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\begin{document}

\begin{center}
{\Large {\bf A Hydrodynamic Approach to\\ \vskip 0.2 truecm Non-Equilibrium Conformal Field Theories}}
\vskip 1.5
 truecm
{\large Denis Bernard${}^{\clubsuit}$ and Benjamin Doyon${}^{\spadesuit}$}

\vspace{0.5cm}
{\small ${}^{\clubsuit}$ Laboratoire de Physique Th\'eorique de l'ENS, CNRS $\&$ Ecole Normale Sup\'erieure de Paris, France.}\\
{\small ${}^{\spadesuit}$ Department of Mathematics, King's College London, London, United Kingdom.}\\
\end{center}

{\centerline \today}

\vspace{1.0cm}
\noindent{\bf Abstract}\\
We develop a hydrodynamic approach to non-equilibrium conformal field theory. We study non-equilibrium steady states in the context of one-dimensional conformal field theory perturbed by the $T\b T$ irrelevant operator. By direct quantum computation, we show, to first order in the coupling, that a relativistic hydrodynamic emerges, which is a simple modification of one-dimensional conformal fluids. We show that it describes the steady state and its approach, and we provide the main characteristics of the steady state, which lies between two shock waves. The velocities of these shocks are modified by the perturbation and equal the sound velocities of the asymptotic baths. Pushing further this approach, we are led to conjecture that the approach to the steady state is generically controlled by the power law $t^{-1/2}$, and that the widths of the shocks increase with time according to $t^{1/3}$.
\vskip 1.5 truecm

\section{Introduction}

One of the most powerful ideas in studying the dynamics of quantum field theory is that emerging from a hydrodynamic description of local averages \cite{Landau}. Hydrodynamics allows to encode in a simple way non-equilibrium states, including states with constant flows and approaches to steady states, by concentrating only on quantities of physical relevance and without having to deal explicitly with an infinite number of degrees of freedom. In the quantum context, the passage from strongly interacting many-body quantum dynamics to classical hydrodynamics involve subtle effects that are sometimes hard to control \cite{Qhydro}.
It is therefore important to have simple models and setups where this passage can be studied with more precision.

The main purpose of this paper is to implement the hydrodynamic approach in the context of perturbed non-equilibrium one-dimensional conformal field theory (Neq-CFT), and show that it emerges from the quantum description.

Recall that Neq-CFT has been introduced to study out-of-equilibrium quantum phenomena using the extensive toolbox of conformal field theory (CFT) combined with an $S$-matrix approach \cite{BD11-12}. It aims at describing non-perturbatively, and beyond the linear response theory, the low-energy sector of gapless one-dimensional quantum systems driven far from equilibrium. The driving is obtained by unitary evolution of independently thermalized subsystems, a setup considered in other contexts before \cite{caro,rub,spo}. Exact results in one-dimensional Neq-CFT \cite{BD11-12} have been verified numerically in the Heisenberg chain \cite{Moore12}; other exact results along this line of thoughts have been obtained for free-fermionic quantum chains \cite{tas,ah,og,aschp,Luca_Viti,delucamv} and for higher-dimensional free models \cite{collura2,doyonKG}, and conjectures proposed for certain integrable systems \cite{doyonint,DeLucaVitiXXZ}. In higher-dimensional non-equilibrium CFT, a hydrodynamic approach was used within this setup in \cite{CKY,Nat-Phys}, leading to conjectured analytic formulas extending some of the results of \cite{BD11-12} and numerically verified within the Gauge/Gravity duality \cite{AY}. As explained in \cite{Nat-Phys}, the lack of integrability plays an important role for the hydrodynamic argument, and, combined with other ideas, the hydrodynamic data give the exact quantum density matrix of the steady state.

The CFT description of gapless systems is never exact but only asymptotically exact at low energy \cite{lowCFT}. The asymptotic approach to the low energy effective dynamics is controlled by the so-called irrelevant operators -- those which, by definition, do not influence the low-energy effective dynamics at equilibrium. Perturbations of CFTs by irrelevant operators are then important, especially far from equilibrium, in order to understand low-energy numerical \cite{Moore12} and experimental  \cite{Exp} results for non-equilibrium steady states and their universality \cite{conductance, Mintchev}, driven systems \cite{irrel-out} and quantum quenches \cite{Cardy-London}. In this paper we study Neq-CFT with a particular irrelevant perturbation related to effects of band curvature; as this breaks integrability, hydrodynamic ideas are expected to be fruitful in this situation.

\section{A hydrodynamic approach}

Before embarking into the hydrodynamic approach, let us recall some facts about Neq-CFTs \cite{BD11-12}. In the simplest case, one generates an out-of-equilibrium state in a CFT by first preparing two copies of this CFT on a semi-infinite half line each at different temperatures $T_l=\beta^{-1}_l$ and $T_r=\beta^{-1}_r$, and then gluing them at some initial time $t=0$ through a contact point, say the origin $x=0$. It was shown that a non-equilibrium steady state emerges on any interval $[-L,L]$ for all times $t>L/v_F$ where $v_F$ is the Fermi velocity; beyond the interval, the system remains in its thermal state. That is, two shock waves are associated to this energy flow, which are sharp within the CFT description and which propagate at the Fermi velocity, along the two branches of the light-cone. The interior region of the light-cone is in a non-equilibrium state carrying a mean energy current $J_E=\frac{c\pi}{12}(T^2_l-T_r^2)$ with $c$ the central of charge of the CFT, whereas the regions at the left and at the right of the light-cone remain at thermal equilibrium at temperature $T_l$ and $T_r$ respectively. Since there are no fixed reservoirs in this setup, in general local energy gradients decrease as time evolves, so that the emerging steady state must be controlled by ballistic transport. A similar framework for generating non-equilibrium steady states was first studied in the context of electronic transport \cite{caro,rub} and in harmonic chains \cite{spo} and originally named the partitioning approach.

Within the CFT approximation, the Hamiltonian is $H_{\rm CFT}=\int {\rm d} x\,(T(x)+\bar T(x))$ where $T(x)$ and $\bar T(x)$ are the two chiral components of the stress tensor \cite{CFT-DiF}. They have scaling dimension $2$, so that the Hamiltonian has the correct dimension $1$ \footnote{By convention, we normalize the Fermi velocity to $v_F=1$. We set $k_B=1$ and $\hbar=1$.}. The most natural irrelevant operators are bilinear in the stress tensor components, that is $T\bar T$, and $T^2$ or $\bar T^2$.  For simplicity, we shall restrict ourselves to the $T \bar T$ perturbation, associated to curvature effects in the dispersion relation of the underlying gapless microscopic model.  The perturbed Hamiltonian we shall consider is thus
\begin{equation}\label{eq:hamilton}
 H= \int {\rm d} x\, (T(x)+\bar T(x)) + g \int {\rm d} x\, T(x)\bar T(x).
 \end{equation}
At equilibrium this perturbation is irrelevant because the operator $T\bar T$ has dimension $4$ and hence $g$ scales like $a^2$ with $a$ the short distance (UV) scale of the microscopic model.

In the following sub-sections we use a hydrodynamic approach to study the effects of the $T\bar T$ perturbation on the non equilibrium steady state, on the approach to the steady regime, and on the structure of the associated shock waves, and we show by a direct perturbative computation that these hydrodynamic considerations are correct.

\subsection{Generalities: hydrodynamic description near criticality}

The hydrodynamic approach is rooted in two principles \cite{ref-book-standard}: the first consists in assuming that there is some kind of local thermalization and the second that the equations of motion follow from conservation laws.

Local thermalization assumes that at every point $x$ and time $t$, the state may be described, locally in a neighborhood $\mathfrak{N}_{x,t}$, by an effective density matrix $\rho_{\rm local}$ of the form of an equilibrium density matrix, but involving all available local conserved charges of the theory. Near criticality in non-integrable models, omitting internal symmetries as we are only considering energy transport, the only charges that may describe transport are the hamiltonian $H$ and the momentum $P$, associated to invariance under time and space translations. In non-integrable systems there are no other local conserved charges. With $h$ and $p$ the hamiltonian and momentum densities respectively, $\rho_{\rm local}$ is of the form
\begin{eqnarray} \label{eq:rholocal}
\rho_{\rm local} \propto \exp\Big( - \beta_h(x,t)\int_{\mathfrak{N}_{x,t}} \hskip -0.4 truecm {\rm d} x'\, h (x')+\beta_p(x,t)\int_{\mathfrak{N}_{x,t}}  \hskip -0.4 truecm {\rm d} x'\, p (x') \Big)
\end{eqnarray}
with $x,t$-dependent parameters (local potentials) $\beta_h$ and $\beta_p$. This implies that the expectations of local observables can be computed as in a generalized thermal state of the form $e^{-\beta_h H + \beta_p P}$ (generalized by the presence of the momentum operator $P$), but with local effective inverse temperature $\beta_h$ and momentum potential $\beta_p$ that depend on space and time.

In quantum chain models, there is of course no continuous space translation, hence no conserved momentum operator. Continuous space translation only emerges near criticality, where low-energy excitations are supported on such large scales that the underlying lattice structure is not important. Without a conserved momentum operator, the density matrix is locally thermal, hence it cannot describe energy transport\footnote{With integrability, this problem can be circumvented by the presence of appropriate non-trivial conserved charges.}. It is the presence of a conserved momentum near criticality that allows, within this hydrodynamic description, for a possibly nonzero energy current to develop at large times in the partitioning approach. This agrees with the intuition according to which only ballistic currents may develop in this approach, while the lattice structure introduces diffusion away from criticality.

The choice of the local observables that should be described by local thermalization is important. For instance, in a thermal state, averages of derivatives of local observables are zero by translation invariance in space and time. On the other hand, in a locally thermalized state the derivative of the average of an observable is generically nonzero. Hence in a locally thermalized state, the choice of putting the derivative inside the average or outside matters, and this is guided by the physical scale on which the derivative is to be taken: either at the microscopic level (inside a hydrodynamic cell, where quantities are effectively constant), or at the hydrodynamic level (there are variations from one hydrodynamic cell to the next). In the hydrodynamic description of QFT one assumes that conserved densities and currents are locally thermalized, and that hydrodynamic equations are determined by the conservation laws.

In the present situation we have two local conservation laws, those of time and space translation invariance:
\begin{eqnarray} \label{eq:conserve}
	\partial_t h + \partial_x j = 0,\quad \partial_t p + \partial_x k = 0.
\end{eqnarray}
Within the (pure) hydrodynamic approximation, expectations of the densities $h$ and $p$ and the currents $j$ and $k$ are evaluated by local thermalization: they are first evaluated in generalized thermal states, then the local potentials are made dependent on space and time. Hence they become functions of the local potentials $\beta_h(x,t)$ and $\beta_p(x,t)$. Let us denote these expectations by ${\tt h}$, ${\tt p}$ for the densities and ${\tt j}$ and ${\tt k}$ for the currents. From eq.(\ref{eq:conserve}) we get two equations
\begin{eqnarray} \label{eq:hydro}
 \partial_t {\tt h} + \partial_x {\tt j}=0,\quad \partial_t {\tt p} + \partial_x {\tt k} =0 .
 \end{eqnarray}
These are non-linear differential equations for $\beta_h$ and $\beta_p$, which are the hydrodynamic equations. Notice that we have two equations for two functions with given initial condition, so that in principle the problem is reduced to solving non-linear PDEs.

In thermal equilibrium, the energy current and momentum density are zero by time-reversal and space-reversal symmetry. Since there is only one parameter (the temperature), there is an equation of state ${\tt k} = F({\tt h})$ relating the pressure ${\tt k}$ to the energy density ${\tt h}$. In the generalized state, the equations of state may be generalized to ${\tt k} = F({\tt h},{\tt p})$, and ${\tt j} = G({\tt h},{\tt p})$. With the generalized equations of state, the hydrodynamic problem is recast into equations for local energy and momentum densities, instead of equations for $\beta_h$ and $\beta_p$

Other parametrizations of the hydrodynamic problem are possible,  and one often uses ``fluid velocities''. In some situations, it is possible to use a parametrization that explicitly separates ``dynamical'' degrees of freedom from ``thermal'' ones. Assume that there is an additional dynamical symmetry of boost: a space-time transformation operator $B$ that transforms energy into momentum, $[B,H] = P$, and that preserves the space spanned by $H$ and $P$. Two immediate examples are the relativistic and the galilean boosts, $[B,P]=H$ (with Casimir $H^2-P^2$) and $[B,P]=m{\bf 1}$ (with Casimir $H-P^2/(2m)$), respectively. The former applies to critical points with dynamical exponent $z=1$, the latter with $z=2$ (see for instance \cite{Joe-Ben} for a hydrodynamic study of non-equilibrium states in this context). Using the boost operator, we can always write $\beta_hH - \beta_p P = \beta_{\rm rest}\, e^{-\theta B} H e^{\theta B}$. The choice of $B$ is not unique, and  for every space-time point $x,t$ we have a boost $B_{x,t}$ that keeps $x,t$ invariant. The local density matrix can be written as a local boost of a local thermal state,
\begin{eqnarray} \label{eq:rholocal}
\rho_{\rm local} \propto \exp\Big( - \beta_{\rm rest}(x,t)\int_{\mathfrak{N}_{x,t}} \hskip -0.4 truecm {\rm d} x'\, e^{-\theta(x,t) B_{x,t}} h (x') e^{\theta(x,t) B_{x,t}}\Big).
\end{eqnarray}
The local boost parameter $\theta(x,t)$ is the dynamical part, and the  local rest-frame inverse temperature $\beta_{\rm rest}(x,t)$ is the thermal part. This hydrodynamic parametrization was used in \cite{Nat-Phys} in order to give a characterization of higher-dimensional non-equilibrium CFT and describe the steady state as a boosted thermal state.

It turns out that the requirement that the momentum density be equal to the energy current up to, possibly, an even derivative term, $p=j + \mu\,\partial_x^{2\ell}j$, is essentially enough to imply relativistic invariance. Further, this implies that, in a thermal state,
\begin{equation}\label{relthermo}
	T\frc{d}{dT} {\tt k} = {\tt h} + {\tt k},
\end{equation}
and that, given the thermal equation of state ${\tt k} = F({\tt h})$, all generalized thermal averages are explicitly expressed as functions of $\theta$ and ${\tt h}$, see eq.\eqref{p1} in Appendix \ref{apprel}. These facts, which we show within the general context of local many-body systems in Appendix \ref{apprel}, can be useful to simplify the description of the hydrodynamic problem.

Finally, the hydrodynamic approximation discussed above is the ``pure'' hydrodynamics. It can be made more accurate by adding {\rm derivative corrections}, including viscosity terms. These are corrections, in the expressions for the densities and currents, involving higher derivatives of the basic quantities (for instance corrections to the equations of state  involving derivatives of energy and momentum densities), and are associated with shrinking the hydrodynamic cell in order to take into account more and more of the microscopic variations.

We will make all these concepts more precise in the example of the perturbed CFT studied in the present paper.

\subsection{The exact first-order hydrodynamic description of $T\b T$ perturbed CFT}

In the present setting of perturbed conformal field theory, the hamiltonian density could be taken as $T(x) + \bar T(x) + g T(x) \bar T(x)$, and the momentum density $p(x) = T(x) - \bar T(x)$. Of course, these are defined up to total derivatives of local densities, and for our purposes it will be convenient to add the term $-gc/(24\pi)\partial_x^2(T(x)+\b T(x))$ to the above energy density. The equations of motion are then determined by the Hamiltonian (\ref{eq:hamilton}), via $\partial_t O= i[H,O]$ for any operator $O$. The basic commutation relations for the stress-energy tensor are
\[ -i[T(x),T(y)]=-(T(x)+T(y))\,\delta'(x-y) +\frac{c}{24\pi}\delta^{'''}(x-y),\]
with $c$ the central charge of the CFT \cite{CFT-DiF}, and similarly for the $\bar T$ commutation relation except for the change of $i \to -i$. The $T$ and $\bar T$ components commute. Hence, we have
\begin{eqnarray}
h(x) &=& T(x) + \bar T(x) - \frc{gc}{24\pi} (T''(x)+ \b T''(x)) + g T (x)\bar T(x)\nonumber\\ \label{eq:defhj1}
 j(x) &=& T(x)-\b T(x) + O(g^2)
\end{eqnarray}
for the energy density and current, and
\begin{eqnarray}
p(x) &=& T(x) - \bar T(x) \nonumber\\
 k(x) &=& T(x) + \bar T(x) - \frc{gc}{24\pi} (T''(x)+ \b T''(x)) + 3g T(x) \b T(x)
 + O(g^2)\label{eq:defhj2}
\end{eqnarray}
for the momentum density and current. Here and below primes denote space derivatives.

Let us briefly analyze the case $g=0$. In this case, the local density matrix is  \begin{eqnarray} \label{eq:rholocal}
\rho_{\rm local} \propto \exp\Big( - \beta(x,t)\int_{\mathfrak{N}_{x,t}} \hskip -0.4 truecm {\rm d} x'\, T (x')-\bar \beta(x,t)\int_{\mathfrak{N}_{x,t}}  \hskip -0.4 truecm {\rm d} x'\,\bar T (x') \Big)
\end{eqnarray}
with $x,t$-dependent parameters $\beta=\beta_h-\beta_p$ and $\bar \beta=\beta_h+\beta_p$, and we have $j=p=T-\b T$ and $k=h= T+\b T$. Hence the averages of $T$ and $\b T$ can be used as hydrodynamic variables, and the expectations of any local observables, say products or powers of $T$ and $\bar T$ at some point $x$ and time $t$, can be computed, within the hydrodynamic approximation, as in equilibrium CFT but with local effective ``temperatures'' $\beta(x,t)$ and $\bar \beta(x,t)$. In particular, there is chiral factorization: right-movers and left-movers are independently thermalized. At $g=0$ the CFT enjoys relativistic invariance, with boost operator $B=-i\int {\rm d} x\,xh(x)$ satisfying $[B,H] = P$ and $[B,P]=H$, and the above can be interpreted as a boosted density matrix with rest-frame inverse temperature $\beta_{\rm rest} = \sqrt{\beta\b\beta}$ and boost parameter given by $\tanh\theta = (\b\beta-\beta)/(\b\beta+\beta)$. The thermal average at temperature $\tau$ is \cite{Cardy, Affleck}
\begin{equation}\label{zeroth}
	\langle T\rangle = \langle \b T\rangle = \frc{c\pi}{12} \tau^2,
\end{equation}
and with the initial condition where left and right halves are independently thermalized at temperatures $T_l$ and $T_r$ respectively, the hydrodynamic problem gives $\beta(x,t) = T_l^{-1} \Theta(x+t)$ and $\b\beta(x,t) = T_r^{-1} \Theta(-x+t)$. As was shown in \cite{BD11-12,Nat-Phys}, this hydrodynamic description is in fact {\em exact}, and does not necessitate any derivative corrections: it is a direct consequence of chiral factorization at the level of the quantum equations of motion. This solution represents sharp shocks emanating from the contact point and propagating at the speed of light in opposite directions, between which the steady states lies.

Let us now consider the first order correction in $g$. We first use the facts that, in a generalized thermal state, there is translation invariance, $\langle T''\rangle = \langle \b T''\rangle=0$, and that to leading order in $g$ there is chiral factorization $g\langle T\b T\rangle = g\langle T \rangle \langle \b T\rangle + O(g^2)$. We can then parametrize averages using the two quantities $w(x,t) = \langle T(x,t)\rangle$ and $\b w(x,t)= \langle \b T(x,t)\rangle$ evaluated within the pure hydrodynamic approximation using $\rho_{\rm local}$, and we get
\begin{equation}\label{noder}
\begin{aligned}
& {\tt h} = w + \bar w + g\, w \bar w + O(g^2),\quad
{\tt j} = w-\b w  + O(g^2),\\
& {\tt p} = w - \bar w,\quad
{\tt k} = w + \bar w + 3g\, w \b w +O(g^2).
\end{aligned}
\end{equation}
Recall that the equations of motion are $\partial_t {\tt h} + \partial_x {\tt j}=0$ and $\partial_t {\tt p} + \partial_x {\tt k} =0$. These are two equations for the two unknowns $w$ and $\bar w$, chosen as a parametrization of the two unknown local potentials $\beta_h$ and $\beta_p$.

This description is {\em not} exact at order $g$: there are derivative corrections. Fortunately, in the present case it is possible to obtain the full, exact derivative corrections at order $g$: they are a direct consequence of the quantum equations of motion. It turns out that these derivative corrections are ``trivial'': they can be entirely absorbed into appropriate definitions of basic hydrodynamic quantities, in such a way that the  hydrodynamic equations remain unchanged. Writing the conservation laws for quantum averages using \eqref{eq:defhj1} and \eqref{eq:defhj2} without assuming local thermalization, using only chiral factorization at $g=0$, the definitions
\begin{equation} \label{eq:defw}
\begin{aligned}
 w(x,t) = \langle T(x,t)\rangle - \frc{gc}{48\pi}\lt(\langle T''(x,t)\rangle + \langle \b T''(x,t)\rangle\rt)\\
 \b w(x,t)= \langle \b T(x,t)\rangle - \frc{gc}{48\pi}\lt(\langle T''(x,t)\rangle + \langle \b T''(x,t)\rangle\rt)
 \end{aligned}
\end{equation}
give rise to the pure-hydrodynamic equations for the quantities \eqref{noder}. That is, in terms of these exact quantum averages, the conservation equations for the densities and currents \eqref{noder} are a consequence of the quantum equations of motion, hence are {\em exact} for all $x$ and $t$ to order $g$. All derivative corrections have been absorbed into the definitions \eqref{eq:defw} and the choice of the energy density in \eqref{eq:defhj1}.

The initial conditions are those reflecting that initially the left and right parts are independently thermalized at temperatures $T_l$ and $T_r$ respectively. At thermal equilibrium with temperature $\tau$, the average momentum density is zero (and averages are space-time independent), hence $w=\b w=:w_{\rm th}(\tau)$. By dimensional analysis
\begin{equation}\label{wth}
	w_{\rm th}(\tau) = \frc{c\pi}{12} \tau^2 f(g\tau^{2})
\end{equation}
for some function $f$ with $f(0)=1$.  We expect $f$ to have an expansion in positive integer powers of $g$, and the normalization of the zeroth order \eqref{zeroth} gives
\begin{equation}\label{f}
	f(u) = 1 + au + O(u^2).
\end{equation}
Hence, at initial time,
\begin{eqnarray}
	&& w=\bar w=\frac{c\pi}{12}T^2_l\lt(1+agT_l^2 + O(g^2)\rt)\quad \mbox{on the left ($x<0$)}\nonumber\\
	&& w=\bar w=\frac{c\pi}{12}T^2_r\lt(1+agT_r^2+O(g^2)\rt)\quad \mbox{on the right ($x>0$)}\label{init}
\end{eqnarray}
with an abrupt crossover in the neighborhood of the contact point (that is, close to the origin). Below we compute the constant $a$ by using relativistic invariance, giving
\begin{equation}\label{a}
	a = -\frc{c \pi}6.
\end{equation}
This fully fixes the hydrodynamic problem (with all derivative corrections) at order $g$.
\medskip

\begin{rema}
Given any solution $w(x,t)$ and $\b w(x,t)$, the quantum averages $\langle T(x,t)\rangle$ and $\langle \b T(x,t)\rangle$, obtained by solving \eqref{eq:defw}, are naively not uniquely fixed:
\[\begin{aligned}
	\langle T(x,t)\rangle &= w(x,t) + \frc{gc}{48\pi}\lt(w''(x,t)+\b w''(x,t)\rt)
	+ \omega(x,t)\\
	\langle \b T(x,t)\rangle &= \b w(x,t) + \frc{gc}{48\pi}\lt(w''(x,t)+\b w''(x,t)\rt)
	+  \omega(x,t)
	\end{aligned}
\]
where $\omega(x,t)$ is solution of the equation $\omega -\frac{gc}{24\pi}\,\omega''=O(g^2)$ whose kernel is of the form $A_+ e^{x/\sqrt{gc/(24\pi)}} + A_- e^{-x/\sqrt{gc/(24\pi)}}$. However, this kernel is non-perturbative in $g$, hence beyond the present order-$g$ calculation.
\end{rema}

\subsection{Equation of state and relativistic structure at order $g$}

The thermal equation of state relating the pressure ${\tt k}$ to the energy density ${\tt h}$ can be obtained from \eqref{noder} by using the fact that $w=\b w$ in a thermal state:
\begin{equation}\label{eos}
	{\tt k} = F({\tt h}) = {\tt h} +\frc g2{\tt h}^2 + O(g^2).
\end{equation}
The generalized equations of state can be obtained similarly,
\begin{equation}\label{geos}
	{\tt k} =  {\tt h} +\frc g2({\tt h}^2 - {\tt p}^2) + O(g^2),\quad {\tt j} = {\tt p} + O(g^2).
\end{equation}
Associated to the equilibrium equation of state \eqref{eos} is the sound velocity (at temperature $\tau$)
\begin{equation}\label{sound}
	v_{\rm s} = \sqrt{F'({\tt h})} = 1+\frc{g}2{\tt h} + O(g^2) = 1 + \frc{gc\pi}{12} \tau^2 + O(g^2)

\end{equation}
Remark from \eqref{noder} that to order $g$, we have ${\tt j} = {\tt p}$. This equality implies propagation of small disturbances of energy, momentum and pressure densities near their equilibrium values occurs through sound-like waves with this sound velocity (see for instance the arguments presented in \cite{Levitov,D_super}). We will see below that the sound velocities also play a role far from equilibrium at order $g$.

Further, observe from \eqref{eq:defhj1} and \eqref{eq:defhj2} that the equality $j = p$ holds at the operator level at order $g$. This implies that relativistic invariance emerges (see Appendix \ref{apprel} for the general theory). Indeed one can verify explicitly that the boost operator $B = -i\int {\rm d} x\,xh(x)$, with $h(x)$ the perturbed energy density \eqref{eq:defhj1}, satisfies $[B,H] = P +O(g^2)$ and $[B,P]=H$. Therefore, the averages in the generalized thermal state are expressed in terms of relativistically boosted observables in a thermal state. Using the result \eqref{p1} along with \eqref{eos}, we may express them as
\begin{eqnarray}
	{\tt h} &=& \cosh 2\theta\,{\tt h}_{\rm rest}
	+ \frc{g}4 (\cosh2\theta-1)\,{\tt h}_{\rm rest}^2 + O(g^2)\nonumber\\
	{\tt j}={\tt p} &=& \sinh2\theta\,\lt({\tt h}_{\rm rest}
	+ \frc g4\, {\tt h}_{\rm rest}^2\rt) +O(g^2)\nonumber\\
	{\tt k} &=& \cosh 2\theta\,{\tt h}_{\rm rest}
	+ \frc{g}4 (\cosh2\theta+1)\,{\tt h}_{\rm rest}^2 +O(g^2).
	\label{relg}
\end{eqnarray}
where ${\tt h}_{\rm rest}$ is the rest-frame thermal energy density ${\tt h}_{\rm rest} = {\tt h}_{\rm th}(T_{\rm rest})$, deduced from
\eqref{noder}, \eqref{wth} and \eqref{f}, given by ${\tt h}_{\rm th}(\tau) = \frc{c\pi}{6} \tau^2\lt(1+g\tau^2\lt(a + \frc{c\pi}{24}\rt) + O(g^2)\rt)$. The relation \eqref{relthermo} can be combined with the thermal equation of state \eqref{eos} and the form \eqref{hth} of the thermal energy density. A simple calculation shows that this fixes $a$ to \eqref{a}, and thus
\begin{equation}\label{hth}
	{\tt h}_{\rm th}(\tau) = \frc{c\pi}{6} \tau^2\lt(1-\frc{gc\pi}{8}\tau^2 + O(g^2)\rt).
\end{equation}
\medskip

\begin{rema} \label{rema2} Formally extending the above thermodynamic equations to large values of $|g|$ one finds that, in some respects, $g<0$ appears to be more natural. For instance, the energy density \eqref{hth} has an unphysical maximum, as a function of temperature, if $g>0$. Further, if $g>0$, the sound velocity \eqref{sound} is greater than the Lorentz speed of light (which is set to unity) associated to the (order-$g$) relativistic invariance. However, these facts do not lead to perturbative inconsistencies, and none of the perturbative calculations below are affected by the sign of $g$. The temperature at which the energy density is maximal is nonperturbative (of order $1/g$); and a Lorentz transformation from a point in the space-time region lying between the light cone and the sound cone to a point at a time $t=0$ would involve a very large boost parameter (non-perturbative in $g$), thus precluding, in perturbative considerations, the conclusion of nonzero equal-time correlations (breaking of causality). In fact, the Lieb-Robinson velocity $v_{LR}$ can be shown perturbatively to be greater than $1+Ag$ for any constant $A>0$, and thus greater than the sound velocity no matter the sign of $g$. Below we keep $g$ arbitrary, and all conclusions stay valid perturbatively in $g$ independently of its sign.
\end{rema}

\subsection{Light cone effects, shocks and steady states}

In this section, we solve the hydrodynamic problem (\ref{eq:hydro}) associated to the $T\bar T$ perturbation, using the parametrization in terms of the functions $w$ and $\b w$ in Eqs. \eqref{noder}. We decipher how the $T\bar T$ perturbation modifies the shock propagation, the structure of the shocks as well as the approach to the steady states and the steady state itself. We shall work to first order in $g$ only.

\subsubsection{Shock wave assumption}\label{sssectshocks}

The simplest analysis of the hydrodynamic problem is that which assumes that the picture of shocks emanating from the contact point, which holds exactly at $g=0$, is accurate even with $g\neq0$ at large scales. This picture was used in \cite{CKY,Nat-Phys} in order to generalize results to higher-dimensional CFT. More precisely, we make the assumption that, with exponential accuracy, averages of local observables as functions of the space-time point $(x,t)$ take steady and uniform values whenever $|x+v_lt| \gg \xi_l(t)$ and $|x-v_rt| \gg \xi_r(t)$, for some $\xi_{l,r}(t)$ bounded from above by a power law $\propto t^a$ for all $t$ with some exponent $a<1$. This represents shocks at speeds $v_l>0$ and $v_r>0$, propagating towards the left and the right respectively, with time-dependent widths $\xi_l(t)$ and $\xi_r(t)$, respectively, which grow sublinearly at large times. By the initial conditions, two the left (right) of both shocks, averages are evaluated in the initial left (right) reservoir, and the steady state lies between both shocks, described by a generalized thermal state.

It turns out that this shock assumption, along with the knowledge of the averages in the generalized thermal states, is sufficient to fix both the shock velocities and the steady-state parameters. It does not, however, give any information on the approach to the steady state and on the structure of the shocks themselves.

Consider the conservation laws \eqref{eq:conserve} integrated along a rectangular contour crossing a shock and whose diagonal goes from $(x_1,t_1)$ to $(x_2,t_2)$. That is,
\[
	\int_{x_1}^{x_2} {\rm d} x\, (h(x,t_2) - h(x,t_1))
	+ \int_{t_1}^{t_2} {\rm d} t\, (j(x_2,t)-j(x_1,t)) = 0,
\]
with $(x_1,t_1)$ and $(x_2,t_2)$ both lying inside the shock, and similarly for $p$ and $k$. These equations can be written for the left and the right shocks. We can choose the rectangle such that $|x_2-x_1|=v_{l,r}|t_2-t_1|$ with $|x_2-x_1|$ much larger than the width of the shock. Using the assumptions that the densities and the currents are asymptotically uniform and steady away from the shocks, that the shocks are of extent that grows sublinearly in time, and that the current vanishes outside the light-cone, this yields two sets of two equations, one for the left shock the other for the right shock,
\begin{equation}\label{shockseq}
	v_l({\tt h}_l - {\tt h}_{s}) = {\tt j}_{s},\quad
	v_l {\tt p}_{s} = {\tt k}_{l} - {\tt k}_{s},\quad
	v_r({\tt h}_{s}-{\tt h}_{r}) = {\tt j}_{s},\quad
	v_r {\tt p}_{s} = {\tt k}_{s} - {\tt k}_{r},
\end{equation}
where indices $l,\,r,\,s$ indicate the averages in the left reservoir, right reservoir and in the steady-state region, respectively.
Adding them allows to eliminate ${\tt h}_s$ and ${\tt k}_s$, keeping only the densities ${\tt h}_{l,r}$ and the pressures ${\tt k}_{l,r}$ evaluated in the asymptotic region at equilibrium,
\begin{equation}\label{eexx}
 (v_l+v_r)\, {\tt p}_s= {\tt k}_l-{\tt k}_r,\quad (v_l^{-1}+v_r^{-1})\, {\tt j}_s = {\tt h}_l-{\tt h}_r.
 \end{equation}
Combining them, we find the two equations
\begin{equation}\label{ee}
	({\tt h}_l-{\tt h}_s)({\tt k}_{l}-{\tt k}_{s}) =
	({\tt h}_r-{\tt h}_s)({\tt k}_r-{\tt k}_s)
	= {\tt j}_{s}{\tt p}_{s}.
\end{equation}
These are two equations for two steady-state unknowns.
The steady-state unknowns can be taken as $w_s$ and $\b w_s$ using the parametrization \eqref{noder}, the boost and rest-frame temperature $\theta$ and $T_{\rm rest}$ using the relativistic parametrization \eqref{relg}, or simply the steady-state energy density and current ${\tt h}_s$ and ${\tt j}_s$ using the generalized equations of state \eqref{geos}. A calculation up to order $g$ gives, within the latter parametrization,
\begin{eqnarray}
	{\tt h}_s &=& \frc{{\tt h}_l + {\tt h}_r}2 + \frc g8 ({\tt h}_l-{\tt h}_r)^2+ O(g^2)\nonumber\\
	{\tt j}_s &=&
	\frc{{\tt h}_l-{\tt h}_r}2 + \frc{g}8
	({\tt h}_l^2-{\tt h}_r^2) +O(g^2).\label{twoshockssol}
\end{eqnarray}
The thermal averages ${\tt h}_{l,r}$ can be evaluated in terms of the temperatures $T_{l,r}$ using \eqref{hth}, and in particular we obtain
\begin{equation}
	{\tt j}_s =
	\frc{c\pi}{12} T_l^2\lt(1-\frc{gc\pi}{12} T_l^2 \rt)
	-
	\frc{c\pi}{12} T_r^2\lt(1-\frc{gc\pi}{12} T_r^2 \rt)
	+ O(g^2).
\end{equation}
The shock velocities can now be evaluated through
\begin{equation}
	v_{l,r} =
	\lt|\frc{{\tt j}_s}{{\tt h}_{l,r} - {\tt h}_s}\rt|
	= 1+\frc{gc\pi}{12} T_{l,r}^2 + O(g^2).
\end{equation}

We make several observations.
\begin{itemize}
\item First, to this order in $g$, the steady-state current is still a difference of a function of the left-reservoir temperature minus the same function of the right-reservoir temperature, ${\tt j}_s = J(T_l) - J(T_r)$. In the unperturbed case this had important echoes on the full counting statistics, and in general this implies that the non-equilibrium current can be obtained purely from the linear-response conductivity $G(\tau) = d{\tt j}_s/dT_l\big|_{T_l=T_r=\tau} =  dJ(\tau)/d\tau$ as ${\tt j}_s = \int_{T_r}^{T_l}d\tau\,G(\tau)$ \cite{Moore12}.
\item Second, the shock velocities $v_l$ and $v_r$ are exactly the sound velocities \eqref{sound} of the left and right reservoir, respectively. That is, it is the linear sound velocities of the reservoirs that control the speed of the shocks describing the far-from-equilibrium steady state.
\item Third, parametrizing the reservoirs using $w$ (with the thermal result \eqref{wth}, \eqref{f}, \eqref{a}) we note that the current takes the simple form
\begin{equation}\label{jw}
	{\tt j}_s = v_lw_l - v_rw_r  + O(g^2).
\end{equation}
This can naturally be interpreted as the ballistic transport of energy by independent left/right movers, with chiral energy densities $w_{l,r}$ and velocities $v_{l,r}$ that are solely determined by the reservoirs' temperatures. One may understand this as a generalization of the case $g=0$, where $w$ and $\b w$ are right- and left-moving quantities at the speed 1: to order $g$, $w$ and $\b w$ may still be seen as independent right- and left-moving energy densities, but their velocity is ``dressed'' into the sound velocity by the reservoirs' state. This is also an extension of the picture that holds near equilibrium. With $T_l\approx T_r$, there is linear wave propagation with the sound velocity $v$. The natural independent right- and left-moving energy densities are the combinations of space-time dependent small variations (near the equilibrium values) $\delta\varepsilon_\pm = (\delta{\tt h}\pm v^{-1}\delta{\tt j})/2$, and the current is $v(\delta\varepsilon_+-\delta\varepsilon_-)$.
\item Fourth, near equilibrium (i.e. with $T_l\sim T_r$), linear wave propagation implies that \cite{D_super} ${\tt j}_s \sim ({\tt k}_l - {\tt k}_r)/(2v)$ where $v$ is the equilibrium sound velocity. One can indeed check that this agrees with \eqref{jw}, using $d(vw)/d\tau = (1/2v)\,d{\tt k}/d\tau$ (where $v,\,w,\,{\tt k}$ are at equilibrium with temperature $\tau$).
\item Fifth, one can verify that the inequality ${\tt j}_s > ({\tt k}_l - {\tt k}_r)/(2v_{LR})$ \cite{D_super} is verified, where $v_{LR}$ is the Lieb-Robinson velocity (see Remark \ref{rema2} concerning $v_{LR}$).
\item Finally, we quote the steady-state rest-frame temperature and boost velocity from its relativistic parametrization:
\begin{equation}
	T_{\rm rest} = \sqrt{T_lT_r}\lt(1-\frc{gc\pi}{48}(T_l-T_r)^2 + O(g^2)\rt),\quad
	\tanh\theta = \frc{T_l-T_r}{T_l+T_r}\lt(1-\frc{gc\pi}{12}T_lT_r
	+O(g^2)\rt).
\end{equation}
In particular, this means that in the steady state, the potential $\beta_h$ associated with the hamiltonian in the density matrix as in \eqref{eq:rholocal} takes the simple form
\begin{equation}
	\beta_h = \frc{\beta_l+\beta_r}2.
\end{equation}
\end{itemize}

\subsubsection{Solution to the hydrodynamic problem: appearance of shocks}

In the previous paragraph, we have made the assumption that two shocks emanate from the connection space-time point. In the present paragraph, we provide the solution to the full order-$g$ hydrodynamic problem. The main observations from the calculation below are as follows:
\begin{itemize}
\item The problem does not have a unique solution. The space of solution includes the two-shock solution above, but also contains solutions with additional ``remnant'' shocks at $x=\pm t$.
\item Nevertheless, the steady state is unique: every solution gives rise to the same state in the central region (this state is in agreement with that found from the two-shock assumption).
\end{itemize}

Let us discuss briefly the first point above. The non-uniqueness of the weak solutions to the Riemann problem (the step initial conditions) for pure hydrodynamic equations has been observed before \cite{Nat-Phys}. In standard hydrodynamic problems, it is usually possible to lift this non-uniqueness by considering the entropy current or any other inequality condition \cite{Bressan}. With the requirement that the entropy cannot decrease at the shocks (a local implementation of the $2^{\rm nd}$ law of thermodynamics), this may forbid certain shocks and lead to rarefaction waves (transition regions of extent growing linearly with $t$). Positivity of local entropy production occurs with viscosity terms, and weak solutions to pure hydrodynamics should be seen as emerging, at large scales, from viscous hydrodynamics; thus positive local entropy production at the shocks becomes a condition for selecting weak pure hydrodynamics solutions. In the present case, higher-derivative corrections are completely absorbed into redefinitions, hence at order $g$ there is no positivity condition that can fix the solution. We hope to come back to this problem in the future by analyzing higher orders in $g$.

Let us proceed with the calculation. The full order-$g$ hydrodynamic problem is given, in terms of the parametrization $w$, $\b w$ as defined in \eqref{eq:defw}, by the conservation equations \eqref{eq:hydro} with \eqref{noder}, along with the initial conditions \eqref{init}.

At $g=0$, denoting $w=w_0$, these equations simplify to $\partial_xw_0 = -\partial_t w_0$ and $\partial_x \b w_0 = \partial_t \b w_0$. With the initial conditions, the solution is immediately given by
\begin{equation}
	w_0 = \frc{c\pi}{12} \lt(T_l^2 \Theta(t-x) + T_r^2\Theta(x-t)\rt),\quad
	\b w_0 = \frc{c\pi}{12} \lt(T_l^2 \Theta(-t-x) + T_r^2\Theta(x+t)\rt).
\end{equation}
This indeed represents two sharp shock waves both at velocity 1.

Up to first order in $g$, we may solve the hydrodynamic problem by using this zeroth-order solution in order to simplify the bilinar terms proportional to $w\b w$ in \eqref{noder}. The main idea is that, at order $g$, we may replace $gw$ and $g\b w$ by $gw_0$ and $g\b w_0$ respectively. We first divide space-time (with positive times $t>0$) into three regions: the left (L) $x<t$, the center (C) $-t<x<t$ and the right (R) $x>t$. Within each of these regions, $w_0$ and $\b w_0$ are both constant, with $w_0 = \frc{c\pi}{12} T_l^2$ in the center and left, $w_0 =\frc{c\pi}{12} T_r^2$ in the right; and $\b w_0 = \frc{c\pi}{12} T_r^2$ in the center and right, $\b w_0 = \frc{c\pi}{12} T_l^2$ in the left. Hence we may write $g\partial_t(w\b w) \approx gw_0 \partial_t \b w + g\b w_0 \partial_t w$ with $w_0$ and $\b w_0$ piecewise constant, and similarly for $g\partial_x (w\b w)$. These transform the bilinear terms into linear terms with piecewise constant coefficients. The equations then become
\begin{equation}  \label{eq:linear}
\begin{aligned}
	(\partial_x +(1+g\b w_0)\partial_t)w- (\partial_x-(1+g w_0)\partial_t)\b w  &= 0 \\
	(\partial_t+(1+3g\b w_0)\partial_x) w -(\partial_t -(1+3g w_0) \partial_x) \b w  &= 0.
\end{aligned}
\end{equation}

Eqs. \eqref{eq:linear} are linear equations which we can solve, within each region, by looking for solutions of the form $e^{i(px-E_pt)}$. A straightforward calculation shows that these plane waves have dispersion relations with two branches:
\begin{equation}
	E_p = \lt\{\begin{aligned}
	&p(1+g\b w_0 + O(g^2))\\
	& -p(1+gw_0 + O(g^2)).
	\end{aligned}
	\rt.
\end{equation}
We observe that the dispersion relation, to order $g$, is still {\em linear}, and hence only changes the propagation velocities.
This means that the solutions for $w(x,t)$ and $\b w(x,t)$ have the form, up to $O(g^2)$ terms,
\begin{equation}\label{solw}
	w(x,t) = w^+(x-v^+t) + w^-(x+v^-t),\quad
	\b w(x,t) = \b w^+(x-v^+t) + \b w^-(x+v^-t)
\end{equation}
where $v^+ = 1+g\b w_0 $ and $v^- = 1+g w_0$. In the left ($L$), center ($C$) and right ($R$) regions we obtain the velocities:
\begin{equation}\label{vpm}
	v^\pm_L =v_C^-=  1 + \frc{gc\pi}{12} T_l^2,\quad
	v_R^\pm = v^+_C = 1 + \frc{gc\pi}{12} T_r^2,
\end{equation}
and within these regions we denote the left- and right-moving waves by $w^\pm_{L,C,R}$ and $\b w^\pm_{L,C,R}$. Note that the propagation velocities \eqref{vpm} are the shock velocities $v_l$ and $v_r$ obtained in paragraph \ref{sssectshocks} from assuming the presence of two shocks.

For definiteness we assume that $g>0$, so that all velocities are greater than one; a similar calculation can be done for $g<0$, leading to the same conclusions.

The functions $w^+$, $w^-$, $\b w^+$ and $\b w^-$ in \eqref{solw} are related to each other. This can be obtained by putting \eqref{solw} into any one of the two equations \eqref{eq:linear}, for instance the first one. Since the arguments of the functions are either $x-v^+t$ or $x+v^-t$ and the equation holds for all $x$ and $t$ (within a given region), this single equation leads to two relations, one for the right movers the other for the left movers. These two relations contain single-derivative terms with constant coefficients, as in \eqref{eq:linear}. Integrating, one obtains similar relations but without derivatives. The integration constants may be set to zero, which accounts for the ``gauge symmetry'' in \eqref{solw},  $w^\pm\mapsto w^\pm \pm A$ and $\b w^\pm\mapsto \b w^\pm \pm B$ for constants $A$ and $B$.
After these operations, we find that the relations are
\begin{equation}\label{relation}
	(1 \mp v^+v^\pm) w^\pm = (1\pm v^+v^\mp) \b w^\pm + O(g^2).
\end{equation}
Since the velocities are of the form $v^\pm = 1+O(g)$, we see that $\b w^+ = O(g)$ and $w^- = O(g)$. It is then convenient to work solely with $w^+$ and $\b w^-$ instead, and using \eqref{relation} and the form of $v^\pm$ we find that \eqref{solw} becomes
\begin{equation}\label{wbw}
	w = w^+ - gw_0 \b w^- + O(g^2),\quad
	\b w = \b w^- - g\b w_0 w^+  + O(g^2).
\end{equation}

The boundary conditions are conditions on the functions $w_{R,L}$ and $\b w_{R,L}$. On the right, we have $w_R(x,0) = \b w_R(x,0) = w_r$ for all $x>0$, and on the left we have $w_L(x,0) = \b w_L(x,0) = w_l$ for all $x<0$, where we recall that $w_{l,r}$ are the thermal values $w_{\rm th}(T_{l,r})$. Using the decomposition \eqref{wbw} and the velocities \eqref{vpm}, as well as the leading part of the thermal values $w_{l,r} = (c\pi/12) T_{l,r} + O(g)$, this gives
\begin{equation}\label{initcond}\begin{aligned}
	w_R^+(x) &= \b w_R^-(x) = w_r + g w_r^2 +O(g^2)\qquad (x>0)\\
	w_L^+(x) &= \b w_L^-(x) = w_l + g w_l^2 +O(g^2)\qquad (x<0).
	\end{aligned}
\end{equation}

The full solution to the problem is obtained by supplementing the boundary conditions with the continuity conditions between the regions, at the interfaces $x=\pm t$. The continuity conditions are obtained by considering the integrated version of the equations \eqref{eq:linear}, which, we recall, are the full order-$g$ equations and are equivalent to \eqref{eq:hydro}.
Consider first the jumps at the interface $x=t$ between the center and right regions. For a shock of velocity 1 between the center and right regions across the interface $x=t$, this is (using ${\tt j}={\tt p}+O(g^2)$)
\[
	{\tt h}_C-{\tt h}_R = {\tt j}_C - {\tt j}_R = {\tt k}_C - {\tt k}_R.
\]
From \eqref{noder}, the first equation is equivalent to
\[
	\b w_R-\b w_C = \frc{g}2 (w_C - w_R) \b w_R + O(g^2),
\]
while the second is similar but with $g$ replaced by $3g$. Combining, one possible solution is $\b w_R = O(g)$; but it turns out that this is inconsistent with the boundary condition. Hence the only consistent solution is
\begin{equation}\label{bwcont}
	w_R-w_C = O(g),\quad \b w_R - \b w_C = O(g^2)\qquad
	\mbox{(at $x=t$).}
\end{equation}
That is, to order $g$ there is no jump in $\b w$ at that interface, but there may be an $O(g)$ jump in $w$. A similar calculation at the interface $x=-t$ gives
\begin{equation}\label{wcont}
	\b w_L - \b w_C = O(g),\quad w_L - w_C = O(g^2)
	\qquad
	\mbox{(at $x=-t$).}
\end{equation}

Let us now combine the continuity condition at the interface $x=t$ with the boundary conditions. Using the decomposition \eqref{wbw} and the velocities \eqref{vpm}, the second relation of the continuity condition \eqref{bwcont} is
\[
	\b w_C^-((1+v_l)t) - g w_r w_C^+((1-v_r)t)
	= \b w_R^-((1+v_r)t) - gw_r w_R^+((1-v_r)t)  + O(g^2)
\]
while the first is $w_C^+((1-v_r)t) = w_R^+((1-v_r)t) + O(g)$, thus leading to
\begin{equation}
	\b w_C^-((1+v_l)t) = \b w_R^-((1+v_r)t) + O(g^2).
\end{equation}
With the initial conditions \eqref{initcond} we then find
\begin{equation}\label{bwc}
	\b w_C^-(x) = w_r + g w_r^2 + O(g^2) \quad (x>0).
\end{equation}
A similar calculation combining the initial conditions with the continuity conditions \eqref{wcont} at the interface $x=-t$ gives
\begin{equation}\label{wc}
	w_C^+(x) = w_l + g w_l^2 +O(g^2) \quad (x<0).
\end{equation}
These then imply, using the above assumption that the velocities are greater than 1,
\begin{equation}\label{restRL}
	w_R^+(x) = w_l +O(g) \quad (x<0),\qquad
	\b w_L^-(x) = w_r + O(g) \quad (x>0).
\end{equation}

Equations \eqref{initcond} fully fix $w_R$ to the right of the right shock, $x>v_rt$, and $w_L$ to the left of the left shock, $x < -v_l t$. Further, since the velocities are greater than 1, equations \eqref{bwc} and \eqref{wc} fully fix $w_C$ in the center region, $-t<x<t$. The remaining parts are the areas between the regions' interfaces and the shocks, $t<x<v_r t$ and $-v_l t<x<-t$. These are fixed by equations \eqref{restRL}, but only up to $O(g)$ terms. Hence, the order $g$ problem does have a unique solution. The space of solutions include the two-shock solution considered in paragraph \ref{sssectshocks}, which is that obtained by setting $w_R^+(x) = w_l +gw_l^2 + O(g^2)\ (x<0)$ and $\b w_L^-(x) = w_r + g w_r^2 + O(g^2)\ (x>0)$. But it also includes a continuum of solutions, which in general present four shocks, at $x=v_rt$, $x=-v_lt$ and also at $x=\pm t$.

Despite the non-uniqueness of the solution to order $g$, it is clear that the steady state is unique. Indeed, the steady state is completely determined by the center region, fixed by equations \eqref{bwc} and \eqref{wc} and the decomposition \eqref{wbw}. We find
\begin{equation}
	w_C = w_l -g w_l(w_r-w_l),\quad \b w_C = w_r + gw_r(w_r-w_l),
\end{equation}
and this leads to the following steady-state energy current and density:
\begin{eqnarray}\label{eq:steady}
j_s &=& w_C - \b w_C = (w_l-w_r)(1+ g(w_l+w_r)+O(g^2)),\\
h_s &=& w_C+\b w_C + g w_C\b w_C = w_l+w_r + g(w_l^2-w_lw_r+w_r^2) +O(g^2),
\end{eqnarray}
in agreement with the two-shock solution \eqref{twoshockssol}.

\subsubsection{Approach to the steady state and spreading of the shocks}

We now consider how the steady states is approached, and study the internal structure of the shocks. In order to do so, we extend beyond the first order in $g$ the analysis by using properties we expect to be valid at higher orders. The following results are conjectural.

{\em Approach to the steady state.} The steady state emerges at large time in the center region. In order to understand how the system approaches steadiness we have to look at the large time behavior of $w$ and $\bar w$ in this center region, say at $x\approx 0$. Let us solve \eqref{eq:linear} to all order in $g$. Continuing the solution in the central region all the way beyond the region, at $t=0$ we find an initial condition that is a sharp profile (modified by the jumps through the shocks). Because of this sharp profile, it is better to consider the space derivatives $w'$ and $\bar w'$, whose initial conditions are then of the form $w'(x,t=0)=\bar w'(x,t=0)\propto\delta(x)$. These have smooth Fourier transforms, and the solution can be written as
\[ w'(x\approx 0, t) \simeq \int dp \big[ U_p e^{i(px-E^+_pt)} + V_p e^{i(px-E^-_pt)}\big] ,\]
with $E^\pm_p$ the two branches of the dispersion relation in the center region and $U_p$ and $V_p$ smooth functions of $p$. In the bulk of the steady state, we assume that we can solve recursively, order by order in $g$, the system of differential equations by bringing it to linear equations. Thus the above form should also be valid, but with modified dispersion relation. This is our working hypothesis. Assuming that $U_p$ and $V_p$ stay smooth (because we are dealing with the derivative of $w$), for $x$ fixed the large time behavior of the integral representing $w(x,t)$ can be evaluated by a saddle point approximation. The saddle is at some $p_*$ and the saddle point integral is then of the form $\int dp\, e^{i\,{\rm const}\,(p-p_*)^2t}$. As a consequence, $w'(x,t)$ decreases as $1/\sqrt{t}$. This behavior has been seen in the approach to the steady state of the non-equilibrium Ising model \cite{Luca_Viti} and in Ising quantum quenches \cite{CEF}.

{\em Estimating the width of the shocks.} We consider a similar type of calculation in order to estimate the width of the shocks. For this purpose, let us assume that $E_p^\pm$ is of the form $E^\pm_p= \pm p(\alpha^\pm + \gamma^\pm p^2+\cdots)$, where the coefficients $\gamma^\pm$ are of order $g^2$. Although parity constrains the possible dispersion relations, we do not know how to justify within this picture the absence of a $p^2$ term in full generality, except that it is valid in the Ising case, but we will see that the result agrees with an independent argument based on quantum flows.

We want to probe points at the ``shoulders'' of the shocks, so that we set $x\pm v_C^\pm t=\epsilon\, t$ and we look at the large time behavior with $\epsilon$ fixed but large enough. Again, we perform a saddle point approximation. A simple calculation gives the saddle point to be $p_\star\sim \sqrt{\epsilon}$, and the saddle phase is $e^{i\,{\rm const}\,\epsilon^{3/2}t}$. Replacing $\epsilon t$ by $x\pm v_C^\pm t$ leads to the phase $\exp\lt[i\,{\rm const}\lt(\frc{(x\pm v_C^\pm t)^3}t\rt)^{1/2}\rt]$, and thus we estimate the width of the shocks to be $\delta\xi_{\rm shock}\simeq (gt)^{1/3}$. This scaling relation is reminiscent to that appearing in KP or KdV equations \cite{Aba}.

Of course this result is conjectural because the argument is based on hypothesis that are not fully supported by calculations: the use of the linear approximation near the shock, and the form of the dispersion relation. Nevertheless we find the same estimate using a quite different approach in the following section.

\section{Quantum Virasoro flows and quantum hydrodynamics}

In this short Section we would like to make a step towards connecting the previous hydrodynamic formalism to possible quantum flows on Virasoro modules\footnote{This tentative formulation was motivated by a seminar given by J. Cardy in December 2014, at the conference ``Mathematical physics of non-equilibrium quantum systems", King's College London, in which he presented a possible connection between $T\bar T$ perturbations and CFTs in random metrics \cite{Cardy-London}. The approach we follow is different from that of J. Cardy, as we use a purely algebraic setup without appealing to Hubbard-Stratonovich transformations.}.
The first step consists in reformulating the $T\bar T$ perturbations as random diffeomorphisms. This connection, based on the fact that the stress tensor is the generator of diffeomorphisms, is made here only to first order in $g$. We will use it to recover some of the previous results from the quantum flow perspective.

To be more specific, let $\phi(x)$ be a chiral primary field on the line -- with scaling dimension $h$ -- and consider its time evolution $\phi(x,t)$ with respect to the perturbed hamiltonian (\ref{eq:hamilton}). Because $\bar T$ commutes with $\phi$ and since $T$ is the generator of diffeomorphisms, we claim that the hamiltonian flow on chiral fields such as $\phi$  is that induced by the ``vector field'' $v(x) :=1+ g \bar T(x)$, that is:
\begin{eqnarray} \label{eq:H-flow}
 \phi(x,t) :=e^{-itH}\,\phi(x)\,e^{+itH} = [X_t'(x)]^h\ \phi(X_t(x)),
 \end{eqnarray}
with $X_t$ the flow line generated by $v$ and started at $x$ at time $t=0$, i.e.
\begin{eqnarray} \label{eq:Vir-flow}
 \dot X_t(x)=v(X_t(x))=1+ g \bar T(X_t(x)) ,\quad \mathrm{with}\quad X_{t=0}(x)=x.
  \end{eqnarray}
The statement (\ref{eq:H-flow}) is actually valid to first order in $g$ only, because claiming that $\phi(x,t)$ commutes with $\bar T$ is true only to first order in $g$.

To prove eq.(\ref{eq:H-flow}) we have to prove that $\phi(x,t)=[X_t'(x)]^h\ \phi(X_t(x))$ is solution of $\partial_t\phi(x,t)=-i[H,\phi(x,t)]$. Let us first recall well-known relations between diffeomorphisms, flow lines and the stress tensor. Since $T$ is the generator of diffeomorphisms, its commutation relation with a primary field $\phi$ of conformal dimension $h$ read
\[ -i\big[ \int dy\, \xi(y)T(y)\,  ,\,\phi(x)\, \big] = h\, \xi'(x)\,\phi(x)+ \xi(x)\, \phi'(x).\]
Hence, for $\phi(x,t)=[X_t']^h\, \phi(X_t)$ we have
\[ -i\Big[\int dy\, \xi(y)T(y)\,,\,[X_t']^h\, \phi(X_t)\Big]= h\,\xi'(X_t)\, [X_t']^h\phi(X_t) + \xi(X_t)\, [X_t']^h\phi'(X_t),\]
provided that $X_t$ commutes with $T$. The latter commutator coincides with the time derivative $\partial_t\phi(x,t)$ provided we choose $X_t$ such that $\dot X_t=\xi(X_t)$, because then $\dot X_t'=X_t'\,\xi'(X_t)$ by the chain rule. That is: to solve for the time evolution we have to choose $X_t$ as the flow line started at $x$ associated to the vector field $\xi$. Now, our perturbed CFT hamiltonian is $H=\int dy \big( T(y)+\bar T(y) + g T(y)\bar T(y)\big)$. Since $\bar T$ commutes with any chiral field, we can apply the above remark to first order in $g$ provided we identify the vector field $\xi$ with $1+g\bar T$.

Thus, at least to first order in $g$, the dynamics of chiral fields is encoded into properties of the flows (\ref{eq:Vir-flow}) that we would like to call quantum Virasoro flows. These flows are random because $\bar T$ is a fluctuating data. We do not yet know whether these quantum flows can be made mathematically rigorous -- of course defining the probability distribution functions of the flow trajectories from data on the Virasoro algebra requires appropriate regularization.

Let us nevertheless use this flow to deduce properties that were found in the previous sections.

The temperature dependent deformation of the light cone is easy to understand using quantum Virasoro flows. The light cone velocity
is simply the mean velocity  $\langle 1+g\bar T(X_t)\rangle$, evaluated in the initial state. It is equal to $v_{l,r}=1 + g\frac{c\pi}{12} T^2_{l,r}$, with $T_{l,r}$ the left/right temperatures, depending in which region the shock front is moving.

The quantum Virasoro flows also give an alternative understanding (based on scaling arguments) for the $t^{1/3}$-scaling behavior we have been arguing for in the previous Section. Indeed let us consider two (quantum) trajectories started at neighborhood points $x$ and $x+\delta x$, and let $\delta X_t$ be their separation after a time duration $t$. By construction, its time evolution is $\dot{(\delta X_t)}= g \, \delta \bar T(X_t)$. Since $\bar T$ is of scaling dimension $2$, $\delta \bar T$ typically scales like $1/(\delta X_t)^2$, or equivalently $\dot{(\delta X_t)}\sim g/(\delta X_t)^2$. By integration this yields that $(\delta X_t)^3$ scales linearly in time,  $(\delta X_{t})^3\sim g\, t$, which indicates that the correlation length in quantum Virasoro flows scales with time as $(g\, t)^{1/3}$, which thus gives the natural width of the shocks. It would be interesting to make such arguments less conjectural.

\section{Conclusion}

In this paper, we reviewed the general hydrodynamic approach to non-equilibrium quantum field theory. In order to study non-equilibrium effects it was important to keep two conserved quantities, including a parity-odd conserved charge, in the local fluid description. We then applied these ideas to develop a hydrodynamic approach for conformal field theory perturbed by the $T\b T$ operator. We verified by direct quantum computations that the main aspects of hydrodynamics emerge at first nontrivial order in the coupling $g$.

An immediate question is about the higher orders in perturbation theory. Clearly there were many simplifications at first order, in particular the emergence of Lorentz invariance, which are not present at higher orders. At higher orders, the fact that the perturbation is irrelevant may become more crucial and generate non-universal contributions.

It will also be interesting to study the internal structure of the shocks, already at first order in $g$. We believe that the conjectured scaling relations are correct and universal. An important further question concerns the existence or not of universal structures inside the shocks, in a way similar to what has been revealed in classical fluctuating hydrodynamics \cite{Spohn}.

In this paper we have focussed on  mean transport phenomena and it will be very interesting to describe fluctuations and their large deviation functions. In particular one may wonder whether the extended fluctuation relations \cite{BD13xx}, proved in pure non-equilibrium conformal field theory and in integrable systems (and conjectured to hold as well in higher-dimensional CFT \cite{Nat-Phys}), remain valid once the $T\bar T$ perturbation has been turned on.

\bigskip

{\bf Acknowledgements:}
We are pleased to thank J. Cardy for motivating discussions as well as A. Abanov, E. Akkermans, E. Boulat and A. Lamacraft for sharing with us their insights. D.B. thanks the Kadanov Center for Theoretical Physics at the University of Chicago for hospitality where part of this work was done. B.D. thanks J. Bhaseen, A. Lucas and K. Schalm for discussions and collaborations on this subject. D.B. and B.D. thank the Institute of Physics of the University of Amsterdam for hospitality during the LDQCM 2015 Workshop, 29 June - 3 July 2015. This work was supported in part by the ANR contracts ANR-2010-BLANC-0414 and ANR-14-CE25-0003-01.

redpajama_set_name: RedPajamaArXiv


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text:
The PFL 11 mixed martial arts event for the 2018 season of the Professional Fighters League, the PFL 2018 Championship, was held on December 31, 2018, at the Hulu Theater at Madison Square Garden in New York City, New York.

Background
The event was the eleventh and final event of the 2018 season. Each champion in the six weight classes was crowned and won a championship prize of $1 million each.

Results

2018 PFL Heavyweight playoffs

2018 PFL Light Heavyweight playoffs

2018 PFL Middleweight playoffs

2018 PFL Welterweight playoffs

2018 PFL Lightweight playoffs

2018 PFL Featherweight playoffs

See also
List of PFL events
List of current PFL fighters

References

Professional Fighters League
2018 in mixed martial arts
Mixed martial arts in New York (state)
Sports in Manhattan
2018 in sports in New York (state)
December 2018 sports events in the United States
Sporting events in New York City
Mixed martial arts events

redpajama_set_name: RedPajamaWikipedia


Carolina tassel-rue is a small foam-flowery like plant with large dissected leaves and puffy airy flowers. It grows in moist rich sun-dappled woods or open sunny wetlands. White flowers in July-August. Grows 2-4' tall. Zone 5-8. VA. ecotype.


text:
Nasdaq 7HANDL Index ETF
Strategy Shares Newfound/ReSolve Robust Momentum ETF
Literature & Forms
Fund Family Overview
NASDAQ 7HANDL™ Index ETF
The NASDAQ 7HANDL™ Index ETF is a first-of-its-kind target distribution ETF designed to seek investment results that correlate generally, before fees and expenses, to the price and yield performance of the NASDAQ 7HANDL™ Index.
The NASDAQ 7HANDL Index seeks to represent an allocation to a balanced portfolio of low-cost ETFs that provide exposure to U.S. equities, bonds and alternative investments and employs leverage in an amount equal to 23% of the total portfolio. To achieve this leverage, the NASDAQ 7HANDL ETF holds a total return swap on an unleveraged version of the Index, the NASDAQ 7HANDL™ Base Index.
The NASDAQ 7HANDL ETF has adopted a policy to pay monthly distributions on Fund shares at a target rate that represents an annualized payout of approximately 7.0% on the Fund's per-share net asset value on the date of a distribution's declaration. All or a portion of a distribution may consist of a return of capital.
The Fund's investment objective is to seek to provide investment returns that correspond, before fees and expenses, to the performance of the Newfound/ReSolve Robust Equity Momentum Index.
The Index uses a quantitative, rules-based methodology to provide exposure to broad U.S. equity, international equity, and emerging market equity indices, to the extent that such equity indices are exhibiting positive momentum relative to U.S. Treasury market indices.
The Index generally consists of exchange traded funds ("ETFs") that track regional equity indices, representative of U.S. equities, developed international equities and emerging market equities, as well as ETFs that track U.S. Treasury market indices.
The allocations to equity ETFs within the Index are determined by proprietary quantitative models that include momentum and trend following factors that are evaluated over various time horizons. The weightings within the Index are determined based on measures of the momentum of the global equity markets relative to the momentum of U.S. Treasury securities. The Fund may have 100% portfolio exposure to U.S. equity or developed international ETFs, and up to 25% portfolio exposure in emerging market equity ETFs.<//li>
The Index will include U.S. Treasury ETFs when equity ETFs exhibit negative momentum and trend following characteristics versus the U.S. Treasury markets. The Fund may have 100% portfolio exposure to U.S. Treasury ETFs under such circumstances.
The Index is rebalanced weekly.
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© 2020 STRATEGY SHARES. ALL RIGHTS RESERVED. | Questions/Comments - Contact Strategy Shares Shareholder Services: 1-800-253-0412 Monday - Friday between 8:30 a.m. and 6:00 p.m. EST.
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redpajama_set_name: RedPajamaCommonCrawl


Social sanctions will be necessary alongside legal measures if domestic violence is to be curbed in Turkey, according to experts from a number of fields who gathered at a conference of the Journalists and Writers Foundation (GYV) in İstanbul on Sunday, marking the International Day for the Elimination of Violence against Women.
The international conference, titled "Family and Community Violence," kicked off with a reception at the Harbiye Congress Center on Friday night, where the event will be held from Nov. 24 to 26. Thirty academics from 15 countries are expected to attend the three-day conference. Six sessions will be held during the conference with the titles "Family and Violence: International Perspectives," "Domestic Violence and its Legal Aspects," "Violence against Children and its Consequences," "Family and Violence: Religious Perspectives," and "Domestic Violence and its Social Aspects."
Speaking during the session "Domestic Violence and its Legal Aspects," on Sunday, Lawyer Fatma Benli pointed to the role of social sanctions in the prevention of violence in Turkey. She stated if people who witness a domestic violence incident keep silent, people who carry out violence will believe they have the right to do so. "Yes, many legal measures have been adopted in order to prevent domestic violence or protect its victims in the recent years in our country. Law 6284 on the Prevention of Violence against Women and the Protection of the Family, which came into force in March, 2012, is the widest one. However, it should not be forgotten that adopting legislative measures are not sufficient. The prevention of domestic violence is not a goal that can be achieved with just legal processes. The government can not prevent domestic violence alone; every person in the community has the responsibility in achieving this goal. Social sanctions should be adopted in a society. People who are close to those who commit acts of domestic violence must react harshly to this behavior in order to deter them from doing so again. Sometimes social responses are more effective than legal ones in a society," Benli noted.
Making a presentation during the conference on Sunday, Hilal Elver -- a research professor of Global Studies at the University of California, Santa Barbara -- said that headscarved women are subject to serious discrimination both in public and at work in Turkey. "Women who wear a headscarf are unable to participate in many places of work. They also face great discrimination at work. For example, many headscarved lawyers are not accepted into courtrooms, simply because of their headscarf. Because those women can not enter legal hearings, many people avoid hiring them. These discriminative acts can be described as economical violence. Many bar associations such as İstanbul, Ankara and İzmir Bar Associations refuse to accept headscarved lawyers into their associations. The right of defense for accused are violated by such acts," Elver noted. Elver also described the discrimination headscarved women face in universities, calling this a violation of the right to education. She said that discrimination against headscarved women should be prevented. "Necessary measures should be adopted while Turkey is working on constructing a new constitution," Elver commented.
Opening speeches of the conference were given by Family and Social Policy Minister Fatma Şahin, GYV Chairman Mustafa Yeşil and Ministry of Family and Social Policies Undersecretary Ahmet Zahteroğulları on Saturday.
During her opening speech, Minister Şahin said family and violence are two terms that should never be voiced together. She also stated that the institution of family is generally associated with love, compassion, respect, confidence and togetherness, adding that it is a great contradiction that violence exists within such an institution.
Arguing that greater awareness in the decision to tie the knot will contribute to strengthening the institution of family, Minister Şahin announced a pilot program launched by the Ministry to counsel to married couples in Burdur. Şahin noted increasing rates of divorce in Turkey and world, and described the services her ministry's initiative would offer: "The experts from the centers will meet with couples seeking divorce and try to find out the problems that pushed them to consider divorce. If possible, the counselors will help the couple work past their issues. If the problems cannot be solved, then the couples will get a divorce. In Burdur, where the pilot scheme for the project is underway, out of 25 couples that filed for divorce, five couples changed their mind thanks to contributions of experts from the marriage counseling centers."
During his speech, GYV Chairman Yeşil said: "An ideal community can be formed with ideal individuals. The first place in which a person acquires their personality and values is the family. Think of the trauma that a child can suffer due to domestic violence in his or her family during the period when personality is developing most critically. Such traumas can lead to behavior disorders later in life. We should do whatever is possible to prevent this; that is why we have gathered here today."


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text:
Q: Why is forwarding reference constructor called instead of copy constructor? Given the following code
#include <iostream>

using namespace std;

template <typename Type>
struct Something {
    Something() {
        cout << "Something()" << endl;
    }

    template <typename SomethingType>
    Something(SomethingType&&) {
        cout << "Something(SomethingType&&)" << endl;
    }
};

int main() {
    Something<int> something_else{Something<int>{}};
    auto something = Something<int>{};
    Something<int>{something};
    return 0;
}

I get the following output
Something()
Something()
Something(SomethingType&&)

Why is the copy constructor being resolved to the templated forwarding reference constructor but not the move constructor?  I am guessing that it's because the move constructor was implicitly defined but not the copy constructor.  But I am still confused after reading the cases for where the copy constructor is not implicitly defined in stack overflow.

A:
I am guessing that it's because the move constructor was implicitly defined but not the copy constructor.

No, both are implicitly defined for class Something.

Why is the copy constructor being resolved to the templated forwarding reference constructor

Because the copy constructor takes const Something& as its parameter. That means for the copy constructor to be called, implicit conversion is needed for adding const qualifier. But the forwarding reference constructor could be instantiated to take Something& as its parameter, then it's an exact match and wins in the overload resolution.
So if you make something const, the implicitly defined copy constructor will be invoked for the 3rd case instead of the forwarding reference constructor.
LIVE

but not the move constructor?

Because for the move constructor the above issue doesn't matter. For the invocation of the 1st and 2nd case, both implicitly defined move constructor and forwarding reference constructor are exact match, then non-template move constructor wins.

redpajama_set_name: RedPajamaStackExchange


BISHOP, Autumn singled to left field.
ROBERTSON, Sarah singled to right center; BISHOP, Autumn advanced to second.
Perez, Briana singled to catcher, bunt.
Jordan, Aaliyah doubled down the rf line, RBI; Nickles, Bubba scored.
CANETTO, Kara singled to third base.
BISHOP, Amber doubled to center field, RBI; CANETTO, Kara scored.
ROBERTSON, Sarah reached first on a fielding error by 2b, SAC; BISHOP, Amber advanced to third, scored on the error, unearned.
Vines, Anna to 2b for Azevedo, Holly.
ERICH, Alexis hit by pitch; ROBERTSON, Sarah advanced to second.
Godin, Kelli reached first on a fielding error by 3b.
VIA, Madison singled to second base.
CANETTO, Kara singled to left center, advanced to second on the throw, RBI; VIA, Madison scored.
Garcia, Rachel doubled to center field.
Pack, Taylor singled to center field, RBI; Garcia, Rachel scored.
Washington, Kinsley singled to center field; Tautalafua, Brianna advanced to second.
DIMARTINO, Julia to p for CASSADY, Chase.
Godin, Kelli singled to catcher, bunt; Washington, Kinsley advanced to second; Tautalafua, Brianna advanced to third.
Nickles, Bubba homered to right center, RBI.
Jordan, Aaliyah doubled to center field.
COOPER, Braelyn to p for DIMARTINO, Julia.
Garcia, Rachel advanced to second on a passed ball; Jordan, Aaliyah advanced to third on a passed ball.
Pack, Taylor reached on a fielder's choice to shortstop; Snow, Alana advanced to third; Jordan, Aaliyah out at home ss to c to 3b to 1b.
Tautalafua, Brianna reached first on a fielding error by 3b; Pack, Taylor advanced to second.
Tautalafua, Brianna advanced to second; Pack, Taylor advanced to third; Snow, Alana out at home c to 2b to c to 3b to c.
AMADER, Jaclyn hit by pitch.
BISHOP, Amber singled to pitcher.
ROBERTSON, Sarah reached on a fielder's choice to first base, SAC; BISHOP, Amber advanced to second.
CLICK, Jasmeen pinch ran for BISHOP, Amber.
SWEAT, Emily pinch hit for GRIFFIN, Denay.
BISHOP, Amber to 2b for CLICK, Jasmeen.
Jordan, Aaliyah doubled to left center.
Garcia, Rachel doubled to right center, RBI; Jordan, Aaliyah scored.


text:
Don't Need Much is the debut album from English punk rock band Great Cynics. It was released through Household Name Records in June 2011.

Track listing

Personnel
Great Cynics
Giles Bidder - Vocals/Guitar
Bob Barrett - Drums

Other Musicians
Peter Miles - Bass

References

2011 debut albums
Great Cynics albums
Household Name Records albums


The recording of real-time generative sound work originating from the artwork Baltic Sea Radio by Varvara Guljajeva & Mar Canet. The composition was generated by tracking local boats in Gdansk Bay and applying this data as a score in real-time.
Baltic Sea Radio is a sound installation that makes use of the sea traffic for artistic purposes in Baltic Sea and offers a novel sonic experience to the audience. The artwork reflects the local ships' movements in real-time and applies marine traffic as a score of a sound composition. In short, the boats that can be reached by the AIS-receiver are affecting and determining the behavior of the artwork.


text:
package com.maths22.laundryviewapi.data;

import java.io.Serializable;

/**
 * Created by Jacob on 1/21/2016.
 */
public class School implements Serializable {
    private String id;
    private String name;

    public School(String id, String name) {
        this.id = id;
        this.name = name;
    }

    public School() {
        this.id = "";
        this.name = "";
    }

    public String getId() {
        return id;
    }

    public void setId(String id) {
        this.id = id;
    }

    public String getName() {
        return name;
    }

    public void setName(String name) {
        this.name = name;
    }
}

redpajama_set_name: RedPajamaGithub


text:
Vidya Balan Excited To ...
Vidya Balan Excited To Play 'Human Computer'
| 9 May 2019 7:02 AM GMT
Actress Vidya Balan has confirmed she is playing the lead role in a film on mathematics genius Shakuntala Devi, who earned the moniker of 'human computer' for her ability to make swift calculations. Set for a summer 2020 release, the film is to be directed by Anu Menon of "London, Paris, New York" and "Four More Shots Please!" fame, and produced by Vikram Malhotra-led Abundantia Entertainment.
"I am extremely excited to play the human computer, Shakuntala Devi, on the big screen. She was truly someone who embraced her individuality, had a strong feminist voice and braved many a naysayer to reach the pinnacle of success," Vidya Balan said in a statement.
"But what truly fascinates me is that you wouldn't normally associate a fun person with math... and she completely turns that perception on its head."
"I am thrilled that Vikram, who I've worked with on 'Kahaani', and his team are producing the film. Vikram, Anu and I are proud to bring to life the story of one of the most inspiring women of this country," said Vidya, who has been widely lauded for her versatility.
"How incredible that 'The Math Genius', 'The Human Computer', was a small-town Indian girl, who took the world by storm," she added. (IANS)
Tags: Vidya Balan


Global components of its strategy include partnering with cities on integrated waste management programs and collaborating with the UN and other agencies to host workshops on good governance.
But it has an initial focus is on South and Southeast Asian countries, where the ocean plastic waste problem is especially acute, accounting for about half of the plastic that flows into oceans annually. This focus is seen through AEPW's investment of USD 10 million in a new initiative of impact investment management firm Circulate Capital and innovation agency SecondMuse that supports waste management technologies and start-up entrepreneurs. The initiative, called the Incubator Network, is designed to help coordinate work of several existing efforts towards achieving similar plastic waste reduction goals.
The Incubator Network is a targeted initiative within Circulate Capital's broader goals to move plastic waste out of the environment and into the recycling value chain and improve the plastic industry's circularity, beginning in five countries – India, Indonesia, the Philippines, Thailand and Vietnam – some of which have been replacing China as the world's dumping ground for solid recyclable waste. Circulate Capital is partnered with the Ocean Conservancy and, separate from the AEPW's infusion, expects to receive USD 90 million from initial investors including PepsiCo, Procter & Gamble, Dow, Danone, Unilever and the Coca-Cola Company.
The concentration of plastic waste in South and Southeast Asia is largely due to an extreme shortage of investment in waste management infrastructure, coupled with a lack of taxes that pay for waste collection and processing, as do garbage taxes in North America or Europe. Instead, waste collection is often handled by an informal sector composed of people who scavenge for sellable waste.
According to the Ocean Conservancy, solving the ocean plastics problem would be helped considerably by three measures: reducing plastic in the waste stream, improving solid waste management, and increasing capture and re-use of plastics.
More than 100 businesses, large and small, in plastic waste management have already responded to Circulate Capital's initial request for investment proposals, says Kaplan. Their pitches include new technologies for organizing plastic waste collection and sorting; innovations in chemical recycling at the waste-processing stage; and new ideas for re-incorporating plastic waste materials into manufacturing products such as packaging, furniture and other durables. The fund is now performing due diligence on several proposals and has said that funding could begin in the first quarter of 2019.
Circulate Capital also hopes its investments will help local entrepreneurs build solid track records, collectively generating an encouraging atmosphere for other investors to buy in along the plastic waste management chain. This can, in turn, boost local economic growth and job creation.
The Incubator Network currently includes WeWork Labs in India, working on early-stage startups; management consulting firm McKinsey & Company's non-profit McKinsey.org; and the Ocean Plastic Prevention Accelerator (OPPA), which highlights startups tackling packaging and recycling in Indonesia.
"We're seeing a pathway toward preventing plastic pollution – or at least ocean plastic pollution – in the next five to 10 years," says Kaplan.


The house has all of the classic Second Empire goodies: the general shape, mansard roof, the decorative slate tiles, and the handsome cupola.
Architect Eric Safyan rearranged the floor plan of this Park Slope row house and retrofitted it to meet passive house energy standards.
Architect and Brooklyn native Eric Safyan incorporated a mix of unconventional, salvaged and found materials in the renovation of his family's Greenwood Heights home.


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Living in electric dreams
Lancashire Electric will be screened from November 1-7 in The Storey Gallery.
Images of lives spent at work and play in Lancashire will be moving in more ways than one when they are screened in Lancaster this November.
Lancashire Electric will see rare archive footage from a century of life in the county projected into the Storey Gallery combined with music and lighting from November 1-7.
The last Morecambe Bay shrimp fisherman to use a horse will be seen in Lancashire Electric. Clip courtesy of the North West Film Archive.
Among the clips due to be shown will be holidaying in Morecambe in the Forties, a Heysham boy's sixth birthday celebrations in 1978, the opening of the motorway through Lancaster, and the last Morecambe Bay shrimp fisherman to use a horse.
West Coast Holiday shows many of the attractions which drew thousands of visitors to Morecambe in its heyday including the mechanical 'jumbo ride' as well as simple pleasures such as children building sandcastles on the beach.
Nikil Kapur's sixth birthday celebrations are captured in a charming clip including shots of bathing in the River Lune at Kirkby Lonsdale.
Another film from the Seventies is Netting The Tide, a student documentary profiling Tom Smith as he prepares for a day's shrimp fishing, along with his horse in Morecambe Bay.
The birthday celebrations of Nikil Kapur from Heysham in 1978 feature in Lancashire Electric. Clip courtesy of the North West Film Archive.
Images from these films and many more including scenes of Silverdale, Preston and Blackpool will be projected into the gallery and screened on deckchairs hung in the space.
Themes include the changing relationship between people and the landscape – from rural Lancashire to the coast – particularly focusing on those who worked the land and those who enjoyed spending leisure time there.
It will also depict the changing face of the landscape including developments in housing and the impact of environmental change.
Visitors will be able to move around and immerse themselves in moving images of childhood, work and play chosen by Imitating the Dog, the company co-founded by co-artistic director, Andrew Quick, professor of performance and theatre at Lancaster University's Institute of Contemporary Arts.
Women at work in this blast from the county's industrial past captured in Lancashire Electric. Clip courtesy of the North West Film Archive.
For Lancashire Electric, Imitating the Dog has also worked with lighting designer, Phil Supple whose previous projects include lighting the Blood Swept Lands and Seas of Red ceramic poppy installation at the Tower of London in 2014.
The clips will be accompanied by a stunning soundtrack by Leeds-based band, Hope & Social whose members include Simon Wainwright, also an Imitating the Dog co-artistic director.
"We wanted to make something which looked back and celebrated the wonderful characters of this part of the world, but also to add a new, creative layer," said Simon.
"We spent plenty of time looking at the amazing films held at the North West Film Archive and it was a tough job to whittle them down to the clips we finally used," Simon added.
A Blackpool promenade scene which features in Lancashire Electric. Clip courtesy of the North West Film Archive.
"We wanted to look at change over time, particularly how people over the last hundred years or so, have worked and played in their environments. We selected film clips which showed peoples' relationship with Lancashire's glorious countryside and the miles of coastline."
Geoff Senior, collections assistant at the North West Film Archive said: "We are always keen to work with others to help bring our collection to the people the films were made by and for, and as such this Lancashire Electric project fits neatly within those aims."
"The films range from cinema newsreels to educational and promotional films, but also home movies shot by members of the public."
Lancashire Electric will be launched on November 1, as part of Light Up Lancaster and continues until November 7. Each screening lasts 15 minutes.
Three free talks will further explore its themes.
On November 2 at 2pm chief executive of Morecambe Bay Partnership, Susannah Bleakley will explain the complex relationship with Morecambe Bay and on November 4 at 6pm, Sarah Hughes of Northumbria University will explore the legacies of sanctuary in Lancashire.
Farm work in the county is reflected in Lancashire Electric. Clip courtesy of the North West Film Archive.
Both talks take place in The Storey Lecture Theatre.
In The Storey Gallery at 6pm on November 6, Simon Wainwright will talk about the making of Lancashire Electric. Admission to Lancashire Electric is free. It is open from 5.30-10pm on November 1 and 5.30pm-7.30pm on November 2.
Visitors are also welcome on November 4-6 from 10am-8pm and November 7 from 10am-5pm.
Lancashire Electric is a partnership project with The Dukes.
It is supported by the BFI Film Audience Network as part of Changing Times: Shifting Ground, and North West Film Archive at Manchester Metropolitan University.
Anyone inspired by Lancashire Electric to find out more about the North West Film Archive's collection can visit www.nwfa.mmu.ac.uk.
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Alstom Finalizes Bombardier Acquisition
By Jack Burke23 September 2020
Alstom has finalized its purchase of Bombardier's rail unit, whose products range from trains to sub-systems and signaling to complete turnkey transport systems, e-mobility technology and data-driven maintenance services
"The acquisition of Bombardier Transportation represents a transformational change for Alstom," said Henri Poupart-Lafarge chairman and CEO of Alstom. "It will enable the group to accelerate on its strategic roadmap and strengthen its leadership in the context of a dynamic market, at a time where sustainable transportation is at the heart of the global agenda."
For Bombardier, the sale marks a significant milestone towards achieving the company's near-term priorities and repositioning Bombardier as a pure-play business jet company, said Éric Martel, president and CEO of Bombardier Inc.
"The proceeds from this transaction will allow us to begin reshaping our capital structure and start addressing our balance sheet through debt paydown so that we can achieve the full potential of our incredibly talented employees and our industry-leading business jet portfolio," Martel said.
Bombardier Transportation adds a complementary geographical presence to broaden Alstom's commercial reach in key growing markets, strong product complementarities in rolling stock, strategic scale in services and signaling, industrial capacity in key countries, a leading portfolio offering and additional R&D capabilities to invest in green and smart innovation, Alstom said in a news release. Alstom will also welcome new talent and expertise, with the arrival of Bombardier Transportation employees.
Terms of the agreement have been adapted to the current market situation. A €300 million reduction in the price range has been agreed with Bombardier Inc and CDPQ. Excluding any further downward adjustments linked to the net cash protection mechanism, the price range for the acquisition of 100% of Bombardier Transportation shares will be therefore €5.5 to €5.9 billion.
The financing structure remains the same as communicated previously, in particular the size and terms of the rights issue and the reserved capital increases of CDPQ and Bombardier Inc. As announced last February, pursuant to the new terms of the acquisition, CDPQ will become the largest shareholder of Alstom with approximately 18% of the share capital and voting rights.
In a news release, Alstom said it is convinced of the strong strategic rationale for the acquisition and is confident in its ability to recover Bombardier Transportation's commercial, operational and profitability potential. Alstom confirms its objectives to deliver €400 million run rate cost synergies in year 4 to 5 and to restore Bombardier Transportation margin towards standard level in the medium term.
Alstom's Board of Directors approved these new terms and is confident in Alstom's capacity to deliver strong value creation for all stakeholders through this acquisition, the company said.
Following positive progress on antitrust regulation process, the closing of the transaction is now expected for Q1 2021 subject to regulatory approvals and customary closing conditions, with an extraordinary shareholders' meeting to be held on Oct. 29.
Headquartered in Berlin, Germany, Bombardier Transportation employs around 36 000 people and its products and services operate in over 60 countries.
Engines, Hydraulics & Powertrains Trains/Rail