text:
\section{Introduction}

Potts models~\cite{potts} have been often considered in literature to reproduce the critical
properties of more complex physical
systems~\cite{wu,svya,gavai,fuku,alves,schmidt,Janke,Karsch,caselle1,pottsim1,pottsim2,caselle2,papa,babe,babedu}.
In the present work we are interested in the 3-state Potts model defined on a three dimensional cubic lattice.
The generic $q$-state Potts model is defined by the following partition function
\begin{eqnarray}
Z(\beta, H) = \sum_{\{ \sigma_i \}} {\rm e}^{-\beta (E - H M)} \, ,
\label{Z}
\end{eqnarray}
where the spin variable $\sigma_i$ lives on lattice site $i$ and can
take $q$ different possible values,
{\it e.g.\ } $\sigma_i \in \{ 0,1, \dots ,(q-1) \}$, while $\beta = 1/(k_B T)$.
$E$ and $M$ denote respectively the energy and magnetization with respect
to a chosen reference spin value $\bar\sigma$ ({\it e.g.\ } $\bar\sigma = 0$):
\begin{eqnarray}
E &=&  - J \sum_{\langle i,j \rangle} \delta_{\sigma_i, \sigma_j} \label{energy} \\
M &=& \sum_i \delta_{\sigma_i, \bar\sigma}
\label{EM}
\end{eqnarray}
where $J$ is the coupling constant, $H$ is an external applied field
and $\langle i,j \rangle$ in the sum denotes all pairs of nearest neighbor
lattice sites.
In the following,
as usual, we shall set $J = 1$ and make use of the normalized magnetic field
$h \equiv \beta H$.

For $h = 0$ the system has an exact symmetry, corresponding
to all possible global permutations of the $q$ spin values,
{\it i.e.\ } the symmetry group is the group of permutations
$S_q$. Such symmetry gets spontaneously broken below
a given critical temperature,
where the spin variables align themselves along a given direction.
The corresponding phase transition is first order, in three dimensions,
for $q \geq 3$, and second order for $q = 2$ (the system coincides
with the Ising model in this case). For $q = 3$ the critical temperature
is given by $\beta_c(h=0) = 0.550565(10)$ \cite{Janke}.

The critical properties of
3D Potts models with $q = 2$ or 3 at $h = 0$, have often been associated with those
met at the finite $T$ phase transition of QCD (with 2 or 3 colors) in
the pure gauge limit, via the well known Svetitsky-Yaffe conjecture~\cite{svya}.
The symmetry group which is spontaneously broken in the high T, deconfined phase of
$SU(N)$ pure gauge theories is that associated with center symmetry, $Z_N$,
corresponding to local gauge transformations which are periodic
in the Euclidean time direction only up to a global group element belonging
to the center of the gauge group.
The symmetry group coincides with the
permutation group for $N = 2$, while for $N = 3$ one has to add charge conjugation to
center transformations to recover the full permutation group $S_3$.
The corresponding order parameter in pure gauge theories, playing the role
of magnetization and signalling the spontaneous breaking of center symmetry
in the deconfined phase, is the Polyakov loop, {\it i.e.\ }  a closed parallel transport in the Euclidean
time direction.

For $h \neq 0$ the symmetry $S_q$ is explicitly broken to a residual
$S_{q-1}$, corresponding to permutations among spin values other than
the chosen direction $\bar\sigma$. The case $h > 0$, in which alignment of
spin variables along $\bar\sigma$ is favoured, has been extensively studied
in the literature: one can still distinguish two phases in which the system
is more (low T) or less (high T) aligned along $\bar\sigma$, however
in both phases the residual $S_{q-1}$ symmetry stays unbroken, so that
no real phase transition is expected a priori. However, for $q \geq 3$,
the first order transition present at $h = 0$ persists also for
non zero positive values of $h$, till a critical endpoint is met, after
which the transition disappears. Such critical endpoint is expected to be
in the Ising 3D universality class and for $q = 3$ it has been located at
$(\beta_c, h_c) =(0.54938(2), 0.000775(10))$~\cite{Karsch, pottsim2}.

In the analogy with finite $T$ pure gauge theories, the case $h > 0$ corresponds to
adding dynamical fermions of mass $m$ and in the fundamental representation of the gauge group
(the limit $h \to 0$ corresponding to $m \to \infty$): that induces an effective
coupling to the Polyakov loop, which breaks center symmetry and aligns the Polyakov
loop along the positive real direction, while the residual charge conjugation symmetry
stays unbroken for all values of $T$.

The case $h < 0$ is quite different. Indeed in this case the reference state
$\bar\sigma$ is disfavoured, so that low T ordering happens along one of the remaining
$q-1$ states: there is therefore an order/disorder transition associated with the
spontaneous breaking of the unbroken $S_{q-1}$ symmetry group. Actually, in the limit
of large $|h|$, the system becomes completely equivalent to a $(q - 1)$-state
Potts model at zero magnetic field, since the disfavoured state disappears from the
statistical ensemble. Therefore a true phase transition is expected for every value
of $|h|$, coinciding with the transition of $q$- or $(q-1)$-state Potts model in the limit
of zero or infinite field respectively.

In the present paper we shall discuss the case $q = 3$ in three dimensions,
which is particularly interesting (as well as the case $q = 5$ in two dimensions),
since in this case the transition
at $h = 0$ is first order, while the transition at $h = -\infty$ is second order in the 3D Ising
universality class. Hence the expectation is that the first order continues
for small values of $|h|$, until a tricritical point $h_{\rm tric}$ is met, governed by mean field
indexes, after which the transition becomes second order in the 3D Ising universality class.
An accurate verification of this scenario and the precise location of the tricritical point
is the aim of our study.
Notice that, for small $|h|$, we expect an interesting example of system which may
be naively believed in the Ising 3D universality class because of symmetry reasons
(the relevant symmetry being $Z_2$), but has instead a first order transition because
of the interplay with different dynamical degrees of freedom, corresponding to
the disfavoured state $\bar\sigma$ in this case.

Going back to the correspondence with the critical properties of $SU(3)$ lattice gauge
theories, switching the sign of $h$ is like turning the boundary
conditions of dynamical fermions in the temporal direction from anti-periodic
to periodic: fermions are not thermal any more and the Euclidean temporal direction
can then be viewed as a compactified spatial direction; the effective coupling
to the Polyakov line changes sign, so that the Polyakov line tends to align along
one of the complex center elements below a given compactification radius, thus
breaking spontaneously the residual charge symmetry (see {\it e.g.\ } Refs.~\cite{degrand0,degrand,lucini,lucini2}
for early lattice studies of the associated transition, which has been studied in the context of orientfold planar equivalence
\cite{asv, uy2006}). Alternatively, one can interpret
the system as the usual thermal theory in presence of a purely imaginary quark number
chemical potential such that ${\rm Im}(\mu)/T = \pi$: in that case the $Z_2$ breaking
transition is interpreted as the endpoint of the high $T$ Roberge-Weiss (RW) transitions
which are met in the $T$-${\rm Im}(\mu)$ plane~\cite{rw}.

The importance of this $Z_2$ transition and of its order for the general features of the QCD phase diagram has been
discussed extensively in recent literature~\cite{sqgp,Kouno:2009bm,rwep,rwep2,aarts,ssky}.
In particular, its order has been investigated by lattice simulations in QCD with
two degenerate flavors in Ref.~\cite{rwep}, and more recently also for the
three flavor theory~\cite{rwep2}: in both cases one finds a non-trivial phase structure,
with the transition being first order both for small and high quark masses, and second order
in the middle. Such phase structure can be mapped to that of the Potts model with a negative
magnetic field, which is the subject of our study, on the large mass side; on the other hand,
on the small quark mass side, chiral degrees of freedom come into play, requiring a different
effective model description. In the context of the investigation of the QCD phase diagram,
it is of course particularly important to give precise estimates of the tricritical values of the quark
mass, separating the second order from the first order regions.

The study of the 3D three-state Potts model in a negative magnetic field can be placed in the more
general context of studies of the same model in complex magnetic fields~\cite{pottsim1,pottsim2,rwep2},
aimed at mimicking the dynamics of QCD in presence of a quark chemical potential, which have
also considered the properties of the tricritical point~\cite{rwep2}. Our purpose is that of performing
a detailed study of the critical behavior of the system as a function of $h$, with the specific
aim of determining the location of the tricritical field $h_{\rm tric}$.
We will make use of different and consistent approaches in order to do that: the strategy
developed for this model and the corresponding results can then be taken as a guideline
for the analogous determination of the tricritical masses for the endpoint of the
Roberge-Weiss transition in QCD \cite{bcds}.

The paper is organized as follows: in Sec.~\ref{setup} we present and discuss the different
strategies used to investigate the critical properties of the system; in Sec.~\ref{results}
we present our numerical results and finally, in Sec~\ref{concl}, we give our conclusions.

\section{Observables and numerical analysis setup}\label{setup}

\begin{table}[bt!]
\begin{tabular}{|c|c|c|c|c|c|}
\hline & $\nu$ & $\gamma$ & $\alpha$ & $\gamma/\nu$ & $\alpha/\nu$\\
\hline $3D$ Ising & 0.6301(4) & $1.2372(5)$ & 0.110(1) & $\sim 1.963$ & $\sim 0.175$ \\
\hline Tricritical & 1/2 & 1 & 1/2 & 2 & 1\\
\hline $1^{st}$ Order & 1/3 & 1 & 1 & 3 & 3\\
\hline
\end{tabular}
\caption{Critical exponents (see {\it e.g.\ } \cite{landau, pelissvic}).}\label{CRITEXP}
\end{table}

Natural observables for the Potts model are the energy $E$, which is defined in Eq.~(\ref{energy}),
and the magnetization. As for the latter, we replace it by a new quantity $P$ which, in the analogy with
QCD, plays the role of the average Polyakov line. In order to define $P$, let us associate with each spin
variable a complex number on the unit circle as follows
\begin{eqnarray}
s_i = \exp\left( \frac{i 2 \pi \sigma_i}{3} \right) \, ;
\end{eqnarray}
then we define
\begin{eqnarray}
P = \frac{1}{V} \sum_i s_i \, ,
\end{eqnarray}
where $V = L^3$ is the lattice volume. Assuming that the state coupled to the magnetic field is
$\bar\sigma = 0$, the residual $Z_2$ symmetry of the model corresponds to an exchange of the
states $1$ and $2$, {\it i.e.\ } to complex conjugation for the complex spin variables $s_i$ and for $P$.
Therefore, while $E$ and ${\rm Re} (P)$ are even under the residual $Z_2$ symmetry,
${\rm Im} (P)$ is odd and plays the role of the order parameter for the realization
of this symmetry.

The purpose of our investigation is that of determining the location of the phase transition,
its order and universality class, as a function of the magnetic field $h$, which is taken to
be negative. In order to do that, we shall consider at first the susceptibility of the order
parameter
\begin{eqnarray}
\chi \equiv L^3\ (\langle {\rm Im}(P)^2 \rangle - \langle |{\rm Im}(P)| \rangle^2) \label{susc}
\end{eqnarray}
and the specific heat of the system
\begin{eqnarray}
C \equiv L^3\ (\langle E^2 \rangle - \langle E \rangle^2) \, .
\end{eqnarray}
The scaling of the two quantities around the phase transition, as a function of the size
of the system, is fixed by the respective critical indexes
\begin{eqnarray}
\chi = L^{\gamma/\nu}\ \phi_1 (t L^{1/\nu}) \, \label{fss1}
\end{eqnarray}
and
\begin{eqnarray}
C = C_0 + L^{\alpha/\nu}\ \phi_2 (t L^{1/\nu}) \, ,  \label{fss2}
\end{eqnarray}
where $C_0$ is a regular contribution and $t \equiv (T - T_c)/T_c$ is the reduced temperature.
In the following we will be interested in particular in the scaling of the height of the peaks,
which is regulated by $\gamma/\nu$ and $\alpha/\nu$ respectively, and in the scaling of the width
of the peaks, which is regulated by $1/\nu$ in both cases.
The critical indexes which are relevant to the different possibilities which may take place ({\it i.e.\ }
first order, second order in the 3D Ising universality class and mean field tricritical) are listed
in Table~\ref{CRITEXP}.

Another interesting quantity is the modulus of $P$, however it takes contribution both from
the order parameter and from the spin state coupled to the magnetic field, which is
$Z_2$ even, hence its susceptibility is expected to be the mixing of different contributions
scaling like $\chi$ or $C$ respectively, apart from the limit $h \to \infty$,
in which case the contribution of the state coupled to $h$ is completely
suppressed and the system can be mapped exactly to a 3D Ising model.
We shall not consider such quantity in the following.

\begin{figure}[h!]
\includegraphics*[width=0.46\textwidth]{triscal.eps}
\caption{On finite volumes tricritical scaling is expected to dominate
a finite range of $h$ values around $h_{\rm tric}$, which shrinks to zero as $L \to \infty$.}
\label{triscal}
\end{figure}

It is interesting to notice that, while first order scaling and 3D Ising scaling are expected
to take place for a continuous range of values of $h$, tricritical scaling is in principle expected
only for a specific value $h_{\rm tric}$, which we want to determine, at the boundary between the
first order and the second order region.
However the correct expectation is to have tricritical scaling regulating a neighborhood
of $h_{\rm tric}$, with the size of the neighborhood going to zero as $L \to \infty$. Putting the question
the other way around, we expect to need increasingly large volumes to discriminate between
first order and 3D Ising second order as we approach the tricritical field $h_{\rm tric}$ from either side,
since a fictitious tricritical scaling will mask the true thermodynamical limit for not large enough
volumes (see Fig.~\ref{triscal} for a graphical representation of that).

This reasoning can be made more quantitative by use of the so-called crossover exponents: in the thermodynamical
limit, the true critical behaviour of the system can be seen only for $|t|\lesssim p^{1/\phi}$, where $t$ is the
reduced temperature, $p$ is the parameter that controls the change of critical behaviour and $\phi$ is the
crossover exponent (see {\it e.g.\ } \cite{Cardy, BinderDeutsch, pelissvic}), which is by definition $\phi=y_p/y_t$ ($y_t$ and $y_p$
are the renormalization group eigenvalues of the relevant variables $t$ and $p$). In our case $p\propto h-h_{\rm tric}$
and $\phi=1/2$~\cite{LawSarb}. On a finite lattice of typical size $L$,
$|t|$ can be traded for $L^{-1/\nu}$ and the previous condition becomes
$L\gtrsim |h-h_{\rm tric}|^{-\nu/\phi}$; in particular, according to the known tricritical indexes in Table~\ref{CRITEXP},
one expects tricritical behaviour to dominate up to a critical size
\begin{eqnarray}
L_c \simeq A\ |h - h_{\rm tric}|^{-1}
\label{critsize}
\end{eqnarray}
where $A$ is some unknown
constant which may be different on the first order and on the second order side; a numerical check of this behaviour will
be reported in Sec.~\ref{resultsA} (see in particular Fig.~\ref{triscal_data}).
That implies that a correct and precise determination of $h_{\rm tric}$ may be quite
difficult if one looks at the finite size scaling of susceptibilities or other quantities alone.

\begin{figure}[t!]
\includegraphics*[width=0.47\textwidth]{histo_energy_0.0025.eps}
\caption{Probability distribution of the energy density for different lattice sizes $L$ and $h = -0.0025$, where the transition is first order.}
\label{histo0.0025}
\end{figure}

\begin{figure}[t!]
\includegraphics*[width=0.47\textwidth]{histo_energy_0.01.eps}
\caption{Probability distribution of the energy density for different lattice sizes and $h = -0.01$,
where the transition is second order.}
\label{histo0.01}
\end{figure}

As an alternative and easiest way to determine $h_{\rm tric}$, we shall determine quantities which
give a measure of the strength of the first order transition, such as the latent heat or the
gap of the order parameter at the transition, and study the variation of such quantities as a
function of $h$, in order to extrapolate the point $h_{\rm tric}$ where they vanish, {\it i.e.\ } where the first
order disappears, without the need of making simulations very close to $h_{\rm tric}$.

We shall consider in particular the Binder-Challa-Landau cumulant~\cite{Challa} of the energy,
which is defined as \(B_4=1-\langle E^4\rangle/(3\langle E^2\rangle ^2)\). It
can be shown (see {\it e.g.\ }~\cite{LeeKosterlitz}) that near a transition \(B_4\) develops minima
whose depth scales as
\begin{eqnarray}
B_4|_{min} &=&
\frac{2}{3}-\frac{1}{12}\left(\frac{E_+}{E_-}-\frac{E_-}{E_+}\right)^2+O(L^{-3}) \nonumber \\
&=&
\frac{2}{3}-\frac{1}{3}\left(\frac{\Delta_E}{\epsilon}\right)^2+O(\Delta_E^3)+O(L^{-3})
\label{Bmin}
\end{eqnarray}
where \(E_{\pm}=\lim_{\beta\to\beta_c^{\pm}}\langle E\rangle\), \(\Delta_E=E_+-E_-\) and
\(\epsilon=\frac{1}{2}(E_++E_-)\). In particular the thermodynamical limit of  \(B|_{min}\) is less
than \(2/3\) if and only if a latent heat is present; to simplify the notation in the following we will use
the shorthand $B=\frac{2}{3}-B_4|_{min}$.

A different, but analogous quantity is the gap of the order parameter, $\Delta$, which can be extracted by
looking at the scaling of the maximum of its susceptibility, $\chi$, and using the relation, valid in the
large volume limit for a first order transition,
\begin{eqnarray}
\chi_{\rm max} \sim A +  \frac{L^3}{4} \Delta^2 \, . \label{suscmax}
\end{eqnarray}

Both $\Delta_E$ and $\Delta$ are expected to vanish as we approach the tricritical field
$h_{\rm tric}$ from the first order side. In particular, the leading order expected behaviour is the following
(see \cite{LawSarb} or \cite{Sheehy} for a brief summary)
\begin{eqnarray}
\Delta_E \propto \sqrt{h - h_{\rm tric}} \label{deltae_beh}
\end{eqnarray}
and
\begin{eqnarray}
\Delta \propto \sqrt{|(h - h_{\rm tric})\log(h-h_{\rm tric})|} \label{delta_beh}
\end{eqnarray}

Another useful quantity is the fourth-order cumulant of the order parameter. This is usually  defined by
$\langle M^4\rangle/\langle M^2\rangle^2$ (\cite{Binder1, Binder2}), where $M$ is the order parameter, and is
typically used in the study of second order transitions. Since in this work we will analyze mainly region of the parameter
space in which first order transitions are present, the connected form
\begin{eqnarray}
U_4=\frac{\langle (\delta M)^4 \rangle}{\langle (\delta M)^2\rangle ^2} \qquad
\delta M=| {\rm Im}P|- \langle|{\rm Im}P|\rangle  \label{z2_binder}
\end{eqnarray}
appears to be best suited to disentangle the fluctuations inside a thermodynamical phase from the tunneling between the
two sectors with different $Z_2$ magnetization.
Another reason to prefer the connected form is that at the critical point it develops a minumum, making thus possible to obtain the value of the cumulant
at the transition, without introducing cross-correlations with other observables, or between different lattice size data. As a last point, we note that the relative
error of the cumulant value at transition turned out to be smaller by a factor ~2 for the connected cumulant than for the usual one.

For a first order transition it is simple to show that in the thermodynamic limit $U_4\to 1$, by using a double gaussian
approximation for the distribution of the order parameter.
For a second order transition it can be shown that the value of $U_4$ at the transition is a renormalization
group invariant (\cite{Binder1, Binder2}), so that the intersection point of $U_4$ calculated on two
lattices of different size can be used as an estimator of the transition point. It can also be shown
that the slope of $U_4$ at the transition point $U_4^*$ satisfies the relation
\begin{eqnarray}
\left. \frac{\partial U_4(bL)}{\partial U_4(L)}\right|_{U_4^*}=b^{1/\nu} \label{binder_prime_scal}
\end{eqnarray}
thus giving an estimate of the $\nu$ critical index.

\section{Numerical Results}\label{results}

The first order transition, which is already quite weak at $h = 0$, gets weaker for negative $h$
values, so that we do not need to use algorithms specifically designed for strong first orders, like the multicanonical one.
While approaching the tricritical point autocorrelation times grow up, however, since we will perform our simulations mainly
in the first order region, this slowing down is not expected to be too significant for our study. Numerical
simulations have thus been performed using a standard Metropolis algorithm.

Collected statistics have been of the order of $10^7\div10^8$ independent configurations
for all volumes and parameter sets explored; numerical simulations have been performed on GRID resources provided
by INFN.

\subsection{Discerning the critical behaviour from finite size scaling.}
\label{resultsA}

One way to discern between a first order and a second order critical behaviour\footnote{For the sake of simplicity we will speak
of ``critical behaviour'' also for the case of first order transitions, although this is not completely appropriate.}
is to look at the distribution of physical observables, like the energy,
at the transition point: that is expected to develop a double peak structure,
in the thermodynamical limit, for a first order transition, while it stays single peaked in
the second order case. In Figs.~\ref{histo0.0025} and \ref{histo0.01} we show two examples,
for $h = -0.0025$ and $h = -0.01$ respectively, where the situation is quite clear:
$h = -0.0025$ clearly belongs to the first order region, while
$h = -0.01$ appears to be on the second order side.

\begin{figure}[h!]
\includegraphics*[width=0.48\textwidth]{energysusc0.0025.eps}
\caption{Scaling of the specific heat peak with $L$ for $h = -0.0025$. $\chi^2/{\rm d.o.f.} \simeq 0.4$
(range of fit: $L>50$).}
\label{specheat00025}
\end{figure}

\begin{figure}[h!]
\includegraphics*[width=0.48\textwidth]{energysusc0.01.eps}
\caption{Scaling of the specific heat peak with $L$ for $h = -0.01$. In this
case the correct scaling with 3D Ising critical indexes is visible already
from moderate size lattices. $\chi^2/{\rm d.o.f.} \simeq 0.95 $ (range of fit: $L>20$)}
\label{specheat0010}
\end{figure}

\begin{figure}[h!]
\includegraphics*[width=0.48\textwidth]{energysusc0.005.eps}
\caption{Scaling of the specific heat peak with $L$ for $h = -0.005$. According
to our determination of $h_{\rm tric}$, the system belongs to the 3D Ising universality
class, however data scale according to tricritical indexes, with small
deviations appearing only on the largest available volumes. $\chi^2/{\rm d.o.f.} \simeq 1.4$ (lattices with $L>100$
are not included in the fit).}
\label{specheat0005}
\end{figure}

Such conclusions are confirmed by looking at the scaling of the height of the specific heat peak. For
$h = -0.025$ (see Fig.~\ref{specheat00025}) a cubic term in $L$, which is characteristic of first order,
nicely fits the behaviour on the larger volumes.  For $h = -0.01$ (see Fig.~\ref{specheat0010}) the situation
is also quite clear and data correctly scale according to 3D Ising critical indexes.

What is less clear is the critical behaviour of the specific heat peak for $h = -0.005$, which
is shown in Fig.~\ref{specheat0005}. Data scale linearly with $L$, {\it i.e.\ } according to tricritical
indexes, for a large range of lattice sizes, with small deviations, going in the direction
of a smallest value of $\alpha/\nu$ (hence in the direction of the 3D Ising class), visible only on
the largest sizes explored, $L > 100$. Our subsequent analysis will clearly show that for this value
of $h$ the system  belongs to the 3D Ising universality class, however it would have been difficult
to state that clearly from the scaling of the specific heat alone: tricritical indexes regulate
the system behavior till $L \sim 100$, completely masking the correct thermodynamical limit,
which would be evident only on much larger lattices. We expect the situation to be worse
and worse as one gets closer to $h_{\rm tric}$ (see the previous discussion in Sec.~\ref{setup}).

\begin{figure}[h!]
\includegraphics*[width=0.48\textwidth]{z2susc0.0025.eps}
\caption{Scaling with $L$ of the peak of the order parameter susceptibility for $h = -0.0025$. The contribution
of two different terms, corresponding respectively to first order scaling and tricritical scaling, is needed to
correctly fit our data. $\chi^2/{\rm d.o.f.} \simeq 0.45$ (range of fit: $L > 50$).}
\label{z2susc00025}
\end{figure}

\begin{figure}[h!]
\includegraphics*[width=0.48\textwidth]{z2susc0.005.eps}
\caption{Scaling with $L$ of the peak of the order parameter susceptibility
for $h = -0.005$. In this case it is hard to disentangle tricritical
from 3D Ising scaling, since $\gamma/\nu = 2$ in the first case and
$\gamma/\nu \simeq 1.968$ in the second case.  $\chi^2/{\rm d.o.f.} \simeq 1.4$ (range of fit: $L < 130$).}
\label{z2susc0005}
\end{figure}

\begin{figure}[h!]
\includegraphics*[width=0.48\textwidth]{z2suscFWHM0.005.eps}
\caption{Scaling with $L$ of the half height width of the peak of the order parameter susceptibility
for $h = -0.005$. While till moderate size the scaling is compatible with a tricritical one,
on larger volumes a deviation from the tricritical behaviour is clearly seen. $\chi^2/{\rm d.o.f.} \simeq 1.3$
for the small volumes ($L < 100$), $\chi^2/{\rm d.o.f.} \simeq 0.3$ for the larger ones ($L > 80$).}
\label{z2suscFWHM0005}
\end{figure}

The situation is even more difficult when studying the scaling of the peak of the order parameter
susceptibility, $\chi$, since in this case the relevant critical index, $\gamma/\nu$, practically
coincides for the tricritical ($\gamma/\nu = 2$) and 3D Ising cases ($\gamma/\nu \simeq 1.963$).

For $h = -0.0025$ one clearly sees a first order contribution (see Fig.~\ref{z2susc00025}). Notice however that,
in order to correctly fit data, it is necessary to take into account also a small but non-zero contribution
proportional to $L^2$; this is the dominant term in the case of tricritical scaling, therefore
we can interpret that as evidence for a non-negligible influence from a possibly close tricritical point.
The fit with the functional form $a+bL^3+cL^2$ gives the estimates for the parameters $a=710(70)$, $b=0.0139(3)$ and
$c=-0.37(3)$, meaning that the tricritical corrections to first order is about $20\%$ on the largest lattices explored
for this $h$ value.
For $h = -0.005$ there is no hope to discern between mean field and 3D Ising (see Fig.~\ref{z2susc0005}).

A better probe in this case is furnished by the width of the susceptibility peak at half height, which is
expected to scale like $L^{-1/\nu}$, since $1/\nu$ changes appreciably from mean field tricritical to 3D Ising
(see Table~\ref{CRITEXP}). Data for the width are shown in Fig.~\ref{z2suscFWHM0005}: two different
regimes are visible, the first for $L < 100$ regulated by the tricritical exponent ($\nu = 1/2$) and the second,
for larger lattices, where this index is sensibly larger, $\nu=0.526(3)$. We explicitly note that this value
for $\nu$ is to be regarded just as an ``effective'', size dependent,  index interpolating between the
tricritical ($\nu=1/2$) and Ising one ($\nu=0.63$).
That confirms what already found by looking at the scaling of the specific heat.

Let us summarize and comment the results contained in this subsection. Discerning the correct
critical behaviour from the finite size scaling analysis of susceptibilities or other quantities
may be a difficult task since, as expected, tricritical behavior masks the correct asymptotic scaling behaviour
for some range of lattice sizes, which increases as we get closer to the tricritical field $h_{\rm tric}$
according to the tricritical crossover exponents, as summarized in Eq.~(\ref{critsize}). We have tried to
verify quantitatively the prediction reported in Eq.~(\ref{critsize}) by estimating, for each value
of $h$ on the first order side, the critical size $L_c$ such that for $L > L_c$ the scaling of the maxima
of the specific heat is well described by a first order scaling. Results are reported in
Fig.~\ref{triscal_data}: Eq.~(\ref{critsize}) is well verified by
using the value of $h_{\rm tric}$ obtained and reported
in Section~\ref{tric_pos_section}.

\begin{figure}[h!]
\includegraphics*[width=0.46\textwidth]{triscal_data.eps}
\caption{On the vertical axis it is plotted the size $L_c$ such that for $L>L_c$ the scaling of the maxima of the energy
susceptibility is well described by a first-order scaling. The red line is a fit of the form $a+b/|h-h_{\rm tric}|$
where $h_{\rm tric}=-0.00415(3)$ (see Section \ref{tric_pos_section}).}
\label{triscal_data}
\end{figure}

The difficulties are generally larger on the second order side than on the first order one,
and we can easily understand why: the growth of susceptibilities is larger for
mean field tricritical indexes than for 3D Ising critical indexes, hence a fake tricritical behavior
can mask 3D Ising indexes for a large range of lattice sizes; on the other hand a first
order behaviour, which implies a faster growth of susceptibilities with respect to the
tricritical one, is in general more easily detectable as a correction to tricritical behaviour
starting from smaller lattice sizes.
As a last comment, we note that such difficulties make it preferable to look at the scaling of the specific
heat rather than at that of the order parameter, since the critical index regulating the growth
of the specific heat with $L$, $\alpha/\nu$, changes more drastically when going from first
order ($\alpha/\nu = 3$) to tricritical ($\alpha/\nu = 2$) and to 3D Ising ($\alpha\nu \simeq 0.175$).

\subsection{The latent heat, the order parameter gap and determination of $h_{\rm tric}$}
\label{tric_pos_section}

As explained in Section~\ref{setup}, we will now determine the parameters which fix
the strength of the first order transition taking place for small values of $|h|$, in order
to extrapolate the critical value $h_{\rm tric}$ at which the first order disappears. The parameters
are the latent heat, or equivalently the minimum of the Challa-Landau-Binder cumulant
defined in Eq.~(\ref{Bmin}), and the gap of the order parameter, which can be extracted
from the large volume limit of the maximum of its susceptibility $\chi$, see Eq.~(\ref{suscmax}).

In Fig.~\ref{Binder_0.0025} we show the quantity $B$ (see Eq.~(\ref{Bmin}) and the related discussion) as a function of $1/V$
for $h = -0.0025$. It clearly extrapolates to a non-zero value for $V \to \infty$, $a=8.35(4)\times 10^{-4}$, with
both $1/V$ and $1/V^2$ corrections visible in the range of explored volumes. For the same
value of $h$ and using the same fit shown in Fig.~\ref{z2susc00025}, from the coefficient
of the cubic term in $L$ we extract the order parameter gap, $\Delta^2 = 1.689(5)\times 10^{-3}$.
In Fig.~\ref{Binder_0.005} instead we show the case $h = -0.005$, together with a power law fit $B\propto L^{a}$. If we try instead
 $B=b_0+L^a$ we get for $b_0$ the result consistent with zero shown in Tab.~\ref{table_gap}, indicating that no latent heat is present.

\begin{figure}[h!]
\includegraphics*[width=0.47\textwidth]{binder0.0025.eps}
\caption{Binder-Challa-Landau cumulant of the energy
for $h = -0.0025 > h_{\rm tric}$. $B$ extrapolates to a non zero value
as $V \to \infty$. $\chi^2/{\rm d.o.f.} \simeq 1.4$.}
\label{Binder_0.0025}
\end{figure}

\begin{figure}[h!]
\includegraphics*[width=0.47\textwidth]{binder0.005.eps}
\caption{As in Fig.~\ref{Binder_0.0025},  for $h = - 0.005 < h_{\rm tric}$. $B$ goes to zero but
the power law changes beyond a given size separating tricritical from 3D Ising scaling: $a=-1.77(1)$ and $b=-1.98(1)$.
$\chi^2/{\rm d.o.f.} \simeq 1.4$ ($L < 100$) and $\chi^2/{\rm d.o.f.} \simeq 0.4$ ($L > 80$) respectively.}
\label{Binder_0.005}
\end{figure}

\begin{table}
\begin{tabular}{|l|l|l|}
\hline
\rule{0mm}{3.4mm}\hfill $h$ \hfill{}     & \hfill $B$ \hfill{} & \hfill $\Delta^2$ \hfill{} \\ \hline
\rule{0mm}{3.4mm}  $-0.002$   & $1.17(2)\times 10^{-3}$  &  $1.45(2)\times 10^{-2}$   \\ \hline
\rule{0mm}{3.4mm}  $-0.0025$  & $8.29(6)\times 10^{-4}$  &  $1.28(7)\times 10^{-2}$   \\ \hline
\rule{0mm}{3.4mm}  $-0.003$   & $5.1(1) \times 10^{-4}$  &  $7.8(3) \times 10^{-3}$   \\ \hline
\rule{0mm}{3.4mm}  $-0.0035$  & $2.78(1)\times 10^{-4}$  &  $5.6(3) \times 10^{-3}$   \\ \hline
\rule{0mm}{3.4mm}  $-0.0038$  & $1.7(2) \times 10^{-4}$  &  $2.7(4) \times 10^{-3}$   \\ \hline
\rule{0mm}{3.4mm}  $-0.005$   & $5(11) \times 10^{-6}$   &  $-2(2.8) \times 10^{-4}$   \\ \hline
\end{tabular}
\caption{Estimated values for the thermodynamical limit of \(B\) and \(\Delta^2\).}\label{table_gap}
\end{table}

We have applied the same procedure to all values of $h$ where the first order transition
is clearly detectable on the explored volumes, obtaining the values for $B$
and the gap reported in Table~\ref{table_gap}. From those values, and using the expected
behaviors reported in Eqs.~(\ref{deltae_beh}) and (\ref{delta_beh}), we can fit the value
of $h_{\rm tric}$ from both quantities. Results are reported in Fig.~\ref{endpoint}: we obtain
$h_{\rm tric} = -0.00410(5)$ from the extrapolated minimum of the cumulant, and
$h_{\rm tric} = -0.00412(7)$ from the order parameter gap. The two values are in perfect
agreement with each other and with the outcome of the finite size scaling analysis reported above;
however the finite size scaling analysis alone would have not been able to locate
$h_{\rm tric}$ with such precision.
We also notice that our determination for $h_{\rm tric}$ is in good agreement with the results reported
in Refs.~\cite{pottsim2} and \cite{rwep2} (Fig.~5 in both references), whose estimate was\footnote{P.~de Forcrand, private communication.}
$h_{\rm tric}=-0.00445(20)$.

\begin{figure}[h!]
\includegraphics*[width=0.5\textwidth]{endpoint.eps}
\caption{Extrapolation of the Binder cumulant and of the order parameter gap
in order to extract the tricritical value of the magnetic field $h_{\rm tric}$.}
\label{endpoint}
\end{figure}

As an alternative, independent way to locate $h_{\rm tric}$, we have studied the cumulant $U_4$ defined in
Eq.~(\ref{z2_binder}). The theoretical expectation is that increasing the lattice size $U_4\to 1$ for
$h>h_{\rm tric}$, $U_4\to U_4^{\rm tric}$ for $h=h_{\rm tric}$ and $U_4\to U_4^{\rm Ising}$ for $h<h_{\rm tric}$.
In particular the cumulants calculated on different lattices are expected to intersect at the tricritical point,
with a  slope increasing as $L^{1/\nu}$ (see Eq. \ref{binder_prime_scal}), with $\nu=1/2$.

Numerical results for $U_4$ are reported in Fig.~\ref{z2binder} and Table \ref{table_z2binder}; the location of the
intersection point is determined by using the method exposed in \cite{crossing}, \S III.B: a scaling law of the form
$U_4=f((h-h_{\rm tric})L^y)$ is assumed and, since we are sufficiently close to the tricritical point, we can develop
$f(x)$ in power series around $x=0$ (also scaling corrections are usually to be taken into account, see the discussion
in \cite{crossing}); a fit is then performed in the expansion parameters taking into account the data measured at different
$L$ values. By using data for $-0.005\le h < -0.002$ and $L\ge 40$ the fit has 19 d.o.f. and $\chi^2/{\rm d.o.f.}\simeq
0.8$ The estimated location of the tricritical point is $h_{\rm tric}=-0.00415(3)$. In Fig.~\ref{z2binderprime} it is
shown that the derivative of the cumulant scales with the expected critical index.

\begin{figure}[h!]
\includegraphics*[width=0.49\textwidth]{z2_binder.eps}
\caption{Binder cumulant of the order parameter at the critical temperature,
as defined in Eq.~\ref{z2_binder}. The curves at different volumes
intersect at $h_{\rm tric}$.}
\label{z2binder}
\end{figure}

\begin{figure}[h!]
\includegraphics*[width=0.49\textwidth]{z2_binder_prime.eps}
\caption{Slope of the Binder cumulant of the order parameter at the critical temperature. The line is a linear fit.}
\label{z2binderprime}
\end{figure}

\begin{table}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\hfill $h$ \hfill{} & \hfill $L=40$ \hfill{} &  \hfill $L=50$ \hfill{} &  \hfill $L=60$ \hfill{} &
\hfill $L=70$ \hfill{} & \hfill $L=80$ \hfill{}  \\ \hline
$-0.0015$  &  $1.496(2)$ &   $1.410(2)$ &   $1.325(2)$ &   $1.244(2)$ &   \hfill $\cdots$ \hfill {} \\ \hline
$-0.002$   &  $1.573(4)$ &   $1.512(5)$ &   $1.435(4)$ &   $1.367(2)$ &   $1.301(3)$ \\ \hline
$-0.0025$  &  $1.648(4)$ &   $1.598(4)$ &   $1.542(5)$ &   $1.496(4)$ &   $1.431(5)$ \\ \hline
$-0.003$   &  $1.710(2)$ &   $1.677(2)$ &   $1.646(3)$ &   $1.612(3)$ &   $1.573(4)$ \\ \hline
$-0.0035$  &  $1.770(3)$ &   $1.745(4)$ &   $1.739(7)$ &   $1.719(6)$ &   $1.691(7)$ \\ \hline
$-0.0038$  &  $1.794(4)$ &   $1.781(5)$ &   $1.782(6)$ &   $1.769(5)$ &   $1.761(6)$ \\ \hline
$-0.005$   &  $1.880(4)$ &   $1.906(4)$ &   $1.913(7)$ &   $1.932(6)$ &   $1.955(5)$ \\ \hline
$-0.01$    &  $2.080(6)$ &   $2.071(6)$ &   $2.121(7)$ &   $2.116(8)$ &   $2.123(9)$ \\ \hline
\end{tabular}
\caption{\(U_4\) values at the transition for different lattice sizes and magnetic field.}\label{table_z2binder}
\end{table}

\subsection{Critical temperature}

We conclude the presentation of our numerical results by analyzing the behaviour
of the critical temperature as a function of $h$, $\beta_c(h)$. Our determinations of
$\beta_c$ are summarized in Table~\ref{betac} and they have been obtained by using the
number of phase criterion for first order transitions (\cite{Janke}), while for the
second order ones the crossing point of the order parameter cumulant was used
(see {\it e.g.\ } \cite{Binder2}).

\begin{table}[!t]
\begin{tabular}{|l|l|}
\hline
\rule{0mm}{3.4mm}\hfill $h$ \hfill{}  & \hfill $\beta_c$ \hfill{} \\ \hline
\rule{0mm}{3.4mm}  $-0.0015$  & $0.549537(4)$   \\ \hline
\rule{0mm}{3.4mm}  $-0.002$   & $0.549237(2)$   \\ \hline
\rule{0mm}{3.4mm}  $-0.0025$  & $0.548942(1)$   \\ \hline
\rule{0mm}{3.4mm}  $-0.003$   & $0.548652(1)$   \\ \hline
\rule{0mm}{3.4mm}  $-0.0035$  & $0.548358(3)$   \\ \hline
\rule{0mm}{3.4mm}  $-0.0038$  & $0.548199(2)$   \\ \hline
\rule{0mm}{3.4mm}  $-0.005$   & $0.5475152(6)$  \\ \hline
\rule{0mm}{3.4mm}  $-0.01$    & $0.545071(8)$   \\ \hline
\rule{0mm}{3.4mm}  $-0.5$     & $0.484166(5)$   \\ \hline
\rule{0mm}{3.4mm}  $-1.0$     & $0.465188(4)$   \\ \hline
\rule{0mm}{3.4mm}  $-1.5$     & $0.45576(1)$    \\ \hline
\rule{0mm}{3.4mm}  $-2.0$     & $0.450591(4)$   \\ \hline
\end{tabular}
\caption{Estimated values for $\beta_c$ at fixed $h$.}\label{betac}
\end{table}

We expect that for large negative values of $h$ the state coupled to the magnetic
field disappears from the system dynamics, which then becomes completely equivalent to that
of a 3D Ising system. That must be visible from the behaviour of $\beta_c(h)$ which should
approach two times\footnote{This multiplicative factor is caused by a different normalization in the Hamiltonians:
the Ising one is usually written as a sum of terms $\beta \sigma_i \sigma_j=\beta (2\delta_{\sigma_i \sigma_j}-1)$.}
$\beta_c({\rm Ising}) = 0.2216546(10)$ (\cite{betaising}) as $h \to -\infty$. In Fig.~\ref{betac_large} we
show the quantity $\beta_c(h) - 2\, \beta_c({\rm Ising})$, in the regime of large $|h|$, which is expected to vanish
in the same limit; indeed we have verified that the functional behaviour expected from a strong coupling expansion
\begin{eqnarray}
\beta_c(h) - 2\beta_c({\rm Ising}) = b_1 e^h + b_2 e^{2h} + b_3 e^{3h}  \label{high_beta_fit}
\end{eqnarray}
fits our data with $b_1 = 0.0514(1)$, $b_2 = 0.0185(5)$, $b_3 = 0.0150(5) $ and
$\tilde\chi^2/{\rm d.o.f.} = 2.8$. The agreement is reasonable taking into account that our data are very accurate and
we truncate the strong coupling series just to third order.

In the opposite limit of small values of $h$, as shown in Fig.~\ref{betac_small},
we have been able to fit the
$\beta(h)$ dependence by a third order polynomial in $h$
\begin{eqnarray}
\beta_c(h) = \tilde{b}_0 + \tilde{b}_1 h + \tilde{b}_2h^2 + \tilde{b}_3h^3 \label{low_beta_fit}
\end{eqnarray}
with $\tilde{b}_0=0.550500(45)$ $\tilde{b}_1 = 0.676(44)$, $\tilde{b}_2 = 26(13)$, $\tilde{b}_3 = 2100(1400) $ and
$\tilde\chi^2/{\rm d.o.f.} = 2.5$. We notice that $b_0$ gives an estimate of the critical point position of the Potts
model without external field compatible with the known result $\beta_c(h=0)=0.550565(10)$ obtained in \cite{Janke}
and that the slope of $\beta_c (h)$ at $h = 0^-$ is different from the one observed on the positive $h$ side \cite{Karsch}.

Finally, fitting data for $\beta_c(h)$ around $h_{\rm tric}$,
we can estimate also the temperature location of the tricritical point and state
$(\beta_{\rm tric},h_{\rm tric}) = (0.5480(1),-0.00415(3))$.

\begin{figure}[h!]
\includegraphics*[width=0.49\textwidth]{beta_c_big.eps}
\caption{Plot of $\beta_c(h)-2\beta_c({\rm Ising})$. The line is the result of a fit with the function in
Eq.~(\ref{high_beta_fit}).}
\label{betac_large}
\end{figure}

\begin{figure}[h!]
\includegraphics*[width=0.49\textwidth]{beta_c_small.eps}
\caption{Plot of $\beta_c(h)$. The line is the result of a fit with the function in Eq.~\ref{low_beta_fit}. The point at
$h=0$ is the result of \cite{Janke} and is not included in the fit. The (red in color) spot on the fitted line corresponds
to our location of the tricritical point.}
\label{betac_small}
\end{figure}

\section{Conclusions}\label{concl}

We have investigated the critical properties of the three-dimensional three-state Potts
model as a function of a negative magnetic field coupled to one of the three spin states.
In this case the system possesses a residual exact symmetry, which gets spontaneously
broken at a critical coupling $\beta_c(h)$, which approaches twice the critical
coupling of the 3D Ising model for $|h| \to \infty$.
In particular, we have determined the tricritical value $h_{\rm tric}$ at which
the finite temperature first order transition, taking place for null or small values
of $|h|$, turns into a second order transition in the universality class of the
3D Ising model.

We have shown that, in proximity of the tricritical field
$h_{\rm crit}$, it is difficult to determine the critical behavior from
the finite size scaling of susceptibilities alone, since,
at a given distance from $h_{\rm crit}$, tricritical scaling
masks the correct critical indexes up to a given lattice size $L_{\rm max}$, which is regulated
by tricritical crossover exponents ($L_{\rm max} \propto |h - h_{\rm tric}|^{-1}$ in our case).
A better strategy is to determine parameters which fix the strength of the first
order region, like the order parameter gap $\Delta$ or the latent heat
$\Delta_E$, and to determine $h_{\rm tric}$ as the value of $h$ at which these
parameters extrapolate to zero. We also showed that the order parameter cumulant
is another very useful quantity to look at. In this way we have obtained the quite accurate
estimate $h_{\rm tric} = -0.00415(3)$, which is in agreement with previous
determinations reported in Refs.~\cite{pottsim2,rwep2}. Although in this work the cumulant method has proved to be the
most efficient, which of the studied methods is to be preferred to locate a tricritical point
should be model dependent.

Our results may be useful in lattice QCD studies aimed at
determining the order and universality class of the Roberge-Weiss endpoint,
and the values of the tricritical masses separating the second
order from the first order regions both for two~\cite{rwep, bcds} and
three~\cite{rwep2} degenerate flavors. In particular we expect that distinguishing
the correct critical behavior on feasible lattice sizes will be increasingly difficult
as we approach the tricritical masses, the specific heat being anyway more sensitive
than the order parameter susceptibility. An accurate determination of the tricritical
masses may be based, for instance, on a careful study of the transition strength as a function of $h$
in the first order regions.

\section*{Acknowledgments}

We thank V.~Alba, G.~Cossu, P.~de~Forcrand, A.~Di Giacomo, F.~Sanfilippo and E.~Vicari for useful discussions.
Numerical simulations have been performed on GRID
resources provided by INFN.

redpajama_set_name: RedPajamaArXiv


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text:
James Weatherall
The Department of Logic and Philosophy of Science
PhD, 2012
Irvine, California, United States of America
All publications (50)
Scientific polarization
with Cailin O'Connor
European Journal for Philosophy of Science 8 (3): 855-875. 2017.
Contemporary societies are often "polarized", in the sense that sub-groups within these societies hold stably opposing beliefs, even when there is a fact of the matter. Extant models of polarization do not capture the idea that some beliefs are true and others false. Here we present a model, based on the network epistemology framework of Bala and Goyal, 784–811 1998), in which polarization emerges even though agents gather evidence about their beliefs, and true belief yields a pay-off advantage.…Read more
Contemporary societies are often "polarized", in the sense that sub-groups within these societies hold stably opposing beliefs, even when there is a fact of the matter. Extant models of polarization do not capture the idea that some beliefs are true and others false. Here we present a model, based on the network epistemology framework of Bala and Goyal, 784–811 1998), in which polarization emerges even though agents gather evidence about their beliefs, and true belief yields a pay-off advantage. As we discuss, these results are especially relevant to polarization in scientific communities, for these reasons. The key mechanism that generates polarization involves treating evidence generated by other agents as uncertain when their beliefs are relatively different from one's own.
Fiber bundles, Yang–Mills theory, and general relativity
Synthese 193 (8). 2016.
I articulate and discuss a geometrical interpretation of Yang–Mills theory. Analogies and disanalogies between Yang–Mills theory and general relativity are also considered.
Gauge Theories
On G.E. Moore's 'Proof of an External World'
Pacific Philosophical Quarterly 98 (2). 2017.
A new reading of G.E. Moore's 'Proof of an External World' is offered, on which the Proof is understood as a unique and essential part of an anti-sceptical strategy that Moore worked out early in his career and developed in various forms, from 1909 until his death in 1958. I begin by ignoring the Proof and by developing a reading of Moore's broader response to scepticism. The bulk of the article is then devoted to understanding what role the Proof plays in Moore's strategy, and how that role is …Read more
A new reading of G.E. Moore's 'Proof of an External World' is offered, on which the Proof is understood as a unique and essential part of an anti-sceptical strategy that Moore worked out early in his career and developed in various forms, from 1909 until his death in 1958. I begin by ignoring the Proof and by developing a reading of Moore's broader response to scepticism. The bulk of the article is then devoted to understanding what role the Proof plays in Moore's strategy, and how that role is played.
Dogmatist and Moorean Replies to Skepticism
The Geometry of Conventionality
with John Byron Manchak
Philosophy of Science 81 (2): 233-247. 2014.
There is a venerable position in the philosophy of space and time that holds that the geometry of spacetime is conventional, provided one is willing to postulate a "universal force field." Here we ask a more focused question, inspired by this literature: in the context of our best classical theories of space and time, if one understands "force" in the standard way, can one accommodate different geometries by postulating a new force field? We argue that the answer depends on one's theory. In Newt…Read more
There is a venerable position in the philosophy of space and time that holds that the geometry of spacetime is conventional, provided one is willing to postulate a "universal force field." Here we ask a more focused question, inspired by this literature: in the context of our best classical theories of space and time, if one understands "force" in the standard way, can one accommodate different geometries by postulating a new force field? We argue that the answer depends on one's theory. In Newtonian gravitation the answer is yes; in relativity theory, it is no
Science, Logic, and MathematicsSpace and Time
Are Newtonian Gravitation and Geometrized Newtonian Gravitation Theoretically Equivalent?
Erkenntnis 81 (5): 1073-1091. 2016.
I argue that a criterion of theoretical equivalence due to Glymour :227–251, 1977) does not capture an important sense in which two theories may be equivalent. I then motivate and state an alternative criterion that does capture the sense of equivalence I have in mind. The principal claim of the paper is that relative to this second criterion, the answer to the question posed in the title is "yes", at least on one natural understanding of Newtonian gravitation.
The Scope and Generality of Bell's Theorem
Foundations of Physics 43 (9): 1153-1169. 2013.
I present a local, deterministic model of the EPR-Bohm experiment, inspired by recent work by Joy Christian, that appears at first blush to be in tension with Bell-type theorems. I argue that the model ultimately fails to do what a hidden variable theory needs to do, but that it is interesting nonetheless because the way it fails helps clarify the scope and generality of Bell-type theorems. I formulate and prove a minor proposition that makes explicit how Bell-type theorems rule out models of th…Read more
I present a local, deterministic model of the EPR-Bohm experiment, inspired by recent work by Joy Christian, that appears at first blush to be in tension with Bell-type theorems. I argue that the model ultimately fails to do what a hidden variable theory needs to do, but that it is interesting nonetheless because the way it fails helps clarify the scope and generality of Bell-type theorems. I formulate and prove a minor proposition that makes explicit how Bell-type theorems rule out models of the sort I describe here.
Quantum NonlocalityBell's Theorem
On Explanations in Physics
I offer one possible explanation of why inertial and gravitational mass are equal in Newtonian gravitation. I then argue that the explanation given is an example of a kind of explanation that is not captured by standard philosophical accounts of scientific explanation. Moreover, this form of explanation is particularly important, at least in physics, because demands for this kind of explanation are used to motivate and shape research into the next generation of physical theories.
Classical MechanicsTheories of Explanation, Misc
On the status of the geodesic principle in Newtonian and relativistic physics
Studies in History and Philosophy of Science Part A 42 (4): 276-281. 2011.
A theorem due to Bob Geroch and Pong Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16, ] provides a sense in which the geodesic principle has the status of a theorem in General Relativity. I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation [Weatherall, J. O. "The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52, ]. Here I compare the interpretations of these two theorems. I argue …Read more
A theorem due to Bob Geroch and Pong Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16, ] provides a sense in which the geodesic principle has the status of a theorem in General Relativity. I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation [Weatherall, J. O. "The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52, ]. Here I compare the interpretations of these two theorems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case.
Science, Logic, and MathematicsClassical MechanicsSpace and Time
Regarding the 'Hole Argument'
British Journal for the Philosophy of Science. 2016.
I argue that the Hole Argument is based on a misleading use of the mathematical formalism of general relativity. If one is attentive to mathematical practice, I will argue, the Hole Argument is blocked.
The Hole Argument
Understanding Gauge
Philosophy of Science 83 (5): 1039-1049. 2016.
I consider two usages of the expression "gauge theory". On one, a gauge theory is a theory with excess structure; on the other, a gauge theory is any theory appropriately related to classical electromagnetism. I make precise one sense in which one formulation of electromagnetism, the paradigmatic gauge theory on both usages, may be understood to have excess structure, and then argue that gauge theories on the second usage, including Yang-Mills theory and general relativity, do not generally have…Read more
I consider two usages of the expression "gauge theory". On one, a gauge theory is a theory with excess structure; on the other, a gauge theory is any theory appropriately related to classical electromagnetism. I make precise one sense in which one formulation of electromagnetism, the paradigmatic gauge theory on both usages, may be understood to have excess structure, and then argue that gauge theories on the second usage, including Yang-Mills theory and general relativity, do not generally have excess structure in this sense.
Categories and the Foundations of Classical Field Theories
In Elaine Landry (ed.), Categories for the Working Philosopher, Oxford University Press. forthcoming.
I review some recent work on applications of category theory to questions concerning theoretical structure and theoretical equivalence of classical field theories, including Newtonian gravitation, general relativity, and Yang-Mills theories.
Market crashes as critical phenomena? Explanation, idealization, and universality in econophysics
with Jennifer Jhun and Patricia Palacios
Synthese 195 (10): 4477-4505. 2018.
We study the Johansen–Ledoit–Sornette model of financial market crashes :219–255, 2000). On our view, the JLS model is a curious case from the perspective of the recent philosophy of science literature, as it is naturally construed as a "minimal model" in the sense of Batterman and Rice :349–376, 2014) that nonetheless provides a causal explanation of market crashes, in the sense of Woodward's interventionist account of causation.
On Representational Redundancy, Surplus Structure, and the Hole Argument
with Clara Bradley
Foundations of Physics 50 (4): 270-293. 2020.
We address a recent proposal concerning 'surplus structure' due to Nguyen et al.. We argue that the sense of 'surplus structure' captured by their formal criterion is importantly different from—and in a sense, opposite to—another sense of 'surplus structure' used by philosophers. We argue that minimizing structure in one sense is generally incompatible with minimizing structure in the other sense. We then show how these distinctions bear on Nguyen et al.'s arguments about Yang-Mills theory and o…Read more
We address a recent proposal concerning 'surplus structure' due to Nguyen et al.. We argue that the sense of 'surplus structure' captured by their formal criterion is importantly different from—and in a sense, opposite to—another sense of 'surplus structure' used by philosophers. We argue that minimizing structure in one sense is generally incompatible with minimizing structure in the other sense. We then show how these distinctions bear on Nguyen et al.'s arguments about Yang-Mills theory and on the hole argument.
Theoretical Equivalence in Physics
I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and "interpretational" equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the rec…Read more
I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and "interpretational" equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including definitional equivalence and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality.
Collective Belief, Kuhn, and the String Theory Community
with Margaret Gilbert
In Michael S. Brady & Miranda Fricker (eds.), The Epistemic Life of Groups, Oxford University Press. pp. 191-217. 2016.
One of us [Gilbert, M.. "Collective Belief and Scientific Change." Sociality and Responsibility. Lanham, MD: Rowman & Littlefield. 37-49.] has proposed that ascriptions of beliefs to scientific communities generally involve a common notion of collective belief described by her in numerous places. A given collective belief involves a joint commitment of the parties, who thereby constitute what Gilbert refers to as a plural subject. Assuming that this interpretive hypothesis is correct, and that s…Read more
One of us [Gilbert, M.. "Collective Belief and Scientific Change." Sociality and Responsibility. Lanham, MD: Rowman & Littlefield. 37-49.] has proposed that ascriptions of beliefs to scientific communities generally involve a common notion of collective belief described by her in numerous places. A given collective belief involves a joint commitment of the parties, who thereby constitute what Gilbert refers to as a plural subject. Assuming that this interpretive hypothesis is correct, and that some of the belief ascriptions in question are true, then the members of some scientific communities have obligations that may act as barriers both to the generation and, hence, the fair evaluation of new ideas and to changes in their community's beliefs. We argue that this may help to explain Thomas Kuhn's observations on "normal science", and go on to develop the relationship between Gilbert's proposal and several features of a group of physicists working on a fundamental physical theory called "string theory", as described by physicist Lee Smolin [Smolin, L.. The Trouble with Physics. Mariner Books: New York.]. We argue that the features of the string theory community that Smolin cites are well explained by the hypothesis that the community is a plural subject of belief.
Collective EpistemologyCollective Belief
A brief remark on energy conditions and the Geroch-Jang Theorem
The status of the geodesic principle in General Relativity has been a topic of some interest in the recent literature on the foundations of spacetime theories. Part of this discussion has focused on the role that a certain energy condition plays in the proof of a theorem due to Bob Geroch and Pong-Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1) (1975)] that can be taken to make precise the claim that the geodesic principle is a theorem, rather than a pos…Read more
The status of the geodesic principle in General Relativity has been a topic of some interest in the recent literature on the foundations of spacetime theories. Part of this discussion has focused on the role that a certain energy condition plays in the proof of a theorem due to Bob Geroch and Pong-Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1) (1975)] that can be taken to make precise the claim that the geodesic principle is a theorem, rather than a postulate, of General Relativity. In this brief note, I show, by explicit counterexample, that not only is a weaker energy condition than the one Geroch and Jang state insufficient to prove the theorem, but in fact a condition still stronger than the one that they assume is necessary
Interpreting Quantum Theories
Philosophical Review 124 (2): 275-278. 2015.
Quantum Theories
Paradox Regained? A Brief Comment on Maudlin on Black Hole Information Loss
with J. B. Manchak
We discuss some recent work by Tim Maudlin concerning Black Hole Information Loss. We argue, contra Maudlin, that there is a paradox, in the straightforward sense that there are propositions that appear true, but which are incompatible with one another. We discuss the significance of the paradox and Maudlin's response to it.
Some Philosophical Prehistory of the (Earman-Norton) hole argument
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 70 79-87. 2020.
The celu of the philosophical literature on the hole argument is the 1987 paper by Earman \& Norton ["What Price Space-time Substantivalism? The Hole Story" Br. J. Phil. Sci.]. This paper has a well-known back-story, concerning work by Stachel and Norton on Einstein's thinking in the years 1913-15. Less well-known is a connection between the hole argument and Earman's work on Leibniz in the 1970s and 1980s, which in turn can be traced to an argument first presented in 1975 by Howard Stein. Remar…Read more
The celu of the philosophical literature on the hole argument is the 1987 paper by Earman \& Norton ["What Price Space-time Substantivalism? The Hole Story" Br. J. Phil. Sci.]. This paper has a well-known back-story, concerning work by Stachel and Norton on Einstein's thinking in the years 1913-15. Less well-known is a connection between the hole argument and Earman's work on Leibniz in the 1970s and 1980s, which in turn can be traced to an argument first presented in 1975 by Howard Stein. Remarkably, this thread originates with a misattribution: the argument Earman attributes to Stein, which ultimately morphs into the hole argument, was not the argument Stein gave. The present paper explores this episode and presents some reflections on how it bears on the subsequent literature.
Why Be regular?, part I
with Benjamin Feintzeig, J. B. Le Manchak, and Sarita Rosenstock
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 65 122-132. 2019.
Maxwell-Huygens, Newton-Cartan, and Saunders-Knox Spacetimes
I address a question recently raised by Simon Saunders [Phil. Sci. 80: 22-48 ] concerning the relationship between the spacetime structure of Newton-Cartan theory and that of what I will call "Maxwell-Huygens spacetime". This discussion will also clarify a connection between Saunders' work and a recent paper by Eleanor Knox [Brit. J. Phil. Sci. 65: 863-880 ].
Against dogma: On superluminal propagation in classical electromagnetism
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 48 (2): 109-123. 2014.
It is deeply entrenched dogma that relativity theory prohibits superluminal propagation. It is also experimentally well-established that under some circumstances, classical electromagnetic fields propagate through a dielectric medium with superluminal group velocities and superluminal phase velocities. But it is usually claimed that these superluminal velocities do not violate the relativistic prohibition. Here I analyze electromagnetic fields in a dielectric medium within a framework for unders…Read more
It is deeply entrenched dogma that relativity theory prohibits superluminal propagation. It is also experimentally well-established that under some circumstances, classical electromagnetic fields propagate through a dielectric medium with superluminal group velocities and superluminal phase velocities. But it is usually claimed that these superluminal velocities do not violate the relativistic prohibition. Here I analyze electromagnetic fields in a dielectric medium within a framework for understanding superluminal propagation recently developed by Geroch and elaborated by Earman. I will argue that for some parameter values, electromagnetic fields do propagate superluminally in the Geroch-Earman sense
Inertial motion, explanation, and the foundations of classical spacetime theories
In Dennis Lehmkuhl, Gregor Schiemann & Erhard Scholz (eds.), Towards a Theory of Spacetime Theories, Birkhauser. pp. 13-42. 2017.
I begin by reviewing some recent work on the status of the geodesic principle in general relativity and the geometrized formulation of Newtonian gravitation. I then turn to the question of whether either of these theories might be said to ``explain'' inertial motion. I argue that there is a sense in which both theories may be understood to explain inertial motion, but that the sense of ``explain'' is rather different from what one might have expected. This sense of explanation is connected with …Read more
I begin by reviewing some recent work on the status of the geodesic principle in general relativity and the geometrized formulation of Newtonian gravitation. I then turn to the question of whether either of these theories might be said to ``explain'' inertial motion. I argue that there is a sense in which both theories may be understood to explain inertial motion, but that the sense of ``explain'' is rather different from what one might have expected. This sense of explanation is connected with a view of theories---I call it the ``puzzleball view''---on which the foundations of a physical theory are best understood as a network of mutually interdependent principles and assumptions.
Would two dimensions be world enough for spacetime?
with Samuel C. Fletcher, J. B. Manchak, and Mike D. Schneider
We consider various curious features of general relativity, and relativistic field theory, in two spacetime dimensions. In particular, we discuss: the vanishing of the Einstein tensor; the failure of an initial-value formulation for vacuum spacetimes; the status of singularity theorems; the non-existence of a Newtonian limit; the status of the cosmological constant; and the character of matter fields, including perfect fluids and electromagnetic fields. We conclude with a discussion of what cons…Read more
We consider various curious features of general relativity, and relativistic field theory, in two spacetime dimensions. In particular, we discuss: the vanishing of the Einstein tensor; the failure of an initial-value formulation for vacuum spacetimes; the status of singularity theorems; the non-existence of a Newtonian limit; the status of the cosmological constant; and the character of matter fields, including perfect fluids and electromagnetic fields. We conclude with a discussion of what constrains our understanding of physics in different dimensions.
Modeling How False Beliefs Spread
with Cailin O'Connor
The Logic in Philosophy of Science, by Hans Halvorson
Mind. forthcoming.
_ The Logic in Philosophy of Science _, by HalvorsonHans. Cambridge: Cambridge University Press, 2019. Pp. viii + 296.
British Journal for the Philosophy of Science 69 (2): 329-350. 2018.
I argue that the hole argument is based on a misleading use of the mathematical formalism of general relativity. If one is attentive to mathematical practice, I will argue, the hole argument is blocked. _1._ Introduction _2._ A Warmup Exercise _3._ The Hole Argument _4._ An Argument from Classical Spacetime Theory _5._ The Hole Argument Revisited
A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold
with Sarita Rosenstock
Journal of Mathematical Physics 57 102902. 2016.
A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30, ], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies …Read more
A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30, ], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation.
What Is a Singularity in Geometrized Newtonian Gravitation?
I discuss singular space-times in the context of the geometrized formulation of Newtonian gravitation. I argue first that geodesic incompleteness is a natural criterion for when a model of geometrized Newtonian gravitation is singular, and then I show that singularities in this sense arise naturally in classical physics by stating and proving a classical version of the Raychaudhuri-Komar singularity theorem
Space and Time
Endogenous epistemic factionalization
Synthese 1-22. forthcoming.
Why do people who disagree about one subject tend to disagree about other subjects as well? In this paper, we introduce a model to explore this phenomenon of 'epistemic factionization'. Agents attempt to discover the truth about multiple propositions by testing the world and sharing evidence gathered. But agents tend to mistrust evidence shared by those who do not hold similar beliefs. This mistrust leads to the endogenous emergence of factions of agents with multiple, highly correlated, polariz…Read more
Why do people who disagree about one subject tend to disagree about other subjects as well? In this paper, we introduce a model to explore this phenomenon of 'epistemic factionization'. Agents attempt to discover the truth about multiple propositions by testing the world and sharing evidence gathered. But agents tend to mistrust evidence shared by those who do not hold similar beliefs. This mistrust leads to the endogenous emergence of factions of agents with multiple, highly correlated, polarized beliefs.
Find a philosopher
The Radar
Index of professional philosophers
Index of departments
The PhilPapers Foundation
The American Philosophical Association
Centre for Digital Philosophy, Western University

redpajama_set_name: RedPajamaCommonCrawl


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text:
package bftsmart.tom.leaderchange;

import java.util.Iterator;
import java.util.LinkedList;
import java.util.ListIterator;
import java.util.Timer;
import java.util.TimerTask;
import java.util.TreeSet;
import java.util.concurrent.locks.ReentrantReadWriteLock;
import java.util.HashMap;
import java.util.Set;

import bftsmart.communication.ServerCommunicationSystem;
import bftsmart.reconfiguration.ServerViewController;
import bftsmart.tom.core.TOMLayer;
import bftsmart.tom.core.messages.TOMMessage;
import bftsmart.tom.util.TOMUtil;

import org.slf4j.Logger;
import org.slf4j.LoggerFactory;

/**
 * This thread serves as a manager for all timers of pending requests.
 *
 */
public class RequestsTimer {

    private Logger logger = LoggerFactory.getLogger(this.getClass());

    private Timer timer = new Timer("request timer");
    private RequestTimerTask rtTask = null;
    private TOMLayer tomLayer; // TOM layer
    private long timeout;
    private long shortTimeout;
    private TreeSet<TOMMessage> watched = new TreeSet<TOMMessage>();
    private ReentrantReadWriteLock rwLock = new ReentrantReadWriteLock();

    private boolean enabled = true;

    private ServerCommunicationSystem communication; // Communication system between replicas
    private ServerViewController controller; // Reconfiguration manager

    private HashMap <Integer, Timer> stopTimers = new HashMap<>();

    //private Storage st1 = new Storage(100000);
    //private Storage st2 = new Storage(10000);
    /**
     * Creates a new instance of RequestsTimer
     * @param tomLayer TOM layer
     */
    public RequestsTimer(TOMLayer tomLayer, ServerCommunicationSystem communication, ServerViewController controller) {
        this.tomLayer = tomLayer;

        this.communication = communication;
        this.controller = controller;

        this.timeout = this.controller.getStaticConf().getRequestTimeout();
        this.shortTimeout = -1;
    }

    public void setShortTimeout(long shortTimeout) {
        this.shortTimeout = shortTimeout;
    }

    public void startTimer() {
        if (rtTask == null) {
            long t = (shortTimeout > -1 ? shortTimeout : timeout);
            //shortTimeout = -1;
            rtTask = new RequestTimerTask();
            if (controller.getCurrentViewN() > 1) timer.schedule(rtTask, t);
        }
    }

    public void stopTimer() {
        if (rtTask != null) {
            rtTask.cancel();
            rtTask = null;
        }
    }

    public void Enabled(boolean phase) {

        enabled = phase;
    }

    public boolean isEnabled() {
    	return enabled;
    }

    /**
     * Creates a timer for the given request
     * @param request Request to which the timer is being createf for
     */
    public void watch(TOMMessage request) {
        //long startInstant = System.nanoTime();
        rwLock.writeLock().lock();
        watched.add(request);
        if (watched.size() >= 1 && enabled) startTimer();
        rwLock.writeLock().unlock();
    }

    /**
     * Cancels a timer for a given request
     * @param request Request whose timer is to be canceled
     */
    public void unwatch(TOMMessage request) {
        //long startInstant = System.nanoTime();
        rwLock.writeLock().lock();
        if (watched.remove(request) && watched.isEmpty()) stopTimer();
        rwLock.writeLock().unlock();
    }

    /**
     * Cancels all timers for all messages
     */
    public void clearAll() {
        TOMMessage[] requests = new TOMMessage[watched.size()];
        rwLock.writeLock().lock();

        watched.toArray(requests);

        for (TOMMessage request : requests) {
            if (request != null && watched.remove(request) && watched.isEmpty() && rtTask != null) {
                rtTask.cancel();
                rtTask = null;
            }
        }
        rwLock.writeLock().unlock();
    }

    public void run_lc_protocol() {

        long t = (shortTimeout > -1 ? shortTimeout : timeout);

        //System.out.println("(RequestTimerTask.run) I SOULD NEVER RUN WHEN THERE IS NO TIMEOUT");

        LinkedList<TOMMessage> pendingRequests = new LinkedList<>();

        try {

            rwLock.readLock().lock();

            for (Iterator<TOMMessage> i = watched.iterator(); i.hasNext();) {
                TOMMessage request = i.next();
                if ((System.currentTimeMillis() - request.receptionTimestamp ) > t) {
                    pendingRequests.add(request);
                }
            }

        } finally {

            rwLock.readLock().unlock();
        }

        if (!pendingRequests.isEmpty()) {

            logger.info("The following requests timed out: " + pendingRequests);

            for (ListIterator<TOMMessage> li = pendingRequests.listIterator(); li.hasNext(); ) {
                TOMMessage request = li.next();
                if (!request.timeout) {

                    logger.info("Forwarding requests {} to leader", request);

                    request.signed = request.serializedMessageSignature != null;
                    tomLayer.forwardRequestToLeader(request);
                    request.timeout = true;
                    li.remove();
                }
            }

            if (!pendingRequests.isEmpty()) {
                logger.info("Attempting to start leader change for requests {}", pendingRequests);
                //Logger.debug = true;
                //tomLayer.requestTimeout(pendingRequests);
                //if (reconfManager.getStaticConf().getProcessId() == 4) Logger.debug = true;
                tomLayer.getSynchronizer().triggerTimeout(pendingRequests);
            }
            else {
                rtTask = new RequestTimerTask();
                timer.schedule(rtTask, t);
            }
        } else {

            logger.debug("Timeout triggered with no expired requests");

            rtTask = new RequestTimerTask();
            timer.schedule(rtTask, t);
        }

    }

    public void setSTOP(int regency, LCMessage stop) {

        stopSTOP(regency);

        SendStopTask stopTask = new SendStopTask(stop);
        Timer stopTimer = new Timer("Stop message");

        stopTimer.schedule(stopTask, timeout);

       stopTimers.put(regency, stopTimer);

    }

    public void stopAllSTOPs() {
        Iterator stops = getTimers().iterator();
        while (stops.hasNext()) {
            stopSTOP((Integer) stops.next());
        }
    }

    public void stopSTOP(int regency){

        Timer stopTimer = stopTimers.remove(regency);
        if (stopTimer != null) stopTimer.cancel();

    }

    public Set<Integer> getTimers() {

        return ((HashMap <Integer,Timer>) stopTimers.clone()).keySet();

    }

    public void shutdown() {
        timer.cancel();
        stopAllSTOPs();
        LoggerFactory.getLogger(this.getClass()).info("RequestsTimer stopped.");

    }

    class RequestTimerTask extends TimerTask {

        @Override
        /**
         * This is the code for the TimerTask. It executes the timeout for the first
         * message on the watched list.
         */
        public void run() {

            int[] myself = new int[1];
            myself[0] = controller.getStaticConf().getProcessId();

            communication.send(myself, new LCMessage(-1, TOMUtil.TRIGGER_LC_LOCALLY, -1, null));

        }
    }

    class SendStopTask extends TimerTask {

        private LCMessage stop;

        public SendStopTask(LCMessage stop) {
            this.stop = stop;
        }

        @Override
        /**
         * This is the code for the TimerTask. It sends a STOP
         * message to the other replicas
         */
        public void run() {

                logger.info("Re-transmitting STOP message to install regency " + stop.getReg());
                communication.send(controller.getCurrentViewOtherAcceptors(),this.stop);

                setSTOP(stop.getReg(), stop); //repeat
        }

    }
}

redpajama_set_name: RedPajamaGithub


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A computer, internet, and a headset. A laptop or desktop with a mouse would likely work best for projects. However, kids can connect to the virtual classroom with a tablet and headset.
Will kids get feedback on their work from the teacher?
Yes! During virtual classroom sessions, the students will receive guidance on their projects and have the chance to share their ideas and questions with the teacher and group. Throughout the week, the kids will collaborate with their peers and post reflections online. The teacher will view and respond to online posts as well.
Yes, teachers are certified in the state in which the program is being held.
What if my child misses a class session?
Missing the occasional class session should not have a negative impact on the student experience. But, make up sessions will not be held.
Are the projects relevant to what the kids will be doing next year in school?
The projects are based on the Common Core Curriculum being implemented in the public school systems. The content will give kids exposure to next year's curriculum, while maintaining what they've learned the previous year.
What if we have to cancel after we've registered?
You can receive a full refund, if you cancel 8 weeks in advance. You will receive a 50% refund if you cancel between 2 to 8 weeks. There are no refunds if the session is cancelled within two weeks of the session starting. In some cases, we may be able to allow your child to take a different session than the one registered for.
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text:
Kids sued by RIAA seek to rope Sharman, AOL into P2P lawsuit
Two children being sued by the RIAA for copyright infringement are trying to …
Eric Bangeman - Aug 17, 2007 12:02 am UTC
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Although Patti Santangelo came out on top in the lawsuit filed against her by the RIAA, two of her children remain embroiled in a legal fight against the music industry. Her kids, Michelle and Robert Santangelo, are now trying to bring Sharman Networks—creators of KaZaA—and AOL, the Santangelos' ISP at the time of the alleged infringement, into the lawsuit as well.
The Santangelos' attorney filed a motion yesterday seeking permission to name Sharman and AOL as third-party defendants in Elektra v. Santangelo. If the judge approves, the Santangelo children would then become third-party plaintiffs in the case. By drawing Sharman and AOL into the case as third-party plaintiffs, the Santangelos are in effect saying that those two companies are ultimately liable for any harm done to the recording industry.
Sharman, is, of course, the creator of KaZaA, the P2P application allegedly used by Michelle and Robert when they were 15 and 11 years old, respectively (and installed by one of Robert's friends, who is also named in the motion). The motion argues that KaZaA "operates in the background of one's computer" while "recklessly disregarding copyright law" in order to ensnare "unsuspecting users into unintended sharing." KaZaA also allegedly blocked the any copyright infringement warning notices from the RIAA, according to the Santangelos' motion, preventing them from being warned about the possible illegality of their actions.
The proposed summons also points out that Sharman settled the RIAA's copyright infringement claims for over $100 million, but failed to take Sharman's "other victims" into account—KaZaA users who weren't aware that their music collections were being shared over the network.
AOL is targeted for "reckless endangerment of children," among other things. The summons accuses AOL of being aware that there was illegal P2P traffic on its network, saying that the ISP had the ability to prevent children from accessing files over P2P networks, but failed to do so. The Santangelos also accuse AOL of not using its heavily-advertised parental controls to stop file-sharing, "even though it had the information, superior knowledge, ability, skill, techniques, tools, power and authority to prevent such downloading."
Copyright attorney Ray Beckerman points out that this is likely the first case where one of the RIAA's targets has attempted to drag KaZaA or another third party into the legal proceedings. If the judge goes along with the motion, it could lessen the impact for the Santangelo kids should they be found liable for copyright infringement. Sharman Networks' multimillion-dollar copyright infringement settlement will be one black mark against it, but the allegations that AOL's parental controls not functioning as advertised may have a harder time passing legal muster.
One thing for sure: if the judge approves of the Santangelos' motion and the other parties are added to the lawsuit, this is going to become a very expensive case for the RIAA to litigate.
Eric Bangeman Eric has been using personal computers since 1980 and writing about them at Ars Technica since 2003, where he currently serves as Managing Editor.