\bf r}_2) , \qquad & (r_1 < R) , \label{4.7a} \\
( \Delta_1 - m_0^2) G_{++}({\bf r}_1,{\bf r}_2) = 0 ,
& (r_1 > R) . \label{4.7b}
\end{eqnarray}
\endnumparts
In terms of polar coordinates, the solution of these equations
is written as an expansion of the form
\numparts
\begin{eqnarray}
\fl G_{++}({\bf r}_1,{\bf r}_2) = \frac{m}{2\pi} \sum_{l=-\infty}^{\infty}
\left[ I_l(m r_<) K_l(m r_>) \right. \nonumber \\
\left. + a_l I_l(m r_1) I_l(m r_2) \right]
\exp[{\rm i}l(\varphi_1-\varphi_2)], & ( r_1 < R), \label{4.8a} \\
\fl G_{++}({\bf r}_1,{\bf r}_2) = \frac{m}{2\pi} \sum_{l=-\infty}^{\infty}
b_l K_l(m_0 r_1) I_l(m r_2) \exp[{\rm i}l(\varphi_1-\varphi_2)], \qquad
& ( r_1 > R). \label{4.8b}
\end{eqnarray}
\endnumparts
The polar version of relation (\ref{4.2}) for $G_{-+}$ takes the form
\begin{equation} \label{4.9}
G_{-+}({\bf r}_1,{\bf r}_2) = - \frac{\rme^{\rmi \varphi_1}}{m({\bf r}_1)}
\left( \frac{\partial}{\partial r_1} + \frac{\rmi}{r_1}
\frac{\partial}{\partial\varphi_1} \right) G_{++}({\bf r}_1,{\bf r}_2) .
\end{equation}
The coefficients $a_l$ and $b_l$ are determined by the continuity
conditions for $G_{++}({\bf r}_1,{\bf r}_2)$ and
$G_{-+}({\bf r}_1,{\bf r}_2)$ at the disc boundary $r_1=R$:
\numparts
\begin{eqnarray}
K_l(mR) + a_l I_l(mR) = b_l K_l(m_0 R) , \label{4.10a} \\
K_{l+1}(mR) - a_l I_{l+1}(mR) = b_l K_{l+1}(m_0 R) . \label{4.10b}
\end{eqnarray}
\endnumparts
The solution reads
\numparts
\begin{eqnarray}
a_l = - \frac{K_l(mR)}{I_l(mR)} + \frac{1}{m R V_l}
\frac{K_l(m_0 R)}{I_l(mR)} , \label{4.11a} \\
b_l = \frac{1}{m R V_l} ,
\end{eqnarray}
\endnumparts
where
\begin{equation} \label{4.12}
V_l = I_l(mR) K_{l+1}(m_0 R) + I_{l+1}(m R) K_l(m_0 R) .
\end{equation}
Note the symmetry $V_l = V_{-l-1}$.
It can be readily shown that the auxiliary quantity $V_l$ fulfills
the equation
\begin{equation} \label{4.13}
\frac{\partial}{\partial R} {\rm ln}\, ( R V_l) = \frac{m-m_0}{V_l}
\left[ I_l(m R) K_l(m_0 R) + I_{l+1}(m R) K_{l+1}(m_0 R) \right] .
\end{equation}
One proceeds analogously when the source point ${\bf r}_2$ lies
in the wall, i.e. $r_2 > R$.
For the case $r_1 > R$, one gets
\begin{eqnarray}
\fl G_{++}({\bf r}_1,{\bf r}_2) = \frac{m_0}{2\pi} \sum_{l=-\infty}^{\infty}
\left[ I_l(m_0 r_<) K_l(m_0 r_>) \right. \nonumber \\
\left. + c_l K_l(m_0 r_1) K_l(m_0 r_2) \right]
\exp[\rmi l(\varphi_1-\varphi_2)], \qquad ( r_1, r_2 > R) . \label{4.14}
\end{eqnarray}
The coefficients $c_l$ are given by the boundary conditions at
$r_1=R$ as follows
\begin{equation} \label{4.15}
c_l = - \frac{I_l(m_0 R)}{K_l(m_0 R)} + \frac{1}{m_0 R V_l}
\frac{I_l(m R)}{K_l(m_0 R)} .
\end{equation}

The density at $r\to R^-$, determined by (\ref{4.4}), (\ref{4.8a})
and (\ref{4.11a}), reads
\begin{equation} \label{4.16}
n(R^-) = \frac{m}{\pi R} \sum_{l=-\infty}^{\infty}
\frac{1}{V_l} I_l(m R) K_l(m_0 R) .
\end{equation}
The sum in (\ref{4.16}) is divergent, and so it has to be formally
regularized by taking an upper cutoff on $l$.
Similarly, the density at $r\to R^+$, determined by (\ref{4.4}),
(\ref{4.14}) and (\ref{4.15}), is expressible as follows
\begin{equation} \label{4.17}
n(R^+) = \frac{m_0}{m} n(R^-) .
\end{equation}
The grand canonical partition function $\Xi_R(m,m_0)$ is again
determined by the differential equation (\ref{3.19}), in particular
\begin{equation} \label{4.18}
\fl \frac{\partial}{\partial R} {\rm ln}\, \Xi_R =
2 (m - m_0) \sum_{l=0}^{\infty} \frac{1}{V_l}
\left[ I_l(m R) K_l(m_0 R) + I_{l+1}(m R) K_{l+1}(m_0 R) \right] .
\end{equation}
Due to the equality (\ref{4.13}),
this equation can be integrated to the form
\begin{equation} \label{4.19}
{\rm ln}\, \Xi_R(m,m_0) = {\rm const} +
2 \sum_{l=0}^{\infty} {\rm ln}(R V_l) .
\end{equation}
The integration constant is fixed by the $R\to 0$ limit.
Since $\lim_{R\to 0} R V_l = m_0^{-1} (m/m_0)^l$ for $l\ge 0$,
one gets
\begin{eqnarray}
\fl {\rm ln}\, \Xi_R(m,m_0) = {\rm ln}\, \Xi_{R=0} +
2 \sum_{l=0}^{\infty} {\rm ln}\Bigg\{ \left( \frac{m_0}{m} \right)^l
m_0 R  \nonumber \\
\times \left[ I_l(m R) K_{l+1}(m_0 R) + I_{l+1}(m R) K_l(m_0 R) \right]
\Bigg\} . \label{4.20}
\end{eqnarray}
Here, $\Xi_{R=0}$ is the grand canonical partition function of
the system with zero disc radius, i.e.
$\lim_{\Lambda\to\infty} {\rm ln}\, \Xi_{R=0}/\vert \Lambda\vert
= \beta p(n_0)$.

Although (\ref{4.20}) was derived under the assumption
that both rescaled fugacities $m$ and $m_0$ are nonzero, there is
no problem to consider the zero limit of either of fugacities
directly in (\ref{4.20}).
One obtains
\begin{equation} \label{4.21}
{\rm ln}\, \Xi_R(m,m_0=0) = 2 \sum_{l=0}^{\infty} {\rm ln}
\left[ l! \left( \frac{2}{mR}\right)^l I_l(m R) \right]
\end{equation}
and
\begin{equation} \label{4.22}
{\rm ln}\, \Xi_R(m=0,m_0) = {\rm ln} \Xi_{R=0}
+ 2 \sum_{l=0}^{\infty} {\rm ln}\left[
\frac{2}{l!} \left( \frac{m_0 R}{2} \right)^{l+1}
K_{l+1}(m_0 R) \right] .
\end{equation}

\subsection{Large-$R$ analysis}
The result (\ref{4.21}) for the 2D two-component plasma at
$\beta=2$, in the disc surrounded by vacuum, has already been
derived in \cite{Jancovici1}.
There, the large-$R$ asymptotic form of the grand potential
was derived in the form
\begin{equation} \label{4.23}
\beta \Omega_R(m,m_0=0) = -\beta p(m) \pi R^2 + \beta \gamma(m) 2\pi R
+ \frac{1}{6} {\rm ln}( m R ) + O(1) ,
\end{equation}
where $p(m)$ is the regularized pressure and the surface tension
$\gamma(m)$ is given by
\begin{equation} \label{4.24}
\beta \gamma(m) = \int_0^{\infty} \frac{\rmd l}{2\pi}
{\rm ln}\left( \frac{2\sqrt{m^2+l^2}}{l+\sqrt{m^2+l^2}} \right)
= m \left( \frac{1}{4} - \frac{1}{2\pi} \right) .
\end{equation}

One can apply the standard procedure to formula (\ref{4.22})
corresponding to the plasma outside of the empty disc, to obtain
the large-$R$ asymptotic behaviour
\begin{equation} \label{4.25}
\fl \beta \Omega_R(m=0,m_0) = -\beta p(m_0)(\vert \Lambda\vert - \pi R^2)
+ \beta \gamma(m_0) 2\pi R - \frac{1}{6} {\rm ln}( m_0 R ) + O(1) .
\end{equation}
One sees that, similarly as in the Debye-H\"uckel limit, in comparison
with (\ref{4.23}) the universal logarithmic term has the opposite sign.

Technically, it is simpler to derive (\ref{4.25}) by summing up
the two basic expressions (\ref{4.21}) and (\ref{4.22}) taken
at the same rescaled fugacity, say ${\bar m}$:
\begin{eqnarray}
\fl \beta \Omega_R(m={\bar m},m_0=0) + \beta \Omega_R(m=0,m_0={\bar m})
\nonumber \\ \lo= \beta \Omega_{R=0} - 2 \sum_{l=0}^{\infty}
{\rm ln}\left[ {\bar m} R I_l({\bar m}R) K_{l+1}({\bar m}R) \right] .
\label{4.26}
\end{eqnarray}
The standard asymptotic procedure for the modified Bessel functions
leads to the following asymptotic behaviour
\begin{equation} \label{4.27}
\fl \beta \Omega_R(m={\bar m},m_0=0) + \beta \Omega_R(m=0,m_0={\bar m})
= \beta \Omega_{R=0} + 2 \beta \gamma({\bar m}) 2\pi R + O(1) ,
\end{equation}
which proves the consistency of relations (\ref{4.23}) and (\ref{4.25}).

Finally, considering the special combination of grand potentials
\begin{eqnarray}
\fl \beta \Omega_R(m,m_0) - \left[ \beta \Omega_R(m,m_0=0)
+ \beta \Omega_R(m=0,m_0) \right] \nonumber \\
\lo= - 2 \sum_{l=0}^{\infty} {\rm ln} \left[
1 + \frac{I_{l+1}(m R) K_l(m_0 R)}{I_l(m R) K_{l+1}(m_0 R)} \right] ,
\label{4.28}
\end{eqnarray}
and using the standard asymptotic procedure, one gets
\begin{eqnarray}
\fl \beta \Omega_R(m,m_0) - \left[ \beta \Omega_R(m,m_0=0)
+ \beta \Omega_R(m=0,m_0) \right] \nonumber \\
\lo= R \int_0^{\infty} \rmd l\, {\rm ln}\left[
\frac{\left( l+\sqrt{m^2+l^2}\right)\left( l+\sqrt{m_0^2+l^2}\right)}{2
\left( m m_0 + \sqrt{m^2+l^2} \sqrt{m_0^2+l^2} +l^2\right)} \right] + O(1) .
\label{4.29}
\end{eqnarray}
With regard to the asymptotic relations (\ref{4.23}) and (\ref{4.25}),
one concludes that
\begin{equation} \label{4.30}
\fl \beta \Omega_R(m,m_0) = - \beta p(m) \pi R^2 - \beta p(m_0)
(\vert \Lambda\vert - \pi R^2) + \beta \gamma(m,m_0) 2\pi R + O(1) ,
\end{equation}
where
\begin{equation} \label{4.31}
\beta \gamma(m,m_0) = \int_0^{\infty} \frac{\rmd l}{2\pi}
{\rm ln}\left( \frac{2 \sqrt{m^2+l^2} \sqrt{m_0^2+l^2}}{m m_0
+ \sqrt{m^2+l^2} \sqrt{m_0^2+l^2} + l^2}\right)
\end{equation}
defines the surface tension of the two 2D plasmas at $\beta=2$,
see equation (4.12) of paper I.
As was expected, the universal logarithmic term disappears once again.

We have verified that the basic features of the mean-field behaviour,
predicted by the Debye-H\"uckel analysis in the previous section,
persist also at the specific inverse temperature $\beta=2$.

\section{Conclusion}
The name ``ideal-conductor'' usually means that the electric potential has
some constant value (for instance zero) inside the conductor, without any
fluctuations.
More precisely, we call such a model of a conductor
``inert ideal-conductor''.
The present paper deals with microscopic models of conductors, in
which there are fluctuations of the electric potential.
They are ``living'' conductors.
In the limit when the microscopic lengths, such as the Debye length,
go to zero (in practice, are small compared to the macroscopic length,
here such as the radius $R$), the fluctuations of the electric
potential survive.
We call a conductor in that high-density limit ``living ideal-conductor''.

In a previous publication \cite{Jancovici2}, Coulomb systems with
inert ideal-conductor boundary conditions were studied\footnote{In
\cite{Jancovici2}, the Coulomb potential in $\nu$ dimensions $(\nu>2)$
was defined as $r^{2-\nu}$ while in I it was defined as $r^{2-\nu}/(\nu-2)$.
These different definitions do not change the Casimir terms under
consideration.}.
In this model, for a Coulomb system in a slab of width $d$, a
repulsive Casimir force was found, which is just the opposite of
(\ref{2.8}) which holds for living ideal-conductor plates
separated by vacuum.
For a Coulomb system in a slab with living ideal-conductor walls,
the two contributions cancel each other, leaving only a
short-range attraction, as shown in I.

Similarly, in ref. \cite{Jancovici2}, a 2D Coulomb system in a disc
of radius $R$ with inert ideal-conductor walls was found to have
a logarithmic universal contribution $(1/6)\ln R$ to $\beta$ times
its grand potential.
This is just the opposite of the $-(1/6)\ln R$ found here for an empty
circular hole surrounded by a Coulomb gas.
Again, for a Coulomb system in a disc with living
conductor walls, the two contributions cancel each other.

\ack
The authors acknowledge support from the CNRS-SAS agreement,
Project No. 14439.
A partial support of L. {\v S}amaj by a VEGA grant is acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Unitat de superfície, patrons per facilitar l'intercanvi de dades en els mesuraments quotidians o científics
 Superfície o àrea una quantitat que expressa l'extensió d'una superfície o forma de dues dimensions al pla
 Matemàtiques:
 Superfície (matemàtiques), varietat diferenciable en dues dimensions.
 Superfície corba, la que no és plana ni està composta de superfícies planes,
 Superfície de revolució, l'engendrada pel moviment d'una corba que gira al voltant d'una recta fixa,
 superfície cilíndrica, la generada per una recta que es mou paral·lelament sobre una corba donada,
 Superfície cònica, la generada per una recta que, passant per un punt fix, recorre una corba donada,
 superfície desenvolupable, la superfície reglada que es pot estendre sobre un pla, conservant la distància entre els seus punts, vegeu Superfície reglada # Superfícies desenvolupables
 Superfície reglada, la qual pot contenir línies rectes en determinades direccions.
 Superfície de Riemann
 Integral de superfície, integral definida calculada sobre una superfície la qual pot ser corbada
 Superfície reglada, superfície engendrada per una família infinita de rectes que depenen d'un paràmetre
 Superfície cilíndrica, superfície il·limitada formada per la unió de les rectes paral·leles que passen per cada un dels punts d'una corba fixa
 Àrea de superfície, mesura de la quantitat de superfície exposada que presenta un sòlid expressada en unitats quadrades
 Superfície de revolució
 superfície equipotencial, lloc geomètric dels punts d'un camp escalar en els quals el potencial de camp
 Superfície específica, superfície real d'un objecte per oposició a la superfície aparent
 Física:
 Superfície (física), és el límit d'un medi continu en contacte amb un altre mitjà de propietats físiques diferenciades,
 superfície d'ona, la formada pels punts que es troben en la mateixa fase, en un moment donat, en un moviment ondulatori vegeu Front d'ona
 Superfície equipotencial, el lloc geomètric dels punts d'un camp de força que tenen el mateix potencial.
 Micel·lania:
 Gran superfície, centre comercial de grans dimensions
 Míssil aire-superfície

redpajama_set_name: RedPajamaWikipedia


Old books! On your wall! With large paintings by Ekaterina Panikanova! I'm a little more excited about the books than the paintings (nothing against Ms. Panikanova, the paintings are lovely) but, hey, I'll take 'em as intended. The wallpaper is heavy-duty, and is even washable with a soft cloth, so don't be too terribly precious about where it is installed. Also, pay attention to the explanation of how the mural is laid out; it can be a tiny bit confusing at first glance.
Available from NLXL, $153.00 .


Alright, in my last post I asked which mini you wanted to see in an unboxing next. I got a few responses between this site, Facebook, and Reddit, and the results were unanimous. Everyone wanted Grim Ironjaw for the next unboxing. So… here we go!
When the first images of the Grim Ironjaw were released, I read a lot of negative comments online about the mini. Honestly, I am not really sure what all people have against this mini. Other than the fact that it is yet another mini standing on one foot (seriously… does EVERY mini from Forge World have to be on one foot?!), I really have no issue with the new Grim. I might even say that I like him. He is definitely a different style than the last Grim Ironjaw, but he fits the new style of the game quite well.
Time to dig in and take a look.
Front and back views of the package.
Here are the obligatory shots of the insert.
As with the other Star Players, the insert opens up to lots of good fluff about Grim. It is always worth taking a few minutes to read up on your Star Players.
The back of the insert, as always, has assembly instructions. At only five pieces, Grim is, by far, the least complex Star Player released by Forge World so far. But… what is that piece listed as number 5? …More on that in a minute.
Again, as with every other Star Player from Forge World, we get both a standard base, and a slotted base.
I decided to stick to the using the base tab. At this point I kind of figured it could not hurt to use it. While drilling the top of the base tab, my bit slipped slightly and you can see where it broke through the front of the tab. I did not worry about that as once the tab was inserted into the base, the base itself would prevent the pin from sliding out of the tab. Luckily Grim's foot was angled in such a way that I was able to embed the pin fairly deep into his ankle for added strength.
Here Grim is attached to his base. Only three more pieces to go.
Next up is his pony tail braid. Unfortunately, I was not careful enough trimming the braid off of the sprue, and I did not leave enough of a nub on the end to seat into the small hole on the back of his head. Looks like I will have to pin another piece on this mini.
Pinning the braid was easy enough. The braid itself had more than enough thickness to hold the pin, and the attachment point on his head already had a small hole. All I needed to do was drill one small hole, and slightly enlarge another. No sweat.
And just like that, the braid is attached. Over halfway done.
One hand down, one hand to go.
And just like that, we are done. Other than the weirdness with the base tab (what the heck is up with that?) this mini went together pretty easily. I honestly have no real assembly complaints on this one.
Here are a couple more angles on the mini.
Lets do a quick size comparison.
Here is Grim next to a boxed set Blitzer, and a 3rd edition Grim Ironjaw. His one footed lurching pose not withstanding, he is pretty much the same size as the 3rd edition Grim. Maybe just slightly larger. The main difference is that he is much more "realistic" scale as to his proportions.
My final verdict on the new Grim Ironjaw is a positive one. I genuinely like the mini. It is not the best Blood Bowl mini I have ever seen, but if you like the style of the newer releases, this Grim Ironjaw is a very good mini to add to your collection.
Next up will be the White Dwarf/Black Gobbo combo. Check back soon!
This entry was posted in Blood Bowl, Gaming and tagged Blood Bowl, Forge World, Forgeworld, Games Workshop, gaming, Grim Ironjaw, Miniatures, Star Players, table-top games, tabletop wargames, wargames, Wargaming by Mike C. Bookmark the permalink.
He looks pretty good. I like the large Mohawk and it's nice that they didn't make him yet another leaping troll-slayer.
I'm definitely not crazy about that 3e version of Grim.
My ni picks for the new Grim are, price, the ponytail, and that weird looking thing in his hand. Reminds of when I was converting Ogre Bills for BB. There was some piece in one of their hands and after I got done converting it ended up looking something like that.
I guess like the Blitzer, it's a stylized take on the same kinda fist weapon you see on 3rd Ed. Chaos Beastmen?
Yeah, I would say that it is a modernized version of the "knuckle dusters" that were fairly prevalent in 3rd edition minis.
The Chaos Beastmen have a spiked gauntlet or a studded gauntlet. Both are really similar to some medieval or gladiatorial weapons. I guess to me, Grim's weapon looks a bit too fantastical, maybe too elf-like. Yea, it's silly, but that's why I said it's just a nit-pick for me.
I get that. The 3rd edition minis have LOTS of examples of similar things. Morg has his mail fist, with added giant punch dagger, Black Orcs have their chain wrapped fists, Jordell has his giant punch dagger… they really are everywhere. This is just the next evolution… even if it is a little bit on the weird side.
I really like the mini. But the tab really puzzles me? Why afix to a tab when you can just glue straight to the base.
Yeah, it was definitely strange. I used it because I wanted to test it out as it was something I had never seen before, and I had lots of experience with just gluing the minis straight to bases.
Though it means another one-foot-on-the-ground mini, his pose mits the more dynamic 5th ed Dwarf team much better than the old stand-and-roar model!
Overall a solid model, lots of little details that'll make it easy to paint for a wash-loving noob or a fine details pro.
It is a more dynamic pose, for sure. I just wish once or twice they could manage a "dynamic" pose that involved having both feet on the ground. My reasons are purely practical as two feet touching the base means more stability.
Man… those Pro Elf Catchers were a REAL pain. Especially after the base tabs broke off.
I never tracked down the Pro Elves, but I can imagine. I think the one I was struggling with might have been a Jordell Freshbreeze or Hubris Rakarth. Both look precariously balanced on one foot.


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How Lafarge's mining, corruption cause untold hardship to Mfamosing community (Part 1)
Health and EnvironmentEnvironment
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By Ekemini Simon
WHILE Lafarge Africa PLC boasts of a humane environmental policy in her mining operations, this investigation by TheMail unravels the nexus between illegal and unhealthy practices by the company, the government's gross regulatory failure and untold hardship in Mfamosing, a solid mineral-rich community in Akamkpa, Cross River State.
Idaraobong Udo a mother of seven woke up on the morning of October 23, 2020 in Mfamosing community, located less than 100 kilometres south-east of Calabar, capital of Cross River State, a property owner but by night she was homeless wandering with her family from house to house to find who would shelter her and her seven children. After she had gone outside to carry out house chores, without a prior sign, her mud house caved in. She lost all her property and was only lucky that the ravaging landslide did not strike when she was inside with her children.
The night before the day the disaster struck, Lafarge Africa, a limestone mining company operating close to her house had blasted rocks in their limestone quarry. This was not the first time Lafarge blasted for extraction nevertheless. Their blasting only kept drawing closer to residential areas.
Now, where Udo and her family once had a home has been overgrown with weeds while she and her seven children patch up with a good Samaritan. Udo, with a voice saturated with agony, told TheMail that although officials of Lafarge in company with community leaders visited to see the havoc their operations had caused, that was the last time she ever saw and heard from the company officials.
Into the Crack Houses
Udo mentioned in the outset is not alone in the unfortunate experience of the resource curse in Mfamosing. Many houses in Abimfam, a sizable community within the Mfamosing community are either on the verge of caving in or falling apart.
Since it began operations in Mfamosing through different consortiums, Lafarge Africa PLC has been blasting rocks to source limestone beneath it at least thrice weekly. The blasting is accompanied by heavy vibration underneath which causes cracks and compromises the structural integrity of houses and sometimes landslides.
"Anytime they blast the rocks, it is like a bomb thrown inside the ground. The whole house would be shaking like an earthquake has occurred. It has caused cracks in all our houses here; be it mud or houses built with blocks. None is safe from the damage," says the community head, Abimfam, Joshua Ntuen.
With his face drenched in anguish, Ntuen showed our reporter his house which he built to lintel level before he noticed serious cracks around the house.
Joshua Ntuen shows TheMail reporter the crack around his house. Photo credit: Ekemini Simon.
TheMail also discovered that the mining operations of Lafarge are responsible for several landslides recorded in the community. Most residents have abandoned the houses they built with their hard-earned resources to become tenants in other communities where mining operations are not taking place. Their fear? One day, their house may collapse and crush them to death.
Daniel Udobong is quite popular in the Abimfam community. He spent six years building a five-bedroom bungalow with the hope that he will cease being a tenant. This was cut short since he had to vacate the house to a safe area after cracks began to surface.
Insight into Mining Asset in Mfamosing
Lafarge has six communities of Mfamosing as its host communities. They are Ekong Anaku, Mbobui, Akansoko, Akwa Ikot Effanga, Diamond and Abimfam (Camp II). Among these communities, Abimfam is where Lafarge mining operations exist. The community has an estimated limestone reserve of 30 million tonnes. For about two decades, mining companies with different nationalistic interests have scrambled to explore the million tonnes of limestone beneath Mfamosing's land.
This mineral resource used for the production of cement is extracted in the Abimfam community to feed its 5 million metric tonnes per year cement plant located at Lafarge's single largest cement production site, in Mfamosing.
Although information at the Corporate Affairs Commission shows that Lafarge Africa registered to do business in Nigeria on February 24, 1959, with registration number 1858, it started exploring Mfamosing's limestone in 2002.
According to Lafarge 2020 Accounts and Annual Report, the company was originally established in 2002 as United Cement Company Nigeria (UNICEM) Limited, after the acquisition of the moribund Calabar Cement Company (CalCemCo). The company, UNICEM, a joint venture amongst Lafarge, Holcim, Dangote and Flour Mills.
Eventually, Lafarge Africa, a subsidiary of Lafarge Holcim which is based in Zurich, Switzerland, acquired all the assets.
Flying Rocks and the Pangs of Distress and Death
The tales of sheer horror, fear of impending doom and forced evacuation from one's house and community are not limited to the cracks. Another distressing effect of Lafarge's operations on the community is flying rocks that stem from rock blasting. This has made residents of Mfamosing turn part-time occupants even in their homes.
In rain or blazing sun, asleep or awake, sick or dying; residents of Mfamosing must scramble out of their homes and head towards the main road once they hear the company's alarm indicating that blasting is about to commence.
Residents disclosed that the company does not have a specific rock blasting routine for the community to plan their activities. Day or night, anytime convenient for the company, blasting can take place and they have few minutes after the third siren and in some cases, orders from the company's military officers to get out of the blasting area.
Being alert for the sound of a siren is a very important survival skill in Mfamosing.
" When we hear the alarm, we all run to the main road close to the market to be safe from the flying rocks. After that, we return when the thick dust subsides. There are days that they blast twice daily and sometimes even at night", Iquo Akpan, a resident disclosed.
The community head, Abimfam, Joshua Ntuen laments the discomfort caused by the blasting operation even to animals. "The blasting comes with a heavy vibration. Even a dog won't be able to stay. Our domestic animals often join us to run".
Another resident, Uduak Saviour says: "Even if you were cooking at hand, you just have to leave your food and run. My main problem is that I have an aged mother who is 83 years of age staying here with me. You can imagine the stress of taking her to run almost every day. I don't want the flying rocks to kill her for me".
Saviour's mother may be lucky to have someone take her to safety zones when the blasting commences but Okon Okoko was not that lucky when rocks from the blast led to his early grave. The Abimfam community council briefed TheMail of Okoko's death when he was hit by a flying rock in late 2020.
"One night, Lafarge blasted their rock as usual. We all ran out and went to the main road. We didn't know Okoko was still inside. It was in the morning that we saw that his leg was crushed by a rock that hit through his thatched house and met him", says the Council Secretary, Simon Effiong.
He added that Okoko was a man without a family who probably wasn't sound in health the day the disaster struck thus limiting his ability to run for safety as usual. Effiong said tetanus later affected the crushed leg and this led to his death because of his inability to afford treatment.
The flying rocks from Lafarge do not spare the roofs. Nkoyo Nkennang is 60 years old and also a widow. She has had countless encounters with rocks destroying her roof.
one of the roofs in the community was destroyed by flying rocks. Photo credit: Ekemini Simon.
Holes on Nkoyo Nkennang's roof. Photo credit: Ekemini Simon.
" You can see the holes on our roof yourself. This is caused by the rocks they blast. When it rains, our house is usually flooded with water. If it rains at night, I just have to wake up and stand throughout the night and the sleepless nights usually cause me severe health issues during the day".
Health challenges attributed to limestone mining in Mfamosing
The environmental situation in Mfamosing is far worse and more pathetic than what one could ever imagine. Members of the Mfamosing community say that when limestone mining started in their domain, they gradually became victims of strange diseases.
They identify the dust that hovers after blasting and their water bodies which have been polluted as the main source of their plague.
Findings showed that Lafarge constantly drains water held up inside the mine into the surrounding environment. The water eventually finds its way to the community, where it pollutes the major source of water for residents and their livestock.
In their stream, to the ordinary eyes, the water sparkles as it flows and it doesn't give concern to the taste bud that much except for the clay-like taste experienced, residents said.
Water in Lafarge's mine. Photo credit: Ekemini Simon.
A resident of the area also stressed that owing to the contamination, the water does not lather easily for washing, and often requires more detergents before a good wash can be achieved.
Earlier, this reporter had sighted boreholes branded as being sponsored by Lafarge. Members of the community admit that the borehole was a relief until all the taps ran dry. The three boreholes they have were not functioning as of early December 2021 when the reporter visited. Residents say none of the boreholes lasted beyond a year.
Borehole provided by Lafarge but overgrown with weeds. Photo credit: Ekemini Simon.
" None is working again. The borehole stopped functioning because of the blasting. The level at which the water was has gone down hence when we pump it, we can't have water," says Youth Secretary of Abimfam Community, Okon Asuquo.
Privileged members of the community now resort to getting to a nearby community to buy water at an exorbitant rate. Three 25 litres sell for N500.
Sadly, those who cannot afford the essential commodity are left with a Hobson's Choice but to rely on the polluted stream for drinking and domestic purposes.
Community members reported that health problems such as cough, catarrh, eye irritation and heart disease, miscarriages, among other gastrointestinal ailments are now common among them.
Destruction of Crops and livelihood
Mfamosing community is largely agrarian. The environmental impact of limestone mining in the community is often cited by villagers for their poor farm yields. The vegetation is covered with dust and plants destroyed by blasted rocks. In separate interviews, farmers recount how heavy limestone dust covers their farmlands in the dry season leading to poor growth of their crops.
Farmland is covered by dust caused by limestone blasting. Photo credit: Ekemini Simon.
" During the rainy season, limestone particles from the extraction site flow with rainwater into our farms. This has made our cassava not grow well. We now have poor yield. It wasn't like this before the days Lafarge began extraction. Past companies were more responsible than Lafarge", says the Community Secretary, Effiong who is also a farmer.
Controversies over Encroachment from the mining area
While the company continues to extend its mining operations into the community forest and also residential areas, the people of Mfamosing feel short-changed that the company is encroaching into lands that are not part of the sites given to them by the government for mining.
The Secretary of Abimfam Community Council, Effiong notes " Lafarge by the day keeps mining into our forest area. Although we have not seen the area covered by the mining lease given to them by the government. We strongly doubt they have kept within the approved limit".
This is corroborated by the Village Head of Mfamosing, Mfam Clement Ekpe Emayip, who is also the Paramount Ruler of Akamkpa local government area. " They have encroached into our forest reserve. We did not give them the approval to mine where they are mining. If we had given, I would be the one to sign".
A freedom of information, FOI request to the Director-General, Nigerian Mining Cadastre Office, Ministry of Mines and Steel Development, which sought information on the mining lease to Lafarge Africa although acknowledged was not acceded to.
The request specifically demanded information on a copy of the mining lease, duration and area covered by the mining lease. A reminder sent to the office was not also attended to. According to section 5 (b) of the Nigerian Minerals Mining Act (2007), the Mining Cadastre Office is responsible for the relinquishment of mineral titles or extension of areas for mining.
This same request was also contained in an FOI request to the Plant Manager of Lafarge Africa in Mfamosing, Idara Uyok. Although the request and a reminder sent at the expiration of seven working days provided for in the Act were acknowledged, the request was not responded to.
Lafarge Breaches Environmental and Mining Laws
The Nigerian Minerals and Mining Act, which came into effect in 2007, does not only seek to protect the environment but also communities where solid mineral resources are found. The law protects the rights of host communities and their environment, provides for rehabilitation and penalties for defaulters.
Yet, Lafarge Africa in their operations has failed to make recourse to environmental and mining laws of the land. Analysis of these laws showed that Lafarge is culpable.
According to the National Environmental (quarrying and blasting operations) Regulations, 2013, section 20 states "a person shall not locate a quarry or engage in blasting within three kilometres (3km) of any existing residential, commercial or industrial area".
In addition, the Nigerian Minerals and Mining Act, 2007, section 3 (c) identifies lands excluded from mineral exploration and exploitation to include those occupied by any town or village.
Google Earth, an open-source app using satellite imagery, aided TheMail's investigation in ascertaining the distance between the blasting site and residences. From the blasting site to the last residential house in the Mfamosing community, it is 0.38kilometers. Measurement to the road which residents run to for safety is 0.58 kilometres, a limit that falls within the threat zone stipulated by law.
The measurement from the blasting site to the last residential house
Measurement to the road which residents run to for safety.
Besides, section 22 of the National Environmental Regulations adds that " A person shall not blast in such a way that the impact of such blast will cause any form of discomfort or nuisance to the public and residents within 1,000 metres from the epicentre of the site or users of the road thereof"
Regarding water pollution, not only host communities are protected by the law but also surrounding communities near and far.
"No person shall in the course of mining or exploration for minerals pollute or cause to be polluted any water or watercourse in the area within mining lease or beyond that area," Section 123 of the Mining Act states.
As highlighted earlier, Lafarge operations have destroyed houses, made residents be strangers in their homes, polluted their water bodies, destroyed farmlands among other discomforts.
Section 23 (2) of the National Environmental Regulations further specifies the time the blasting must not take place. "Blasting operations shall not be carried out at the rush hours of 7 am-10 am and 5 pm and beyond." Regardless, residents report that the company does not observe these restricted times. They blast even at night.
In addition, section 23 (6) notes that the adjourning community shall be initially informed at least 48 hours prior to any type of blasting operations through the mass media and augmented locally by the use of siren, announcement, letters, warning signs and other means deemed appropriate to convey an impending operation. Members of the community report that besides blaring a siren thrice during the day signifying that blasting will take place coupled with orders from military officers working with the company, Lafarge does not give warnings aforehand.
In addition, the Environmental Impact Assessment (EIA) Act of 1992 frowns at the pollution experienced in Mfamosing. Though Lafarge Africa is duly registered with the Corporate Affairs Commission (CAC), it failed to comply with the guiding principles of the EIA Act.
(1) The public or private sector of the economy shall not undertake or embark on or authorise projects or activities without prior consideration, at an early stage, of their environmental effects.
"(2) Where the extent, nature or location of a proposed project or activity is such that it is likely to significantly affect the environment, its environmental impact assessment shall be undertaken in accordance with the provisions of this Act," Section 2 (1)(2) of the Act states.
TheMail in a freedom of information (FOI) request to Lafarge in December 2021 and a reminder sent at the expiration of seven working days provided by the FOI Act requested a report on Environment Impact Assessment of the company's mining operations in Mfamosing. Although the company acknowledged receipt of the letters, it did not respond. TheMail further wrote to the Country Chief Executive Officer of Lafarge Africa in January 2020. The company acknowledged receipt of the request but did not respond.
TheMail also wrote to the minister, federal ministry of mines and solid minerals development for a copy of the EIA. In a response dated December 18, 2021 but sent to The Mail on January 17, 2022, the Minister through the Director of Mines Environmental Compliance Department, Dr Okono Vivian assured of addressing the request. " I am to further inform you that the ministry is working on your request and will get back to you in due course".
FOI response from the federal ministry of mines and steel development.
As of the time of filing this report, the ministry did not respond despite a reminder sent to the office thus raising suspicion about the non-availability of the EIA.
The EIA is a compulsory study about the social and environmental impacts of a proposed project. It is usually a joint study conducted by environmental consultants with inputs from experts, host communities and relevant government officials to examine whatever impacts a mining project may have on the people socially, environmentally as well as remedial measures to be taken.
Section 70 (g) of the Mining Act, (2007) states the obligation of Lafarge on EIA. " Every holder of a mining lease shall comply with all requirements for Environmental Impact Assessment studies and protection plans".
Most importantly, Section 71 (1) notes that the holder of a mining lease shall not commence any development work or extraction of minerals resources on the mining lease area until after "the submission and approval by the mines Environmental compliance Department of all Environmental Impact Assessment Studies and Mitigation Plans required under applicable environmental laws and regulations".
Interestingly, section 4 (2) of the National Environmental (quarrying and blasting operations) Regulations stipulates that Environmental Audit (EA) shall be conducted on all existing quarries every three (3) years.
Members of the Mfamosing community denied knowledge and involvement in the EIA process. " We have never been contacted for any Environmental Impact Assessment before or after mining operations started. But we have heard from our lawyers that it should have been done and should always be done periodically", the Secretary of Abimfam community says.
Read part 2 here and Part 3 here.
*This investigation republished from TheMail is supported by Policy Alert with funding from Open Society Initiative for West Africa (OSIWA)*
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