text:
\section{Introduction}
\label{sec:introduction}

One of the main challenges of statistical mechanics is to understand the thermodynamics of particle systems that are not in detailed balance. A violation of detailed balance means that even in a steady state, there can be non-trivial net flows. It is therefore natural to study flows, currents and their corresponding large deviations, which is the basis of macroscopic fluctuation theory~\cite{BDSGJLL2015MFT}.
Various limits and large deviations of flows and currents have been studied in the literature, e.g. steady-state or pathwise large deviations as the number $N$ of particles and/or lattice sites goes to infinity, e.g. \cite{Derrida2007,BDSGJLL2015MFT}, or large deviations of time-averaged flows as time $T$ goes to infinity, e.g.~\cite{BertiniFaggionatoGabrielli2016Flows,BertiniFaggionatoGabrielli2016RigLDP}. In this work we are concerned with large deviations of time-averaged flows $\tfrac1T\int_0^T\!Q^N(t)\,dt$, where \emph{first} the number $N$ of particles and \emph{then} the time horizon $T$ is sent to infinity, as for example in~\cite{MR2227084,BDSGJLL2005CurFluct,BDSGJLL2007LdpEmpCur,BodineauDerrida2004,BodineauDerrida2005}. This yields a large deviation principle of the type:
\begin{align*}
  \Prob\!\left( {\textstyle \tfrac1T\int_0^T\!Q^N(t)\,dt \approx \overline q} \right) \stackrel{\substack{N\to\infty\\T\to\infty}}{\sim} e^{-NT\Psi(\overline q)},\notag\\
\intertext{with}
  \Psi(\overline q):=\lim_{T\to\infty} \inf_{\substack{(\rho,q):\\ \tfrac1T\int_0^T\!q(t)\,dt=\overline q,\\ \dot\rho(t)+\div q(t)=0}} \mfrac{1}{T}\int_0^T\!\L(\rho(t),q(t))\,dt,
\end{align*}
where the infimum ranges over paths of particle densities $\rho(t)$ and particle flows $q(t)$, and $\L$ is some non-negative cost function.

In many cases, the infimum over paths is attained or approximated by constant paths, i.e. $\rho(t)$ constant and $q(t)$ constant and divergence free. In this case the above expression simplifies to $\Psi(\overline q)=\inf_{\rho}\L(\rho,\overline q)$ ~\cite{BodineauDerrida2004}. Such simplification will fail if a time-dependent flow $q(t)$ is significantly less costly than its corresponding time-averaged flow $\overline q$. In that case we say that a \emph{dynamical phase transition} occurs (the transition being in parameter $\overline q$), see Definition~\ref{def:DFT}.

Dynamical phase transitions have been shown to occur for systems like exclusion processes with an external bulk force~\cite{BodineauDerrida2005} or the KMP model~\cite{MR2227084}, in the hydrodynamic scaling limit. See also \cite{PhysRevLett.107.180601} for numerial simulations and \cite{PhysRevLett.118.115702}, \cite{Garrahan_2009} and references therein for problems of this type for different models. In this case the mesh of the underlying graph goes to zero in the many-particle limit, yielding quadratic large-deviation cost functions, i.e. $\L(\rho,j)=\tfrac12\lVert j - k(\rho)\rVert^2_{L^2(1/\chi(\rho))}$ for some mobility $\chi(\rho)$ and expected flow $k(\rho)$. By contrast, we consider a finite graph $(\X,E)$ that is not rescaled but remains discrete in the limit; in that case the cost function is typically \emph{entropic}, i.e. $\L(\rho,q)=\sum_{(x,y)\in E} s\big(q_{x,y}\mid k_{x,y}(\rho)\big)$, where $s$ is a relative entropy like functional of $q_{x,y}$ with respect to $k_{x,y}(\rho)$. The precise model, assumptions, and large-deviations statements are introduced in Section~\ref{sec:particle system}.

Entropic cost functions are in many aspects more challenging than quadratic ones. For example, they can only induce generalised gradient flows~\cite{MielkePeletierRenger2014}, they can only be decomposed using a generalised notion of orthogonality~\cite{KaiserJackZimmer2018,RengerZimmer2019TR}, and it is not clear whether they relate to some manifold or more general geometry.

In this work we focus on jump processes on finite graphs with corresponding entropic cost functions, and study whether and when dynamical phase transitions occur.

The paper is organised as follows. In Section~\ref{sec:particle system} we introduce the general setting that we work in; in Section~\ref{sec:convexity} we present general properties of dynamical phase transitions, and in particular sufficient assumptions to rule them out. In Section~\ref{sec:discrete ring} we restrict to zero-range processes on a discrete ring and give sufficient conditions under which a dynamical phase transition does occur.

\section{Particle systems, limits and large deviations}
\label{sec:particle system}

In Section~\ref{subsec:particle system} we introduce the general particle systems that we work with; the limiting behavior and large deviations as $N\to\infty$ are discussed in Section~\ref{subsec:limit and ldp N},  and the limit and large deviations of average flows as $N\to\infty,T\to\infty$ are discussed in Section~\ref{subsec:ldp time-average}.

\subsection{Particle systems}
\label{subsec:particle system}

Let $(\X, E)$ be a finite directed graph, where the directed edges are $E\subseteq \X\times\X$, excluding the diagonal. The particle system consists of $N$ identical particles $X_1(t),\hdots,X_N(t)$ on $\X$.

The initial states $(x_i)_{i=1}^N$ of the particles are chosen deterministically so that the empirical measure converges, as $N\to +\infty$, to some fixed measure:
\begin{equation*}
  \rho^N_x(0):=\frac mN\sum_{i=1}^N\mathds1_{x_i}(x) = \frac mN\#\{\text{particles at site } x \text{ at time } 0\} \to \rho^*_x.
\end{equation*}
Each single particle carries a mass of $m/N$, so that the total mass in the system can be tuned by the parameter $m>0$, and $\rho^N(0),\rho^*\in\M_m(\X)$, the set of non-negative measures on $\X$ of total mass $m$. The precise choice of the initial condition does not play a key role in this paper, since from the next section on we are only interested in the long-time behaviour.

For the dynamics of the system, one particle randomly hops from site $x$ to $y$ with model-specific rate $N k_{x,y}(\rho^N(t))$ where the \emph{empirical measure} is:
\begin{equation}
  \rho^N_x(t):=\frac mN\sum_{i=1}^N\mathds1_{X_i(t)}(x) = \frac mN\#\{\text{particles at site } x \text{ and time } t\}.
\label{eq:discr emp measure}
\end{equation}
The empirical measure $\rho^N(t)$ is a Markov process in $\M_m(\X)$ with generator
\begin{equation}
  \hat L^N f(\rho) = \sum_{(x,y)\in E}\, Nk_{x,y}(\rho) \big\lbrack f(\rho + \tfrac mN(\mathds1_y -\mathds1_{x}) - f(\rho)\big\rbrack.
\label{eq:discr dens generator}
\end{equation}

For this model, we introduce the \emph{integrated flow} on an edge $(x,y)\in E$  as
\begin{equation}
  W^N_{x,y}(t):=\frac mN \#\big\{\text{particles jumped from $x$ to $y$ in time interval } (0,t\rbrack \big\}\,.
\label{eq:discr emp flow}
\end{equation}
This definition also fixes the initial condition $W^N(0)\equiv0$. The integrated flow $W^N(t)\in\M(E)$ is a non-negative measure on edges, related to the empirical measure through the discrete continuity equation:
\begin{align}\label{eq:discr cont eq}
  \rho^N(t) = \rho^N(0) - \div W^N(t)\, &&\text{where}&&
  \div W_x:= \sum_{y:(x,y)\in E} W_{x,y} - \sum_{y:(y,x)\in E}W_{y,x}
\end{align}

The pair $(\rho^N(t),W^N(t))$ is again a Markov process, now in $\M_m(\X)\times \M(E)$, with generator
\begin{equation*}
  L^Nf(\rho,w) = \sum_{(x,y)\in E}\, Nk_{x,y}(\rho)\left[f\Big(\rho +\tfrac mN\left(\mathds1_{y}-\mathds1_{x}\right), w + \tfrac mN\mathds1_{(x,y)}\Big) - f\Big(\rho,w\Big)\right].
\end{equation*}
The initial condition will be always of the form $W^N(0)=0$ and $\rho^N_x(0)=\frac{n_xm}{N}$ for some integers $n_x$ such that $\sum_{x\in \X}n_x=N$.
Since the intensities depend on the density $\rho$ only, ignoring the integrated flows reduces this process to the process with generator~\eqref{eq:discr dens generator}.

Note that the rates $k_{x,y}(\rho)$ may depend on all coordinates $(\rho_z)_{z\in\X}$, allowing for example for attraction or catalysis. Here we recall the technical assumtions on $k$ used in \cite{PattersonRenger2019} to prove a large deviations principle. The exact assumptions on $k$ do not play a key role in the current paper, but we briefly mention them here for completeness:
\begin{enumerate}
\item each $k_{x,y}(\rho)\to0$ sufficiently fast when $\rho_x\to0$;
\item each $k_{x,y}(\rho)$ is Lipschitz continuous and non-decreasing in $\rho$ with respect to the natural partial ordering on $\M_m(\X)$;
\item there exists a non-decreasing bijection $\phi:\lbrack0,1\rbrack\to\lbrack0,1\rbrack$ so that
$k_{x,y}(\delta \rho)\geq \phi(\delta) k_{x,y}(\rho)$
for all $\rho\in\M_m(\X),\delta\in\lbrack0,1\rbrack$ and $(x,y)\in E$.
\end{enumerate}
The first condition is needed to prevent negative densities. The continuity implies in particular that the process does not blow up and that the random paths are almost surely of bounded variation, so that the time derivatives $(\dot\rho^N(dt),\dot W^N(dt))$ always exist as measures in time. We call $Q^N(dt):=\dot W^N(dt)$. The monotonicity and the lower bound are rather technical assumptions needed in the proof of the large-deviation principle Theorem~\ref{th:discr path ldp} that we discuss in the next section.

\begin{remark}
In the limit $N\to\infty$ and the corresponding large-deviation regime, the paths become more regular and their derivatives exists as $L^1$-densities $(\dot\rho(t),\dot w(t))$. The precise topology on the path space and rigorous proofs of scaling limits and large-deviations principles will not be discussed here. We will use the notation $q(t):=\dot w(t)$.
\label{rem:BV}
\end{remark}

\begin{remark}
We believe that our results can be extended to more general models. For example, the microscopic jump rates can be $Nk_{x,y}(\rho)$ in approximation only, and the continuity equation~\eqref{eq:discr cont eq} could include more general linear operators than the discrete divergence. This would include the classical Kurtz model for chemical reactions~\cite{Kurtz1970}, allowing also for annihilation and creation of mass. We could consider even models with mass randomly flowing  across each edge and not necessarily transported by particles, like for example in the KMP model \cite{MR656869}. The monotonicity of the rates $k_{x,y}$ should also be generalisable as to include repulsion effects, but the best of our knowledge this has not yet been carried out in the literature.

\end{remark}

\subsection{Limit and large deviations as $N\to\infty$}
\label{subsec:limit and ldp N}

The many-particle limit follows by a similar argument as the classic results by Kurtz:
\begin{theorem}[{\cite{Kurtz1970,Kurtz1972,PattersonRenger2019}}]
The random paths $(\rho^N, W^N)\in\BV(0,T;\M_m(\X)\times\M(E)$ converge narrowly as $N\to\infty$ to the solution $(\rho,w)\in W^{1,1}(0,T;\M_m(\X)\times\M(E))$ of the problem
\begin{align}
&\dot w(t) = k(\rho(t)),
&
&\dot\rho(t) + \div\dot w(t)=0, \notag\\
&w(0)=0,
&
&\rho(0)=\rho^*.
\label{eq:ODE}
\end{align}
\end{theorem}

Fluctuations around this limit are quantified by the following large-deviation principle. Recall that we use the notation $w(t)=:\int_0^t\!q(s)ds$.

\begin{theorem}[{\cite{PattersonRenger2019}}]
The random paths $(\rho^N, W^N)\in\BV(0,T;\M_m(\X)\times\M(E))$ satisfy a large-deviation principle as $N\to\infty$ with speed $N$:
\begin{equation*}
  \Prob\big( (\rho^N, W^N) \approx (\rho,w)\big) \sim e^{-N \I_{\lbrack 0,T\rbrack}(\rho,\dot w\mid\rho^*)}
\end{equation*}
with rate functional
\begin{multline}
\I_{\lbrack0,T\rbrack}(\rho,q \mid \rho^*)=\\
\begin{cases}
	\sum_{(x,y)\in E}\int_0^T\!s\big( q_{x,y}(t) \mid k_{x,y}(\rho(t)) \big)\,dt,
            & \textrm{if } \rho\in W^{1,1}(0,T;\M_m(\X)), q \in L^1(0,T;\M(E)),\\
            & \quad \dot\rho+\div q=0 \text{ and } \rho(0)=\rho^*\,,\\
	+\infty,  & \textrm{otherwise}\,,
\end{cases}
\label{eq:LDP N}
\end{multline}
	where
\begin{align*
s(a\mid b):=
\begin{cases}
a\log\mfrac{a}{b} -a +b,  &a,b>0,\\
b,                        &a=0,\\
+\infty,                   &\text{otherwise}.
\end{cases}
\end{align*}
\label{th:discr path ldp}
\end{theorem}
The reader is referred to \cite{Renger2018a,PattersonRenger2019} for a proof of the above Theorem. Note that the integrand in~\eqref{eq:LDP N} can be interpreted as a kind of Lagrangian with corresponding Hamiltonian $\mathcal H(\rho,p):=\sum_{(x,y)\in E} k_{x,y}(\rho)\big(e^{p_{x,y}}-1\big)$, where $p$ are the variables dual to the variables $q$ \cite{Monthus2019}. See also \cite{MR3851785} for a similar rate functional obtained in the framework of time periodic Markov chains and  Appendix \ref{app:ldp proof} for the outline of a simple derivation in the case of independent particles using a Sanov Theorem for paths.

\begin{remark}
Clearly, the limit densities~\eqref{eq:ODE} satisfy the closed equation:
\begin{equation}
  \dot \rho(t)+\div k(\rho(t))=0\,.
\label{eq:closedrho}
\end{equation}
In general it is possible to construct models for which the equations \eqref{eq:closedrho} have several stable and unstable equilibrium points or even limit cycles (see for example \cite{lazar}). We are however mostly interested in dynamical phase transitions for models for which \eqref{eq:closedrho} has a unique and globally attractive equilibrium. This corresponds to the behavior of a thermodynamic system relaxing to equilibrium.
\label{rem:single ss}
\end{remark}

\subsection{Limit and large deviations as $N\to\infty$ and $T\to\infty$}
\label{subsec:ldp time-average}

We now study the time-averaged flow (recall the notation introduced just before Remark~\ref{rem:BV}),
\begin{equation*}
  \overline Q^N(T):=\mfrac1T\int_0^T\! Q^N(dt):=\mfrac1T W^N(T),
\end{equation*}
in the limit where $N\to\infty$ while $T=T_N$ for some function of $N$ such that $T_N\to\infty$ sufficiently slow. To simplify the arguments we always take $N\to\infty$ first and subsequently $T\to\infty$. Under sufficient conditions, the average flow $\overline Q^N(T)$ converges to $k(\overline\rho)$ where $\overline\rho$ is the unique attractive steady state of~\eqref{eq:closedrho}.

In order to state the corresponding large-deviation principle we first need to introduce some notation and facts. Let
\begin{equation*
  \I_{\lbrack0,T\rbrack}(\rho,q):=\I_{\lbrack0,T\rbrack}(\rho,q\mid\rho(0))
\end{equation*}
be the functional that coincides with \eqref{eq:LDP N} but without the constraint on the initial condition. We then define
\begin{align}
  \Psi_{T,m}(\bar q\mid\rho^*)&:=\frac 1T\inf_{(\rho,q)\in \A_{T,m}(\bar q)} \I_{\lbrack0,T\rbrack}(\rho,q\mid\rho^*)\,,
\label{eq:PsiT}\\
  \Psi_{T,m}(\bar q)&:=\frac 1T\inf_{(\rho,q)\in \A_{T,m}(\bar q)} \I_{\lbrack0,T\rbrack}(\rho,q)\,,
\notag
\end{align}
where the infima are taken over the set,
\begin{multline}
\mathcal A_{T,m}(\overline q):= \Big\{(\rho, q)\in W^{1,1}(\lbrack 0,T\rbrack;\M_m(\X))\times L^1(\lbrack 0,T\rbrack;\M(E)) \;:\; \\
\dot\rho+\div q=0,\, \mfrac 1T\int_0^Tq(t)\,dt=\overline q \Big\}.
\label{eq:defA}
\end{multline}
Note there is a slight ambiguity in the notation~\eqref{eq:PsiT} since $\rho^*$ fixes the total mass $m$, but we include the index $m$ for consistency with the other function.
The following auxiliary proposition allows to show that the functions
\begin{align*}
  \Psi_m(\bar q\mid\rho^*):=\lim_{T\to +\infty}\Psi_{T,m}(\bar q\mid\rho^*) &&\text{and}&&
  \Psi_m(\bar q):=\lim_{T\to +\infty}\Psi_{T,m}(\bar q)
\end{align*}
are well defined and have certain properties. In particular, in this limit for large times, the dependence on the initial condition vanishes.

\begin{proposition}[Well-posedness of $\Psi$] For any $\overline q\in\M(E)$ and $\rho^*\in \mathcal M_m(\X)$;
	\begin{enumerate}[(i)]
		\item $\Psi_m(\overline q\mid\rho^*)=\inf_{T>0} \Psi_{T,m}(\overline q\mid\rho^*)$ and $\Psi_m(\overline q)=\inf_{T>0} \Psi_{T,m}(\overline q)$ and hence the limits are well-defined;
		\item $\Psi_m(\overline q|\rho^*)$ and $\Psi_m(\overline q)$ are convex in $\overline q$;
		\item $\Psi_m(\overline q)=\infty$ if $\div\bar q\neq0$;
		\item $\Psi_m(\overline q\mid\rho^*)=\Psi_m(\overline q)$ \,\,(i.e. $\Psi_m$ does not depend on $\rho^*$);
		\item the convergence in item $(i)$ is a pointwise and a Gamma-convergence. \label{it:Gamma-convergence}
		\end{enumerate}
	\label{prop:well-posedness of Psi}
\end{proposition}

\begin{proof}[Sketch of proof]
The proof follows the same arguments of \cite{BDSGJLL2005CurFluct, MR2227084, BDSGJLL2007LdpEmpCur}, and we do not discuss the full details. The subadditivity that implies existence and item $(i)$ is obtained by glueing paths together one after the other. Convexity follows by a similar construction.  If $\div \bar q\neq 0$ we obtain after a long enough time a negative mass somewhere that is impossible. The independence of the initial condition follows by the fact that one can construct a finite-time finite-cost path between any two initial conditions. The fact that the convergence is also a Gamma-convergence is obtained by combining item $(i)$ and $(ii)$ along the lines of \cite[Sec.~4]{BDSGJLL2007LdpEmpCur}.
\end{proof}

We can now state the main statement of this section.
\begin{theorem}\quad
\begin{enumerate}[(i)]
\item The average flow $\overline Q^N(T)$ satisfies a large-deviation principle with speed $N$ and rate functional
$$
\bar q \to T\Psi_{T,m}(\bar q\mid \rho^*).
$$
\item The average flow $\overline Q^N(T)$ satisfies a large-deviation principle (where first $N\to\infty$ and then $T\to\infty$) with speed $NT$ and rate functional $\Psi_m(\bar q)$.
\end{enumerate}
\label{th:average flux LDP}
\end{theorem}

\begin{proof}[Sketch of proof]
The first statement follows from a straight-forward contraction principle~\cite[Th.~4.2.1]{Dembo1998}.
The second item can be written formally as
\begin{equation*
  -\mfrac{1}{NT}\log\PP^N\big(\{\overline Q^N(T) \approx \overline q\}\big) \xrightarrow[N\to\infty]{} \mfrac1T \Psi_{T,m}(\overline q\mid\rho^*) \xrightarrow[T\to\infty]{} \Psi_m(\overline q).
\end{equation*}
 In order for the statement to be interpretable as a large-deviation principle, the second convergence must be a Gamma limit, which holds by Proposition~\ref{prop:well-posedness of Psi}\eqref{it:Gamma-convergence}.
\end{proof}

\section{Dynamical phase transitions I: absence and occurrence}
\label{sec:convexity}

We now consider a system that consists of the graph $(\X,E)$, jump rates $k$ and total mass $m$ as explained in the previous section. In this section we study dynamical phase transitions for the general function $\Psi_m$, and study when phase transitions can~\emph{not} occur, and when they do occur. We then illustrate the conditions for a simple two-state example.

Let us first provide a more precise definition of dynamical phase transitions, following the general framework of \cite{BDSGJLL2005CurFluct,MR2227084,BDSGJLL2007LdpEmpCur,BDSGJLL2015MFT}:
\begin{definition}
We say that the system undergoes a dynamical phase transition in the average flow $\overline q\in\M(E)$ if
\begin{equation*}
  \Psi_m\left(\overline q\right) < \Psi^+_m\left(\overline q\right),
\end{equation*}
where
\begin{equation}
  \Psi^+_m\left(\overline q\right) =
  \begin{cases}
    \inf_{\rho\in \M_m(\X)}\sum_{(x,y)\in E}s\left(\overline q_{x,y}\mid k_{x,y}(\rho)\right),  & \div\overline{q}=0,\\
    \infty,                                                                                &\text{otherwise}.
  \end{cases}
\label{eq:defft}
\end{equation}
\label{def:DFT}
\end{definition}
Recall that $\Psi_m(\overline q)$ is the large-deviation cost to observe an atypical average flow $\overline q$. If the system does not undergo a dynamical phase transition for this $\overline q$, then the infimum over $\A_{T,m}$ can be taken over paths that are constant in time, which corresponds to an (atypical) stationary steady state. On the other hand, if a dynamical phase transition occurs at $\overline q$, then the system will have time-dependent behaviour when constrained to have an atypical flow $\overline q$. Furthermore, recall from~Proposition~\ref{prop:well-posedness of Psi} that when $\div\overline{q}\neq0$ then $\Psi_m(\overline q)=\infty$ and hence there can not be a dynamical phase transition.

\subsection{Dynamical phase transitions and lower and upper bounds}
\label{subsec:rule out}

We now study upper and lower bounds for $\Psi_m$, from which we deduce a sufficient condition under which dynamical phase transition can not occur, and a sufficient condition under which it does.
The upper bound $\Psi^+_m$ is already introduced in~\eqref{eq:defft}. A lower bound is constructed as follows. Considering the rate $k$ as an operator $\M_m(\X)\to \mathbb R_+^E$ with range
\begin{equation}
  \Ran_m(k):=\{(k_{x,y}(\rho))_{(x,y)\in E}\in \RR_+^E : \rho\in\M_m(\X) \},
\label{eq:Ran}
\end{equation}
one can rewrite the upper bound as
\begin{equation*}
  \Psi^+_m(\overline q)=\inf_{K\in \Ran_m(k)} \sum_{(x,y)\in E} s(\overline q_{x,y} \mid K_{x,y})\,.
\label{eq:equipsi+}
\end{equation*}
For the lower bound we introduce, denoting by $\co$ the convex hull,
\begin{equation*}
\label{eq:equipsi-}
  \Psi^-_m(\overline q):=\inf_{K\in \textrm{co}(\Ran_m(k))} \sum_{(x,y)\in E}s(\overline q_{x,y}\mid K_{x,y})\,.
\end{equation*}

\begin{proposition} For any divergence-free $\overline q$,
	\begin{equation*}
  	\Psi^-_m(\overline q)\leq \Psi_m(\overline q)\leq \Psi^+_m(\overline q)\,.
	\end{equation*}
	\label{prop:Psi bounds}
\end{proposition}

\begin{proof}
	Take an arbitrary $\rho\in\M_m(\X)$. Since $\overline q$ is divergence-free, the constant path $(\rho,\overline q)$ is an element of $\A_{T,m}(\overline q)$. Therefore
	\begin{align*}
  	\Psi_m(\overline q) \leq \lim_{T\to\infty} \mfrac{1}{T}\sum_{(x,y)\in E}\int_0^T\!s\big(\overline q_{x,y} \mid k_{x,y}(\rho)\big)\,dt = \sum_{(x,y)\in E} s\big(\overline q_{x,y} \mid k_{x,y}(\rho)\big).
	\end{align*}
	Optimising over all possible $\rho\in\M_m(\X)$ yields the upper bound.

  Now take an arbitrary path $(\rho,q)\in \mathcal A_{T,m}(\overline q)$. Applying Jensen's inequality to the jointly convex function $s(\cdot\mid\cdot)$ yields
	\begin{equation*}
	\sum_{(x,y)\in E}\frac 1T\int_0^T\!s\big(q_{x,y}(t)\mid k_{x,y}(\rho(t)\big)\,dt\geq \sum_{(x,y)\in E} s\Big(\overline q_{x,y}\mid {\textstyle \frac 1T\int_0^T\!k_{x,y}(\rho(t))\,dt} \Big)\geq \Psi^-_m(\overline q)\,,

	\end{equation*}
	because $T^{-1}\int_0^T\!k(\rho(t))\,dt \in \co(\Ran_m(k))$. The claimed lower bound is obtained by taking the infimum over all paths $(\rho(t), q(t))_{t\in\lbrack0,T\rbrack}\in \mathcal A_{T,m}(\overline q)$ and then letting $T\to\infty$.
\end{proof}

As a direct consequence we obtain the first condition to rule out phase transitions.
\begin{corollary}
If the set $\Ran_m(k)$ is convex then there are no dynamical phase transitions for any value of $\overline q$.
\label{cor:convex Ran}
\end{corollary}
\begin{proof}
  If $\Ran_m(k)$ is convex then $\Psi_m^-(\overline q)=\Psi_m(\overline q)=\Psi_m^+(\overline q)$.
\end{proof}

\begin{remark}
In this paper we are restricting to conservative models for which the total mass is preserved. The arguments are however easily extendible to non conservative models. In this case $\Ran(k)$ may be unbounded and the validity for example of condition of Corollary \ref{cor:convex Ran} are more likely to be verified.
This seems to confirm the phenomenon observed in \cite{PhysRevLett.102.250601} for which the addition of boundary conditions non preserving mass may inhibit dynamic phase transitions.
\end{remark}

The upper bound in Proposition \ref{prop:Psi bounds} can be improved, and the lower bound can be replaced by another one. To describe the lower bound, recall from~\eqref{eq:defft} that $\Psi_m^+(\overline{q})=+\infty$ if $\div\overline{q}\neq 0$. By contrast, we define another function
\begin{equation*}
  \Psi_m^\#(\overline{q}):=\inf_{\rho\in \M_m(\X)}\sum_{(x,y)\in E}s\left(\overline q_{x,y}\mid k_{x,y}(\rho)\right),
\end{equation*}
defined even in the case $\div \overline q\neq 0$, so that that $\Psi_m^\#(\overline q)=\Psi_m^+(\overline q)$ when $\div \overline q=0$. In the following lower and upper bounds ``$\co$'' now denotes the convex envelope.

\begin{proposition}\label{disturba}
For any $\overline{q}$,
\begin{equation*}
  \left(\co\Psi_m^\#\right)(\overline{q})\leq\Psi_m(\overline{q}) \leq \left(\co\Psi_m^+\right)(\overline{q})\,.
\end{equation*}
\end{proposition}
\begin{proof}
The upper bound follows by the upper bound in \ref{prop:Psi bounds} and the convexity of $\Psi_m$ (property $(ii)$ of Proposition~\ref{prop:well-posedness of Psi}).

The lower bound is obtained by the following argument. For any $T>0$ and any pair $(\rho,q)\in \A_{T,m}(\upbar q)$,
\begin{align*}
\frac 1T \mathcal I_{[0,T]}(\rho,q)&=\frac 1T\int_0^T\!\sum_{(x,y)\in E} s\big(q_{x,y}(t)\mid k_{x,y}(\rho(t))\big)\,dt\\
&\geq \frac 1T\int_0^T\!\Psi^\#_m(q(t))\,dt \geq \frac 1T\int_0^T\!\left(\co\Psi^\#_m\right)(q(t))\,dt\stackrel{\text{(Jensen)}}{\geq} \left(\co\Psi^\#_m\right)(\overline{q})\,.
\end{align*}
The result follows by minimising over $(\rho,q)\in \A_{T,m}(\upbar q)$ and taking $T\to\infty$.
\end{proof}

Together with the theory from Section~\ref{subsec:ldp time-average} we obtain a condition under which a phase transition occurs.
\begin{corollary}
If $\Psi^+_m$ is not convex, then the system undergoes a dynamical phase transition.
\label{cor:Psiplus not convex}
\end{corollary}
\begin{proof}
In this case there exists a $\overline q\in \mathcal M(E)$ such that we have for the convex envelope $(\co\Psi^+_m)(\overline{q})<\Psi^+_m(\overline{q})$. By the upper bound in Proposition~\ref{disturba} we deduce that $\Psi_m(\overline{q})<\Psi^+_m(\overline{q})$ so that we have a dynamic phase transition at $\overline q$.
\end{proof}
When $\Psi_m^+$ is not convex and $\Psi_m=\co\Psi_m^+$ the typical behaviour of the system is to spend fractions of times with different typical densities and flows (see \cite{BDSGJLL2005CurFluct,MR2227084} for an extended discussion).

\subsection{Dynamical phase transitions and joint convexity}
\label{subsec:joint convexity}

In this section we derive an alternative condition to rule out dynamical phase transitions, again by a convexity argument, but now by joint convexity of the function,
\begin{equation}
  \M_m(\X)\times\M(E) \ni (\rho,q) \mapsto \sum_{(x,y)\in E} s(q_{x,y} \mid k_{x,y}(\rho)) \in \RR_+\,.
\label{eq:sk}
\end{equation}

\begin{proposition}
If the jump rates $k_{x,y}$ are such that the function~\eqref{eq:sk} is jointly convex in $(\rho,q)\in \M_m(\X)\times\M(E)$, then there are no dynamical phase transitions.
	\label{prop:jointly convex}
\end{proposition}
\begin{proof}
  Consider a pair $(\rho,q)\in \A_{T,m}(\upbar q)$. By Jensen's inequality:
	\begin{align*}
	\frac 1T \mathcal I_{[0,T]}(\rho,q)&=\frac 1T\int_0^T\!\sum_{(x,y)\in E} s\big(q_{x,y}(t)\mid k_{x,y}(\rho(t))\big)\,dt\\
	&\geq \sum_{(x,y)\in E} s\!\left({\textstyle \frac 1T\int_0^T\! q_{x,y}(t)\,dt} \mid {\textstyle k_{x,y}\big(\frac 1T \int_0^T\!\rho(t)\,dt\big)}\right)\\
	&=:\sum_{(x,y)\in E} s\left(\overline q_{x,y}\mid k_{x,y}(\tilde \rho_T)\right)\geq \Psi^+_m(\upbar q)\,,
	\end{align*}
	where we used the definition of $\mathcal A_{T,m}(\overline q)$ and we called  $\tilde \rho_T:=\frac 1T\int_0^T\rho_s ds\in \M_m(\X)$. Taking the infimum over all $(\rho, q)\in \A_{T,m}(\overline q)$ and then letting $T\to\infty$ yields $\Psi_m(\overline q) \geq \Psi^+_m(\overline q)$, which shows that there are no dynamical phase transitions.
\end{proof}
As we will illustrate in Section~\ref{subsec:2d example} the above proposition is not very powerful, but can be strengthened by requiring joint convexity in a small neighbourhood only.

\begin{remark}
In~\cite{BodineauDerrida2005}, the authors construct a dynamical phase transition by considering small perturbations around time independent trials $(\tilde\rho, \bar q)$. We consider $(\rho_\epsilon(t), q_\epsilon(t))$ where $q_\epsilon(t)= \bar q + \epsilon v(t)$, for some periodic perturbation $v$ such that $\int_0^Tv(s)ds=0$ and $\rho_\epsilon$ is determined by the continuity equation. The first variation $\tfrac{d}{d\epsilon}\I_{(0,T)}(\rho_\epsilon, q_\epsilon)\rvert_{\epsilon=0}=0$ since $(\tilde\rho,\bar q)$ is a critical point for the time independent trials. If one can find a $v(t)$ for which the second variation becomes negative, this shows that a small periodic perturbation can decrease the cost of a constant state, which corresponds to a dynamical phase transition. For a general local cost function $\L$ (as in the introduction), the second variation is
\begin{equation*}
  \tfrac{d^2}{d\epsilon^2}\I_{(0,T)}(\rho_\epsilon, q_\epsilon)\rvert_{\epsilon=0} =
    \int_0^T\! \begin{bmatrix} m(t)\\v(t)\end{bmatrix}^\mathsf{T} \nabla^2\L(\tilde \rho, \overline q) \begin{bmatrix} m(t)\\ v(t)\end{bmatrix} \, dt.
\end{equation*}
where at first order in $\epsilon$ we have $\rho_\epsilon(t)=\tilde\rho+\epsilon m(t)$.
Therefore, the second variation cannot become negative if $\L$ is jointly convex, which is related to our Proposition~\ref{prop:jointly convex}.
\label{rem:second variation}
\end{remark}

\subsection{A simple two-state example}
\label{subsec:2d example}

We give a geometric illustration of the conditions from the previous section in the  simplest network with $|\X|=2$, i.e. $\X=\{1,2\}$. In this case $\M_m(\X)=\{(\rho_1,m-\rho_1):\rho_1\in\lbrack0,m\rbrack\}$ is one dimensional, and any choice of the jump rates can be rewritten as a zero-range process; this means that the rate at which the mass flows exiting from a given site only depends on the amount of mass present at that site. By Proposition~\ref{prop:well-posedness of Psi} we only need to consider divergence-free flows $\div \bar q=0$; this corresponds to $\bar q_{1,2}=\bar q_{2,1}$ both of which we shall denote as $\bar q$. For simplicity (and with a slight abuse of notation) we consider an homogeneous model, i.e. we assume that $k_{1,2}(\rho)=k(\rho_1)$ and $k_{2,1}(\rho)=k(m-\rho_1)$ for some function $k:\RR_+\to\RR_+$.

We first study the implication of the range condition from Corollary~\ref{cor:convex Ran} for the two-state example. The range~\eqref{eq:Ran} can now be parametrised by
\begin{equation*}
  \Ran_m(k):= \left\{\Big(\alpha, k(m-k^{-1}(\alpha))\Big) :\alpha\in\lbrack 0,k(m)\rbrack\right\}.
\end{equation*}
Clearly for a linear $k$ the function $\alpha\to k(m-k^{-1}(\alpha))$ is linear and hence its graph is a convex set, see Figure~\ref{fig:phase space}(A), which rules out dynamical phase transitions by Corollary~\ref{cor:convex Ran}. This special case corresponds to independent particles jumping on the network. If $k$ is a convex  function, then $\alpha\to k(m-k^{-1}(\alpha))$ is a convex function, in which case its graph $\Ran_m(k)$ is not a convex set, see Figure~\ref{fig:phase space}(B). However, for a small $\overline q$ the gradient of $K\mapsto\sum_{(x,y)\in E}s(\overline q_{x,y}\mid K_{x,y})$ points left/downwards, and so the minimiser $K\in\co(\Ran_m(k))$ in $\Psi^-(\overline q)$ will still lie on $\Ran(k)$, implying that $\Psi^-(\overline q)=\Psi^+(\overline q)$. So for a convex function $k$ dynamical phase transitions can only occur for flows $\overline q$ that lie above the set $\Ran_m(k)$, see Figure~\ref{fig:phase space}(B). Similarly, for a concave function $k$, dynamical phase transitions are only possible for flows that lie below the set $\Ran_m(k)$, see Figure~\ref{fig:phase space}(C).

\begin{figure}[h!]
  \centering
  \subfloat[$k$ linear]
    \begin{tikzpicture}[scale=0.75]
      \tikzstyle{every node}=[font=\scriptsize];
      \draw[->](0,0)--(0,4) node[anchor=east]{$k_{2,1}$};
      \draw[->](0,0)--(4,0) node[anchor=north]{$k_{1,2}$};
      \draw(0,3) node[anchor=east]{$k(m)$} --node[above,pos=0.2,sloped]{$\Ran_m(k)$} (3,0) node[anchor=north]{$k(m)$};
      \draw[->](0,0)--(3,3) node[anchor=south west]{$\overline q$};
    \end{tikzpicture}
  }
  \subfloat[$k$ convex]
    \begin{tikzpicture}[scale=0.75]
      \tikzstyle{every node}=[font=\scriptsize];
      \draw[->](0,0)--(0,4) node[anchor=east]{$k_{2,1}$};
      \draw[->](0,0)--(4,0) node[anchor=north]{$k_{1,2}$};
      \filldraw[lightgray](0,3) .. controls (0,1.7) and (1.7,0) .. (3,0)--(0,3);
      \draw(0,3) node[anchor=east]{$k(m)$} .. controls (0,1.7) and (1.7,0) .. (3,0) node[below=-1,pos=0.3,sloped]{$\Ran_m(k)$} node[anchor=north]{$k(m)$};
      \draw[dotted](0,3)--(3,0);
      \draw(0,0)--(1.02,1.02);
      \draw[double](1.02,1.02)--(2.97,2.97);\draw[->](2.99,2.99)--(3,3) node[anchor=south west]{$\overline q$};
    \end{tikzpicture}
  }
  \subfloat[$k$ concave]
    \begin{tikzpicture}[scale=0.75]
      \tikzstyle{every node}=[font=\scriptsize];
      \draw[->](0,0)--(0,4) node[anchor=east]{$k_{2,1}$};
      \draw[->](0,0)--(4,0) node[anchor=north]{$k_{1,2}$};

      \filldraw[lightgray](0,3) .. controls (1.3,3) and (3,1.3) .. (3,0)--(0,3);
      \draw(0,3) node[anchor=east]{$k(m)$} .. controls (1.3,3) and (3,1.3) .. (3,0) node[above,pos=0.3,sloped]{$\Ran_m(k)$} node[anchor=north]{$k(m)$};
      \draw[dotted](0,3)--(3,0);
      \draw[double](0,0) --(1.98,1.98);
      \draw[->](1.98,1.98)--(3,3) node[anchor=south west]{$\overline q$};
    \end{tikzpicture}
  }
  \caption{Three possible phase space plots. The grey areas depict the convex hull $\co(\Ran_m(k))$, and double stroked lines depict divergence-free flows $\overline q$ for which dynamical phase transition are not ruled out by Corollary~\ref{cor:convex Ran}.}
  \label{fig:phase space}
\end{figure}

\begin{remark} From Figure~\ref{fig:phase space} one sees that by changing the total mass $m$ in the system the set $\Ran_m(k)$ can be moved upward or downward, which influences the region of possible dynamical phase transitions. The flexibility to play with $m$ will become more apparant and crucial in the next section where we construct an example of a dynamical phase transition.
\end{remark}

We now study what the joint convexity condition of Proposition~\ref{prop:jointly convex} implies for the two-state homogeneous example. In that case joint convexity of the sum~\eqref{eq:sk} is implied by the joint convexity of the function $\RR^2\ni(\rho,q) \mapsto s(q\mid k(\rho))$, which corresponds to positive semidefiniteness of its Hessian matrix:
\begin{equation*}
	\begin{bmatrix}
	- q(\log k)''(\rho) + k''(\rho)  & -(\log k)'(\rho) \\
	-(\log k)'(\rho)            & 1/q
\end{bmatrix}.
\end{equation*}
By Sylvester's criterion, this matrix is positive semidefinite if and only if all its principal minors are non-negative, i.e.:
$$
\left\{
\begin{array}{l}
- q (\log k)''(\rho) + k''(\rho)\geq 0,\\
1/q \geq 0 ,\\
- (\log k)''(\rho) + k''(\rho)/q - \left((\log k)'(\rho)\right)^2 \geq 0\,.
\end{array}
\right.$$
The second inequality is trivial since $q\geq 0$, and the first inequality is contained in the third. Proposition~\ref{prop:jointly convex} thus means that dynamical phase transitions do not occur when the third inequality holds, that is:
\begin{equation}
	k''(\rho)\left(\frac1q- \frac1{k(\rho)}\right) \geq 0.
\label{eq:jointly convex condition}
\end{equation}
Actually this inequality holds for \emph{all} $\rho,q$ if and only if $k''(\rho)\geq0$ and $-k''(\rho)/k(\rho)\geq0$, which can only be true for linear functions $k$. In this sense Proposition~\ref{prop:jointly convex} is just ruling out the independent particles case. However, the same argument can be used to rule out phase transitions obtained by small oscillatory perturbations within a small neighbourhood of a critical time-independent $(\rho,\overline{q})$ (see Remark \ref{rem:second variation}). With this interpretation, we can deduce from~\eqref{eq:jointly convex condition} that for a convex $k$ dynamical phase transitions (with small oscillations) can only occur for large flows $\overline q$, and the other way around for concave $k$. Indeed, this statement is consistent with the phase space diagrams from Figure~\ref{fig:phase space}.

An exact characterisation of when a dynamical phase transition indeed occurs is computationally not easy, even in the simple two-state case. Instead of using a direct computational approach we will show the existence of a dynamical phase transition using a limiting procedure and a comparison with a continuous framework where the computations are easier.

\section{Dynamical phase transition II: an example}
\label{sec:discrete ring}

In the previous section we mostly focussed on sufficient conditions under which a dynamical phase transition can not occur; in this section we present an example where it does. The dynamics corresponds to a totally asymmetric zero range dynamics on a discrete ring with $L$ sites. The dynamics is spatially homogeneous and with strictly increasing jump rates
\begin{equation}
  k_{x,y}(\rho):=L\kappa(\rho_x)\mathds1_{x+1}(y),
\label{eq:k on ring}
\end{equation}
that we discuss in more detail in Section~\ref{subsec:discrete ring}, see also Figure~\ref{fig:directed ring}. Note that the rates $k$ on the left hand side of \eqref{eq:k on ring} depend on the size $L$ even if we are not writing explicitely such a dependence. We will fix the function $\kappa$ on the right hand side of \eqref{eq:k on ring} in such a way that  \eqref{eq:sk} is strictly concave on a region and such that there is just one single globally attractive stationary solution to equation \eqref{eq:closedrho}, which is natural in light of Remark~\ref{rem:single ss}.
We refer to \cite{MR3619540, MR3862457}
for discussions of dynamical phase transitions for asymmetric dynamics of particle systems in the hydrodynamic rescaling. We stress again that the limit considered here is very different from the hydrodynamic one.

More precisely we have the following result that allows to deduce the existence of a dynamical phase transition.

\begin{theorem} Consider a family of models on rings with $L$ sites and defined by the rates \eqref{eq:k on ring}. Assume that there exist $\overline\Phi,\tau,\delta>0$ and $\mu\geq0$ such that the function
\begin{equation}
  \phi \mapsto s\big(\phi \mid \kappa(\mu+\tau\phi)\big)
\label{eq:concavity}
\end{equation}
is strictly concave on $\lbrack \overline\Phi-\delta,\overline\Phi+\delta\rbrack$. If for average flow $\overline{q}_{x,x+1}\equiv \overline\Phi$ and any large $L>0$, the minimisation problem in the definition~\eqref{eq:defft} of $\Psi^+(\overline{q})$ has a unique minimiser, then for $L$ sufficiently large and $m:=L(\mu+\tau\overline\Phi)$ the system with rates $L\kappa$ undergoes a dynamical phase transition in $\overline{q}$.
\label{th:DPT}
\end{theorem}
The dynamical phase transition will be constructed in two stages. In Section~\ref{cont-comp} we perform an auxiliary computation on a continuous ring. We introduce a cost functional that is a continuous version of the discrete one and show that for this functional a travelling wave is less costly than a constant flow. In Section~\ref{subsec:approximation} we discretize the continuous travelling wave to show that the same result holds on the discrete ring, for $L$ sufficiently large but finite. This will conclude the proof of the result. In Section~\ref{subsec:examples} we present an example of a rate $k$ and flow $\overline{q}$ that satisfy the conditions in Theorem \ref{th:DPT} so that a dynamical phase transition takes place.

\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=0.4]
\tikzstyle{every node}=[font=\scriptsize]
\def \n {10}
\def \m {9}
\def \radius {3}
\def \margin {10}

  \node[draw, fill=lightgray,circle,inner sep=1] at ({360/\n * (1 - 1)}:\radius){1};
  \draw[->, >=latex] ({360/\n * (1 - 1)+\margin}:\radius)
    arc ({360/\n * (1 - 1)+\margin}:{360/\n * (1)-\margin}:\radius) node[midway,anchor=west]{$L\kappa(\rho_1)$};
\foreach \s in {2,...,3}
{
  \node[draw, fill=lightgray,circle,inner sep=1] at ({360/\n * (\s - 1)}:\radius){\s};
  \draw[->, >=latex] ({360/\n * (\s - 1)+\margin}:\radius)
    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}
\foreach \s in {4,...,\m}
{
  \node[draw, fill=lightgray,circle,inner sep=1] at ({360/\n * (\s - 1)}:\radius){\phantom{1}};
  \draw[->, >=latex] ({360/\n * (\s - 1)+\margin}:\radius)
    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}
  \node[draw, fill=lightgray,circle,inner sep=1] at ({360/\n * (\n - 1)}:\radius){L};
  \draw[->, >=latex] ({360/\n * (\n - 1)+\margin}:\radius)
    arc ({360/\n * (\n - 1)+\margin}:{360/\n * (\n)-\margin}:\radius);
\end{tikzpicture}
\caption{A directed discrete ring}
\label{fig:directed ring}
\end{figure}

\subsection{Totally asymmetric spatially homogeneous zero-range processes on a discrete ring.}
\label{subsec:discrete ring}

The graph now consists of a discrete ring $\X:=\{1,2,\dots ,L\}$ with directed nearest-neighbour edges $E=\{(x,x+1)\,, x=1,\dots , L\}$ where the sum is modulo $L$, see Figure~\ref{fig:directed ring}. This means that mass can flow through the directed ring in one direction only, and that the divergence-free condition from Proposition~\ref{prop:well-posedness of Psi} imposes that $\overline q$ is a constant.

We assume that the jump rates are zero-range and spatially homogeneous, i.e. \eqref{eq:k on ring} for some smooth strictly increasing function $\kappa: \mathbb R_+\to \mathbb R_+$ satisfying the assumptions mentioned in Section~\ref{subsec:particle system}. Indeed, this setting is similar to the two-state model from Section~\ref{subsec:2d example}, where the rates are now rescaled with a factor  $L$.

The limiting system of equations~\eqref{eq:closedrho} associated to this class is given by
\begin{equation}\label{idring}
  \dot\rho_x(t)=L\kappa(\rho_{x-1}(t))-L\kappa(\rho_x(t))\,, \qquad x\in \X\,.
\end{equation}
with corresponding large-deviation rate functional for the paths,
\begin{equation*}
  \I_{\lbrack0,T\rbrack}(\rho,q)=\sum_{x=1}^L \int_0^T\!s\big(q_{x,x+1}(t)\mid L\kappa(\rho_x(t))\big)\,dt.
\end{equation*}
Since $\kappa$ is strictly increasing, for any fixed $m$ there exists a unique stationary solution $\overline\rho\in \M_m(\X)$ of \eqref{idring}, given by $\overline\rho_x=\frac mL$. Similarly, if the minimiser in the Definition~\eqref{eq:defft} of $\Psi_m^+(\overline{q})$ is unique, then by symmetry,
\begin{equation}\label{psipiu}
  \Psi^+_m(\overline q)= L s\big(\overline q \mid L\kappa(m/L)\big)\,.
\end{equation}

As mentioned in Remark~\ref{rem:single ss}, for systems with multiple equilibria and unstable behaviours one can easily imagine the occurrence of dynamical phase transitions, especially if the limit equation~\eqref{idring} itself already has a time-dependent solution. Therefore it is important that the equilibrium $\overline\rho$ is unique and globally attractive, so that the typical behaviour of the system is to relax to this equilibrium. This follows automatically from the fact that the equation is dissipative, which we will now derive from the monotonicity of $\kappa$.

The invariant measure of the zero range model is known to be:
  $$\nu^N\left(\rho^N\right)=\prod_{x=1}^L\frac{1}{Z\widehat{L \kappa}(\rho^N_x)}\,,
  $$
when $\rho^N\in\M_m(\X)\cap (\tfrac{m}{N}\NN)^L$. Here $\widehat{L\kappa}\left(\frac {m i}{N}\right):=\prod_{j=1}^i L\kappa\left(\frac{m j}{N}\right)$ for $i\geq 1$ and $\widehat{L\kappa} (0):=1$, and $Z$ is a normalisation factor, see~\cite[Sec.~2.3]{Kipnis1999}. A direct computation shows that when distributed according to $\nu_N$ and conditioned on the total mass $\sum_x\rho^N_x=m$, then $\rho^N$ satisfies a large-deviation principle with speed $N$ in $\M_m(\X)$ with rate functional:
\begin{equation}\label{clacdir}
  V_m(\rho)=\frac 1m \sum_{x=1}^L\int_{\overline\rho}^{\rho_x}\!\log \kappa(\alpha)\,d\alpha\,,
\end{equation}
where the equilibrium $\overline\rho$ now serves as a normalisation constant such that $V_m(\overline\rho)=0$. Note that due to monotonicity of $\kappa$, the function $V_m$ is convex and has unique minimiser $\overline\rho$. By macroscopic fluctuation techniques~\cite[Prop.~5.3]{RengerZimmer2019TR} it can be proved that the functional $V_m$ decreases along solutions $\rho(t)$ of~\eqref{idring}, which can also be seen from the direct computation:
\begin{multline*}
  \frac{d V_m(\rho(t))}{dt}=\sum_{x\in L} \frac Lm\Big[\kappa(\rho_{x-i}(t))-\kappa(\rho_x(t))\Big]\log \kappa(\rho_x(t))\\=-\frac Lm\sum_{x\in L} s\Big(\kappa(\rho_x(t))\Big|\kappa(\rho_{x+1}(t)\Big)\leq 0\,.
\end{multline*}
This shows that $V_m$ is indeed a Lyapunov functional, from which we deduce the asymptotic stability, that is, the limit equation~\eqref{idring} is dissipative and so $\overline{\rho}$ is globally attractive.

\begin{remark}
For models for which the invariant measure cannot be computed explicitly, \eqref{clacdir} and similar expressions can often still be derived from a Hamilton-Jacobi equation along the lines of a discrete version of Macroscopic Fluctuation Theory.
\end{remark}

\subsection{An inequality on the continuous torus}
\label{cont-comp}

Here we introduce an auxiliary functional on smooth density and flows that is related to the rate functional for a large ring under suitable conditions.
It may correspond heuristically to a model on a \emph{continuous} ring where mass can flow anticlockwise only. Although it is not obvious whether this framework actually corresponds to a large-deviation principle (flows may develop shocks), the computation will be useful to prove and to understand the origin of the dynamical phase transition. Indeed the scaling limit of the discrete functionals is an interesting problem. Quantities on the continuous ring will be denoted by a hat (\;$\hat{\,}$\;).

On the continuous torus, $[0,1]$ with periodic boundary conditions, we consider smooth non-negative flows and densities $\hat q(x,t),\hat\rho(x,t)\geq 0$ such that
the continuous version of the continuity equation is satisfied:
\begin{equation}\label{contc}
  \partial_t\hat \rho(x,t)+\partial_x \hat q(x,t)=0\,, \qquad x\in [0,1]\,.
\end{equation}
For any such pair the corresponding continuous version of the rate functional is given by
\begin{equation}\label{func}
  \hat\I_{[0,T]}(\hat q,\hat \rho):=\int_0^T\!\int_0^1\! s\big(\hat q(x,t) \mid \kappa(\hat\rho(x,t)\big)\,dx\,dt.
\end{equation}
Note that in this expression appears $\kappa$ and this functional will be approximated by discrete models with a large number $L$ of sites and having rates $L\kappa$.

We first construct a specific travelling wave solution of equation~\eqref{contc}, under the assumption of Theorem~\ref{th:DPT} with given $\overline\Phi,\tau,\delta>0$ and $\mu\geq0$. Let $\Phi:[0,1]\to\lbrack\overline\Phi-\delta,\overline\Phi+\delta\rbrack$ be a smooth and periodic non-constant function with average $\int_0^1\!\Phi(x)\,dx=\overline\Phi$, and define the travelling wave ($\TW$) as:
\begin{align}
  \hat \rho^\TW(x,t):=\mu + \tau\Phi(x-t/\tau).
    &&\text{and}&&
  \hat q^\TW(x,t):=\Phi(x-t/\tau)
\label{eq:cont travelling wave}
\end{align}
Then clearly, this pair $(\hat q^\TW,\hat \rho^\TW)$ solves \eqref{contc}, it is $\tau$-periodic in $t$ and $1$-periodic in $x$. Without loss of generality we may assume that $T$ is a multiple of $\tau$, so that the average flow is
$$
  \frac 1T \int_0^T\!\hat q^\TW(x,t)\,dt=\int_0^1 \Phi(x) dx:=\overline \Phi,
$$
and the corresponding total mass is given by
$$
  m:=\int_0^1\!\hat\rho^\TW(x,t)\,dx=\mu+\tau\overline\Phi\,.
$$
We may therefore write $(\hat\rho^\TW,\hat q^\TW)\in\M_m([0,1])\times\M([0,1])$. The functional~\eqref{func} for this special pair is
$$
  \hat\I_{[0,T]}(\hat\rho^\TW,\hat q^\TW)=\int_0^T\!\int_0^1\!s\big(\Phi(x-t/\tau) \mid \kappa( \mu + \tau\Phi(x-t/\tau))\big)\,dx\,dt.
$$
Since the function $\Phi$ is periodic of period one, exchanging the order of integrations, the integrand of the integral on $dx$ does not depend on $x$ and the integral can be removed computing the integrand on $0$. With the change of variable $u=-t/\tau$ and again assuming that $T$ is a multiple of $\tau$,
\begin{equation*}
  \frac1T\hat\I_{[0,T]}(\hat \rho^\TW,\hat q^\TW)=\int_0^1\!s\big(\Phi(u) \mid  \kappa( \mu + \tau\Phi(u))\big)\,du.
\end{equation*}

Next, let us compare the travelling wave to its corresponding constant profile
\begin{align}
  \hat \rho^\const(x,t):=\mu + \tau\overline\Phi.
    &&\text{and}&&
  \hat q^\const(x,t):=\overline\Phi
\label{eq:cont const}
\end{align}
Then clearly $\int_0^1\!\hat\rho^\const(x,t)\,dx=m=\int_0^1\!\hat\rho^\TW(x,t)\,dx$ and $\tfrac1T\int_0^T\!\hat q^\const(x,t)\,dt=\overline\Phi=\tfrac1T\int_0^1\!\hat q^\TW(x,t)\,dt$. By the strict concavity of~\eqref{eq:concavity} and the fact that $\Phi$ is not constant, Jensen's inequality yields that
\begin{multline}
  \frac1T\hat\I_{[0,T]}(\hat \rho^\TW,\hat q^\TW) =\int_0^1\!s\big(\Phi(u) \mid  \kappa( \mu + \tau\Phi(u))\big)\,du \\
    < \!s\big(\overline \Phi \mid  \kappa( \mu + \tau\overline\Phi)\big) = \frac1T\hat\I_{[0,T]}(\hat\rho^\const,\hat q^\const),
\label{eq:cont DPT}
\end{multline}
which can be heuristically interpreted as a dynamical phase transition on the continuous torus. We will use this strict inequality to prove the existence of a dynamical phase transition for a finite ring.

\subsection{Discretisation of density-flow pairs}
\label{subsec:approximation}

The construction of the dynamical phase transition on the finite ring is obtained by discretising any smooth density and flow pair $(\hat\rho,\hat q)\in\M_m([0,1])\times\M([0,1])$ as follows. We embed the ring on the continuous torus associating the vertex $x\in \left\{1,\dots ,L\right\}$ to the point $x/L\in [0.1]$. We consider discretized masses and flows
\begin{align*}
  \rho_x^L(t):=L\int_{\frac{x-1/2}L}^{\frac{x+1/2}L}\hat\rho(y,t)\,dy
    &&\text{and}&&
  q_{x,x+1}^L(t):= L\hat q\big(\tfrac{x+1/2}L,t\big).
\end{align*}
It is easy to see that if $(\hat\rho,\hat q)$ satisfies the continuous continuity equation~\eqref{contc} then $(\rho_x^L,q_x^L)$ satisfies the discrete continuity equation~\eqref{eq:discr cont eq}.

We apply this discretisation to the two different profiles from the previous section. For the travelling wave $(\hat\rho^\TW,\hat q^\TW)$ from~\eqref{eq:cont travelling wave} with discretisation $(\rho^{L,\TW},q^{L,\TW})$. Recalling that the rates are $L\kappa$, we compute,
\begin{align}
  \frac1T\I_{[0,T]}(\rho^{L,\TW}, q^{L,\TW}) &=
\sum_{x=1}^L\frac LT\int_0^T\! s\!\left( \textstyle \hat q^\TW\big(\tfrac{x+1/2}L,t) \,\Big\vert\, \kappa\Big( L \int_{\frac{x-1/2}L}^{\frac{x+1/2}L}\!\hat\rho^\TW(y,t)\,dy\Big)\right)dt \notag\\
&=
L^2 \int_0^1\! s\!\left(\textstyle\Phi(u) \,\Big\vert\, \kappa\Big(\mu + \tau L \int_{u-1/L}^u\!\Phi(z)\,dz\Big)\right)\,du.
\label{euna}
\end{align}
For the constant profile $(\hat\rho^\const,\hat q^\const)$ from~\eqref{eq:cont const} with discretisation~$(\rho^{L,\const},q^{L,\const})$,
\begin{equation}
  \frac1T\I_{[0,T]}(\rho^{L,\const}, q^{L,\const}) = L^2 s(\overline\Phi \mid \kappa\big(\mu+\tau\overline\Phi)\big) \stackrel{\eqref{psipiu}}{=} \Psi^+_{L(\mu+\tau\overline\Phi)}(L\overline\Phi).
\label{edue}
\end{equation}
Observe that both discretised paths $(\rho^{L,\TW},q^{L,\TW}),(\rho^{L,\const},q^{L,\const})$ are feasible in the sense that they lie in the set $\A_{T,L(\mu+\tau\overline\Phi)}(L\overline\Phi)$ from~\eqref{eq:defA}.

Without loss of generality we may assume that $\delta$ is sufficiently small so that $\Phi\geq\overline\Phi-\delta>0$ and so under the assumptions on $\kappa$, the function $\kappa\big(\mu + \tau L \int_{u-1/L}^u\!\Phi(z)\,dz\big)$ is bounded from above and from below away from zero. Therefore from formulas~\eqref{euna} and \eqref{edue}
\begin{align*}
  &\lim_{L\to +\infty}\frac{1}{L^2T}\I_{[0,T]}(\rho^{L,\TW}, q^{L,\TW})=\int_0^1\!s\big(\Phi(u)\mid \kappa(\mu +\tau\Phi(u))\big)\,du, \quad\text{and}\\
  &\lim_{L\to +\infty}\frac{1}{L^2T}\I_{[0,T]}(\rho^{L,\const}, q^{L,\const})= s(\overline \Phi \mid \kappa(\mu+\tau\overline\Phi))\,.
\end{align*}
From the strict inequality \eqref{eq:cont DPT} we thus find that for sufficiently large but finite $L$, \eqref{euna} is strictly less than \eqref{edue}. This proves the claim of Theorem~\ref{th:DPT} that for this $L$ and $m:=L(\mu+\tau\overline\Phi)$ the system undergoes a dynamical phase transition for the average flow $\overline\Phi$.

\subsection{An example}
\label{subsec:examples}

For this example we take the classical Young function, see Figure~\ref{fig:Hplot}(A),
\begin{align*}
  \kappa(\rho)&:=(\rho+1)\log(\rho+1)-\rho,
\end{align*}
and $\mu=0$ and $\tau=1$.

Given $\overline q$, the second condition in Theorem~\ref{th:DPT} is easily checked by rewriting,
\begin{align}
  L s\big(\overline q\mid L\kappa(\rho)\big) = L \phi\log\mfrac{\overline q}{L} -L\overline q - L\overline q\log \kappa(\rho) + L^2 \kappa(\rho).
\label{eq:s expand}
\end{align}
Since $\kappa$ is strictly convex and strictly log-concave, this function is clearly strictly convex, which shows the uniqueness of the minimiser in the Definition of $\Psi^+(\overline q)$. Note that this strict convexity is related but a bit different from the global attractiveness of the limit equation that we derived in Subsection~\ref{subsec:discrete ring}, since in this case we force a possible non-typical flow $\overline{q}$ on the system.

To show the strict concavity of~\eqref{eq:concavity} we calculate the second derivative:
\begin{align}
  H(\phi)&:=\mfrac{d^2}{d\phi^2} s\big(\phi\mid \kappa(\phi)\big)\notag\\
    &= \frac{1}{\phi} + \frac{1}{\phi+1} - 2\frac{\log(\phi+1)}{(\phi+1)\log(\phi+1)-\phi}\notag\\
    &\qquad - \frac{\phi}{(\phi+1)\big((\phi+1)\log(\phi+1)-\phi\big)} + \frac{\phi(\log(\phi+1))^2}{\big((\phi+1)\log(\phi+1)-\phi\big)^2}
\label{eq:H}
\end{align}
It is easily computed that $H(1)<0$, and in fact $\lim_{\phi\downarrow0}H(\phi)=-\infty$, so we may choose any $\overline\Phi\leq 1$ and $\delta>0$ sufficiently small, see Figure~\ref{fig:Hplot}. It  follows from Theorem~\ref{th:DPT} that for any $\overline{q}\equiv\overline{\Phi}\leq 1$, there exists a sufficiently large $L$ so that a dynamical phase transition occurs
with mass $m=L\overline \Phi$ and rates $L\kappa$.

\begin{figure}[h!]
\captionsetup{margin=2pt}
\centering
  \subfloat[The Young function]
    \begin{tikzpicture}[scale=0.75]
      \tikzstyle{every node}=[font=\scriptsize];
      \draw[->](0,0)--(4,0) node[anchor=south]{$\rho$};
      \draw[->](0,0)--(0,4);
      \draw(0,0) .. controls (0.5,0) and (2,0.5)..(4,4) node[anchor=east]{$\kappa(\rho)$};
    \end{tikzpicture}
    \label{subfig:Young}
  }\qquad\qquad\qquad
  \subfloat[$H(\phi)$]
    \begin{tikzpicture}[scale=0.75]
      \tikzstyle{every node}=[font=\scriptsize];
      \draw[->](0,0) --(4,0) node[anchor=north]{$\phi$};
      \draw[->](0,0.2)--(0,-3.8);
      \draw(1,0.-0.07)--(1,0.07) node[anchor=south]{$1$};
      \draw(0.2,-3.8) .. controls (0.3,0.2) and (0.5,0.2).. (4,0.1);
      \draw[->](2,-1.8)--(1.2,-0.3);
      \draw[->](2,-1.9) node[anchor=west,align=left]{Dynamical\\ phase\\ transition}--(0.6,-1.3);
      \draw[->](2,-2)--(0.4,-3);
    \end{tikzpicture}
    \label{subfig:H}
  }
\caption{On the left, the Young function $\kappa(\rho)$, and on the right the second derivative of $s\big(\phi\mid \kappa(\phi)\big)$.}
\label{fig:Hplot}
\end{figure}

We mention that the choice of $\kappa$ for which a dynamical phase transition can occur is subtle. For example, a similar calculation as~\eqref{eq:H} can be done for the concave function
$\kappa(\rho):= 2\log(\rho + 1)$, showing that $\lim_{\phi\downarrow0}H(\phi)=-1$. However, this function $\kappa(\rho)$ is strictly log-concave but not convex, and so \eqref{eq:s expand} ceases to be strictly convex for large $L$. Therefore, the minimiser in the definition of $\Psi_m^+(\overline{q})$ may no longer be unique, so possibly $\Psi_m^+(\overline{q})<Ls(\overline q\mid Lk(m/l))$, which breaks down the argument.

\section{Discussion}

We studied dynamical phase transitions on a finite graph, where the graph is fixed, and the number of particles and the time interval are simultaneously sent to infinity. For these models the action functional~\eqref{eq:LDP N} is entropic rather than quadratic. Therefore the dynamical phase transitions behave differently than previously studied models in the literature. We derived two different convexity conditions to rule out dynamical phase transitions: Corollary~\ref{cor:convex Ran} and Proposition~\ref{prop:jointly convex}. In particular the arguments that are worked out for the two-state example in Figure~\ref{fig:phase space} can be used to narrow down the search for possible dynamical phase transitions. As a byproduct of the bounds of Proposition~\ref{disturba} we also obtain a sufficient condition for a dynamical phase transition to occur, see Corollary~\ref{cor:Psiplus not convex}. Finally, we constructed an example of a dynamical phase transition for a specific zero-range process on a sufficiently large ring. It remains an open question whether dynamical phase transitions can occur on a graph with only two or three states. For smaller graphs, even in the case of two or three nodes, the exact structure of the rate function for the long-time averaged flow is more difficult to understand.

redpajama_set_name: RedPajamaArXiv


While February is known as the month of amor, churches mobilized their congregations late April to love on communities across Dallas.
Park Cities Baptist Church held its Serve Dallas Day on April 21 by providing hands-on help to the church's ministry partners.
"Serve Dallas is an opportunity for us as a church to enter into our city and just love our neighbors," said the Rev. Jeff Warren, PCBC pastor.
Church members cleared out an abandoned home to make church offices for Cornerstone Baptist Church; exercised with and then fed their neighbors with Brother Bill's Helping Hand; and helped Jack Lowe Sr. Elementary school with a carnival.
In partnership with Dallas Leadership Foundation, churchgoers at Highland Park Presbyterian Church gathered late April to plant native species and clean up trash along the Trinity River — some cleaning up the water from their very own canoes.


Moodle in English: Survey questions?
That's because the survey questions are pre-built. The type of survey selected determines the questions that are assigned automatically. As far as I know.
I believe Martin intends to add other survey types at a later date.


Calvin Klein Junior CK Arm Print Hooded Top in Grey. Casual everyday wear with a high fashion luxe and comfortable cotton construction. A lightweight weave fabric and zip up front are accented with a hood. and tonal print work throughout.


Grip, rip and chip in classic style with the Prime Time Pants, a purist's favorite fit that meets modern performance expectations.
Grip, rip and chip in classic style with the Prime Time Pants, a purist's favorite fit that meets modern performance expectations. Featuring all–over plaid and stretch fabric that manages moisture, the Prime Time Pants show your old–school sensibilities while keeping focus on your game.


Artest was ejected with 24 seconds left for this clothesline on J.J. Barea. Barkley thinks it's suspension-worthy; we're confident it's not, if only because LA is a bigger TV market than Dallas for any conference final.
One thing's for certain: Barea should have known it was coming. With the game decided and a frustrated Lakers team looking at an 0-2 hole, Mavs coach Rick Carlisle warned his players to be on the lookout for anything "funny or crazy." If that isn't code for Ron Artest trying to take your head off, I must not know my NBA.


Again we would like to thank Sit and Sleep Interiors for helping us with our Apartment. We are very pleased with the finished outcome. It certainly looks up to date and more appealing for a holiday stay.


Premium South African wines sit comfortably at the higher end of wine lists delivering outstanding quality and perfect food wines, whether reds for meat dishes or whites for seafood. The Pinotage grape, indigenous to South Africa, is producing wines that are clean, fresh, vibrant and characterful. Bush vine Chenins have richness and texture yet are light in body to be plentiful and fun.
If asked to name wine producing regions, it's unlikely that many people would say The Lebanon. With the terrible troubles in Syria, you would be forgiven for thinking that the country's wineries had disappeared into the desert. In fact, they are thriving despite the wine growers risking life and limb to grow and harvest the grapes.
Did you know that English vineyards are now producing an average of four million bottles per year? Even the French recognise the quality of our soils with Champagne giant Taittinger recently purchasing 40 hectares in England to be planted with Chardonnay, Pinot Noir and Pinot Meunier.
Red, white… and green wines!
While we are all familiar with red, white or rosé wines, did you know there were 'green wines' too? Our wine list contains a selection of wines that have been produced organically or using practices that protect the environment or that are suitable for vegetarians or vegans.
It all started a few years ago with our Pandora Inn mugs with their fun slogan 'I'd rather be at the Pandora inn!'. They proved hugely successful and people started asking for postcards and T-shirts and… it has just gone on from there! Now you can Shop @ Pandora Inn for all sorts of gifts!
Cornwall is renowned for its cider so it is always a very popular drink in this part of the world. We stock, and sell, a big range here at the Pandora and among our selection are three ciders from Cornish Orchard – Gold, Blush and Pear.
As part of our quest to keep it local, as well as selling Cornish grown tea from Tregothnan, we also sell Cornish coffee! While the climate in England isn't suitable for growing coffee beans (unsurprisingly!) we do get our coffee from a local supplier, Cornish Coffee, who now hand roast their coffee beans in Cornwall!
Wine flights of what you fancy!
Finding just the right wine to suit your palate or to accompany your meal is a very personal thing. So how would you like to sample different wines before you take the plunge with a whole bottle? Our new 'Wine Flights' for white, red and rosé give you the chance to do just that and find out which one takes your fancy!
We are now stocking local tea – yes, that's tea actually grown in Cornwall! Tregothnan put the 'English' into English tea for the first time in history when they planted the first ever tea in the UK in 1999.


Industrial Soundtrack For The Urban Decay receives its film premiere at London's BFI cinema on 8 May, followed by a further screening at London's ICA.
Documenting the emergence of industrial music in response to the political climate of the 1970s, it includes interviews with Throbbing Gristle, Cabaret Voltaire's Stephen Mallinder, and Test Dept's Paul Jamrozy and Graham Cunningham, plus members of Hula, The Klink and In The Nursery, among others. The film goes on official theatrical release in May. Watch the trailer below. More details online here.


text:
#1 Local and Social Marketplace. Free Social Marketplace, FONOLIVE.COM.
Acronym/Abbreviation meaning of SM (Texting)
The acronym SM(Texting) means : Senior Moment
What is the acronym meaning/definition of SM ?
The abbreviation SM stands for: (Texting) Senior Moment
Acronym meaning of SM (Texting)
SM (Texting) means: Senior Moment
Astrological/numerological analysis and explanation of SM acronym
Astrological/horoscope meaning of SM. SM means: With a Life Path 5, your numbers are (5, 14/5, 23/5, 32/5). Life path 5 shows that you entered life with a progressive attitude, with the attitude and the skills to make the world a better place. The key word in your life path is freedom. In the pursuit of freedom, you are naturally versatile, adventurous, and advanced in your thinking. You are one of those people who are always trying to find answers to the many questions that life presents. 5 for positive life path is change and continuous improvement. You want to be completely unrestrained, as the number most associated with the productive use of freedom. You are one of the most compassionate people. A life path of 5 means you appreciate with certainty the way a freedom-loving and a compassionate life. Your love of freedom extends to humanity in general, and concern for one's neighbor, the freedom and well-being, perhaps especially in the mind. A great path life of 5 is US President Abraham Lincoln who declared slave Emancipation Proclamation, and ended slavery in America. As the ultimate progressive thinkers your potential in the Government give you considerable knowledge to exercise authority on others. You are a good communicator, and you know how to motivate people around you. This can be your feature that is the strongest and be the most valuable. Because of this ability, and your amazing mind, you are a really born leader able to sell his views. This ability to sell and motivate extends to any kind of physical product in all ideas or concepts that you can undertake. They hate routine work because it gets boring, and you are not very good to stay with the daily tasks that must be completed in time. On average, the number 5 personality live for today, and not too much about tomorrow. It is also important that you find a job that provides challenging tasks rather than routine and redundant tasks. They deal better with people, but the important thing is he person. With the ability to express yourself anytime, you have an innate ability to think through complex issues and to analyze it quickly even when it is something new. An adventure can dominate your life. This can take the form of physical or mental manifestation, but in both cases you will have the chance for exploration thrills and new ways. Of course, you belong to a group considered most secular and traveled. Obviously you do not ignore a good adventure. You take lot of risk, but often come out successful. If you do not have your own money at stake, you are even more open for a wide range of risks in your daily life. the conservative approach is simply not found in your nature. In romance, you dislike the limits imposed by love. You would like to be boundless. This does not necessarily mean you are unfaithful or promiscuous, but that does not mean you also make a good partner, you need to understand your style. A relationship that is governed on jealousy and narrow-mindedness will not work for you. You need a partner who understands your needs, is trustworthy, can see your trustworthiness, even if you are not always available and to be fully devoted. It is important that you mix with people who are different, and try those who are too serious and avoid demanding tasks. If you live on the negative side of life path 5, you are likely to be versatile, but suffer from a lack of direction, and there is confusion surrounding your ambition. Restless, discontent, and impulsive, you can choose to jump from one job to another without ever finding satisfaction with any. 5 in a negative way of life can be very irresponsible in tasks and decisions about the life of the home and business. The total pursuit of sensation and adventure in you may make you to completely ignore the feelings of those around you.
What is the meaning/definition of the letters in the abbreviation SM?
Meaning of SM acronym by its letters in astronumerology
SM abbreviation means:
Acronyms, abbreviations and meanings similar to SM
Popular questions of SM people search before coming here
Q: What does SM stand for? A: SM stands for "Senior Moment".
Q: How to abbreviate "Senior Moment"? A: "Senior Moment" can be abbreviated as SM.
Q: What is the meaning of SM acronym? A: The meaning of SM acronym is "Senior Moment".
Q: What is SM abbreviation? A: One of the definitions of SM is "Senior Moment".
Q: What does SM mean? A: SM as abbreviation means "Senior Moment".
Q: What is shorthand of Senior Moment? A: A common acronym of "Senior Moment" is SM.
Famous / Popular results for SM
Look up SM, S M, Sm, sm, .sm, or S. M. in Wiktionary, the free dictionary. SM or sm may refer to: Service mark symbol (℠) Spesmilo ₷, a former international
SM Town (stylized as SMTOWN) is a musical collective for the recording artists under South Korean entertainment company SM Entertainment. SM Town artists
SM Supermalls
SM Supermalls, also simply known as SM is a chain of shopping malls owned by Philippines-based SM Prime. It has 82 shopping malls in the Philippines as
.sm is the Internet country code top-level domain (ccTLD) for San Marino. The domain is occasionally used as a domain hack for sites ending in the -ism
SM Entertainment Co., Ltd. (Korean: SM엔터테인먼트) is a South Korean multinational entertainment agency established in 1995 by record executive and record producer
RIM-66 Standard
The RIM-66 Standard MR (SM-1MR/SM-2MR) is a medium-range surface-to-air missile (SAM), with a secondary role as an anti-ship missile, originally developed
Service mark symbol
The service mark symbol ℠ (the letters ⟨SM⟩ in small capitals and superscript style), is a symbol used in the United States and some other jurisdictions
List of SM Supermalls
This is a list of shopping malls under the SM Supermalls chain managed by the Philippine firm SM Prime. Most malls are in the Philippines, while some
SM Retail
companies were established: SM Retail Inc. and SM Land Inc. The SM Supermarket, SM Hypermarket, and SM Department Store (now The SM Store) retail chains were
SM Investments
SM Investments Corporation (SMIC), also known as SM Group, is a Filipino conglomerate with interests in shopping mall development and management, retail
RIM-161 Standard Missile 3
The RIM-161 Standard Missile 3 (SM-3) is a ship-based surface-to-air missile system used by the United States Navy to intercept short- and intermediate-range
Hashtag images & Search Results for SM
Twitter Search Results for SM
This browser is no longer supported. Please switch to a supported browser to continue using twitter.com. You can see a list of supported browsers in our Help Center. Help Center Terms of Service Privacy Policy Cookie Policy Imprint Ads info © 2023 Twitter, Inc.
Search results for SM
results.dogpile.com Checking if the site connection is secure results.dogpile.com needs to review the security of your connection before proceeding. Ray ID: 792f2946a89f7e95 Performance & security by Cloudflare
Acronyms, abbreviations meanings/definitions and descriptions
Meaning of EA-EOMCC
EA-EOMCC means: Electron Attachment-equation Of Motion Coupled Cluster
Meaning of CP
CP means: Cargo Program Office (KSC)
Meaning of ELINT
ELINT means: Electromagnetic Intelligence
Meaning of IMOO
IMOO means: In My Own Opinion
Meaning of JSOA
JSOA means: Joint Special Operations Area

redpajama_set_name: RedPajamaCommonCrawl


A must read for all of you Economist addicts out there. Commercial media just doesn't hold up well when compared to public organizations that do information literacy right. Do you go to the library? Or do you leaf through the Economist on the way to the tournament? The difference is between the athlete who hits the gym, or the one who jogs to the arena after relaxing all week.
You most likely have access to a good research library if you are reading this. Why don't you use it?


text:
Q: managedObjectContext seems to delete my NSUserDefault saving the managedObjectContext seems to eat the NSUserDefault
NSString *defaultSiteUrl = [[NSUserDefaults standardUserDefaults] objectForKey:kSelectedSiteUrlKey];
NSLog(@"the default site url is: %@", defaultSiteUrl);

if (![[managedObjectContext] save:&error]) {
    NSLog(@"Error saving entity: %@", [error localizedDescription]);
}

NSString *defaultSiteUrl2 = [[NSUserDefaults standardUserDefaults] objectForKey:kSelectedSiteUrlKey];
NSLog(@" the default site url is: %@", defaultSiteUrl2);

Result in the console:
2011-05-10 15:26:40.710 Moodle[44566:207] the default site url is: http://jerome.moodle.local/~jerome/Moodle_iPhone
2011-05-10 15:26:40.717 Moodle[44566:207] the default site url is: (null)

What could cause this issue?
In which way is NSUserDefaults related with ManagedObjectContext?
Note:

*

*To let you know my application: I'm developping a Moodle app. This piece code is on a screen displaying a list of courses. I use [[NSUserDefaults standardUserDefaults] synchronize] when I set the userdefault on a "site settings" screen. So I guess it's well saved

*The piece of code works (the user defaults are not deleted by the "save manageObjectContext") if I restart the simulator before going to the course listing screen straight after setting the user default.

A: You must initialize the NSUserDefaults value by [NSUserDefaults registerDefaults:appDefaults], using synchronize method for instiliaztioan could destroy you user defaults, NOTE, it's nothing to do with core data.

redpajama_set_name: RedPajamaStackExchange


text:
Renee Zellweger Looks Unrecognizable While in Character During Set Visit With BF Ant Anstead
A major transformation! Renée Zellweger was spotted on the set of her upcoming The Thing About Pam TV series, looking very different.
The Judy star, 52, was pictured behind the scenes of the project in New Orleans on Monday, October 18, as she walked alongside boyfriend Ant Anstead.
Zellweger, who stars as convicted murder Pam Hupp in the project, was photographed wearing a fat suit, facial prosthetics and a coiffed blonde bob. Her beau — whom she was first linked to in June — wore a casual outfit and a black face mask as they walked together.
The Bridget Jones' Diary star previously made headlines when set photos surfaced earlier this month, where she was spotted wearing a similar fat suit, prosthetics and hairdo.
In the NBC limited series — based on the true crime podcast of the same name — Zellweger portrays the 62-year-old Missouri native, who is currently serving a life sentence in prison for the murder of Louis Gumpenberger.
"She's notorious, not famous," the Oscar winner explained of her role during an October 1 BlumFest interview. "I'm going to play the person whose story [Dateline's Keith Morrison] told [on the podcast]. I'm going to play the person that [Morrison] sort of illuminated in your podcast and try to understand what makes a person like that tick."
She continued at the time, "I'm just going to keep my head down and do my work, try to research different personality types and try to understand whether or not that psychosis or solipsism or combination of both, whatever it might be. Either way, it's going to be interesting and there's a lot to learn."
Amid the 42-year-old former Wheeler Dealers host's set visit, he gushed over his recent travels to see the Texas native in the Big Easy.
"WOW there is something very special about New Orleans," the Celebrity IOU: Joyride host wrote via Instagram on Monday, October 18, alongside a selfie of the pair. "The late night wandering, the mind blowing history this nerd loves, the architecture, the blend of cultures, the food, the people, the smells and noises! All made [sic] better by magical company to share it with."
The England native previously gushed about exploring the Louisiana city with his current flame.
"We wandered into this little jazz room, and it was really amazing," Anstead told Entertainment Tonight earlier this month. "She's a good traveler and she's been to that city before, so it's nice. Looking forward to getting back."
The couple first went public with their romance in June, shortly after the TV presenter finalized his divorce from Christina Haack — with whom he shares 2-year-old son Hudson. The new duo met earlier this year on the set of his Discovery+ series.
The pair made their relationship Instagram official several months later in August, when Anstead shared a selfie from the set, captioning it, "It's finally here… The first episode of #CelebrityIOUjoyride is available on @discoveryplus Monday the 23rd!! And it's [with] the incredibly magical Renee Zellweger."
Scroll below to see photos of Zellweger's major transformation on set:
Celebrity Kids' Halloween Costumes of 2021: Photos
Celebrities and Their Look-Alike Kids: Goldie Hawn and Kate Hudson, Salma Hayek and Valentina, More


Blue Bird Body Company (Blue Bird) is recalling certain model year 2015 Vision non-school buses manufactured July 8, 2015, and All American non-school buses manufactured September 12, 2014. These buses, equipped with hydraulic brakes, may have an incorrectly manufactured parking brake pedal assembly which could separate and bend causing the parking brake not to engage properly.
Blue Bird Body Company (Blue Bird) is recalling certain model year 2015-2016 Vision and All American school buses manufactured July 8, 2014, to September 22, 2014. These buses, equipped with hydraulic brakes, may have an incorrectly manufactured parking brake pedal assembly which could separate and bend causing the parking brake not to engage properly.
KTM North America, Inc. (KTM) is recalling certain model year 2015 Husqvarna FE 350 S and FE 501 S motorcycles manufactured from July 22, 2014, to July 23, 2014. In the forks of the affected motorcycles, the threaded connection between the piston rod and fork cap may not have been sufficiently tightened and therefore the piston rod could become stuck within the fork leg.
Farber Specialty Vehicles (Farber) is recalling certain model year 2005-2008 Winnebago WFG38S vehicles manufactured January 1, 2008, to December 24, 2009 equipped with certain model S2005, S2010, S5005, S5010, S5505, and S5510 wheelchair lifts manufactured by Ricon Corporation. The platform side plate of the affected wheelchair lifts may crack.
Baxter International Inc. announced today it voluntarily initiated a recall in the United States of two lots of 0.9% Sodium Chloride Injection USP in 100 mL MINI-BAG PLUS Container to the hospital/user level. The recall is being initiated as a result of two complaints (one per lot) of particulate matter that was identified as a fragment of the frangible from the vial adapter.
Flat Creek Farm & Dairy of Swainsboro, GA has recalled 3 lots of cheese with lot codes 140802XAZ (Aztec Cheddar), 140702XAZ (Low Country Gouda voluntary recall) and 140725XGO (Aztec Cheddar voluntary recall) because of potential to be contaminated with Salmonella. Salmonella is an organism which can cause serious and sometimes fatal infections in young children, frail or elderly people, and others with weakened immune systems.


text:
"use strict";

function _typeof(obj) { "@babel/helpers - typeof"; if (typeof Symbol === "function" && typeof Symbol.iterator === "symbol") { _typeof = function _typeof(obj) { return typeof obj; }; } else { _typeof = function _typeof(obj) { return obj && typeof Symbol === "function" && obj.constructor === Symbol && obj !== Symbol.prototype ? "symbol" : typeof obj; }; } return _typeof(obj); }

Object.defineProperty(exports, "__esModule", {
  value: true
});
exports["default"] = void 0;

var _react = _interopRequireDefault(require("react"));

var _reactRedux = require("react-redux");

var _classnames = _interopRequireDefault(require("classnames"));

var actions = _interopRequireWildcard(require("../actions"));

var _Body = _interopRequireDefault(require("./Body"));

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redpajama_set_name: RedPajamaGithub


... right on the cliffs for many walks!!! The soft sound of mooing cows every morning and a pub just a few minutes walk away!! Bring a torch!!! Cottage itself had everything we needed... was super comfy and had easy access to all the info we could possibly want... including maps!!! Fresh flowers in every room, a welcome milk and scones... such a beautiful touch to our week!!! We had the most wonderful holiday and would defo return to this particular cottage!!!
Welcoming tray of scones, jam and cream, plus milk in the fridge! Local pub a couple of minutes walk away and Penzance just a short drive.
It's beautifully located in a small hamlet with the ocean a few hundred yards down the lane and a wonderful gastro pub a couple of hundred yards up the lane. The cottage is nicely equipped and was warm despite the December gales outside (the woodburner is a particularly good one). A comfortable relaxed place yet very close to the amenities of St Ives and Penzance.
Well equipped and extremely comfortable.
Ives and Penzance. Very comfortable stay. Owner popped by to give us her local knowledge and check that we had everything we needed. Would stay again.
A lot of human hair in the toilet, and in the corners. So many spiders web - honestly everywhere inside. It should be information that cottage is not fit for people who has more then 190 cm high !. Everything is smaller then on photo. And in the garden just stones - it was really uncomfortable for my dog.
Wir haben uns sehr wohl gefühlt. Es hat in dem Haus an nichts gefehlt. Die Lage ist perfekt. Von hier aus konnten wir vieles besichtigen und die 14 Tage reichten nicht aus. Für Wanderer/Radfahrer und Hunde ein Traum. Wunderschöne Wege an der Küste entlang. Wir waren in ganz Cornwall mit dem Auto unterwegs, aber hier in dieser Ecke von Cornwall war es am schönsten. Ein Pup (Gunard Head), gerade mal 30 m entfernt, dort konnte man vorzüglich essen (mit mehreren Preisen ausgezeichnet). Es ist nicht billig, aber 1x bis 2x sollte man dort gewesen sein. Hervorragendes Essen. Da wo das Cottage liegt, sind ca. 8 Häuser vertreten und eine Molkerei. Vorne beim Eingang auf der Bank sitzend sieht man das Meer. Die Kühe hörte man hier und da mal tagsüber, ansonsten war es sehr, sehr ruhig. In dieser Zeit konnten wir jeden Abend den Sonnenuntergang bewundern. Ein Ort weiter Richtung Zennor gibt es Pups, wo man draußen sitzen kann mit warmer Küche. Scone-Afternoon-Tea sollte man nicht verpassen. Auch hier in gegengesetzter Richtung gibt es ein Cream-Tea-Haus, wo man sehr schön draußen sitzen kann. Wir hatten in dieser Zeit nur schönes Wetter - 20-25 Grad. Insgesamt ist das Cottage sehr liebevoll und gemütlich eingerichtet. Als Begrüßung hat uns die Eigentümerin Nicki selbstgebackene Scones mit Cream und Marmelade hingestellt. Frische Blumen in jedem Zimmer. Zusammengefasst hat alles gepasst und wir hatten eine wunderschöne Zeit.
And the flowers she left in each room was very thoughtful too. Everything was great except a uncomfortable sofa and chair.
Our only small negative is the 2 steps from the kitchen to the living area, one step would have been better and safer.
Really good communication throughout. The cottage was absolutely lovely and cannot recommend the place enough. Great facilities and spotless throughout. Would certainly go back.
2 excellent pubs within walking distance. owner put fresh flowers in the rooms and left a cream tea and other goodies.
Seaward facing with stunning views across Gurnard's Head towards the Atlantic coastline, this characterful granite cottage nestles in the working farm hamlet of Treen, ¾ mile from Zennor and within six miles of sublime St Ives.
The Cornwall Coast Path is paces away and the Tinners Way one mile south.
Ten minutes walk away, accessed by scrambling across rocks, lies sheltered, sandy Treen Cove, with its towering ruined copper mine.
The end one of four old cottages within a small cluster of properties, Gurnard's Cottage has romantic charm, with its tranquil situation also attracting serious walkers, artists, photographers and writers.
Neighbouring dairy cows idle by twice daily, and a renowned pub restaurant is a short stroll up the lane.
Zennor is a tiny village snuggled in a craggy valley of moorland nooks and rocky hills, scattered with footpaths and antiquities including Zennor Quoit and nearby Chysauster Settlement.
It is home to a rural museum, working water mill, ancient pub and church with its own legendary mermaid.
Cornwall's historic mining landscape lies nearby, including the spectacular Crown Mines at Botallack.
Gurnard's Cottage marries traditional style with homely comfort. Through the stable door is a smart kitchen with bespoke cabinets, Delabole slate flooring and a butler sink.
Three shallow steps lead into the roomy, comfortable sitting/dining room, with ceiling beams, exposed granite walls and a superb fireplace with wood burner.
A door leads into the sunroom, opening into the gravelled courtyard patio - a tranquil spot for alfresco dining.
The pretty main bedroom is comfy and cosy with exposed stone wall.
The open garden at the front has a convenient bench for a sun-downer drink and a tap for muddy paws or sandy toes. This wonderful area, adored by the St Ives painters, was famously frequented by writers D.
It later features in the novels of Rosamund Pilcher.
There are super beaches at Gwithian, Praa Sands and Sennen and a breathtakingly scenic coastal drive to the famed St Ives.
Here, discover swimming and surfing beaches, the picturesque harbour and arc of restaurants, galleries and traditional inns peppering the charismatic streets.
Over two floors. Two bedrooms: 1 x double, 1 x single. Ground floor Bathroom, bath with shower over basin and WC. Ground floor kitchen. Utility area. Sitting room with dining area and wood burner. Sun room.
Note: Stairs are steeper and narrower than average.


State Farm provides insurance and bank products to individuals and small businesses. We will be giving away basketballs and allowing people to sign up to receive a quote on their insurance at NW Kids Fair. Everyone who provides information for a quote will be entered to win a pair of Seahawks tickets with no purchase necessary!


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$reply		= array(
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Despite all the critical acclaim and commercial success, I've never been a huge fan of path-drawing games. As with pretty much anything there have certainly been exceptions, but it's never been a particularly exciting thing for me. Though I have to admit, Slashy Hero does a pretty good job of making it more interesting.
Slashy Hero is basically a not-quite-roguelike dungeon crawler where you have to trace a path along the screen in order to attack. Defeated enemies drop candy (because Halloween, I guess) and the occasional piece of equipment, with candy being used back at your base camp to permanently upgrade attack power, speed, or health. Equipment behaves similarly, except it can be swapped out on the fly and will affect the physical look of your character. That, and finding doubles of gear you already have will automatically upgrade it.
There's a tiny bit of a story here about an evil house that's stolen all the candy and the hero setting out to get it all back, but that's really about all there is to know. It's a backdrop for lots of loot and candy gathering. Though I do have to say the environments and enemies definitely look the part for the obviously Halloweeny theme. Even the loot is basically Halloween costumes like Frankenstein's Monster, a teddy bear, ghosts, and so on.
It's a pity that the animations are so simple, though. The various characters, rooms, and items have a nice look to them but nothing really moves. I mean… it moves — but it's more like stretching and hovering than actual walking or attacking. Although to be fair the hero at least spins his sword around after an attack. That's something, right?
Getting into the rhythm of cutting swaths through hordes of enemies while hoovering up tons of candy can certainly be fun. It takes some getting used to before the whole "tap and drag outside the ring to move, tap and drag from inside the ring to slash" thing starts to feel natural, though. And there are quite a few times where a slash will fail to kill (or even damage) an enemy, resulting in your character taking unavoidable damage. Still it works pretty well. And seeing massive piles of candy getting pulled towards your character is an almost Pavlovian joy.
I only have one major issue with Slashy Hero, and its a strange one. Exit portals have a really annoying habit of popping up in the worst possible spots. Almost without fail one will appear right in front of the hero while I'm in the process of wandering around picking up all the remaining candy. I've lost count of how many times I've inadvertently exited a room because of this. I wouldn't call it a game-breaking issue, but it can be super-irritating in a game that encourages searching for secrets and stockpiling resources.
Slashy Hero is fun – it's just also a little irritating at times. But it does make for a good dungeon-crawling (ish) romp and makes good use of the kind of swiping mechanics you don't see often outside of games involving air traffic control or harbor management.


text:
Q: PayPal OAuth 2.0 set up I refereed below video to learn how to integrate paypal with springboot.
https://www.youtube.com/watch?v=GykDyG0ELms
According to the video no need to set up OAuth 2.0 in our development environment (Spring Boot).
but in PayPal documentation is it mentioned that "Before you can integrate a PayPal product or solution, you must set up your development environment to get OAuth 2.0 client ID and secret credentials for the sandbox and live environments."
https://developer.paypal.com/docs/api/overview/
I want to clarify whether I want to set up OAuth 2.0 or not in order to smoothly using PayPal.

A: To use any of PayPal's REST APIs you need the client:secret credentials to do an oauth2 request to obtain an access_token.  Then, you use this access_token in other API calls.
How to do this is documented at the page you linked: https://developer.paypal.com/docs/api/overview/#get-an-access-token

As far as integrating with springboot, I am not familiar, but you won't need to set up or install your own OAuth 2.0 service if that's what your question is asking.  Connecting to PayPal's REST APIs simply requires a connection to PayPal's /oauth2 endpoint to obtain the aforementioned access_token .  If you are using an SDK or other pre-integrated code path, this might already be implemented for you -- but if you are doing your own HTTPS calls then you need to implement the HTTPS call to /oauth2 yourself.


text:
"If a State Party wishes to significantly modify the boundary of a property already on the World Heritage List, the State Party shall submit this proposal as if it were a new nomination (including the requirement to be previously included on the Tentative List – see paragraph 63 and 65). This re-nomination shall be presented by 1 February and will be evaluated in the full year and a half cycle of evaluation according to the procedures and timetable outlined in paragraph 168. This provision applies to extensions, as well as reductions."
Theme: 2.2.6.4 - Significant modifications to the boundaries
35. (i) "States Parties may propose in one single nomination several individual cultural properties, which may be in different geographical locations but which should:
- be linked because they belong to the same historic-cultural group, or
- be the subject of a single safeguarding project, or
- belong to the same type of property characteristic of the zone
Each State Party submits only the cultural properties situated on its territory (even if these properties belong to an ensemble which goes beyond its borders) but it may come to an agreement with another State Party in order to make a joint submission".
Decision of the World Heritage Committee CONF 016 VI.18-20
19. "(...)
(e) States Parties may propose in a single nomination a series of cultural properties in different geographical locations, provided that they are related because they belong : (i) to the same historico-cultural group or (ii) to the same type of property which is characteristic of the geographical zone and provided that it is the series as such and not its components taken individually, which is of outstanding universal value."
Decision: 4 COM VI.18-20
Decision of the World Heritage Committee 41 COM 8B.50
2. "[The World Heritage Committee notes] that some large complex serial transnational nominations may benefit from an agreed nomination strategy before their official submission, (…);
5. [The World Heritage Committee] emphasizes that, if and when, it takes note of a nomination strategy, this is not prejudicial and does not imply that the complex serial transnational nominations proposed would necessarily lead to an inscription on the World Heritage List."
Decision: 41 COM 8B.50