text:
\section{Introduction}
\label{sec_intro}

\paragraph*{Background.}
It is common practice in computational condensed matter physics
\cite{AllenTildesleyBook,FrenkelSmitBook,ThijssenBook}
to truncate a pair interaction potential $U(r)$ at a conveniently chosen cutoff distance $\ensuremath{r_\mathrm{c}}$
with $r$ being the distance between two particles $i$ and $j$.
This allows to reduce the number of interactions to be computed
--- energy or force calculations become thus ${\cal O}(n)$-processes with $n$ being the particle number ---
but introduces some technical difficulties, e.g., instabilities in the numerical solution of differential equations
as well-studied in the past especially for the molecular dynamics (MD) method \cite{AllenTildesleyBook,ToxvaerdDyre11}.
Let us label the interaction between two beads $i < j$ by an index $l$.
For simplicity of the presentation and without restricting much in practice
the generality of our results, it is assumed below that
\begin{itemize}
\item
the pair potential scales as $U(r) \equiv u(s)$ with the reduced dimensionless distance
$s=r/\ensuremath{\sigma_{l}}$ where $\ensuremath{\sigma_{l}}$ characterizes the range of the interaction $l$ and
\item
the same reduced cutoff $\ensuremath{s_\mathrm{c}} = \ensuremath{r_\mathrm{c}}/\ensuremath{\sigma_{l}}$ is set for all interactions $l$.
\end{itemize}
For monodisperse beads with constant bead diameter $\sigma$,
as for the standard Lennard-Jones (LJ) potential \cite{AllenTildesleyBook}
\begin{equation}
\ensuremath{u_\mathrm{LJ}}(s) = 4 \epsilon \left(\frac{1}{s^{12}} - \frac{1}{s^6} \right),
\label{eq_LJ}
\end{equation}
the scaling variable becomes simply $s=r/\sigma$ and the reduced cutoff $\ensuremath{s_\mathrm{c}} = \ensuremath{r_\mathrm{c}}/\sigma$.
Even if the truncated potential
\begin{equation}
\ensuremath{u_\mathrm{t}}(s) = u(s) H(\ensuremath{s_\mathrm{c}} -s)
\label{eq_utrunc}
\end{equation}
with $H(s)$ being the Heaviside function \cite{abramowitz} is taken {\em by definition} as the new system Hamiltonian,
it is well known that impulsive corrections at the cutoff have to be taken into account in general
for the pressure $P$ and other moments of the first derivatives of the potential
\cite{FrenkelSmitBook}.
These corrections can be avoided of course by considering a properly shifted potential \cite{FrenkelSmitBook}
\begin{equation}
\ensuremath{u_\mathrm{s}}(s) = \left( u(s) - u(\ensuremath{s_\mathrm{c}}) \right) H(\ensuremath{s_\mathrm{c}} -s)
\label{eq_ushift}
\end{equation}
as emphasized below (Sec.~\ref{theo_shifted}).

\paragraph*{Goal of presented work.}
In this report we wish to remind that the standard shifting of a truncated potential is insufficient
in general to avoid impulsive corrections for moments of second (and higher) derivatives of the potential.
We demonstrate here that this is particulary the case for the Born contribution $\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$ (defined below)
to the elastic moduli computed using the stress-fluctuation formalism described in great detail in the literature
\cite{RowlinsonBook,FrenkelSmitBook,Hoover69,Ray84,Ray88,Lutsko88,Lutsko89,Lemaitre04,Pablo04,Pablo05,Barrat06,SBM11,papthermoflex}.
This should be of importance for the precise localization of the transition between
different thermodynamic phases using the elastic moduli, especially for liquid ($G=0$) to solid ($G > 0$)
transitions in network forming systems where the shear modulus $G$ plays the role of an order parameter
\cite{Zippelius06}. This is the case, e.g. for colloidal gels \cite{Kob08},
hyperbranched polymer chains with sticky end-groups \cite{Friedrich10} or
bridged networks of telechelic polymers in water-oil emulsions \cite{Porte01b,Porte03} or
living polymer-like micellar systems \cite{Ligoure07}.

\paragraph*{Outline.}
The paper is organized as follows: After reminding first in Sec.~\ref{theo_reminder} the known corrections
for the pressure and similar first derivates of the potential, the impulsive correction for the general
Born contribution $\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$ is stated in Sec.~\ref{theo_key}.
We describe then in Sec.~\ref{theo_isotropic} the corrections
for the compression modulus $K$ and the shear modulus $G$ in isotropic systems.
We comment on polydispersity effects and mixed potentials in Sec.~\ref{theo_mixtures}.
Our results are rephrased in terms of the radial pair distribution function $g(s)$ in Sec.~\ref{theo_gr}
which allows to predict the asymptotic behavior for large $\ensuremath{s_\mathrm{c}}$.
Section~\ref{sec_algo} gives some technical details on the two numerical model systems \cite{Kob95,TWLB02}
in $d=3$ and $d=2$ dimensions used to test our predictions in Sec.~\ref{sec_res}.
We consider in this paper the liquid high-temperature regime of both (in principle glass-forming)
models where the shear modulus $G$ must vanish \cite{RowlinsonBook,HansenBook},
since this provides a simple reference for testing the predicted corrections.

\section{Theoretical predictions}
\label{sec_theo}

\subsection{Reminder}
\label{theo_reminder}

\paragraph*{Truncated potential.}
As usual for pairwise additive interactions the mean pressure $P = \ensuremath{P_\mathrm{id}} + \ensuremath{P_\mathrm{ex}}$ may
be obtained as the sum of the ideal kinetic contribution $\ensuremath{P_\mathrm{id}} = \mbox{$k_{\rm B}T$} \rho$
and the excess pressure contribution \cite{HansenBook}
\begin{equation}
\ensuremath{P_\mathrm{ex}} = - \frac{1}{d V} \sum_{l} \left< \ensuremath{s_{l}} \ensuremath{u^{\prime}_\mathrm{t}}(\ensuremath{s_{l}}) \right>
\label{eq_Pex}
\end{equation}
with $\rho=n/V$ being the number density, $n$ the particle number,
$V$ the $d$-dimensional volume and $\left< \ldots \right>$ indicating the usual
thermal average over the configuration ensemble.
(A prime denotes a derivative of a function with respect to its argument.)
By taking the derivative of the truncated potential
\begin{equation}
\ensuremath{u^{\prime}_\mathrm{t}}(s) = \ensuremath{u^{\prime}}(s) H(\ensuremath{s_\mathrm{c}} -s) - u(s) \delta(s-\ensuremath{s_\mathrm{c}})
\label{eq_utruncprime}
\end{equation}
the excess pressure may be written as the sum $\ensuremath{P_\mathrm{ex}} = \ensuremath{\tilde{P}_\mathrm{ex}} + \ensuremath{\Delta P_\mathrm{ex}}$ of an uncorrected (bare)
contribution $\ensuremath{\tilde{P}_\mathrm{ex}}$ and an impulsive correction $\ensuremath{\Delta P_\mathrm{ex}}$. The latter correction is obtained
numerically from \cite{FrenkelSmitBook}
\begin{eqnarray}
\ensuremath{\Delta P_\mathrm{ex}} & = & \lim_{s\to \ensuremath{s_\mathrm{c}^{-}}} \ensuremath{h_\mathrm{1}}(s) \mbox{ with } \nonumber\\
\ensuremath{h_\mathrm{1}}(s) & \equiv & \frac{1}{d V} \sum_{l} \left< \ensuremath{s_{l}} u(\ensuremath{s_{l}}) \ \delta(\ensuremath{s_{l}}-s) \right>
\label{eq_histoP}
\end{eqnarray}
being a weighted histogram.
In practice, the proper limit $s \to \ensuremath{s_\mathrm{c}^{-}}$ may be replaced by setting $s = \ensuremath{s_\mathrm{c}}$ for the histogram.

\paragraph*{Shifted potential.}
\label{theo_shifted}
The impulsive correction related to first derivatives of the truncated potential can be avoided
by considering the shifted potential $\ensuremath{u_\mathrm{s}}(s)$, Eq.~(\ref{eq_ushift}), since
$\ensuremath{u^{\prime}_\mathrm{s}}(s) = \ensuremath{u^{\prime}}(s) H(\ensuremath{s_\mathrm{c}} -s)$.
With this choice {\em no} impulsive correction arises either for similar observables such
as, e.g., moments of the instantaneous excess pressure tensor
\begin{eqnarray}
\ensuremath{\hat{P}_\mathrm{ex}^{\alpha\beta}} & = & -\frac{1}{V} \sum_{l} \ensuremath{s_{l}^{\alpha}} \frac{\partial \ensuremath{u_\mathrm{s}}(\ensuremath{s_{l}})}{\partial \ensuremath{s_{l}^{\beta}}} \nonumber \\
& = & -\frac{1}{V} \sum_{l} \ensuremath{s_{l}} \ensuremath{u^{\prime}_\mathrm{s}}(\ensuremath{s_{l}}) \ \ensuremath{n_{l}^{\alpha}} \ensuremath{n_{l}^{\beta}}.
\label{eq_pressTen}
\end{eqnarray}
Here $\ensuremath{s_{l}^{\alpha}}$, $\ldots$ stand for the spatial components of the reduced distance between the particles,
$\ensuremath{n_{l}^{\alpha}} = \ensuremath{s_{l}^{\alpha}}/\ensuremath{s_{l}}$, $\ldots$ for the corresponding components of the normalized distance vector and
Greek letters are used for the spatial coordinates $\alpha,\beta,\gamma,\delta = 1,\ldots,d$.
(We remind that the mean excess pressure $\ensuremath{P_\mathrm{ex}}$ is the averaged trace over the
instantaneous excess pressure tensor, $\ensuremath{P_\mathrm{ex}} = < \ensuremath{\text{Tr}}[\ensuremath{\hat{P}_\mathrm{ex}^{\alpha\beta}}] >/d$.)
Specifically, if the potential is shifted, all impulsive corrections are avoided for the
excess pressure fluctuations
\begin{equation}
\ensuremath{C_\mathrm{F}^{\alpha\beta\gamma\delta}} \equiv - \beta V \left( \left< \hat{P}_\mathrm{ex}^{\alpha\beta} \hat{P}_\mathrm{ex}^{\gamma\delta} \right> -
\left< \hat{P}_\mathrm{ex}^{\alpha\beta} \right> \left< \hat{P}_\mathrm{ex}^{\gamma\delta} \right> \right)
\label{eq_muF}
\end{equation}
($\beta \equiv 1/\mbox{$k_{\rm B}$} T$ being the inverse temperature)
which give important contributions
--- especially for polymer-type liquids \cite{SBM11,papthermoflex}
and amorphous solids \cite{Lemaitre04,Barrat06} ---
to the elastic moduli computed using the stress-fluctuation formalism
\cite{FrenkelSmitBook}.

\subsection{Key point made}
\label{theo_key}

\paragraph*{Correction to the Born term.}
An even more important contribution to the elastic moduli (especially at high densities) is given
by the Born term $\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$ already mentioned in the Introduction \cite{foot_CKabcd}. Being a moment
of the first and the second derivatives of the potential it is defined as
\cite{FrenkelSmitBook,Lutsko89,SBM11,TWLB02}
\begin{equation}
\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}} = \frac{1}{V} \sum_{l} \left< \left( \ensuremath{s_{l}}^2 \ensuremath{u^{\prime\prime}_\mathrm{s}}(\ensuremath{s_{l}}) - \ensuremath{s_{l}} \ensuremath{u^{\prime}_\mathrm{s}}(\ensuremath{s_{l}}) \right)
\ensuremath{n_{l}^{\alpha}} \ensuremath{n_{l}^{\beta}} \ensuremath{n_{l}^{\gamma}} \ensuremath{n_{l}^{\delta}} \right>
\label{eq_CBabcd}
\end{equation}
using the notations given above.
We remind that for solids with well-defined reference positions and displacement fields the Born
contribution is known to describe the (free) energy change assuming an {\em affine} response to an
imposed homogeneous strain \cite{Lutsko89,Lemaitre04,Barrat06,SBM11,TWLB02}.
Assuming now a truncated and shifted potential the impulsive correction $\Delta \ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$
to $\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}} = \ensuremath{\tilde{C}_\mathrm{B}^{\alpha\beta\gamma\delta}} + \Delta \ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$ is simply obtained using
\begin{equation}
\ensuremath{u^{\prime\prime}_\mathrm{s}}(s) =  \ensuremath{u^{\prime\prime}}(s) H(\ensuremath{s_\mathrm{c}}-s) - \ensuremath{u^{\prime}}(s) \delta(s-\ensuremath{s_\mathrm{c}})
\label{eq_ushiftprimeprime}
\end{equation}
which yields
\begin{eqnarray}
\Delta \ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}} & = & - \lim_{s\to \ensuremath{s_\mathrm{c}^{-}}} \ensuremath{h_\mathrm{2}^{\alpha\beta\gamma\delta}}(s) \mbox{ with }
\label{eq_CBabd_correct} \\
\ensuremath{h_\mathrm{2}^{\alpha\beta\gamma\delta}}(s) & \equiv & \frac{1}{V} \sum_{l} \left<  \ensuremath{s_{l}}^2 \ensuremath{u^{\prime}}(\ensuremath{s_{l}})
\ensuremath{n_{l}^{\alpha}} \ensuremath{n_{l}^{\beta}} \ensuremath{n_{l}^{\gamma}} \ensuremath{n_{l}^{\delta}} \ \delta(\ensuremath{s_{l}}-s) \right>.
\nonumber
\end{eqnarray}

\paragraph*{General impulsive correction.}
More generally, one might consider a property
$$A = \frac{1}{V} \sum_l \left< f(\ensuremath{s_{l}}) \ensuremath{u_\mathrm{s}}^{(n)}(\ensuremath{s_{l}}) \right>$$
with $f(s)$ being a specified function and $(n)$ denoting
the $n$-th derivative of the shifted potential $\ensuremath{u_\mathrm{s}}(s)$.
Let us further suppose that all potential derivatives up to the
$(n-2)$-th one do vanish at the cutoff $\ensuremath{s_\mathrm{c}}$.
It thus follows that $A = \tilde{A} + \Delta A$ takes
an impulsive correction
\begin{eqnarray}
\Delta A & = & -\lim_{s \to \ensuremath{s_\mathrm{c}^{-}}} h_n(s) \mbox{ with } \nonumber \\
h_n(s) & \equiv & \frac{1}{V} \sum_l \left< f(\ensuremath{s_{l}}) u^{(n-1)}(\ensuremath{s_{l}}) \ \delta(\ensuremath{s_{l}} -s) \right>
\label{eq_hn_histo}
\end{eqnarray}
being the relevant histogram.

\paragraph*{Generalized shifting.}
Obviously, the original potential may not only be shifted by a constant $u(\ensuremath{s_\mathrm{c}})$ but by a polynomial of $s$
to make vanish the first and arbitrarily high derivatives of the potential at $s=\ensuremath{s_\mathrm{c}}$. In this way all
impulsive corrections could be avoided in principle.
Since discontinuous forces at the cutoff may cause problems in MD simulations, a number of studies use for
instance a ``shifted-force potential" where a linear term is added to the potential
\cite{AllenTildesleyBook,ToxvaerdDyre11}.
The difference between the original potential and the generalized shifted potential removing the cutoff
discontinuities means, of course, that the computed properties deviate to some extend from the original model.
Only if the generalized shifting is weak, one may recover the correct thermodynamic properties
using a first-order perturbation scheme \cite{AllenTildesleyBook}.
Since the (simply) shifted potential $\ensuremath{u_\mathrm{s}}(s)$, Eq.~(\ref{eq_ushift}), is anyway the most common choice
\cite{Hoover69,Ray84,Ray88,Lemaitre04,Pablo04,Pablo05,Barrat06,SBM11,papthermoflex,Kob95,TWLB02},
we restrict the presentation on this case and demonstrate how the impulsive correction associated
to the non-vanishing $\ensuremath{u^{\prime}_\mathrm{s}}(\ensuremath{s_\mathrm{c}^{-}})$ can readily be computed.

\subsection{Isotropic systems}
\label{theo_isotropic}

\paragraph*{Lam\'e coefficients.}
In order to show that these corrections may be of relevance
we focus now on homogeneous and isotropic systems. We remind first that the two elastic Lam\'e coefficients
$\lambda$ and $\mu$ characterizing the elastic properties of such systems may be computed numerically using
\cite{SBM11,papthermoflex}
\begin{eqnarray}
\lambda & = & \ensuremath{\lambda_\mathrm{F}}  + \ensuremath{\lambda_\mathrm{B}}, \nonumber \\
\mu - \ensuremath{P_\mathrm{id}}  & = & \ensuremath{\mu_\mathrm{F}} + \ensuremath{\mu_\mathrm{B}} \label{eq_lamu2BF}
\end{eqnarray}
where the only contribution due to the kinetic energy of the particles is contained
by the ideal gas pressure $\ensuremath{P_\mathrm{id}}$ indicated for $\mu$ \cite{foot_kinelast}.
The first contributions indicated on the right hand-side of Eq.~(\ref{eq_lamu2BF}) are the excess pressure
fluctuation contributions $\ensuremath{\lambda_\mathrm{F}}$ and $\ensuremath{\mu_\mathrm{F}}$ which may be obtained from the general $\ensuremath{C_\mathrm{F}^{\alpha\beta\gamma\delta}}$ by setting, e.g.,
$\alpha=\beta=1$ and $\gamma=\delta=2$ for $\ensuremath{\lambda_\mathrm{F}}$ and
$\alpha=\gamma=1$ and $\beta=\delta=2$ for $\ensuremath{\mu_\mathrm{F}}$ characterizing the shear stress fluctuations.
The so-called ``Born-Lam\'e coefficients" \cite{SBM11}
\begin{equation}
\ensuremath{\lambda_\mathrm{B}} \equiv \ensuremath{\mu_\mathrm{B}}
\equiv \frac{1}{d(d+2) V} \sum_{l} \left< \left( \ensuremath{s_{l}}^2 u^{\prime\prime}(\ensuremath{s_{l}}) - \ensuremath{s_{l}} u^{\prime}(\ensuremath{s_{l}}) \right) \right>
\label{eq_lameBorn}
\end{equation}
may be obtained from the general Born terms $\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$ by setting,
e.g., $\alpha=\gamma=1$ and $\beta=\delta=2$.
The $d$-dependent prefactor stems from the assumed isotropy of the system
and the mathematical formula
\begin{equation}
\left< \left( \ensuremath{n_{l}^{\alpha}} \ensuremath{n_{l}^{\beta}} \right)^2 \right> =
\frac{1}{d(d+2)} \left(1 + 2 \delta_{\alpha\beta} \right)
\label{eq_dprefactor}
\end{equation}
($\delta_{\alpha\beta}$ being the Kronecker symbol \cite{abramowitz})
for the components of a unit vector in $d$ dimensions pointing into arbitrary directions.
Equation~(\ref{eq_CBabd_correct}) implies then an impulsive correction
\begin{eqnarray}
\ensuremath{\Delta \lambda_\mathrm{B}} & = & \ensuremath{\Delta \mu_\mathrm{B}} = - \lim_{s\to \ensuremath{s_\mathrm{c}^{-}}} \ensuremath{h_\mathrm{2}}(s) \mbox{ with } \nonumber\\
\ensuremath{h_\mathrm{2}}(s) & \equiv & \frac{1}{d (d+2) V} \sum_{l} \left<  \ensuremath{s_{l}}^2 \ensuremath{u^{\prime}}(\ensuremath{s_{l}}) \ \delta(\ensuremath{s_{l}}-s) \right>.
\label{eq_histoLame}
\end{eqnarray}

\paragraph*{Compression and shear modulus.}
Instead of using the Lam\'e coefficients it is from the experimental point of view
more natural to characterize isotropic bodies using the compression modulus $K$
and the shear modulus $G$. The latter moduli may be expressed as
\begin{eqnarray}
K & = & (\lambda + P) + \frac{2}{d} G, \label{eq_lame2K}\\
G & = & \mu - P = \ensuremath{\mu_\mathrm{B}} + \ensuremath{\mu_\mathrm{F}} - \ensuremath{P_\mathrm{ex}}. \label{eq_lame2G}
\end{eqnarray}
We follow here the notation of Ref.~\cite{SBM11} to emphasize the explicit pressure dependence
which is often (incorrectly) omitted as clearly pointed out by Birch \cite{Birch37} and Wallace \cite{Wallace70}.
As one expects, kinetic elastic contributions terms do not enter explicitly for the shear modulus.
Since only the Born contributions $\ensuremath{\lambda_\mathrm{B}}=\ensuremath{\mu_\mathrm{B}}$ cause a cutoff correction,
this implies
$K = \ensuremath{\tilde{K}} + \ensuremath{\Delta K}$ and $G = \ensuremath{\tilde{G}} + \ensuremath{\Delta G}$ with
$\ensuremath{\tilde{K}}$ and $\ensuremath{\tilde{G}}$ being the uncorrected (bare) moduli and
\begin{eqnarray}
\ensuremath{\Delta K} & = & \ensuremath{\Delta \lambda_\mathrm{B}} + \frac{2}{d} \ensuremath{\Delta \mu_\mathrm{B}} =  \frac{2+d}{d} \ensuremath{\Delta \mu_\mathrm{B}}, \label{eq_deltaK} \\
\ensuremath{\Delta G} & = & \ensuremath{\Delta \mu_\mathrm{B}} \label{eq_deltaG}
\end{eqnarray}
the impulsive corrections. We shall test these predictions numerically in Sec.~\ref{sec_res}.

\subsection{Polydispersity and mixed potentials}
\label{theo_mixtures}

As stated in the Introduction we assume throughout this work the scaling $U(r) \equiv u(s)$ of the
pair potential in terms of the reduced distance $s = r/\ensuremath{\sigma_{l}}$. This is done not only for dimensional
reasons but, more importantly, to describe a broad range of model systems for mixtures and polydisperse
systems where the interaction range $\ensuremath{\sigma_{l}}$ may differ for each interaction $l$. Moreover,
the type and/or the parameter set of the pair potential may vary for different interactions.
For such mixed potentials $u(s)$, $\ensuremath{u_\mathrm{t}}(s)$ and $\ensuremath{u_\mathrm{s}}(s)$ and their derivatives take in principal
an explicit index $l$, i.e. one should write $u_l(s)$, $u_{\mathrm{t},l}(s)$, $u_{\mathrm{s},l}(s)$ and so on.
This is only not done here to keep a concise notation. For example one might wish to consider
\begin{itemize}
\item
a generic polymer bead-spring model where some interactions $l$ describe the bonded interactions
between monomers along the chain (which are normally not truncated and need not to be corrected)
and some the excluded volume interactions between the beads \cite{papthermoflex}.
\item
the generalization of the monodisperse LJ potential, Eq.~(\ref{eq_LJ}),
\begin{equation}
u_l(s) = 4\ensuremath{\epsilon_{l}} \left(s^{-12} - s^{-6} \right) \mbox{ with } s = r/\ensuremath{\sigma_{l}}
\label{eq_LJgeneralized}
\end{equation}
where $\ensuremath{\epsilon_{l}}$ and $\ensuremath{\sigma_{l}}$ are fixed for each interaction $l$.
In practise, each particle $i$ may be characterized by an energy scale $E_i$ and a ``diameter" $D_i$.
The interaction parameters $\ensuremath{\epsilon_{l}}(E_i,E_j)$ and $\ensuremath{\sigma_{l}}(D_i,D_j)$ are then given in terms of
specified functions of these properties.
\item
the famous Kob-Andersen (KA) model for binary mixtures of beads of type $A$ and $B$ \cite{Kob95},
a particular case of Eq.~(\ref{eq_LJgeneralized}) with fixed
interaction ranges $\ensuremath{\sigma_\mathrm{AA}}$, $\ensuremath{\sigma_\mathrm{BB}}$ and $\ensuremath{\sigma_\mathrm{AB}}$ and
energy parameters $\ensuremath{\epsilon_\mathrm{AA}}$, $\ensuremath{\epsilon_\mathrm{BB}}$ and $\ensuremath{\epsilon_\mathrm{AB}}$ characterizing, respectively,
$AA$-, $BB$- and $AB$-contacts.
\item
a network forming emulsion of oil droplets in water bridged by telechelic polymers
where the oil droplets are modeled as big LJ spheres, the telechelic polymers by a bead-spring
model with a soluble ``spacer" in the middle of the chain and insoluble end-groups (``stickers")
strongly attracted by the oil droplets \cite{Porte01b,Porte03}.
Assuming sufficiently strong (in strength, number and life-time) sticker-oil interactions,
such a system should behave as a soft solid with a finite shear modulus $G$
(at least for a fixed finite sampling time) which may be probed, at least in principle,
using Eq.~(\ref{eq_lame2G}).
\end{itemize}
The impulsive corrections given in Eq.~(\ref{eq_histoP}) for the pressure $P$,
in Eq.~(\ref{eq_CBabd_correct}) for the general Born term $\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$ and
in Eq.~(\ref{eq_histoLame}) for the Born Lam\'e coefficients $\ensuremath{\lambda_\mathrm{B}}=\ensuremath{\mu_\mathrm{B}}$ in isotropic systems
stated all in terms of, respectively, the histograms $\ensuremath{h_\mathrm{1}}(s)$, $\ensuremath{h_\mathrm{2}^{\alpha\beta\gamma\delta}}(s)$ and $\ensuremath{h_\mathrm{2}}(s)$
remain indeed valid for such explicitly $l$-dependent potentials. From the numerical point of view,
this is all what is needed and the direct computation of these histograms remains in all cases
straightforward as illustrated in Sec.~\ref{res_histo}.

\subsection{Radial pair distribution function $g(r)$}
\label{theo_gr}

\paragraph*{Notations.}
Especially for simple and complex fluids and for all sorts of glass-forming systems
\cite{RowlinsonBook,HansenBook}
it is common practice to reexpress correlations and histograms in terms of the
radial pair distribution function $g(r)$ \cite{AllenTildesleyBook,FrenkelSmitBook}.
This is also of interest here since for large cutoff distances the pair distribution function
must drop out, $g(\ensuremath{r_\mathrm{c}}) \to 1$, allowing thus to predict the corrections in this limit.
Let us remind first that, using the Gamma function $\Gamma(x)$ \cite{abramowitz},
the $(d-1)$-dimensional surface of a $d$-sphere of radius $r$ is given by
\begin{equation}
A(r) = \frac{2 \pi^{d/2}}{\Gamma(d/2)} \ r^{d-1}
\mbox{ for } d = 2,3,\ldots
\label{eq_Adef}
\end{equation}
and similarly for the (dimensionless) surface $A(s)$ using the reduced distance $s$.

\paragraph*{Monodisperse interactions.}
For strictly monodisperse beads and similar interactions of constant interaction range $\sigma$
it is seen that Eq.~(\ref{eq_histoP}) for the pressure correction becomes
\begin{equation}
\ensuremath{\Delta P_\mathrm{ex}} =
\frac{1}{2} \ \frac{1}{d}
\rho^2 \sigma^d A(\ensuremath{s_\mathrm{c}}) \ensuremath{s_\mathrm{c}}   u(\ensuremath{s_\mathrm{c}}) \times g(\ensuremath{s_\mathrm{c}})
\label{eq_Pexcutmono}
\end{equation}
where the factor $1/2$  assures that every interaction is only counted once.
Note that we have set $g(s) \equiv g(r/\sigma)$ without introducing a new symbol.
For the LJ potential, Eq.~(\ref{eq_LJ}), this leads to
\begin{equation}
\ensuremath{\Delta P_\mathrm{ex}} = - \frac{4 \pi^{d/2}}{\Gamma(d/2)d} \rho^2 \sigma^d \epsilon \ensuremath{s_\mathrm{c}}^{d-6} (1-\ensuremath{s_\mathrm{c}}^{-6}) \times g(\ensuremath{s_\mathrm{c}}).
\label{eq_PexcutmonoLJ}
\end{equation}
Please note that the expression given in Ref.~\cite{FrenkelSmitBook} is recovered by setting $d=3$ and
assuming $g(\ensuremath{s_\mathrm{c}}) \approx 1$.
Similarly, one obtains from Eq.~(\ref{eq_histoLame}) the correction
\begin{equation}
\ensuremath{\Delta \mu_\mathrm{B}} =  -
\frac{1}{2} \ \frac{1}{d (d+2)}
\rho^2 \sigma^{d} A(\ensuremath{s_\mathrm{c}}) \ensuremath{s_\mathrm{c}}^{2} \ensuremath{u^{\prime}}(\ensuremath{s_\mathrm{c}}) \times g(\ensuremath{s_\mathrm{c}})
\label{eq_muBcutmono}
\end{equation}
for the Born-Lam\'e coefficient we are mainly interested in.
For a LJ potential this becomes
\begin{equation}
\ensuremath{\Delta \mu_\mathrm{B}} = - \frac{24\pi^{d/2}}{d(d+2)\Gamma(d/2)}
\rho^2 \sigma^d \epsilon \ensuremath{f_\mathrm{LJ}}(\ensuremath{s_\mathrm{c}}) \times g(\ensuremath{s_\mathrm{c}})
\label{eq_muBcutmonoLJ}
\end{equation}
where we have defined
\begin{equation}
\ensuremath{f_\mathrm{LJ}}(s) \equiv (1 - (\ensuremath{s_\mathrm{0}}/s)^6)/s^{6-d}
\label{eq_fLJdef}
\end{equation}
with $\ensuremath{s_\mathrm{0}} = 2^{1/6}$ being the minimum of the potential.
For sufficiently large cutoff distances where $g(\ensuremath{s_\mathrm{c}}) \approx 1$
the correction thus decays as
\begin{equation}
\ensuremath{\Delta \mu_\mathrm{B}} \sim - A(\ensuremath{s_\mathrm{c}}) \ensuremath{s_\mathrm{c}}^{2} \ensuremath{u^{\prime}}(\ensuremath{s_\mathrm{c}}),
\label{eq_muBcut_asymp}
\end{equation}
e.g. $\ensuremath{\Delta \mu_\mathrm{B}} \sim - 1/\ensuremath{s_\mathrm{c}}^{6-d}$ for a LJ potential.
This asymptotic behavior also holds for the more complicated cases discussed below.

\paragraph*{Mixtures.}
Many experimental relevant systems have mixed potentials such as the KA model for binary
colloidal mixtures sketched above. In general the interaction potential $U_{ab}(r)=u_{ab}(s)$ between
beads of two species $a$ and $b$ takes different energy parameters which causes different weights
at the cutoff depending on which particles interact.
The impulsive corrections of such mixtures are readily obtained by linear superposition of
Eq.~(\ref{eq_muBcutmono}) for different contributions $(a,b)$.
Let $\ensuremath{c_{a}} = \rho_a/\rho$ denote the mole fraction of species $a$,
$\sigma_{ab}$ the interaction range between a bead of type $a$ and a bead of type $b$ and
$g_{ab}(s)$ the respective radial pair distribution function.
The impulsive correction to the Born-Lam\'e coefficient thus becomes
\begin{eqnarray}
\ensuremath{\Delta \mu_\mathrm{B}} & = & - \frac{1}{2} \frac{1}{d (d+1)} \rho^2 A(\ensuremath{s_\mathrm{c}}) \ensuremath{s_\mathrm{c}}^{2} \nonumber \\
 & \times &
\sum_{a} \sum_{b} \ensuremath{c_{a}} \ensuremath{c_{b}} \sigma_{ab}^d \ensuremath{u^{\prime}}_{ab}(\ensuremath{s_\mathrm{c}}) \ g_{ab}(\ensuremath{s_\mathrm{c}})
\label{eq_muBcut_mixture}
\end{eqnarray}
where we have used that for all types of interaction we have the {\em same} reduced cutoff $\ensuremath{s_\mathrm{c}}$.

Let us now assume a mixture described by the generalized LJ potential
$u_{ab}(s) = \epsilon_{ab} (1/s^{12}-1/s^6)$ with $s=r/\sigma_{ab}$.
A reference energy $\ensuremath{\epsilon_{\mathrm{ref}}}$ and a reference interaction range $\ensuremath{\sigma_{\mathrm{ref}}}$
may arbitrarily be defined using, say, the interaction of two beads of type $a=b=1$,
i.e. $\ensuremath{\epsilon_{\mathrm{ref}}} \equiv \epsilon_{11}$ and $\ensuremath{\sigma_{\mathrm{ref}}} \equiv \sigma_{11}$.
Defining the dimensionless ratios $w_{ab} \equiv \epsilon_{ab}/\ensuremath{\epsilon_{\mathrm{ref}}}$ and $v_{ab} = (\sigma_{ab}/\ensuremath{\sigma_{\mathrm{ref}}})^d$
we may thus rewrite the general Eq.~(\ref{eq_muBcut_mixture}) as
\begin{eqnarray}
\ensuremath{\Delta \mu_\mathrm{B}} & = & - \frac{24 \pi^{d/2}}{d(d+2)\Gamma(d/2)} \rho^2 \ensuremath{\sigma_{\mathrm{ref}}}^d \ensuremath{\epsilon_{\mathrm{ref}}} \ \ensuremath{f_\mathrm{LJ}}(\ensuremath{s_\mathrm{c}}) \nonumber \\
 & \times & \sum_a \sum_b \ensuremath{c_{a}} \ensuremath{c_{b}} v_{ab} w_{ab} \ g_{ab}(\ensuremath{s_\mathrm{c}}).
\label{eq_muBcut_mixtureLJ}
\end{eqnarray}
Since $g_{ab}(\ensuremath{s_\mathrm{c}}) \to 1$ for large $\ensuremath{s_\mathrm{c}}$, the function $\ensuremath{f_\mathrm{LJ}}(\ensuremath{s_\mathrm{c}})$ determines the scaling
as already stated above, Eq.~(\ref{eq_muBcut_asymp}).

\paragraph*{Continuous polydispersity.}
We turn now to systems with a continuous polydispersity as in the second model investigated numerically below.
Let us assume that each bead is characerized by a bead diameter $D$ which is distributed according to a
well-defined normalized distribution $\ensuremath{c_{t}}$ with $t = D/\ensuremath{\sigma_{\mathrm{ref}}}$ being a reduced bead diameter with
respect to some reference length $\ensuremath{\sigma_{\mathrm{ref}}}$.
To be specific we shall assume a generalized LJ potential, Eq.~(\ref{eq_LJgeneralized}),
where the interaction range $\sigma_{tt^{\prime}}$ and the energy scale $\epsilon_{tt^{\prime}}$
between two beads are uniquely specified by the two reduced diameters $t$ and $t^{\prime}$.
Defining  $\ensuremath{w_{tt^{\prime}}} = \epsilon_{tt^{\prime}}/\ensuremath{\epsilon_{\mathrm{ref}}}$,
$\ensuremath{v_{tt^{\prime}}} = \left( \sigma_{tt^{\prime}}/\ensuremath{\sigma_{\mathrm{ref}}} \right)^d$ and
using the radial pair distribution function $\ensuremath{g_{tt^{\prime}}}(s)$ for two beads of reduced diameter $t$ and $t^{\prime}$,
the double-sum in Eq.~(\ref{eq_muBcut_mixtureLJ}) can be rewritten as the double-integral
\begin{eqnarray}
\ensuremath{\Delta \mu_\mathrm{B}} & = & - \frac{24 \pi^{d/2}}{d(d+2)\Gamma(d/2)} \rho^2 \ensuremath{\sigma_{\mathrm{ref}}}^d \ensuremath{\epsilon_{\mathrm{ref}}} \ \ensuremath{f_\mathrm{LJ}}(\ensuremath{s_\mathrm{c}}) \nonumber\\
 & \times & \int \ensuremath{\ddiff t} \int \ensuremath{\ddiff t^{\prime}} \ \ensuremath{c_{t}} \ensuremath{c_{t^{\prime}}} \ensuremath{v_{tt^{\prime}}} \ensuremath{w_{tt^{\prime}}} \ \ensuremath{g_{tt^{\prime}}}(\ensuremath{s_\mathrm{c}}).
\label{eq_muBcut_polyLJ}
\end{eqnarray}
In order to determine $\ensuremath{\Delta \mu_\mathrm{B}}$ from Eq.~(\ref{eq_muBcut_polyLJ}) one needs to prescribe
the laws for $\ensuremath{c_{t}}$,  $\sigma_{tt^{\prime}}$ and $\epsilon_{tt^{\prime}}$. In the large-$\ensuremath{s_\mathrm{c}}$
limit the double-integral becomes in any case constant, i.e. we have again
$\ensuremath{\Delta \mu_\mathrm{B}} \sim - \ensuremath{f_\mathrm{LJ}}(\ensuremath{s_\mathrm{c}}) \sim - 1/\ensuremath{s_\mathrm{c}}^{6-d}$.

\begin{table}[t]
\begin{tabular}{|c||c|c|c|c|c|c||c|}
\hline
$\ensuremath{s_\mathrm{c}}/\ensuremath{s_\mathrm{0}}$         &
$e \beta$             &
$P \beta/\rho$        &
$\ensuremath{\tilde{K}} \beta/\rho$ &
$K \beta/\rho$        &
$\ensuremath{\tilde{G}} \beta/\rho$ &
$\ensuremath{\Delta \mu_\mathrm{B}} \beta/\rho$  &
$K \beta/\rho$
\\ \hline
0.9 & 0.162 & 4.61 & -19.2& 15.0 &-17.06& 17.08 & 14.5\\
1.0 & 0.329 & 5.39 & 19.0 & 19.0 & 0.05 &  0.03 & 18.8 \\
1.1 & 0.103 & 4.90 & 20.9 & 17.2 & 1.86 & -1.86 & 17.2 \\
1.5 & -1.24 & 3.13 & 14.1 & 13.2 & 0.44 & -0.43 & 13.3 \\
2.0 & -1.69 & 2.71 & 13.0 & 12.4 & 0.28 & -0.30 & 12.3 \\
2.5 & -1.83 & 2.55 & 12.3 & 12.1 & 0.09 & -0.10 & 12.0\\
3.0 & -1.89 & 2.50 & 12.1 & 12.0 & 0.05 & -0.05 & 11.9\\
3.5 & -1.91 & 2.47 & 12.1 & 12.0 & 0.03 & -0.03 & 12.1\\
4.0 & -1.92 & 2.46 & 11.9 & 11.8 & 0.03 & -0.02 & 11.9\\
\hline
\end{tabular}
\vspace*{0.5cm}
\caption[]{Various properties for polydisperse LJ beads at temperature $T=1$ and density $\rho \approx 0.72$
{\em vs.} the reduced cutoff distance $\ensuremath{s_\mathrm{c}}/\ensuremath{s_\mathrm{0}}$ with $\ensuremath{s_\mathrm{0}}=2^{1/6}$ being the minimum of the potential:
energy per bead $e$,
total pressure $P = \ensuremath{P_\mathrm{id}} + \ensuremath{P_\mathrm{ex}}$,
uncorrected compression modulus $\ensuremath{\tilde{K}}$,
corrected compression modulus $K = \ensuremath{\tilde{K}} + 2 \ensuremath{\Delta \mu_\mathrm{B}}$,
bare shear modulus $\ensuremath{\tilde{G}}$ and
impulsive correction $\ensuremath{\Delta \mu_\mathrm{B}}$ obtained from the histogram $\ensuremath{h_\mathrm{2}}(s)$ at $s = \ensuremath{s_\mathrm{c}}$.
The corrected shear modulus $G = \ensuremath{\tilde{G}} + \ensuremath{\Delta G}$ vanishes as it should.
The last column refers to the compression modulus $K$ obtained using Eq.~(\ref{eq_Kisobaric})
for isobaric ensembles kept at the same pressure $P$ (third column).
\label{tab_scut}}
\end{table}

\section{Computational issues}
\label{sec_algo}
To illustrate the above predictions we present computational data using two extremely well studied models
of colloidal liquids at high temperatures which are described in detail elsewhere \cite{Kob95,TWLB02}:
\begin{itemize}
\item
The already mentioned KA model \cite{Kob95} for binary mixtures of LJ beads in $d=3$ has been investigated
by means of Langevin MD simulation  \cite{FrenkelSmitBook} imposing a temperature $T=0.8$
for $n=\ensuremath{n_\mathrm{A}}+\ensuremath{n_\mathrm{B}}=6912$ beads per simulation box, a total density $\rho = 1.0$ and
molar fractions $\ensuremath{c_{a}}=\ensuremath{n_\mathrm{A}}/n=0.8$ and $\ensuremath{c_{b}}=\ensuremath{n_\mathrm{B}}/n=0.2$ for both types of beads $A$ and $B$.
As in Ref.~\cite{Kob95} we set $\ensuremath{\sigma_\mathrm{AA}}=1.0\sigma$, $\ensuremath{\sigma_\mathrm{BB}}=0.88 \sigma$ and $\ensuremath{\sigma_\mathrm{AB}}=0.8 \sigma$
for the interaction range and $\ensuremath{\epsilon_\mathrm{AA}}=1.0 \epsilon$, $\ensuremath{\epsilon_\mathrm{BB}}=0.5\epsilon$ and $\ensuremath{\epsilon_\mathrm{AB}}=1.5 \epsilon$
for the LJ energy scales. Only data for the usual cutoff $\ensuremath{s_\mathrm{c}}=2.5$ is presented.
\item
Using Monte Carlo (MC) simulations \cite{AllenTildesleyBook,FrenkelSmitBook} we have computed in $d=2$
dimensions a specific case of the generalized LJ potential, Eq.~(\ref{eq_LJgeneralized}), where all
interaction energies are identical, $\ensuremath{\epsilon_{l}}= \epsilon$, and the interaction range is set by
the arithmetic mean $\ensuremath{\sigma_{l}}=(D_i+D_j)/2$ of the diameters $D_i$ and $D_j$ of the interacting beads.
Following Ref.~\cite{TWLB02} the bead diameters are uniformly distributed between $0.8\sigma$
and $1.2\sigma$.  For the examples reported here we have used a temperature $T=1.0$,
$n=10000$ beads per box and a density $\rho \approx 0.72$.
\end{itemize}
We use LJ units throughout this work and Boltzmann's constant $\mbox{$k_{\rm B}$}$ is set to unity.
For the indicated parameter choices both systems correspond to isotropic liquids.
The Table summarizes various properties for polydisperse LJ beads for different reduced cutoff
distances $\ensuremath{s_\mathrm{c}}/\ensuremath{s_\mathrm{0}}$.
Considering thermodynamic properties per particle (rather than per volume), we have made the
data dimensionless by rescaling with the inverse temperature $\beta$ and the density $\rho$.

\section{Computational results}
\label{sec_res}

\begin{figure}[t]
\centerline{\resizebox{0.95\columnwidth}{!}{\includegraphics*{fig1}}}
\caption{Weighted radial pair distribution function $\ensuremath{h_\mathrm{2}}(s) \beta/\rho$ with $s = r/\ensuremath{\sigma_{l}}$
being the reduced distance between two beads $i$ and $j$.
Main panel: KA mixtures in $d=3$ (bold line) and polydisperse LJ beads in $d=2$ (open symbols)
for large reduced distances $s > \ensuremath{s_\mathrm{0}}$ where the potential is attractive.
The filled sphere corresponds to the shear modulus $\ensuremath{\tilde{G}}$ computed using Eq.~(\ref{eq_lame2G})
for the KA system not taken into account the impulsive correction.
Inset: Polydisperse LJ beads for $s \le \ensuremath{s_\mathrm{0}}$ where $\ensuremath{h_\mathrm{2}}(s)$ becomes strongly negative.
\label{fig_histo}
}
\end{figure}

\subsection{Weighted pair distribution function $\ensuremath{h_\mathrm{2}}(s)$}
\label{res_histo}
The weighted pair distribution function $\ensuremath{h_\mathrm{2}}(s)$, Eq.~(\ref{eq_histoLame}), is presented in Fig.~\ref{fig_histo}.
Several cutoff distances $\ensuremath{s_\mathrm{c}}$ are given for the polydisperse LJ model, but for clarity only
for distances $s \le \ensuremath{s_\mathrm{c}}$. For the KA model only one cutoff is given,
but this also for $s > \ensuremath{s_\mathrm{c}}$. Note that albeit different $\ensuremath{s_\mathrm{c}}$ for each model correspond strictly speaking
to different state points
--- as better seen from the energies per bead $e$ or total pressures $P$ indicated in the Table ---
the histograms vary only weakly with $\ensuremath{s_\mathrm{c}}$. Strong differences become only apparent for
very small $\ensuremath{s_\mathrm{c}}$ as shown for $\ensuremath{s_\mathrm{c}} = 0.9 \ensuremath{s_\mathrm{0}}$ in the inset.
One can thus use the histogram obtained for one $\ensuremath{s_\mathrm{c}}$ to anticipate the impulsive
correction for a different cutoff.
Note that for large distances corresponding to an attractive interaction we have $\ensuremath{h_\mathrm{2}}(s) > 0$ (main panel).
Obviously, $\ensuremath{h_\mathrm{2}}(s)$ vanishes at the minimum of the potential $s=\ensuremath{s_\mathrm{0}}$ and for very large distances $s$.
Since $g(s) \approx 1$ in the latter limit, the histogram $\ensuremath{h_\mathrm{2}}(s)$ is given
(up to a known prefactor) by $s^{d+1} \ensuremath{u^{\prime}_\mathrm{s}}(s)$.
As one expects the decay is faster for the $d=2$ data than for the KA mixtures in $d=3$,
since the phase volume at the cutoff is larger for the latter systems.
Since all histograms are rather smooth, one may simply set $s = \ensuremath{s_\mathrm{c}}$ for obtaining $\ensuremath{\Delta \mu_\mathrm{B}}$
from $\ensuremath{h_\mathrm{2}}(s)$ instead of properly taking the limit $s \to \ensuremath{s_\mathrm{c}^{-}}$.

\subsection{Compression modulus $K$}
\label{res_compress}
The compression modulus $K$ may be obtained from Eq.~(\ref{eq_lame2K}) or, equivalently,
using the Rowlinson formula given elsewhere \cite{AllenTildesleyBook,papthermoflex,foot_Kt}.
All our systems are highly incompressible, i.e. the compression modulus $K$ is large as usual
in condensed matter systems, and it is thus difficult to demonstrate the small correction predicted by
Eq.~(\ref{eq_deltaK}). For the KA model we obtain, e.g.
$\ensuremath{\Delta K} \beta/\rho \approx (5/4) \times 0.69 \approx - 1.2$
which compared to the uncorrected estimate $\ensuremath{\tilde{K}} \beta/\rho \approx 21.9$ is not very impressive.

More importantly, it is not easy to obtain an independent and precise $K$-value
for canonical ensembles of mixtures and polydisperse systems using, e.g., the total particle structure factor
\cite{HansenBook,papthermoflex}. For polydisperse LJ beads we have thus computed $K$ directly from
the volume fluctuations $\delta V$ in the isobaric ensemble \cite{AllenTildesleyBook}
\begin{equation}
K = \mbox{$k_{\rm B}T$} \frac{\left< V \right>}{\left< \delta^2 V\right>}
\label{eq_Kisobaric}
\end{equation}
where we impose the same (mean) pressure $P$ as for the corresponding canonical ensemble.
As may be seen from the last column indicated in the Table, this yields similar values as the stress-fluctuation formula,
Eq.~(\ref{eq_lame2K}). Unfortunately, for larger cutoffs our error bars become too large to confirm the correction.
The most striking example, where Eq.~(\ref{eq_deltaK}) can be shown to work, is the case of the
small cutoff $\ensuremath{s_\mathrm{c}} = 0.9 \ensuremath{s_\mathrm{0}}$: Using Eq.~(\ref{eq_lame2K}) an impossible {\em negative} value
$\ensuremath{\tilde{K}} \beta/\rho \approx -19.2$ is obtained.
As may be seen from the inset in Fig.~\ref{fig_histo}, one gets
$\ensuremath{\Delta \mu_\mathrm{B}} \beta/\rho \approx 17.1$ from the weighted
histogram $\ensuremath{h_\mathrm{2}}(s)$. Taking the correction Eq.~(\ref{eq_deltaK}) into account this yields
$K \beta/\rho \approx 15$ which is similar to the value obtained using Eq.~(\ref{eq_Kisobaric}).

\begin{figure}[t]
\centerline{\resizebox{0.95\columnwidth}{!}{\includegraphics*{fig2}}}
\caption{Shear modulus $G$ and impulsive correction $-\ensuremath{\Delta \mu_\mathrm{B}}$ for polydisperse LJ beads
{\em vs.} the reduced cutoff distance $\ensuremath{s_\mathrm{c}}/\ensuremath{s_\mathrm{0}}$. The uncorrected shear modulus $\ensuremath{\tilde{G}}$
(open symbols) has been obtained using the stress-fluctuation formula, Eq.~(\ref{eq_lame2G}),
the correction term (stars) from the histogram $\ensuremath{h_\mathrm{2}}(s)$, Eq.~(\ref{eq_histoLame}).
The solid lines indicate Eq.~(\ref{eq_muBcut_polyLJ}) where we have set $\ensuremath{g_{tt^{\prime}}}(\ensuremath{s_\mathrm{c}}) = 1$.
Main panel: Linear representation showing that $G = \ensuremath{\tilde{G}} + \ensuremath{\Delta \mu_\mathrm{B}}$ (filled squares)
vanishes as predicted, Eq.~(\ref{eq_deltaG}).
Inset: Double-logarithmic representation of the same data emphasizing the
asymptotic power-law decay for large $\ensuremath{s_\mathrm{c}}$ as indicated by the bold dashed line.
\label{fig_Gscut}
}
\end{figure}

\subsection{Shear modulus $G$}
\label{res_shear}

\paragraph*{Asymptotic limit for large sampling times.}
Since all our systems are liquids, the shear modulus $G$ should of course vanish
--- at least in the thermodynamic limit for a sufficiently long sampling time.
We have thus a clear reference and for this reason $G$ is highly suitable to test our predictions.
As can be seen from the solid sphere indicated in Fig.~\ref{fig_histo} for the KA mixtures with
$\ensuremath{s_\mathrm{c}}=2.5$ we obtain $\ensuremath{\tilde{G}} \beta/\rho \approx 0.65$ if the impulsive correction for the Born term
is {\em not} taken into account.  As also shown by the figure, this deviation equals
$\ensuremath{h_\mathrm{2}}(\ensuremath{s_\mathrm{c}}) \beta/\rho \approx 0.69$ as predicted.
The same behavior is seen from Fig.~\ref{fig_Gscut} for polydisperse LJ beads
for a broad range of cutoff distances $\ensuremath{s_\mathrm{c}}$ where the open squares refer to the uncorrected $\ensuremath{\tilde{G}}$
and the filled squares to $G$ obtained using Eq.~(\ref{eq_deltaG}).
The solid lines indicated show Eq.~(\ref{eq_muBcut_polyLJ}). Focusing on the scaling for large $\ensuremath{s_\mathrm{c}}$
we have set $\ensuremath{g_{tt^{\prime}}}(\ensuremath{s_\mathrm{c}}) =1$ in the double-integral which (under this assumption) is close to unity.

\begin{figure}[t]
\centerline{\resizebox{0.95\columnwidth}{!}{\includegraphics*{fig3}}}
\caption{Shear modulus $G$ for polydisperse LJ beads in $d=2$ for different $\ensuremath{s_\mathrm{c}}$ as a
function of the sampling time $t$ given in units of MC Steps (MCS) of the local MC jumps used.
The vertical axis is made dimensionless by means of a factor $\beta/\rho$.
Filled symbols refer to the uncorrected $\ensuremath{\tilde{G}}(t)$.
The horizontal lines indicate $-\ensuremath{\Delta \mu_\mathrm{B}}$ obtained from the histograms $\ensuremath{h_\mathrm{2}}(s)$ for three cutoffs
as indicated in the Table.
The dashed slope characterizes the decay of (the corrected) $G(t)$ with time.
\label{fig_Gt}
}
\end{figure}

\paragraph*{Sampling time dependence.}
Figure \ref{fig_Gt} gives additional information for the shear modulus $G(t)$ plotted as a function
of the number $t$ of MC steps (MCS) for polydisperse LJ beads. Note that the sampling time is proportional
to the number of configurations used for the averages.
To smooth the data we have used gliding averages of length $t$ over the total trajectories of length $10^{7}$ MCS.
For $\ensuremath{s_\mathrm{c}}=1.0\ensuremath{s_\mathrm{0}}$ and $\ensuremath{s_\mathrm{c}}=4.0\ensuremath{s_\mathrm{0}}$ there was no need to add a correction while for $\ensuremath{s_\mathrm{c}}=0.9\ensuremath{s_\mathrm{0}}$
where $\ensuremath{\Delta \mu_\mathrm{B}} \beta/\rho \approx 17.1$ the uncorrected data is negative and cannot be represented.
The filled symbols refer to the uncorrected shear modulus $\ensuremath{\tilde{G}}(t)$ for $\ensuremath{s_\mathrm{c}} = 1.1 \ensuremath{s_\mathrm{0}}$,
$\ensuremath{s_\mathrm{c}} = 1.5 \ensuremath{s_\mathrm{0}}$ and $\ensuremath{s_\mathrm{c}} = 2.0 \ensuremath{s_\mathrm{0}}$ which are seen to approach for large times the predicted
correction $-\ensuremath{\Delta \mu_\mathrm{B}}$ taken from the Table (horizontal lines).
If corrected, all data sets vanish properly with time.

Interestingly, neither $\ensuremath{\mu_\mathrm{B}}$ nor $\ensuremath{P_\mathrm{ex}}$ do (essentially) depend on $t$ while the fluctuation
contribution $-\ensuremath{\mu_\mathrm{F}}(t)$ approaches (the corrected) $\ensuremath{\mu_\mathrm{B}}-\ensuremath{P_\mathrm{ex}}$ from below (not shown).
The (corrected) shear modulus $G(t)$ thus decreases monotonously with time.
As can be seen from Fig.~\ref{fig_Gt}, the (corrected) $G(t)$ decays roughly
as the power-law slope $-1$ indicated by the dashed line.
(Note that the noise becomes too large for $G(t)\beta/\rho < 0.1$.)
Exactly the same behavior has been observed for the KA model in $d=3$ (not shown).
Apparently, $G(t)$ decays quite generally inversely
as the mean-square displacement $\ensuremath{h}(t)$ of the beads in the free-diffusion limit, $\ensuremath{h}(t) \sim t$.
We remind that the same scaling $G(t) \sim 1/\ensuremath{h}(t)$ has also been reported for a bead-spring polymer model
without impulsive corrections ($\ensuremath{s_\mathrm{c}} = \ensuremath{s_\mathrm{0}}$) \cite{papthermoflex}.

\section{Conclusion}
\label{sec_conc}

\paragraph*{Summary.}
It has been emphasized in this study that an impulsive correction to the Born contributions $\ensuremath{C_\mathrm{B}^{\alpha\beta\gamma\delta}}$ of
the elastic moduli must arise if the interaction potential is truncated and
shifted, Eq.~(\ref{eq_ushift}), with a non-vanishing first derivative at the cutoff.
To test our theoretical predictions we have computed the elastic moduli of isotropic liquids
in $d=3$ and $d=2$ dimensions. Since for these systems the shear modulus $G$ must vanish by construction,
this allows a precise numerical verification for different reduced cutoff distances $\ensuremath{s_\mathrm{c}}$.
It has been shown how the impulsive correction for mixtures and polydisperse systems may be obtained
from the readily computed weighted histogram $\ensuremath{h_\mathrm{2}}(s)$ which scales
as $\ensuremath{h_\mathrm{2}}(s) \sim s^{d+1}\ensuremath{u^{\prime}}(s)$ for large $s$.
As one expects, the cutoff effect vanishes if $\ensuremath{s_\mathrm{c}}$ is large, Eq.~(\ref{eq_muBcut_asymp}),
or set to a minimum of the potential. It becomes more important with increasing dimension.

\paragraph*{Comment.}
Incidentally, it should be noted that the stress-fluctuation formula $G = \ensuremath{\mu_\mathrm{B}} + \ensuremath{\mu_\mathrm{F}} -\ensuremath{P_\mathrm{ex}}$
and various other relations used in this work for {\em liquid} systems were originally derived
for {\em solids} assuming well-defined reference positions and displacement fields
\cite{Hoover69,Ray84,Ray88,Lutsko89,Barrat06}. It can be shown, however, that
these assumptions can be relaxed and especially Eq.~(\ref{eq_lame2G}) holds quite generally
for isotropic systems.
The aim of the present paper was to show numerically that the stress-fluctuation formalism yields
the right value ($G=0$) {\em once} the impulsive correction has been taken into account.

\paragraph*{Outlook.}
We are currently using the approach presented here to characterize as a function of temperature,
imposed pressure and sampling time the glass transition of the two models presented here.
The generalization of our results
to
\begin{itemize}
\item
other elastic moduli in anisotropic systems using the more general impulsive correction Eq.~(\ref{eq_CBabd_correct}),
\item
observables related to even higher derivatives of the potential, Eq.~(\ref{eq_hn_histo}), and
\item
arbitrary interaction potentials not necessarily scaling simply with $s = r/\ensuremath{\sigma_{l}}$
and not necessarily being pair interactions using the generalization of the Born term
derived by Ray \cite{Ray88}
\end{itemize}
is straightforward and will be considered in the future.

\begin{acknowledgments}
H.X. thanks the CNRS and the IRTG Soft Matter for supporting her sabbathical stay in Strasbourg
and P.P. the  R\'egion Alsace and the IRTG Soft Matter for financial support.
We are indebted to A. Blumen (Freiburg) and H.~Meyer, O.~Benzerara and J. Farago (all ICS, Strasbourg)
for helpful discussions.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Nothing makes me freaking more ecstatic than talking about NJ BMX companies. Got an email today from Dave "pooh bear" Pawlowski that 2011 is in fact the 10 year anniversary of Scarred For Life Racing, the NJ based clothing…and accessory company that he is part of. Besides Dave being part owner-creator of this thing, he also has…are you ready… a flatlander on the team. Chad Catalano, a spry 38 years old rocks s4l apparel, and you should too. Check the panorama below, hit read more for a few pics of the guys and their products, then head over to scarredforliferacing.com to support another NJ based company into BMX…what is that like 100 companies out of NJ or something? NJ, why move to Cali when everything is here.
PREVIOUS POST Previous post: #RememberThis- Nation BMX is back!


text:
Tags: Government, Aleksandar Vucic, politics, Business
Serbia president defends minister in plagiarism dispute
FILE - In this Sunday, April 2, 2017 file photo, Sinisa Mali attends a press conference after presidential election in Belgrade, Serbia. Serbias finance minister faced calls to resign Thursday, Nov. 21, 2019 after a university said he plagiarized parts of his Ph.D. thesis. Belgrade University, one of Serbias most prestigious places of learning, said in a statement that Sinisa Mali literally took over texts, or entire paragraphs, from the texts of other authors without listing the names of the authors. (AP Photo/Darko Vojinovic, file) (Copyright 2018 The Associated Press. All rights reserved.)
BELGRADE – Serbia's president has dismissed calls for the resignation of the Balkan country's finance minister triggered by a university ruling that he plagiarized parts of the doctoral thesis.
Aleksandar Vucic on Friday told Prva TV that Sinisa Mali "is doing an excellent job." Vucic describes the decision on Mali's 2013 paper as "deeply political."
Belgrade University, arguably Serbia's most prestigious place of learning, on Thursday said Mali "literally took over texts, or entire paragraphs, from the texts of other authors without listing the names of the authors."
Mali has denied this. He too has suggested the decision is political, adding "I know how hard I've worked to be a best student."
Opponents of Vucic's populists have called for protests to press for the resignation of Mali, a former mayor of Belgrade.

redpajama_set_name: RedPajamaCommonCrawl


text:
Edith Rebecca Saunders née le , morte le  est une généticienne britannique et anatomiste des plantes. Elle joue un rôle actif dans la redécouverte des lois de Mendel sur l'hérédité, la compréhension de l'héritage des caractéristiques chez les plantes. Elle est la première collaboratrice du généticien William Bateson. Elle développe également de nombreux travaux sur l'anatomie des fleurs, en particulier sur les gynécées, les organes de reproduction féminins des fleurs.

Biographie
Edith Rebecca Saunders est née le 14 octobre 1865 à Brighton, en Angleterre. Elle étudie au Handsworth Ladies College et en 1884, elle entre au Newnham College, une université réservée aux femmes, à Cambridge. Elle assiste aux cours Part I en 1887 et II en 1888 de l'université des sciences naturelles de Cambridge, sans pouvoir passer les examens, les femmes étant exclues de l'université.

Elle poursuit ses recherches post-universitaires et travaille au Laboratoire biologique pour femmes Balfour entre 1888 et 1890 où les étudiantes des collèges de Newnham et de Girton reçoivent une préparation pour les Tripos en sciences naturelles. Elle est la dernière directrice de ce laboratoire entre 1890-1914. Elle est également directrice des études au Girton College (1904-1914) et au Newnham College (1918-1925).

Elle est nommée membre de la Royal Horticultural Society, société dans laquelle elle a reçu la médaille Banksian en 1906. En 1905, elle est l'une des premières femmes élues membres de la Linnean Society of London.

En 1920, elle est présidente de la section botanique de l'Association britannique pour l'avancement des sciences. Elle est également présidente de la Genetics Society, entre 1936 et 1938.

Pendant la Seconde Guerre mondiale, elle est volontaire pour aider les forces alliées. Elle décède peu après son retour en Grande-Bretagne, en 1945, après avoir été blessée dans un accident de vélo.

Recherche
Les premières recherches de Edith Rebecca Saunders sont axées sur la génétique. Plusieurs de ses expériences génétiques l'amène à définir avec William Bateson des termes importants tels que "allélomorphes" (appelés aujourd'hui allèles ), hétérozygote et homozygote. Avec William Bateson et Reginald Punnett, elle découvre la liaison génétique.

Elle effectue de nombreux travaux sur l'anatomie des plantes, en particulier sur les gynécées. Elle publie plusieurs articles sur le sujet (notamment sa série d'articles sur les "Illustrations du polymorphisme du carpelle" publiés dans la revue New Phytologist de 1928 à 1931).

Notes et références

Liens externes

Généticienne
Membre de la Linnean Society of London
Étudiante de Newnham College
Généticien britannique
Décès en juin 1945
Naissance en octobre 1865
Pages avec des traductions non relues
Décès à 79 ans

redpajama_set_name: RedPajamaWikipedia


text:
Group donates blankets for first responders encountering people in need
Gov. Cuomo announces proposal to open low-cost finance, design, construction services to all non-profits, school districts
Why support on Small Business Saturday is more important than ever
by: Mark Gruba
Posted: Nov 23, 2020 / 08:42 AM EST / Updated: Nov 23, 2020 / 04:49 PM EST
ROCHESTER, N.Y. (WROC) — Small Business Saturday is coming up this Saturday, November 28.
CPA Matt Bryant of the New York State Society of CPAs said this year more than ever it's important to support local businesses Monday during News 8 at Sunrise.
"It's a day to encourage people to shop small and bring more holiday shopping to small businesses," said Bryant. "It was started by American Express back in the recession of 2010 to really highlight those local businesses."
Small Business Saturday is strategically located between Black Friday, which highlights the big box stores, and Cyber Monday, which highlights e-retailers. "Small businesses hire people from the local community so when you're spending money on a small business the money stays in the community," noted Bryant. "It doesn't go to a big retailer who might be headquartered in another state or another country."
Bryant said we're helping our neighbors, which this year – during the pandemic – means more than ever. "During the pandemic, a lot of companies were forced to reduce or trim hours, and provide a lot more delivery, so they didn't have the infrastructure to get that done. They're really looking for a good fourth quarter to help them get through the year and into 2021. Supporting your local community is a great way to help them out."
Bryant concluded by saying the hope is Small Business Saturday will help you see the great products and services offered locally and that – combined with the personal satisfaction you have from shopping with your neighbors – will bring you back time and again as a repeat customer, not just on Small Business Saturday.
More Community Stories
by Christian Garzone / Jan 15, 2021
NORTH GREECE, N.Y. (WROC) — Ask Robbie Tennant if anyone will be stopping into her bar Robbie's in the North Greece/Hilton area this weekend for the Buffalo Bills playoff game, and she will tell you.
"No. None of them are coming in. I mean, we'll be a ghost town," she says.
by WROC Staff / Jan 15, 2021
ROCHESTER, N.Y. (WROC) — Monroe County Executive Adam Bello and Public Health Commissioner Dr. Michael Mendoza gave an update on the pandemic during their weekly coronavirus briefing Thursday.
Officials from the Monroe County Department of Public Health confirmed 575 new COVID-19 cases Thursday. The is now averaging 601 new cases per day over the past week and the seven-day rolling average positivity rate is now 8.1%, the lowest number reported since December 27, 2020.
ROCHESTER, N.Y. (WROC) — Rochester City Ballet is offering up a chance to chase the winter blues away.
The ballet will present a winter digital series beginning on Friday with "Dangerous Liaisons." Thses are past favorite performances available to stream a la carte, or through a new subscription series.
Flurries Precip: 30%
Rochester woman arrested for shoplifting at Von Maur
Phase 1B vaccination explained: How to book an appointment and what's the hold-up?
Gov. Cuomo: State's weekly COVID-19 vaccine supply decreased from 300K doses to 250K doses


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Sjednocená náboženská fronta (: חזית דתית מאוחדת, Chazit Datit Meuchedet) byla izraelská politická aliance tvořená čtyřmi hlavními náboženskými stranami: Agudat Jisra'el, Poalej Agudat Jisra'el, ha-Po'el ha-Mizrachi a Mizrachi, a Sdružením zbožných nezávislých. Měla krátkého trvání (existovala v letech 1949 až 1951) a byla jednou ze stran první a druhé koaliční vlády Davida Ben Guriona.

Historie
Sjednocená náboženská fronta byla politickou aliancí, vytvořenou čtyřmi hlavními izraelskými náboženskými politickými stranami (Mizrachi, ha-Po'el ha-Mizrachi, Agudat Jisra'el a Poalej Agudat Jisra'el) a Sdružením zbožných nezávislých před prvními izraelskými parlamentními volbami v roce 1949.

Ve volbách aliance získala 16 poslaneckých mandátů a stala se tak třetím největším uskupením izraelského parlamentu (Kneset). Původní rozdělení mandátů bylo 7 pro ha-Po'el ha-Mizrachi, 4 pro Mizrachi, 3 pro Poalej Agudat Jisra'el a 2 pro Agudat Jisra'el. Byla přizvána do koaliční vlády vytvořené Davidem Ben Gurionem z vítězné strany Mapaj, na níž se dále podílela s Progresivní stranou, Sefardskými a orientálními komunitami a Demokratickou kandidátkou Nazaretu.

Uskupení však způsobilo vážnou vládní krizi v důsledku rozdílných postojů ke vzdělávání v táborech pro nové přistěhovalce (tzv. ma'abarot) a v otázce náboženského školského systému. Po premiérovi Ben Gurionovo dále požadovalo uzavření ministerstva zásobování a jmenování podnikatele do funkce ministra obchodu a průmyslu. V důsledku těchto vládních sporů nakonec Ben Gurion v říjnu 1950 rezignoval. Když se rozpory vyřešily, vytvořil Ben Gurion v listopadu téhož roku svou druhou vládu, v níž si Sjednocená náboženská fronta ponechala svá dřívější místa.

Po vypsání následujících parlamentních voleb na rok 1951 se uskupení rozpadlo a jednotlivé strany ve volbách kandidovaly samostatně.

Poslanci Knesetu

Odkazy

Reference

Externí odkazy
  Kneset – Historie strany

Zaniklé politické strany v Izraeli
Politické strany založené roku 1949
Politické strany zaniklé roku 1951


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text:
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redpajama_set_name: RedPajamaGithub


Looking for a breakfast epiphany? C.A.N. Cafe can sort you out – for lunch and dinner, too – with its 'retro' good vibrations making the squares of Singapore hip, the pancakes stacked high as kites, the coffee strong and black and damn fine, and the Peranakan and fusion cuisine hop to a groovy new tune in the throbbing neon heart of Serangoon. Upper Serangoon Road is where this joint's to be found, in the Yeley Building next door to the Upper Serangoon Shopping centre, a happening spot that all hep cats should know on a first name basis.
Very quaint cafe with little crowd. We stayed there for the whole of afternoon and was able to enjoy our time with little noise. The food was so so and they got our orders wrong twice.
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The man in the middle with the cap is Robert Hendry and the man at the end could be Albert Ross.
Raymond Maclean and Harry Kelly are the other two fishermen and they are working at the Blue Head nets. Date would be probably about 69/70. I first worked at the salmon fishing in 1971 and this was before then.
Robert Henry is my uncle - our name is not spelt with a D.
My apologies on surname Moira, I should have known.


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text:
342 F. 2d 856 - Melo-Sonics Corporation v. Cropp C W
342 F2d 856 Melo-Sonics Corporation v. Cropp C W
MELO-SONICS CORPORATION and Southwest Oil Corporation, Appellants,
David CROPP, Chalmers C. Jordan, Robert W. Lingo and
Glass-Tite Industries, Inc.
United States Court of Appeals Third Circuit.
Argued Oct. 9, 1964.
Decided March 9, 1965.
Robert P. Garbarino, Crumlish & Kania, Philadelphia, Pa. (Donald W. Grieshober, Blackmore & Grieshober, Erie, Pa., on the brief), for appellants.
William F. Illig, Gifford, Graham, Mac-Donald & Illig, Erie, Pa. (John F. Potter, Erie, Pa., on the brief), for appellees David Cropp, Chalmers C. Jordan and Robert W. Lingo.
Walter T. McGough, Reed, Smith, rick E. Dressler, Cummings & Lockwood, Shaw & McClay, Pittsburgh, Pa. (G. Donald Gerlach, Pittsburgh, Pa., Pat-Stamford, Conn., on the brief), for appellee Glass-Tite Industries, Inc.
Before KALODNER, GANEY and FREEDMAN, Circuit Judges.
GANEY, Circuit Judge.
This matter is now before the Court on an appeal from the lower court's order granting the individual defendants' motion to dismiss the complaint for failure to state a cause of action against them.
This Court has jurisdiction of the case by virtue of the diversity of citizenship of the parties, the plaintiffs, Melo-Sonics Corporation and Southwest Oil Corporation, being corporations existing and organized under the laws of the State of Delaware, having their principal place of business and offices in the State of Delaware. Prior to filing this cause of action, the name of the plaintiff, Melo-Sonics Corporation, was changed from Southwest Oil Corporation to Melo-Sonics Corporation, Southwest Oil Corporation being a wholly owned subsidiary of plaintiff, Melo-Sonics Corporation, and the name of the plaintiff, Southwest Oil Corporation, was changed from Southwest Oil Corporation to Melo-Sonics Corporation. The defendants, David Cropp and Robert W. Lingo, are citizens of the State of Pennsylvania, residing in Meadville, Pennsylvania. The defendant, Chalmers C. Jordan, is a citizen of Pennsylvania, residing in Saegertown, Pennsylvania. The defendant, Glass-Tite Industries, Inc., is a corporation organized and existing under the laws of the State of Rhode Island, having its principal place of business and offices in Providence, and doing business in Pennsylvania with an office and place of business in Saegertown, Pennsylvania, Glass-Tite being the successor to Saegertown Glass Seals, Inc., having assumed all assets and liabilities of the Saegertown Glass Seals, Inc., including this claim, and has succeeded to its corporate interest.
The claim involves a suit for breach by the individual defendants and inducement to breach by the corporate defendant of a contract with the plaintiffs for purchase of stock held by the defendants in Saegertown Components, Inc., L.C.B. Tool, Inc., and Allegheny Electronics, Inc., for $1,500,000, and in their complaint seek compensatory damages in the amount of $3,056,700, plus punitive and exemplary damages.
The pertinent facts concerning the alleged purchase of the stock of the defendants are as follows: Sometime prior to February 13, 1960, it would appear from the telegrams adverted to later, there was some discussion by certain representatives of Southwest Oil Corporation with the individual defendants concerning the purchase of stock of Saegertown Components, Inc., L.C.B. Tool, Inc.,-- all of the outstanding stock being owned by the individual defendants-- as well as for their 42 per cent stock interest in Allegheny Electronics, Inc. On that date one Vincent Pepicelli, on behalf of the individual defendants, telegraphed Arthur Kania, attorney for the Southwest Oil Corporation, as follows:
'SVC MEADVILLE PENN 1960 FEB 13
PM 4 34
YR P HMAO19 327 PME RD AND REPEAT BACK ARTHUR J. KANIA ATTY AT LAW 7131 RIDGE AVE BGD VINCENT J. PEPICELLI ATTY FOR 3 SELLING STOCK-HOLDERS QUOTE THE OFFER OF SOUTHWEST OIL CORPORATION TO PURCHASE CAPITAL STOCK OF MY THREE CLIENTS IN SAEGERTOWN COMPONENTS, INC. L.C.B. TOOL CO. INC. AND ALLEGHENY ELECTRONICS, INC. IS NOT ACCEPTABLE AS IT READS. HOWEVER, MY THREE CLIENTS ARE WILLING TO SELL THEIR CAPITAL STOCK IN SAID CORPORATIONS FOR THE TOTAL PRICE OF ONE MILLION FIVE HUNDRED THOUSAND ($1,500,000) DOLLARS SUBJECT TO FORMALIZING A PRELIMINARY AGREEMENT ALONG LINES PREVIOUSLY DISCUSSED. WILL BE IN YOUR OFFICE AT 10:00 AM ON FEBRUARY 15, 1960 WITH MY CLIENTS FOR PURPOSE OF FORMALIZING SUCH AN AGREEMENT. UNQUOTE.
GV PHILA 13
PS'
Kania, in a reply telegram, which was undated, and sent to all of the individual defendants, but in care of David Cropp, was as follows:
'DAVID CROPP, CHALMEY JORDAN, ROBERT LINGO C/O DAVID CROPP R.D. #3 MEADVILLE, PENNA TELEGRAPHIC OFFER BY YOUR ATTORNEY VINCENT J. PEPICELLI CONCERNING SALE OF CAPITAL STOCK OF SAEGERTOWN COMPONENTS, INC., L.C.B. TOOL CO., INC., AND ALLEGHENY ELECTRONICS, INC. AS STATED IN SAID TELEGRAM AND ALONG LINES PREVIOUSLY DISCUSSED ACCEPTED BY SOUTHWEST OIL CORPORATION. THIS ALSO CONFIRMS MEETING DATE AND TIME TO FORMALIZE AGREEMENT.
SOUTHWEST OIL CORPORATION BY ARTHUR J. KANIA'
EXHIBIT 'B'
In accordance with the terms of these two telegrams, on February 15, 1960, and thereafter, representatives of the plaintiffs met with defendants and their counsel to formalize the agreement. However, on April 5, 1960, the individual defendants notified the plaintiffs they would not conform to the agreement. Thereafter, the plaintiffs demanded full performance in accordance with the terms of the agreement alleging they were ready, willing and able to perform the terms of said agreement. Thereafter, Cropp, Jordan and Lingo, individually, and as 100 per cent owners of Saegertown Components, Inc., and L.C.B. Tool, Inc., and as 42 per cent owners of Allegheny Electronics, Inc., sold to Saegertown Glass Seals, Inc., predecessor in interest to defendant, Glass-Tite Industries, Inc., their stock for a total consideration of $4,556,700. Upon the filing of the plaintiffs' complaint, the individual defendants moved to dismiss; the corporate defendants, however, did not join as they indicated they were only secondarily liable. The court below granted the motion to dismiss before answers to certain pre-trial interrogatories filed by the plaintiff had been answered.
In the disposition of this case, it is important to keep in mind the well-settled rule that all facts contained in the complaint and every inference fairly deducible therefrom, taking into consideration any factual information the Court can take judicial notice of, which includes the record herein, should be treated as admitted and proved, viewing the same in the light most favorable to the plaintiff. Galvin v. Jennings, 3 Cir., 289 F.2d 15; Frederick Hart & Co. v. Recordgraph Corporation, 3 Cir., 169 F.2d 580, 581. In the Hart case, supra, Judge Kalodner, speaking for this Court, stated:
'It is also well-settled that on a motion to dismiss the complaint must be viewed in the light most favorable to the plaintiff and that the complaint should not be dismissed unless it appears to a certainty that the plaintiff would not be entitled to relief under any state of facts which could be proved in support of his claim; further, no matter how likely it may seem that the pleader will be unable to prove his case, he is entitled, upon averring a claim, to an opportunity to try to prove it. We so held in Continental Collieries v. Shober, 3 Cir., 1942, 130 F.2d 631, 635. In Carroll v. Morrison Hotel Corp., 7 Cir., 1945, 149 F.2d 404, 406, the rule was stated as follows: 'On a motion to dismiss on the ground that the complaint does not state a claim on which relief can be granted, the rule is that the complaint must be viewed in the light most favorable to plaintiff and the truth of all facts well pleaded, admitted, Galbreath v. Metropolitan Trust Co., 10 Cir., 134 F.2d 569. This includes facts alleged on information and belief. There is no specific provision covering such allegations in the Federal Rules of Civil Procedure, but Rule 8(f) states that 'All pleadings shall be so construed as to do substantial justice'; consequently to refuse to give credence to them on defendants' motion to dismiss would be opposed to the spirit of the Rules."
The Court, in discussing the motion to dismiss in Leimer v. State Mutual Life Assurance Company, 8 Cir., 108 F.2d 302, 305-306, states as follows:
'* * * Such a motion, of course, serves a useful purpose where, for instance, a complaint states a claim based upon a wrong for which there is clearly no remedy, or a claim which the plaintiff is without right or power to assert and for which no relief could possibly be granted to him, or a claim which the averments of the complaint show conclusively to be barred by limitations.'
Additionally, the Court stated that there was no justification for dismissing a complaint, except where it appears to a certainty that the plaintiff would not be entitled to relief under any state of facts which could be proved in support of the claim.
Accordingly, the problem involved here is to determine whether or not, from the exchange of the telegrams involved and every inference to be drawn therefrom, there should be decided, as a matter of law by the Court, the determination as to whether a provable contract could not be made out on the allegations and inferences here asserted, or whether the plaintiff should have an opportunity at trial to prove the existence thereof. It would seem clear that in this regard, from the facts alleged in the complaint and the content of the exchange of the telegrams involved, that the answer to this question is that the matter should be resolved at trial when all facts and circumstances surrounding the alleged offer and acceptance are attempted to be proved. National Gas Appliance Corp. v. Manitowoc Co., 7 Cir., 311 F.2d 896.
In Roadway Express, Inc. v. General Teamsters, 330 F.2d 859, decided in this Circuit in 1964, a similar situation arose involving several proposed agreements, negotiations pursuant thereto and an alleged telegraphic acceptance. This was followed by several meetings which never eventuated in a formal agreement being executed. The District Court, 211 F.Supp. 796 there granted the motion to dismiss and on appeal this Court held that, 'But whether Roadway believed or should not have believed there was an agreement was a question of fact to be resolved by the trier of facts.'
In the asserted acceptance by Saegertown Components, Inc., L.C.B. Tool Co., Inc., and Allegheny Electronics, Inc., wherein Southwest Oil Corporation stated it would accept the offer made by the defendants, it is submitted there was a sufficient factual basis for a jury to make a determination as to the existence of a contract. Whether this exchange of telegrams and the correspondence previously agreed upon created a contract is a question of the intention of the parties and this question of intention is to be a matter for the trier of the fact. National Gas Appliance Corp. v. Manitowoc Co., supra, 311 F.2d at 898, and the cases therein cited. This question of intention of the parties being one for the trier of the fact is further asserted in In re ABC-Federal Oil and Burner Co., 290 F.2d 886, 889 (3rd Cir.).
Further, it is well-settled in Pennsylvania that where the parties have settled upon the essential terms and the only remaining act to be done is the formalization of the agreement, the latter is not inconsistent with the present contract, Taylor v. Stanley Co. of America, 305 Pa. 546, 552, 158 A. 157, where it is stated:
'Where all the terms of a contract are agreed upon, and its reduction to writing is provided for, merely for proof as to its terms, such provision for a written contract is not inconsistent with a present contract, and this is especially true where the thing to be done is provided for in a written memorandum. The minds of the parties having met and reached an accord as to the essential provisions of the contract, such writing would simply exhibit just what they agreed upon and understood.'
Here, the subject matter of the telegrams, the sale of stock, was unambiguous; the price was definite and the defendants had accepted unconditionally the terms in connection therewith which had previously been discussed and the formalization of the contract was merely putting into writing that which had been agreed upon by the parties.
The lower court's reliance on In re ABC-Federal Oil and Burner Co., supra, is not inconsistent with the position here taken, for there, the Referee in Bankruptcy found, as the trier of the fact, that it was the intent of the parties not to be bound until a formal document was executed, and the Court merely sustained the findings of the Referee. However, this is a review of the case after a full and complete hearing had been made by the Referee in Bankruptcy of the facts and circumstances involved in the exchange of correspondence and after the Referee's decision in view of the debtor.
Whether this plaintiff can establish the facts which he has alleged in his complaint and bring himself within the rule above discussed is a matter which only a trial can settle, but it is submitted he has set out a claim which, if he can prove it, would entitle him to recover.
Accordingly, the judgment of the lower court is reversed and the matter remanded for further proceedings not inconsistent with this opinion.


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Home News & Politics Sails and wine: at Brooklyn Navy Yard, Grain de Sail stands out
Sails and wine: at Brooklyn Navy Yard, Grain de Sail stands out
S.jennifer
From left to right: François Le Naourès, Faustine Langevin, Goulwen Josse, members of the crew
Near the New York ferries moored at the Brooklyn Navy Yard, two masts stand proudly facing the Manhattan skyscraper, amid cranes and port equipment. They belong to the Grain de Sail, the sailboat of the Breton company of the same name which wants to develop the transport of goods by sail between the two shores of the Atlantic. The sailboat arrived from Brittany on the night of Tuesday, December 15 with nearly 15,000 bottles of organic French wines intended for sale in New York.
Disembarking in a city stared at by the Covid-19? "It does not matter. We are here to work, not to sightsee", Says the captain, Loīc Briand.
It's not every day that New York sees such a cargo ship arrive. Far from the container ships to which the authorities are accustomed, the Grain de Sail measures twenty-four small meters and uses velic propulsion (sails) as the main mode of movement. It does have an engine, but this is mainly used for maneuvers of arrivals and departures in ports and occasionally during the crossing to ensure that it works. According to Loīc Briand, only 60 liters of diesel were used for the entire journey (almost a month) between Brittany and New York. "I think the Americans think we are nuts. They wonder how we were able to cross the Atlantic on a small 24-meter boat. They are used to container ships that measure 300. For us sailors, that's normal. We find it funny".
In addition to Loïc Briand, originally from around Saint-Malo, the crew is made up of three members: Faustine Langevin, second, Goulwen Josse, lieutenant, and François Le Naourès, sailor. They cut their teeth in fishing, sailing passenger transport and the merchant navy.
The New York stopover is an important step in Grain de Sail's ambitions. This is the first transatlantic crossing of the freighter-sailboat, built by the Morlaix SME to transport the coffee and cocoa from Latin America necessary to supply its chocolate-roasting factory. After New York, the boat will set sail for the Dominican Republic to fill up on cocoa before returning to Brittany. In the long term, Grain de Sail would like to build other, larger sailboats attached to production centers in France and the United States, and make regular transatlantic rotations.
Seduced by the social and environmental dimension of Grain de Sail, Loïc Briand is convinced that the transport of goods by sail has a bright future. "This week again, a container ship lost 200 containers. At some point, you may have to ask yourself questions. A crew of 18 can't do anything against 200 containers coming loose. We are on a human scale. Could we transport Chinese boxer shorts between Asia and Europe by sail? It would not be profitable. Can we transport value-added products at a limited additional cost? It's possible".
In addition to chocolate and coffee, Grain de Sail aims to conquer other markets, such as spices, whiskey and rum.
While waiting for the departure of the boat after Christmas, weather permitting, the curious can visit the Grain de Sail by appointment and taste the wines transported in its refrigerated hold. They come from seven French winegrowers. To avoid unpleasant surprises, the impact of long journeys on the taste of the wines has been assessed by Grain de Sail using fishing boats in France. "The selected winegrowers have never exported their wines", Specifies Matthieu Riou, US wine and spirits director at Grain de Sail. He came to New York to promote it to New York restaurateurs and wine merchants. "It's not the best time for sure, but there is real interest in the project from the start. The public is interested in initiatives that bring hope. It can work in our favor, he says. There are a lot of projects in Brittany around sailing transport. This can compete with the classic freighter. We may make the trip in a little longer. Nature works that way, but that's okay. We know when the boat leaves but not when it arrives".
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