text:
Contributed Photo | Janet Morales and her daughter Sarai place new plants that will serve as food sources for the butterflies in the school garden at Frank Ledesma Elementary in Soledad.
Soledad school establishes first monarch butterfly sanctuary in South County
Next garden set for King City; future sites planned for Greenfield, Gonzales
By: Ryan Cronk
SOLEDAD — South County's first monarch butterfly sanctuary has been created at Frank Ledesma Elementary School in Soledad, and similar gardens are being planned for surrounding communities in the near future.
Students, parents and staff members from Frank Ledesma Elementary hosted a kickoff event Oct. 5 to prepare the school's Larry Newman Memorial Garden for the monarch butterfly migration. Parts of the garden were replanted with milkweed and other California-native plants that serve as food sources for the butterflies.
The school received grant funds for the project from the Save the Whales organization and the National Oceanic and Atmospheric Administration Bay Watershed Education and Training Program, which is managed by the office of National Marine Sanctuaries.
"We have had a strong partnership with Save the Whales over the years. They have been instrumental in enriching our fourth- and fifth-grade NGSS-based curriculum with parent connections and field trips," said Richard Radtke, principal at Frank Ledesma Elementary. "This year we are excited to add the Monarch Butterfly Garden Project through Save the Whales. Every student at our school will benefit from the partnership."
According to Radtke, the revamped Larry Newman Memorial Garden will extend the students' outdoor learning space and establish a living science lab that will cultivate deeper learning in STEAM (science, technology, engineering, arts and math), environmental stewardship, sustainability, nature appreciation and artistic endeavors.
In addition, Save the Whales Biologist Dan Albro will provide training and education to students, teachers and the community to help them host the endangered migrating butterflies.
Each year the North American monarch butterflies travel up to 3,000 miles during their migration period — the only butterfly species to make such an extensive, round-trip movement across the country. Landscape changes, however, often result in destroying plants that are critical to the monarch butterflies' survival.
"Save the Whales hopes students will learn that everything is connected," said Maris Sidenstecker, executive director of Save the Whales. "The use of pesticides in gardens kills not only targeted insects, but often the insects that pollinate our gardens, such as butterflies and bees. We need them to grow our crops and rely on them for our food supply."
Sidenstecker said the project also gets youth outside and working in the dirt, which teaches them stewardship.
"They can watch their garden grow and see how an area can be transformed into a beautiful thriving area. It builds teamwork and school pride to protect an endangered species, such as the monarch butterfly," she added.
Save the Whales' next location for a monarch butterfly sanctuary will be King City Arts Magnet School in King City.
"They have a school garden, and we will augment the garden with plants to attract the monarch butterflies and do educational lessons about the butterflies," Sidenstecker said.
The organization is also working on obtaining grant funds to establish other butterfly gardens in Greenfield and Gonzales.
Ryan Cronk
Ryan Cronk is the managing editor for King City Rustler and Salinas Valley Tribune, a unified publication of Greenfield News, Soledad Bee and Gonzales Tribune. He covers general news for South Monterey County and the surrounding communities.


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Wednesday night was supposed to be the Pittsburgh Penguins' first visit to Verizon Center this season. Because of the NHL lockout, we won't get to see Matt Cooke trolling or Sidney Crosby caterwauling plaintively at the refs. We'll just have to settle for candy instead.
Hating Pittsburgh sports is coded into my DNA. I'm physically sick over this. Gary Bettman is depriving me of my regular Sidney Crosby hate, and now I am indescribably sad.
To cheer myself up, I have compiled these GIFs from the Alex Ovechkin "Sorry, Penguin" commercial. And now I share them with you. You'll laugh. You'll cry. You'll wave $15 goodbye.
Now please leave us some naughty but PG-rated comments about the Penguins below. With your words of hatred, we can all get through this together.


text:
Video: Check out UFC's touching tribute to legendary heavyweight 'Kimbo Slice'
By MMA Junkie Staff June 19, 2016 9:15 am Follow	@MMAjunkie
By MMA Junkie Staff | June 19, 2016 9:15 am	Follow @MMAjunkie
He didn't have the best record, he didn't face the best competition, and he never won a major title, but Kevin "Kimbo Slice" Ferguson made a big impact in MMA, including the UFC.
During Saturday's UFC Fight Night 89 broadcast, the UFC and FOX Sports 1 released a tribute video for the former heavyweight slugger, who died on June 6, reportedly due to heart failure.
Check out the touching two-minute look back on his UFC career above.
(Spike is also airing a one-hour special on Friday at 7 p.m. ET/PT prior to the night's Bellator 157 broadcast.)
Slice (5-2 MMA, 1-0 BMMA), who turned his YouTube stardom as a Miami-based bareknuckle boxer into a professional MMA career in 2007, almost immediately became one of the sport's biggest draws. He headlined a series of highly rated EliteXC shows on CBS and brought mainstream attention to the sport. He eventually signed with the UFC after EliteXC's financial collapse.
After a stint on the heavyweights-only 10th season of "The Ultimate Fighter 10" in 2009, Slice went 1-1 in the UFC but continued to draw massive ratings wherever he went.
The bearded slugger then turned his attention to a legitimate boxing career, where he went 7-0 as a pro against mostly overmatched opponents. He returned to MMA with a Bellator deal in 2015 and went 1-0 with a TKO win over Ken Shamrock. He also had a win over Dhafir "Dada 5000" Harris overturned earlier this year when Slice failed a post-fight drug test due to an anabolic steroid and elevated testosterone level.
Ferguson was already 33 by the time he launched his MMA career, and his skills never quite matched his renowned drawing power. Despite an intimidating and larger-than-life persona that made his fights an easy sell, the usually soft-spoken father of six was a favorite among promoters, media members, fellow fighters and other industry officials.
Feb 19, 2016; Houston, TX, USA; Kimbo Slice (red gloves) celebrates his TKO win against Dada 5000 (not pictured) during their Heavyweight fight at Bellator 149 at Toyota Center. Mandatory Credit: Troy Taormina-USA TODAY Sports
Feb 19, 2016; Houston, TX, USA; Kimbo Slice (red gloves) competes against Dada 5000 (blue gloves) during their Heavyweight fight at Bellator 149 at Toyota Center. Mandatory Credit: Troy Taormina-USA TODAY Sports
Feb 19, 2016; Houston, TX, USA; Kimbo Slice (red gloves) prior to his Heavyweight fight against Dada 5000 (not pictured) during Bellator 149 at Toyota Center. Mandatory Credit: Troy Taormina-USA TODAY Sports
Kimbo Slice, News, UFC, Videos

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
Soft matter systems such as polymer solutions, colloidal suspensions,
membranes, and microemulsions exhibit many interesting dynamical
behaviors, where hydrodynamic flow plays an important role, as do thermal
fluctuations.
The characteristic time and length scales of
soft-matter systems are in the range from nanoseconds to seconds and from
nano- to micrometers, respectively, and are thus
typically much larger than the atomistic scales.
Mesoscale simulation techniques are therefore necessary
to simulate these systems for sufficiently large system sizes with
reasonable computational effort.
Several mesoscale techniques for the simulation of the flow of complex fluids
accompanied by thermal fluctuations have been developed
in the last decades, such as direct simulation Monte Carlo
(DSMC)~\cite{bird76,bird98}, the Lattice Boltzmann method~\cite{succ01,yeom06},
dissipative particle dynamics
(DPD)~\cite{hoog92,groo97,espa98,pago98,shar03,pete04,alle06,mars97,mast99,ripo01,boek97,spen00,niku07,vent06,graf07,nogu07,nogu07a},
and multi-particle collision dynamics (MPC)
\cite{male99,ihle01,lamu01,alla02,kiku03,pool05,ihle03b,ihle05,padd04,padd06,hech05,ripo04,ripo06,lee06,saka02,nogu04,nogu05b,nogu06a,ihle06,ruck07,goet07,gomp08}.
DSMC, DPD, and MPC are off-lattice hydrodynamics methods and share many
properties.
DPD and MPC have been applied to various soft-matter systems such as
colloids~\cite{boek97,padd04,padd06,hech05},
polymers~\cite{groo97,spen00,niku07,yeom06,ripo04,ripo06,lee06},
and surfactants~\cite{vent06,graf07,saka02,nogu04,nogu05b,nogu06a}.

The key features to distinguish DPD and MPC are the application of a
Langevin thermostat to
the relative velocities of particle pairs or multi-particle collisions,
and whether or not to employ collision cells.
To understand and elucidate the relation between DPD and MPC,
two intermediate methods have been proposed in Ref.~\cite{nogu07}, which are
DPD with a multibody thermostat (DPD-MT) and MPC-Langevin dynamics (MPC-LD).
The standard MPC algorithm does not conserve angular momentum. However,
an angular-momentum-conserving version of MPC has also been proposed
in Ref.~\cite{nogu07}.
We denote the versions of a simulation method with or without angular-momentum
conservation by an extension `$+a$' or `$-a$', respectively.
The importance of angular-momentum conservation in MPC fluids has been
studied in Ref.~\cite{goet07}.
In the absence of angular-momentum conservation, an additional torque appears
which depends linearly on the vorticity, whereas the velocity field is
unaffected.
Therefore, it is essential to employ `$+a$' techniques to simulate systems
such as rotating colloids and binary fluids with different viscosities.

In this paper, we investigate the viscosity $\eta$ and self-diffusion
constant $D$ of MPC and DPD methods.
The transport coefficients of `$-a$' versions of MPC
were previously derived analytically, and show good agreements with
numerical results~\cite{kiku03,pool05,ihle03b,ihle05,nogu07}.
We derive here analytically the viscosity and diffusion constant
of all `$+a$' versions of MPC.

The transport coefficients of original version of DPD were derived
analytically for systems with an ideal-gas equation of state
in the small-time-step limit~\cite{mars97} and with finite time
step~\cite{nogu07a},
and phenomenologically for soft-repulsive interactions~\cite{nogu07a}.
Here, we investigate the transport coefficients of DPD$-a$ and DPD-MT
for the ideal-gas equation of state with finite time step.
The viscosity and diffusion constant are also determined from simulations
of simple shear flow with Lees-Edwards boundary conditions
and of the mean square displacement of a particle, respectively.

The outline of this paper is as follows.
In Sec.~\ref{sec:mpc}, we describe several versions of MPC, both
with and without angular momentum conservation,
and calculate their transport coefficients analytically and numerically.
Transport coefficients of several version of DPD are calculated
in Sec.~\ref{sec:dpd}.
In Sec.~\ref{sec:thermo}, we discuss the upper limits of the local
shear rate for which thermostats in MPC and DPD are capable
to provide local-equilibrium condition.

\section{Multi-Particle Collision Dynamics (MPC)}\label{sec:mpc}

\subsection{Simulation Method}

\subsubsection{MPC without angular-momentum conservation}

MPC is a modification of DSMC to include multi-particle collisions,
in order to make the algorithm more efficient in its application~\cite{male99}.
A fluid is described by point-like particles of mass $m$.
The MPC algorithm consists of alternating streaming and collision steps.
In the streaming step, the particles move ballistically,
\begin{equation}
{\bf r}_{i}(t+\Delta t)
    = {\bf r}_{i}(t) + {\bf v}_{i} \Delta t,
\end{equation}
where $\Delta t$
is the time interval between collisions.
In the collision step, the particles are sorted into cubic cells of lattice constant $l_{\rm c}$.
The collision procedure is different for each version of MPC.
For MPC$-a$, it is generally given by
\begin{equation}\label{eq:mpc-a}
{\bf v}_{i}^{\rm {new}}= {\bf v}_{\rm c}^{\rm G} +
           {\bf \Omega}[{\bf v}_{i,{\rm c}}  ],
\end{equation}
where ${\bf v}_{\rm {c}}^{\rm G}$ is the velocity of the center of mass of all particles in the box,
 and ${\bf v}_{i,{\rm c}}={\bf v}_{i}-{\bf v}_{\rm c}^{\rm G}$.
The collision operator ${\bf \Omega}[{\bf v}_{i,{\rm c}}]$
stochastically changes the relative velocity
${\bf v}_{i,{\rm c}}$, with
$\sum_{i \in {\rm cell}}{\bf \Omega}[{\bf v}_{i,{\rm c}}]=0$
to keep the translational momentum constant.
This stochastic process is independent for each cell and each time step,
and the collision operator ${\bf \Omega}[{\bf v}_{i,{\rm c}}]$ depends
on whether a particle is inside a cell,
but not on its position ${\bf r}_i$ within the cell.
To guarantee isotropy, the operator must be symmetric on average,
with $\langle v_{\alpha}{\bf \Omega}[{\bf v}]_{\beta} \rangle =
(1-A)\langle {v_{\alpha}}^2 \rangle\delta_{\alpha\beta}$,
where  the subscripts $\alpha,\beta \in \{x,y,z\}$ indicate the
spatial components.  The constants $A$ and
$B=1-\langle {\bf \Omega}[{\bf v}]_{\alpha} {\bf \Omega}[{\bf v}]_{\beta}\rangle/\langle v_{\alpha} v_{\beta}\rangle$
are characteristic quantities of each version (see Table~\ref{tab:fac}),
which play an essential role in determining the transport coefficients.
The operator ${\bf \Omega}[{\bf v}_{i,{\rm c}}]$ conserves the total
kinetic energy in each cell (local micro-canonical ensemble)
or is coupled to a thermostat (local canonical ensemble).
The collision cells are randomly shifted before each collision step to
ensure Galilean invariance~\cite{ihle01}.

The operator ${\bf \Omega}[{\bf v}]$ of the original version of MPC
is the rotation operator. It is represented by a matrix
${\bf \Omega}_{\rm R}({\bf v})$ which rotates velocities by an
angle $\theta$.  The rotation axis is chosen randomly for each cell,
which requires one integer or two real random numbers
in two- ($2$D) or three-dimensional ($3$D) space, respectively.
In $2$D, the axis is the $\pm z$ direction (out of plane),
{\it i.e} the rotation is clockwise or anticlockwise with the angle
$\theta$ (see Fig.~\ref{fig:ope}).
This original version of MPC is typically denoted MPC or stochastic
rotation dynamics (SRD).  We denote it MPC-SR$-a$ in this paper,
in order to distinguish this particular version clearly from the
whole family of MPC techniques.
In MPC-SR$-a$, the energy in each cell is conserved.
The temperature can be controlled by
an additional rescaling of the relative velocities
${\bf v}_{i,{\rm c}} \to {\bf v}_{i,{\rm c}}\sqrt{d(N-N_{\rm {cell}})k_{\rm B}T/m\sum_i {{\bf v}_{i,{\rm c}}}^2}$,
where  $d$ is the spatial dimension, $N$ is the total number of particles,
and $N_{\rm {cell}}$ is the number of cells occupied by particles.
This corresponds to a velocity-scaling version of the profile-unbiased
thermostat (PUT)~\cite{evan86},
where cells are introduced to thermostat local velocities relative to
the center-of-mass velocity of each cell.
The number $d(N-N_{\rm {cell}})$ of the degrees of freedom should be
sufficiently large for the central-limit theorem to apply.
This usually implies that the number of cells included in the calculation
of the rescaling factor is large.
When the velocity rescaling is performed on the level of single
collision cells, the Monte Carlo scheme proposed in Ref.~\cite{hech05}
should be employed.

In the random angle version of MPC (denoted MPC-RA$-a$)~\cite{alla02},
the same matrix ${\bf \Omega}_{\rm R}({\bf v})$ is employed, but
the rotational angle $\theta$ is also selected stochastically varied in the
interval $0\leq\theta<\theta_0$.
In MPC-RA$-a$, one or three real random numbers are required for each
cell in $2$D or $3$D, respectively.

In the Andersen-thermostat~\cite{alle87,ande80} version of MPC,
denoted MPC-AT~\cite{alla02,nogu07},
the operator completely renews the relative velocities in the cell,
${\bf \Omega}[{\bf v}]= {\bf v}_{i}^{\rm {ran}}   - \sum_{j \in {\rm cell}} {\bf v}_j^{\rm {ran}}/N_{\rm {c}}$,
where $N_{\rm {c}}$ is the number of particles in a cell.
A velocity ${\bf v}_{i}^{\rm {ran}}$ is chosen from a
Maxwell-Boltzmann distribution.
Thus, in MPC-AT$-a$, the velocities of particles are updated by
\begin{equation}\label{eq:mpat}
{\bf v}_{i}^{\rm {new}}= {\bf v}_{\rm c}^{\rm G} + {\bf v}_{i}^{\rm {ran}}
  - \sum_{j \in {\rm cell}} \frac{{\bf v}_j^{\rm {ran}}}{N_{\rm {c}}}.
\end{equation}
Instead of the energy, the temperature is constant in MPC-AT.

In the Langevin version of MPC (MPC-LD$-a$)~\cite{nogu07},
the Langevin thermostat is applied to the relative velocities in a
collision cell.  The particle motion is governed by
\begin{eqnarray}\label{eq:mpld0}
m \frac{d {\bf v}_{i}}{dt} =  - \frac{\partial U}{\partial {\bf r}_i}
- \gamma {\bf v}_{i,{\rm c}}
 + \sqrt{\gamma}  \Big\{ {\mbox{\boldmath $\xi$}}_i(t) - \sum_{j \in {\rm cell}} \frac{{\mbox{\boldmath $\xi$}}_j(t)}{N_{\rm {c}}} \Big\}.
\end{eqnarray}
In order to satisfy the fluctuation-dissipation theorem,
the Gaussian white noise ${\mbox{\boldmath $\xi$}}_{i}(t)$ has to have
the average $\langle \xi_{i,\alpha}(t) \rangle  = 0$ and the variance
$\langle \xi_{i,\alpha}(t) \xi_{j,\beta}(t')\rangle  =
         2 k_{\rm B}T \delta _{ij} \delta _{\alpha\beta} \delta(t-t')$,
where $\alpha, \beta \in \{x,y,z\}$ and $k_{\rm B}T$ is the thermal energy.
We consider in this paper only fluids with an ideal-gas equation state,
{\em i.e.} $U \equiv 0$ in Eq.~(\ref{eq:mpld0}).
The finite time-step version of MPC-LD$-a$ is given by the leapfrog
algorithm,
\begin{eqnarray}\label{eq:mpldlf}
  {\bf r}_{i}(t_{n+1/2}) &=& {\bf r}_{i}(t_{n-1/2}) + {\bf v}_{i,n}\Delta t, \\
{\bf v}_{i}(t_{n+1}) &=& {\bf v}_{\rm c}^{\rm G}
    + a_{\rm {ld}}{\bf v}_{i,{\rm c}}(t_n)
    + b_{\rm {ld}}\Big\{{\mbox{\boldmath $\xi$}}_{i,n} -\sum_{j \in {\rm cell}}
                   \frac{{\mbox{\boldmath $\xi$}}_{j,n}}{N_{\rm {c}}}\Big\}, \nonumber \\
{\rm with \ \ }
a_{\rm {ld}} &=& \frac{1-\gamma \Delta t/2m}{1+\gamma \Delta t/2m},\ \
b_{\rm {ld}}= \frac{\sqrt{\gamma\Delta t}/m}{1+\gamma \Delta t/2m},
\label{eq:lgvab}
\end{eqnarray}
where $\langle \xi_{i,n,\alpha} \rangle  = 0$ and
$\langle \xi_{i,n,\alpha} \xi_{j,n',\beta}\rangle  =
         2 k_{\rm B}T \delta _{ij} \delta_{\alpha\beta} \delta_{nn'}$.
Thus, the collision operator is
${\bf \Omega}[{\bf v}_{i,{\rm c}}]= a_{\rm {ld}}{\bf v}_{i,{\rm c}}
    + b_{\rm {ld}}\{{\mbox{\boldmath $\xi$}}_{i,n} -\sum {\mbox{\boldmath $\xi$}}_{j,n}/N_{\rm {c}}\}$.
MPC-LD with $\gamma \Delta t/2m=1$ coincides with MPC-AT.
In MPC-AT and MPC-LD, the correlations have a simple relation, $(1-B)= (1-A)^2$.
However, MPC-SR and MPC-RA have additional correlations
between $x$ and $y$ components, {\em i.e.} $(1-B) \not= (1-A)^2$ as shown in
Table~\ref{tab:fac}.

\begin{table}
\caption{\label{tab:fac}
Correlation factors
$A=1-\langle v_{\alpha}{\bf \Omega}[{\bf v}]_{\alpha}\rangle/
     \langle{v_{\alpha}}^2 \rangle$ and
$B=1-\langle {\bf \Omega}[{\bf v}]_{\alpha}
  {\bf \Omega}[{\bf v}]_{\beta}\rangle/\langle v_{\alpha} v_{\beta}\rangle$
of various MPC methods,
where  $\alpha,\beta \in \{x,y,z\}$ and $\alpha\not= \beta$.}
\begin{center}
\begin{tabular}{lccl}
\hline
               &  $A$  & $B$ & \\
 \hline
MPC-SR    & \ \  $\frac{2}{d}(1-\cos \theta)$ \ \    &  $1-\cos 2\theta$  & ($d=2$)  \\
\ \    &         &  $\frac{2}{5}(2-\cos \theta -\cos 2\theta)$   & ($d=3$)  \\
MPC-RA & $\frac{2}{d}(1-\frac{\sin \theta_0}{\theta_0})$   & $1-\frac{\sin 2\theta_0}{2\theta_0}$  & ($d=2$)  \\
\ \    &         &  $\frac{2}{5}(2-\frac{\sin \theta_0}{\theta_0} -\frac{\sin 2\theta_0}{2\theta_0})$   & ($d=3$)  \\
MPC-AT & $1$   & $1$ & \\
MPC-LD & $\frac{\gamma\Delta t/m}{1+\gamma\Delta t/2m}$   & $\frac{2\gamma\Delta t/m}{(1+\gamma\Delta t/2m)^2} $ & \\
\hline
\end{tabular}
\end{center}
\end{table}

\subsubsection{MPC with angular-momentum conservation}

Collisions described by Eq.~(\ref{eq:mpc-a}) conserve translational
momentum, but do {\em not} conserve angular momentum.
However, angular-momentum conservation can be imposed by
an additional constraint.
This modification is straightforward for the MPC versions with an
intrinsic thermostat (such as MPC-AT and MPC-LD).
In this case, the collision is given by
\begin{eqnarray} \label{eq:mpc+a}
{\bf v}_{i}^{\rm {new}}&=& {\bf v}_{\rm c}^{\rm G} +
           {\bf \Omega}[{\bf v}_{i,{\rm c}} ] \\ \nonumber
&+&  m{\bf \Pi}^{-1} \sum_{j \in {\rm cell}} \big\{{\bf r}_{j,{\rm c}}\times
    ({\bf v}_{j,{\rm c}}-{\bf \Omega}[{\bf v}_{j,{\rm c}}]) \big\} \times {\bf r}_{i,{\rm c}} ,
\end{eqnarray}
where ${\bf \Pi}$ is the moment-of-inertia tensor of the particles in the cell.
The relative position is
${\bf r}_{i,{\rm c}}={\bf r}_i - {\bf r}_{\rm c}^{\rm G}$
where ${\bf r}_{\rm c}^{\rm G}$ is the center of mass of the particles in
the cell.  The angular momentum of the cell after the collision,
${\bf \Pi}\mbox{\boldmath $\omega$}_{\rm c} =
m\sum {\bf r}_{j,{\rm c}}\times {\bf v}_{j,{\rm c}}$,
is the same as before the collision.
The subtraction of either position or velocity of the center of mass
can be omitted in the last term of Eq.~(\ref{eq:mpc+a}), since
$\sum {\bf r}_{j,{\rm c}}\times {\bf v}_{j,{\rm c}} = \sum {\bf r}_{j}\times {\bf v}_{j,{\rm c}}=\sum {\bf r}_{j,{\rm c}}\times {\bf v}_j$.

For MPC-AT$+a$ or MPC-LD$+a$, the terms
\begin{eqnarray}\label{eq:mpat+a}
{\bf f}_{\rm {AT}+{\rm a}} &=&
   m{\bf \Pi}^{-1} \sum_{j \in {\rm cell}} \Big\{ {\bf r}_{j,{\rm c}}\times
     ({\bf v}_j-{\bf v}_j^{\rm {ran}})\Big\}\times {\bf r}_{i,{\rm c}}, \\
\label{eq:mpld+a}
{\bf f}_{\rm {LD}+{\rm a}} &=&
   m{\bf \Pi}^{-1} \sum_{j \in {\rm cell}} \Big\{ {\bf r}_{j,{\rm c}}\times
     \{\gamma {\bf v}_j - \sqrt{\gamma}{\mbox{\boldmath $\xi$}}_j(t)\}\Big\}\times
        {\bf r}_{i,{\rm c}} \nonumber \\
\end{eqnarray}
are added to Eqs.~(\ref{eq:mpat}) and (\ref{eq:mpld0}),
respectively~\cite{nogu07}.

\begin{figure}
\includegraphics{fig1.eps}
\caption{\label{fig:ope}
(Color online)
Schematic representation of the collision operation for MPC-SR$\pm a$ and
MPC-DR in $2$D in the co-moving reference frame (with $\sum {\bf v}_i=0$)
at $N_{\rm c}=3$ and $\theta=\pi/2$.
Circles represent the positions of particles ($\circ$) and the
center of mass ($\bullet$).
`old'  indicates the velocities before the collision,
`$\pm$a' and 'DR' represent the velocities after the collision for
MPC-SR$\pm a$ and MPC-DR, respectively, and
`$+a$-vs' indicates the velocities after the `$+a$' collision without
velocity rescaling.
}
\end{figure}

\begin{figure}
\includegraphics{fig2.eps}
\caption{\label{fig:rdf}
(Color online)
Radial distribution function $g(r)$ of MPC-SR$+a$ (with $n=1$,
$\Delta t^*=1$ and $n=10$, $\Delta t^*=0.1$)
and MPC-AT$+a$ (with $n=1$, $\Delta t^*=0.1$) in two-dimensional space.
The inset shows the $n$ dependence of $g(r)$ of MPC-SR$+a$ at $\Delta t^*=0.1$.
Error bars are shown at several data points.
}
\end{figure}

When Eq.~(\ref{eq:mpc+a}) is applied to the operator of MPC-SR or MPC-RA,
the kinetic energy is {\em not} conserved.
Thus, the collision process has to be modified by combining it with velocity
rescaling to conserve the energy,
\begin{eqnarray} \label{eq:mpc+a2}
{\bf v}_{i}^{\rm {new}}&=& {\bf v}_{\rm c}^{\rm G}
+  m{\bf \Pi}^{-1} \sum_{j \in {\rm cell}} \big( {\bf r}_{j,{\rm c}}\times
    {\bf v}_{j,{\rm c}} \big) \times {\bf r}_{i,{\rm c}} \\ \nonumber
           &+& \phi\Big\{ {\bf \Omega}[{\bf v}_{i,{\rm c}} ]
-  m{\bf \Pi}^{-1} \sum_{j \in {\rm cell}} \big({\bf r}_{j,{\rm c}}\times
    {\bf \Omega}[{\bf v}_{j,{\rm c}}]\big) \times {\bf r}_{i,{\rm c}}  \Big\},
\end{eqnarray}
where $\phi=\{\sum_{j \in {\rm cell}} ({\bf u}_j^{\rm {old}})^2\}/\{\sum_{j \in {\rm cell}} ({\bf u}_j^{\rm {\Omega}})^2\}$.
Here, the relative velocities before and after the collision,
${\bf u}_j^{\rm {old}}$ and ${\bf u}_j^{\rm {\Omega}}$, respectively, are given
by ${\bf u}_i={\bf v}_{i,{\rm c}} - m{\bf \Pi}^{-1} \sum_{j \in {\rm cell}}
({\bf r}_{j,{\rm c}}\times{\bf v}_{j,{\rm c}}) \times {\bf r}_{i,{\rm c}}$,
where the total translational and angular velocities of the cell are subtracted.
This collision is shown schematically in Fig.~\ref{fig:ope}.
Under the molecular-chaos assumption, this yields the ideal-gas equation
of state. However, the molecular-chaos assumption is not perfectly valid.
Thus, the radial distribution function $g(r)$ of MPC-SR$+a$ exhibits
deviations from the uniform distribution of the ideal gas,
in particular for small $n$ and small $\Delta t$ (see Fig.~\ref{fig:rdf}).
If the velocity rescaling for the energy conservation is done not for each
cell but for the sum of many cells, this deviation becomes larger.
A similar deviation is seen in DPD simulations \cite{pago98} with the modified
velocity-Verlet algorithm \cite{groo97}.
MPC-AT$+a$ and MPC-LD$+a$ and all `$-a$' versions of MPC give the correct
uniform $g(r)$ --- see, e.g., the data of MPC-AT$+a$ in Fig.~\ref{fig:rdf}.
Thus, MPC-SR$+a$ should not be used for small $n$ or small $\Delta t$.
We recommend to check $g(r)$ for any new MPC operator.

An alternative modification of MPC-SR for {\em two-dimensional} fluids to
conserve angular momentum has been proposed recently by Ryder~\cite{ryde05}
(see also Ref.~\cite{gomp08}).
We denote this algorithm MPC-DR (deterministic rotation).
In MPC-DR, a rotational angle is chosen deterministically
to keep the total angular momentum of particles in a collision cell
constant by the requirement $W\{1-\cos(\theta)\}+Q\sin(\theta)=0$,
where $W=\sum_{j \in {\rm cell}} {\bf r}_{j,{\rm c}}\times{\bf v}_{j,{\rm c}}$
and $Q=\sum_{j \in {\rm cell}} {\bf r}_{j,{\rm c}}\cdot {\bf v}_{j,{\rm c}}$.
This implies
\begin{equation} \label{eq:mpc-dr}
\cos(\theta)=\frac{W^2-Q^2}{W^2+Q^2} {\rm \ \ \ and \ \ }
\sin(\theta)= -\frac{2WQ}{W^2+Q^2}.
\end{equation}
The velocities after a collision in MPC-DR are different from those in
MPC-SR$+a$, since the `$+a$' procedure (from `$-a$' to `$+a$-vs' in
Fig.~\ref{fig:ope}) does not change the radial velocities.
MPC-DR gives the correct uniform $g(r)$ and
is less time-consuming than other `$+a$' versions of MPC.
We also checked that MPC-DR yields the correct constant angular velocities
for phase-separated binary fluids with different viscosities in a circular
Couette flow, as described in Sec.~IV.C of Ref.~\cite{goet07}.
However, this algorithm cannot be generalized to three-dimensional
systems.

\subsection{Transport Coefficients}

\subsubsection{Stress tensor}

Angular-momentum conservation implies that
the stress tensor $\sigma_{\alpha\beta}$ for an isotropic Newtonian fluid
is symmetric,
{\it i.e.} $\sigma_{\alpha\beta}=\sigma_{\beta\alpha}$~\cite{land87}.
In contrast,  MPC$-a$ fluids have an asymmetric stress tensor
\begin{eqnarray} \label{eq:strs}
\sigma_{\alpha\beta}&=&\lambda(\nabla \cdot {\bf v})\delta_{\alpha\beta} \\
&+& \bar{\eta} \left(\frac{\partial v_{\alpha}}{\partial x_{\beta}} \nonumber
            +\frac{\partial v_{\beta}}{\partial x_{\alpha}} \right)
+ \check{\eta} \left(\frac{\partial v_{\alpha}}{\partial x_{\beta}}
            -\frac{\partial v_{\beta}}{\partial x_{\alpha}} \right),
\end{eqnarray}
because of the lack of angular-momentum conservation
\cite{pool05,ihle05,goet07}, where $\alpha,\beta \in \{x,y,z\}$ and
$\lambda$ is the second viscosity coefficient.
$\bar{\eta}$ and $\check{\eta}$ are the symmetric and antisymmetric
components of the viscosity, respectively.
The last term in Eq.~(\ref{eq:strs}) implies that the stress depends linearly
on the vorticity $\nabla\times{\bf v}$,
and does not conserve angular momentum.
Thus, this term vanishes ({\em i.e.} $\check{\eta}=0$) in
angular-momentum conserving systems.

The evolution of the velocity field ${\bf v}({\bf r})$ is determined by
\begin{equation} \label{eq:nseq}
\rho\frac{D {\bf v}}{D t} =
  -\nabla P + (\lambda+ \bar{\eta}-\check{\eta})\nabla(\nabla\cdot{\bf v})
  + (\bar{\eta}+\check{\eta})\nabla^2 {\bf v},
\end{equation}
where $D/Dt$ is Lagrange's derivative and $P$ is the pressure field.
When a fluid is incompressible,
Eq.~(\ref{eq:nseq}) is the normal Navier-Stokes equation with viscosity
$\eta=\bar{\eta}+\check{\eta}$.
This is consistent with the usual definition of the shear viscosity
$\eta=\sigma_{xy}/\dot\gamma$
in simple shear flow with velocity field ${\bf v}=\dot\gamma y {\bf e}_x$,
where ${\bf e}_x$ is the unit vector along the $x$ direction.
Since both the equations of continuity and of velocity evolution
are of the same forms in systems with and without angular-momentum
conservation, the absence of angular-momentum conservation
does not affect the velocity field of a fluid when the boundary conditions
are given by velocities. However, it generates an additional torque, as
described in detail in Ref.~\cite{goet07}.
In this paper, we discuss the stress tensor of various MPC and DPD methods.

\subsubsection{MPC without angular-momentum conservation}

The shear viscosity is calculated from
$\sigma_{xy}/\dot\gamma=\eta=\bar{\eta}+\check{\eta}$
in simple shear flow with ${\bf v}=\dot\gamma y {\bf e}_x$.
The viscosity of MPC fluids consists of two contributions,
$\eta=\eta_{\rm {kin}}+\eta_{\rm {col}}$, where
the kinetic viscosity $\eta_{\rm {kin}}$ and the
collisional viscosity $\eta_{\rm {col}}$ result from the momentum transfer
due to particle displacements and collisions, respectively.
The derivation of the viscosity for MPC-SR$-a$ described in
Refs.~\cite{kiku03,pool05,ihle03b,ihle05} can be employed directly for
other `$-a$' versions of MPC, since
the differences appear only in the factors $A$ and $B$
listed in Table~\ref{tab:fac}.

The kinetic stress $\sigma_{xy}^{\rm {kin}}=\eta_{\rm {kin}}\dot\gamma$
is the momentum flux due to particles crossing a $xz$ plane at $y=y_0=0$.
The stress due to streaming in the time interval $[t,t+\Delta t]$ is
written as
\begin{eqnarray}\label{eq:stk0}
\sigma_{xy}^{\rm {kin}} &=& \frac{m}{S\Delta t} \Big\{
\sum_{y_i(t)>0, v_{i,y}<-\frac{y_i(t)}{\Delta t}}  v_{i,x} \nonumber\\
   && \hspace*{1cm}
     - \sum_{y_i(t)<0, v_{i,y}>- \frac{y_i(t)}{\Delta t}} v_{i,x} \Big\},
\end{eqnarray}
where $S$ is the surface area of the considered plane.
The average over equivalent $xz$ planes yields
\begin{equation}\label{eq:stk1}
\sigma_{xy}^{\rm {kin}} = - mn \langle v_x v_y  \rangle_{t+\Delta t/2} =
                         - \frac{m}{V} \sum_i v_{i,x} v_{i,y},
\end{equation}
where $n=\langle N_{\rm c}\rangle$ is the average number of particles
per cell, and $V$ is the volume of the considered region $\cal V$, with
${\bf r}_i \in {\cal V}$; here
the middle position ${\bf r}_i(t+\Delta t/2)= {\bf r}_i(t)+{\bf v}_i\Delta t/2$
during streaming is employed to determine whether the $i$-th particle is
inside the region $\cal V$.
The expression (\ref{eq:stk1}) is symmetric in $x$ and $y$. The symmetry
of the kinetic part of the stress tensor,
{\em i.e.} $\sigma_{yx}^{\rm {kin}}=\sigma_{xy}^{\rm {kin}}$,
implies $\check\eta_{\rm {kin}}=0$ for all versions of MPC and DPD.
Numerically, $\sigma_{xy}^{\rm {kin}}$ and $\eta_{\rm {kin}}$ can be calculated
from Eq.~(\ref{eq:stk0}) or (\ref{eq:stk1}).
The velocity distribution is shifted by particle streaming so that
\begin{eqnarray}\label{eq:stk2}
\langle v_x v_y  \rangle_{t,t+\Delta t} &=& \int d{\bf v}\  v_x v_y P_v({\bf v}\mp \dot\gamma v_y\Delta t {\bf e}_x/2) \nonumber \\
 &=& \langle v_x v_y  \rangle_{t+\Delta t/2} \pm \langle {v_y}^2 \rangle \dot\gamma\Delta t/2,
\end{eqnarray}
where $P_v({\bf v})$ is the velocity probability distribution.
The velocity distribution is modified by the MPC collisions so that
$\langle {v_x}^{\rm {new}} {v_y}^{\rm {new}} \rangle =  (1-c_{\rm m}) \langle v_x v_y \rangle$,
where the factor $c_{\rm m}$ is determined later.
The self-consistency condition of a stationary shear flow is
$\langle v_x v_y \rangle_t =\langle v_x v_y \rangle_{t+\Delta t} =
(1-c_{\rm m}) (\langle v_x v_y \rangle_t -
         \dot\gamma\Delta t\langle {v_y}^2 \rangle)$.
The kinetic viscosity $\eta_{\rm {kin}}$ is then given by~\cite{kiku03}
\begin{equation}\label{eq:kvis0}
\eta_{\rm {kin}} = \frac{n k_{\rm B}T \Delta t}{ {l_{\rm c}}^d }
                     \left( \frac{1}{c_{\rm m}}  - \frac{1}{2}\right)
\end{equation}
Eq.~(\ref{eq:kvis0}) holds for all `$\pm a$' versions of MPC and DPD.

The velocity correlations for MPC$-a$ are calculated by using
Eq.~(\ref{eq:mpc-a}),
\begin{eqnarray}\label{eq:c-a}
\langle v_{i,x}^{\rm {new}} v_{i,y}^{\rm {new}} \rangle
  &=& \Big\{\frac{1}{ {N_{\rm c}}^2}
     + \frac{2}{N_{\rm c}}\Big(1-\frac{1}{N_{\rm c}}\Big)(1-A)
                                             \nonumber \\ \nonumber
  && + \Big(1- \frac{1}{N_{\rm c}}\Big)^2(1-B)\Big\}
                        \langle v_{i,x}v_{i,y} \rangle \\ \nonumber
  && + \frac{2A -B}{ {N_{\rm c}}^2}\sum_{j\not=i}
                                    \langle v_{j,x}v_{j,y} \rangle \\
  &=&  \Big\{1- B\Big(1- \frac{1}{N_{\rm c}}\Big)\Big\}
                                    \langle v_{i,x}v_{i,y} \rangle,
\end{eqnarray}
where molecular chaos is assumed, {\it i.e.}
$\langle v_{i,x}v_{i,y} \rangle=\langle v_{j,x}v_{j,y} \rangle$
and $\langle v_{i,x}v_{j,y} \rangle=0$ for $i\not=j$.
Thus the correlation factor for a cell occupied by $N_{\rm c}$ particles is
$c(N_{\rm c})=B(1-1/N_{\rm c})$.
An MPC fluid is thermodynamically an ideal gas,
so that the cell occupation number $N_{\rm c}$ fluctuates with the Poisson
distribution, $P(N_{\rm c})=e^{-n}n^{N_{\rm c}}/N_{\rm c}!$ with $n=\langle N_{\rm c}\rangle$.
Thus, the average correlation is give by
$c_{\rm m}=\sum_{k=1}^{\infty} c(k) P(k)k/n =B(n-1+e^{-n})/n$.
The kinetic viscosity of MPC$-a$ is then given by
\begin{equation}\label{eq:kvis-a}
\eta_{\rm {kin}} = \frac{n k_{\rm B}T \Delta t}{ {l_{\rm c}}^d }
\Big\{ \frac{n/B}{n- 1 + e^{-n}}- \frac{1}{2} \Big\}.
\end{equation}

The collisional stress $\sigma_{xy}^{\rm {col}}=\eta_{\rm {col}}\dot\gamma$
is the momentum flux due to MPC collisions in cells crossing a plane at $y=y_0$.
It is given by \cite{kiku03}
\begin{equation} \label{eq:stc0}
\sigma_{xy}^{\rm {col}} = - \frac{m}{{l_{\rm c}}^{d-1}\Delta t}
\sum_{y_0<y_i,i \in {\rm cell}} \langle v_{i,x}^{\rm {new}}- v_{i,x} \rangle.
\end{equation}
When Eq.~(\ref{eq:stc0}) is averaged over the planes crossing
the cell, $y_{\rm {cc}}-l_{\rm c}/2<y_0<y_{\rm {cc}}+l_{\rm c}/2$,
the stress reads
\begin{equation} \label{eq:stc1}
\sigma_{xy}^{\rm {col}} = - \frac{m}{{l_{\rm c}}^{d-1}\Delta t}
\sum_{i \in {\rm cell}}
    \Big( \frac{y_{i,\rm {cc}}}{l_{\rm c}}+\frac{1}{2} \Big)
              \langle v_{i,x}^{\rm {new}}- v_{i,x} \rangle,
\end{equation}
where $y_{i,\rm {cc}}=y_i-y_{\rm {cc}}$ and $y_{\rm {cc}}$ is the
$y$ component of the center-of-cell position ${\bf r}_{\rm {cc}}$.
Numerically, $\sigma_{xy}^{\rm {col}}$ and $\eta_{\rm {col}}$ can be
calculated from either Eq.~(\ref{eq:stc0}) or (\ref{eq:stc1}).
The mean velocity difference is
$\langle v_{i,x}^{\rm {new}}- v_{i,x} \rangle=-(1-\frac{1}{N_{\rm c}})A \dot\gamma y_{i,\rm {cc}}$,
because $\langle {\bf v}_{\rm c}^{\rm G} \rangle= {\bf v}_{i}/N_{\rm c}$,
where $y_j$ is averaged over
$-l_{\rm c}/2<y_j<l_{\rm c}/2$ for $j\not=i$ at $y_{\rm {cc}}=0$.
Then the collisional viscosity $\eta_{\rm {col}}$ of MPC$-a$ is given by
\begin{eqnarray} \label{eq:cvis0}
\eta_{\rm {col}} &=&
  \frac{Am}{{l_{\rm c}}^{d}\Delta t}
  \Big\{\sum_{N_{\rm c}=1}^{\infty} (N_{\rm c}-1)P(N_{\rm c}) \Big\}
    \int_{-\frac{l_{\rm c}}{2}}^{\frac{l_{\rm c}}{2}} dy\
          \Big( \frac{y}{l_{\rm c}}+\frac{1}{2} \Big)y  \nonumber  \\
  &=& \frac{Am(n-1+e^{-n})}{12 {l_{\rm c}}^{d-2}\Delta t}.
\end{eqnarray}
The vorticity viscosity
$\check{\eta}_{\rm {col}}\dot\gamma=\sigma_{xy}^{\rm {col}}-\sigma_{yx}^{\rm {col}}$
is proportional to the angular-momentum transfer with respect to the origin
$(x_{\rm {cc}}-l_{\rm c}/2,y_{\rm {cc}}-l_{\rm c}/2)$, see Eq.~(\ref{eq:stc1}).
Thus, the vorticity viscosity of MPC$+a$ vanishes, $\check{\eta}_{\rm {col}}=0$,
because of angular-momentum conservation.
For MPC$-a$, the molecular-chaos assumption gives
$\sigma_{yx}^{\rm {col}}=0$, because
$\langle v_y^{\rm {new}}(x)\rangle=\langle v_y(x)\rangle=0$.
Thus, the viscosities are
$\check{\eta}=\bar{\eta}_{\rm {col}}=\eta_{\rm {col}}/2$ \cite{pool05,goet07}.
This viscosity relation holds for all `$-a$' versions of MPC and DPD
described in this paper.

As an extension of this approach, the angular-momentum constraint can
be applied only partially,
by employing alternatively the MPC-collision algorithms which conserve
[given by Eq.~(\ref{eq:mpc+a})]
and do not conserve [determined by the difference of the right-hand sides of
Eqs.~(\ref{eq:mpc-a}) and (\ref{eq:mpc+a})] angular momentum. In this
way, the viscosity ratio $\check{\eta}/\eta$ can be varied continuously
between $0$ and approximately $1$.

Next, we derive the self-diffusion constant $D$ of MPC$-a$.
Under the molecular-chaos assumption,
the velocity correlation function decays exponentially,
$\langle v_{i,x}(k \Delta t) v_{i,x}(0)\rangle = (1-s_{\rm m})^kk_{\rm B}T/m$
with
$1-s_{\rm m}=\langle v_{i,x}^{\rm {new}} v_{i,x}\rangle/\langle {v_{i,x}}^2\rangle$.
The diffusion constant is thus given by~\cite{ihle03b}
\begin{eqnarray}
D &=&  \frac{\Delta t}{2}\Big\{\langle v_{i,x}(0)^2 \rangle
   + 2\sum_{k=1}^{\infty} \langle v_{i,x}(k \Delta t) v_{i,x}(0)\rangle \Big\}
                               \nonumber  \\
&=& \frac{k_{\rm B}T\Delta t}{m} \Big( \frac{1}{s_{\rm m}} - \frac{1}{2} \Big).
\label{eq:dif0}
\end{eqnarray}
In MPC$-a$, the correlation factor is
$s_{\rm m}= \sum_{k=1}^{\infty} s(k) P(k)k/n =A(n-1+e^{-n})/n$
with $s(N_{\rm c})=A(1-1/N_{\rm c})$; this implies
\begin{equation}\label{eq:dif-a}
D = \frac{k_{\rm B}T\Delta t}{m}
             \Big( \frac{n/A}{n- 1 + e^{-n}} - \frac{1}{2} \Big).
\end{equation}
However, the velocity auto-correlation function
$\langle v_x(k \Delta t) v_x(0)\rangle$ for small mean free path
$l_{\lambda}=\Delta t\sqrt{k_{\rm B}T/m_0}$ has a long-time tail due to
hydrodynamic backflow~\cite{ihle03b,padd06,ripo04}.
This leads to an additional hydrodynamic contribution to the
diffusion constant $D$, which thereby becomes larger
than predicted by Eq.~(\ref{eq:dif-a}).

\subsubsection{MPC with angular-momentum conservation}

To derive expressions for the self-diffusion constant and viscosity
of MPC$+a$, we employ Eqs.~(\ref{eq:kvis0}), (\ref{eq:stc1}), and
(\ref{eq:dif0}), which remain valid with angular-momentum conservation.
However, the correlation factors $s_{\rm m}$ and $c_{\rm m}$ of MPC$+a$
are different from those of MPC$-a$.
First, we consider the limit of large $n$, where $s_{\rm m}=s(n)$ and
$c_{\rm m}=c(n)$, and derive the corrections for small $n$ subsequently.
The velocity correlation is calculated from
$\sum_j ({\bf r}_{j,{\rm c}}\times {\bf v}_{j,{\rm c}})\times {\bf r}_{i,{\rm c}}
= \sum_j ({\bf r}_{i,{\rm c}}\cdot{\bf r}_{j,{\rm c}}){\bf v}_j
- ({\bf v}_j \cdot {\bf r}_{i,{\rm c}}){\bf r}_{j,{\rm c}}$
with the molecular-chaos assumption.
The positions of particles ${\bf r}_i$ are averaged over the cell,
so that ${{\bf r}_{i,{\rm c}}}^2=(1-1/N_{\rm c}){l_{\rm c}}^2d/12$ and
${\bf \Pi}=(N_{\rm c}-1)m{l_{\rm c}}^2{\bf I}/6$ where ${\bf I}$ is the
identity matrix.
Angular-momentum conservation implies additional correlations, which result
in
\begin{eqnarray} \label{eq:s+a}
s(N_{\rm c})  &=&  A\Big(1-\frac{1}{N_{\rm c}}\Big) -\frac{Ad}{2N_{\rm c}}\big(1-\langle{\hat{x}_{i,\rm {cc}}}^2\rangle\big) \nonumber \\
&=& A\Big(1-\frac{d+1}{2N_{\rm c}}\Big),
\end{eqnarray}
where $\hat{x}_{i,\rm {cc}}$ is the $x$ component of unit vector
$\hat{\bf r}_{i,\rm {cc}}={\bf r}_{i,\rm {cc}}/r_{i,\rm {cc}}$ and $\langle{\hat{x}_{i,\rm {cc}}}^2\rangle=1/d$.
The diffusion constant of MPC$+a$ for large $n$ is thus found to be
\begin{equation}\label{eq:dif+a}
D = \frac{k_{\rm B}T\Delta t}{m}
         \Big( \frac{n/A}{n-(d+1)/2} - \frac{1}{2} \Big).
\end{equation}

\begin{figure}
\includegraphics{fig3.eps}
\caption{\label{fig:visats}
(Color online)
Dependence of the viscosity in MPC-AT$\pm a$ on $\Delta t^*$ in two- or
three-dimensional space for (a) $n=5$ and (b) $n=1$.
Symbols represent the numerical data of MPC-AT$+a$ in $2$D
($\circ$, $\triangle$) or $3$D ($\bullet$, $\blacktriangledown$) and
MPC-AT$-a$  in $2$D ($\Box$, $\diamond$) or $3$D ($\times$, $+$), respectively.
Solid and dashed lines represent analytical results for MPC-AT$+a$ and
MPC-AT$-a$, respectively.
Error bars are smaller than the size of symbols.
}
\end{figure}

\begin{figure}
\includegraphics{fig4.eps}
\caption{\label{fig:visatn}
(Color online)
Dependence in MPC-AT$\pm a$ of the viscosities on the particle number
per cell, $n$.
(a),(c) $\eta_{\rm {kin}}$ and (b) $\eta_{\rm {col}}$.
Symbols represent the numerical data of MPC-AT$+a$ ($\circ$, $\triangle$) and
MPC-AT$-a$ ($\Box$, $\diamond$) for $\Delta t^*=0.1$ and $\Delta t^*=1$
in $2$D, respectively, and the numerical data of MPC-AT$+a$ at
$\Delta t^*=1$ ($\times$) in $3$D.
In (a),(b), the viscosity is rescaled by $\Delta t^*$ and $1/\Delta t^*$,
respectively, in order to facilitate a presentation of data for
different $\Delta t^*$ on the same scale.
Solid and dashed lines in (a) and (b) represent analytical results for
MPC-AT$+a$ and  MPC-AT$-a$, respectively.
Solid and dashed lines in (c) represent analytical results with or without
the correction term $h_{\rm m}$, respectively.
Error bars are smaller than the size of symbols.
}
\end{figure}

For the calculation of the kinetic viscosity,
we obtain the $v_xv_y$ correlation
factor
\begin{eqnarray}\label{eq:c+a}
c(N_{\rm c}) = B\Big(1-\frac{3d+2}{4N_{\rm c}}\Big) + \frac{Ad}{2N_{\rm c}} + O({N_{\rm c}}^{-2}).
\end{eqnarray}
The kinetic viscosity $\eta_{\rm {kin}}$ for large $n$ is then given
by Eqs.~(\ref{eq:kvis0}) and (\ref{eq:c+a}) with $c_{\rm m}=c(n)$.
For MPC-AT$+a$ and MPC-LD$+a$, this implies for large $n$ that
\begin{eqnarray} \label{eq:kvisat+a}
\eta_{\rm {kin}}^{\rm {AT}+{\rm a}} &=&
        \frac{n k_{\rm B}T \Delta t}{ {l_{\rm c}}^d }
           \Big\{ \frac{n}{n- (d+2)/4  } - \frac{1}{2} \Big\}, \\
\eta_{\rm {kin}}^{\rm {LD}+{\rm a}} &=&
           \frac{n k_{\rm B}T}{ {l_{\rm c}}^d }
 \Big\{ \frac{mn(1+\gamma\Delta t/2m)^2/\gamma}{2n-d-1+d\gamma\Delta t/4m }
               - \frac{\Delta t}{2} \Big\}. \ \
\end{eqnarray}
Note that $\eta$ and $D$ of MPC-LD$\pm a$ have a different dependence
on the time step $\Delta t$ than other MPC algorithms, since their
correlation factors $A$ and $B$ depend on $\Delta t$ (see Table~\ref{tab:fac}).

The mean velocity difference for MPC$+a$ is given by
\begin{equation}\label{eq:stc+a0}
\langle v_{i,x}^{\rm {new}}- v_{i,x} \rangle =
  -(1-\frac{1}{N_{\rm c}})A (\dot\gamma-\langle\omega\rangle) y_{i,\rm {cc}}.
\end{equation}
The $z$ component of the velocity is pre-averaged,
the angular velocity is in the vorticity direction,
$\mbox{\boldmath $\omega$}=\omega{\bf e}_z$, and
$\langle {\bf v}_j \rangle=\dot\gamma y_{j,\rm {cc}}{\bf e}_x$, so that
\begin{eqnarray}\label{eq:stc+a1}
\langle \mbox{\boldmath $\omega$} \rangle &=&
  \Big\langle\ \frac{\sum_j {\bf r}_{j,{\rm c}}\times {\bf v}_{j,{\rm c}}}
       {\sum_j {x_{j,\rm c}}^2 + {y_{j,\rm c}}^2 } \Big\rangle,  \nonumber \\
\langle \omega \rangle &=&
   \frac{ {y_{i,\rm {cc}}}^2 + (N_{\rm c}-1){l_{\rm c}}^2/12}
      { {y_{i,\rm {cc}}}^2+ (2N_{\rm c}-1){l_{\rm c}}^2/12}\dot\gamma,
\end{eqnarray}
where the numerator and denominator are averaged over
$x_{i,\rm {cc}}$, $x_{j,\rm {cc}}$, and $y_{j,\rm {cc}}$ independently.
When $\langle\omega\rangle$ is also pre-averaged over $y_{i,\rm {cc}}$,
$\langle\omega\rangle=\dot\gamma/2$ is obtained.
However, Eq.~(\ref{eq:stc1}) together with Eq.~(\ref{eq:stc+a0})
contains an integral with ${y_{i,\rm {cc}}}^2$, which yields an
additional correction term of $O(N_{\rm c}^{-1})$,
\begin{eqnarray}\label{eq:stc+a2}
\int_{-\frac{l_{\rm c}}{2}}^{\frac{l_{\rm c}}{2}}
\frac{\langle\omega\rangle{y_{i,\rm {cc}}}^2}{\dot\gamma{l_{\rm c}}^3} \ dy_{i,\rm {cc}}
= \frac{1}{24}\Big(1+\frac{2}{5N_{\rm c}}\Big) + O({N_{\rm c}}^{-2}).
\end{eqnarray}
Then, the collisional viscosity $\eta_{\rm {col}}$ of MPC$+a$ for large $n$
is given by
\begin{equation}\label{eq:cvis+a0}
\eta_{\rm {col}} = \frac{Am(n-7/5)}{24 {l_{\rm c}}^{d-2}\Delta t}
\end{equation}

\begin{figure}
\includegraphics{fig5.eps}
\caption{\label{fig:visld}
(Color online)
Viscosity of MPC-LD$\pm a$ as a function of (a) $\gamma^*$ and
(b), (c) $n$ in three-dimensional space
at (a) $n=3$ and $\Delta t^*=0.1$ and (b),(c) $\gamma^*=1$.
Symbols represent the numerical data of MPC-LD$+a$ ($\circ$, $\triangle$)
and  MPC-LD$-a$ ($\Box$, $\diamond$).
Dashed and solid lines represent analytical results for
(a) $\eta_{\rm {kin}}$ or $\eta_{\rm {col}}$,
and (b), (c) $\Delta t^*=0.1$ or $1$, respectively.
Error bars are smaller than the size of symbols.
}
\end{figure}

\begin{figure}
\includegraphics{fig6.eps}
\caption{\label{fig:vismpc}
(Color online)
Viscosity $\eta$ in two-dimensional space as a function of $\Delta t^*$ for
 MPC-SR$\pm a$ with $\theta=\pi/2$, MPC-RA$\pm a$ with $\theta_0=\pi$,
and MPC-DR with $n=1$ or $n=5$.
Solid and dashed lines represent analytical results for $n=5$ and $n=1$, respectively.
(a) Symbols represent the numerical data of MPC-RA$+a$ ($\blacktriangle$, $\triangle$,$+$),
MPC-DR ($\bullet$,$\circ$,$\triangleleft$), MPC-RA$-a$ ($\diamond$,$\times$), MPC-SR$-a$ ($\Box$), and MPC-SR$+a$ ($\blacksquare$).
Error bars are smaller than the size of symbols.
}
\end{figure}

Next, we derive the correction terms for small $n$.
For $N_{\rm c}=1$ or $2$, Eqs.~(\ref{eq:s+a}) and (\ref{eq:c+a}) do not
give the correct correlation factors
$s(N_{\rm c})$ and $c(N_{\rm c})$ for MPC$+a$ --- unlike for MPC$-a$.
First, there is no velocity transfer for $N_{\rm c}=1$, {\em i.e.}
$s(1)=c(1)=0$.
Second, in the energy-conserving versions of MPC (MPC-SR$+a$ and MPC-RA$+a$),
{\em all} $dN_{\rm c}$ degrees of freedom are determined for $N_{\rm c}=2$
by the conservation of energy (one degree of freedom), and translational
($d$ degrees) and angular ($d-1$ degrees) momentum, so that $s(2)=c(2)=0$.
In the MPC versions with an intrinsic thermostat (MPC-AT$+a$ and MPC-LD$+a$),
one degree of freedom remains for the velocity transfer for $N_{\rm c}=2$,
so that $s(2)=A/2d$ and $c(2)=(A +B/d)/(d+2)$.
Thus, $s_{\rm m}=\sum_{k=3}^{\infty} s(k)P(k)k/n$ for energy-conserving
versions of MPC,
and $s_{\rm m}= P(2)A/dn +\sum_{k=3}^{\infty} s(k)P(k)k/n$ for
MPC versions with an intrinsic thermostat.
For MPC-SR$+a$ and MPC-RA$+a$, the diffusion constant $D$, and the viscosities
$\eta_{\rm {kin}}$ and $\eta_{\rm {col}}$
are given by Eqs.~(\ref{eq:dif0}) and (\ref{eq:kvis0}) with
\begin{eqnarray} \label{eq:c+a1}
s_{\rm m} &=& A\Bigg\{1- \frac{d+1}{2n}
        + \frac{e^{-n}}{2}\Big(\frac{(d-3)n}{2} + d-1 +
                         \frac{d+1}{n}\Big)\Bigg\}, \nonumber \\
c_{\rm m} &=& B\big\{1-e^{-n}(1+n)\big\} \\ \nonumber
   && + \Big\{Ad -\frac{B(3d+2)}{2}\Big\}\frac{1-e^{-n}(1+n+n^2/2 )}{2n},
                                            \\ \nonumber
\eta_{\rm {col}} &=& \frac{Am}{24 {l_{\rm c}}^{d-2}\Delta t}
   \Bigg\{n-\frac{7}{5} +e^{-n}\Big(\frac{7}{5} + \frac{2n}{5}
                                    - \frac{3n^2}{10}\Big) \Bigg\}.
\end{eqnarray}
For MPC-AT$+a$ and MPC-LD$+a$, the diffusion constant $D$ and the viscosity
contributions $\eta_{\rm {kin}}$ and $\eta_{\rm {col}}$
are given by Eqs.~(\ref{eq:dif0}) and (\ref{eq:kvis0}) with
\begin{eqnarray} \label{eq:s+a2}
s_{\rm m} &=& A\Bigg\{1- \frac{d+1}{2n} \\ \nonumber
&&+ \frac{e^{-n}}{2}\Big(\frac{(d-1)(d-2)n}{2d} + d-1
              + \frac{d+1}{n}\Big)\Bigg\},\\ \label{eq:c+a2}
c_{\rm m} &=& B\big\{1-e^{-n}(1+n)\big\}
         + \Big(A+\frac{B}{d}\Big)\frac{n e^{-n}}{d+2}\\ \nonumber
&&+ \Big\{Ad  -\frac{B(3d+2)}{2}\Big\}\frac{1-e^{-n}(1+n+n^2/2 )}{2n}, \\
\label{eq:cvis+a2}
\eta_{\rm {col}} &=& \frac{Am}{24  {l_{\rm c}}^{d-2}\Delta t} \\ \nonumber
&&\times \Bigg(n-\frac{7}{5}
 +e^{-n}\Big\{\frac{7}{5}+ \frac{2n}{5}
               + \big(\frac{1}{d}-\frac{3}{10}\big)n^2\Big\} \Bigg).
\end{eqnarray}

For MPC-DR, the rotation angle $\theta$ is uniformly distributed in
$-\pi\leq \theta<\pi$ under the molecular-chaos assumption.
Thus, the transport coefficients of MPC-DR coincide with those of MPC-RA$+a$
at $\theta_0=\pi$.  Thus, the diffusion constant $D$, and the viscosities
$\eta_{\rm {kin}}$ and $\eta_{\rm {col}}$ of MPC-DR
are given by Eqs.~(\ref{eq:dif0}), (\ref{eq:kvis0}), and (\ref{eq:c+a1})
with $A=B=1$.
Here, the term $c_{\rm m}$ can be written in a simpler form,
$c_{\rm m}=\{n-1+e^{-n}(1-n^2/2)\}/n$.

\begin{figure}
\includegraphics{fig7.eps}
\caption{\label{fig:visyr}
(Color online)
Viscosity difference
$\Delta\eta_{\rm {col}}=\bar{\eta}_{\rm {col}}-\check{\eta}_{\rm {col}}$
of MPC-AT$-a$ in two and three dimensions.
Symbols with dashed or solid lines represent the numerical data in
$2$D ($\circ$, $\Box$) and  $3$D ($\diamond$, $\triangle$) at $n=1$ or $n=5$,
respectively.
The inset shows the dependence of $\Delta\eta_{\rm {col}}$ on the average
particle number $n$ per cell, for $\Delta t^*=0.1$ ($\Box$) and
$\Delta t^*=1$ ($\circ$).
}
\end{figure}

\begin{figure}
\includegraphics{fig8.eps}
\caption{\label{fig:dif}
(Color online)
Dependence of the diffusion constant $D$ of MPC-AT$\pm a$ on $\Delta t$
at $n=5$ in three dimensions.
Symbols and lines represent numerical and analytical data, respectively.
Error bars are smaller than the size of symbols.
}
\end{figure}

\subsection{Numerical Results}

Figs.~\ref{fig:visats}--\ref{fig:vismpc} show the viscosities
$\eta_{\rm {kin}}$ and $\eta_{\rm {col}}$
for five MPC methods with or without the angular-momentum conservation.
The results are displayed in form of dimensionless quantities
with length and time units $l_{\rm c}$ and
$\tau_{\rm 0}=l_{\rm c}\sqrt{m/k_{\rm B}T}$, respectively.
The main parameters which control the properties of MPC fluids, the
time step and friction constant, have the dimensionless form
$\Delta t^*=\Delta t/\tau_{\rm 0}$ and $\gamma^*=\gamma\tau_{\rm 0}/m$.
Similarly,
the viscosity and diffusion constant of a particle are shown in units
of $\eta_{\rm 0}=\sqrt{mk_{\rm B}T}/{l_{\rm c}}^{d-1}$ and
$D_{\rm 0}= l_{\rm c}\sqrt{k_{\rm B}T/m}$, respectively.
The error bars of the simulation results are estimated from three
independent runs.

Analytical results are calculated from Eqs.~(\ref{eq:dif0}) and
(\ref{eq:kvis0}) together with Eq.~(\ref{eq:c+a1}), or from
Eqs.~(\ref{eq:s+a2}) to (\ref{eq:cvis+a2}),
and show generally good agreement with the numerical data,
in particular for $\Delta t\simeq 1$ and large $n$.
For smaller time step $\Delta t^*=0.1$, the most significant
deviations between numerical and analytical results are found for the
kinetic viscosity $\eta_{\rm {kin}}$, both for MPC-AT$-a$ and MPC-AT$+a$,
as shown in Fig.~\ref{fig:visatn}(a).
Similar deviations between analytical and numerical results for
$\eta_{\rm {kin}}$ have been observed for DPD in Refs.~\cite{mast99,nogu07a},
and have been explained by correlation effects between
collisions~\cite{mast99}.
At $\Delta t^*=0.1$, a pair of particles can collide sequentially several
times; in particular for $n\lesssim 1$, pairwise collision occur frequently
without involving any other particles.
Thus, the molecular-chaos assumption is weakly violated.
There are also deviations between analytical and numerical results for
the viscosity difference $\bar{\eta}_{\rm {col}}-\check{\eta}_{\rm {col}}$
of MPC$-a$ at small $\Delta t$ or small $n$ (see Fig.~\ref{fig:visyr}).
This is also caused by a violation of the molecular-chaos
assumption.

Angular-momentum conservation does not affect
the kinetic viscosity $\eta_{\rm {kin}}$ of MPC-AT in $2$D at large $n$,
compare Eqs.~(\ref{eq:kvis-a}) and (\ref{eq:kvisat+a}). Numerical results
are shown in Figs.~\ref{fig:visats}(a) and \ref{fig:visatn}(a).
However, the correction term in Eq.~(\ref{eq:c+a2}) predicts
a small difference of $\eta_{\rm {kin}}$ for MPC-AT$-a$ and MPC-AT$+a$ for
small $n\simeq 1$, see Figs.~\ref{fig:visats}(b) and \ref{fig:visatn}(a).
The viscosities $\eta_{\rm {kin}}$ and $\eta_{\rm {col}}$ of MPC-AT$-a$,
and $\eta_{\rm {col}}$ of MPC-AT$+a$ for large $n$,
show no dependence on the space dimension $d$ (except for the scale factor
$l_{\rm c}^{-d}$); therefore, the
corresponding symbols and lines in Fig.~\ref{fig:visats} coincide.

In two dimensions, MPC-SR with $\theta=\pi/2$ and MPC-RA with $\theta_0=\pi$
are characterized by $A=1$, and by $B=2$ and $B=1$, respectively.
Thus, they have the same collisional viscosity $\eta_{\rm {col}}$ for
both their `$-a$' and `$+a$' version, but a different
kinetic viscosity $\eta_{\rm {kin}}$, see Fig.~\ref{fig:vismpc}.
Although MPC-DR has the same viscosity of MPC-RA$+a$ theoretically,
the numerical data of MPC-DR shown in Fig.~\ref{fig:visats} display a
slightly larger deviation from the theoretical results for $\eta_{\rm {col}}$
and a smaller deviation for $\eta_{\rm {kin}}$ than the data of MPC-RA$+a$.

Eq.~(\ref{eq:kvis0}) together with (\ref{eq:c+a2}) predicts
a minimum of $\eta_{\rm {kin}}$ around $n=1$, as shown in
Figs.~\ref{fig:visatn}(c) and \ref{fig:visld}(b).
However, this minimum is not seen in numerical data and
could be caused by the negligence of higher-order terms in Eq.~(\ref{eq:c+a}).
We therefore investigate the dependence of the next-order term
$h/{N_{\rm c}}^2$, where $h$ is a free parameter.
The average is estimated by
$h_{\rm m}=\sum_{k=3}^{\infty} P(k)h/kn \simeq
                 \{1-e^{-n}(1+n+n^2+n^3/6-n^4/72)\}h/n^2$,
which yields the asymptotic dependence $h_{\rm m}=hn^2/18$ for small $n$
and $h_{\rm m}=h/n^2$ for $n \to \infty$.
The correction term $h_{\rm m}$ is then added to
Eq.~(\ref{eq:c+a2}) with $h$ as a fit parameter.
Fig.~\ref{fig:visatn}(c) shows that this correction term with $h=-0.6$ in $2$D
and $h=-1$ in $3$D removes the minimum and gives better agreement
with the numerical data of MPC-AT$+a$.

Fig.~\ref{fig:dif} shows the self-diffusion constant $D$ of MPC-AT$\pm a$.
The `$+a$' fluid displays faster diffusion than the `$-a$' fluid.
The diffusion constant $D$ is numerically calculated
from the mean square displacement of a particle,
$\langle \{{\bf r}_i(t)-{\bf r}_i(0)\}^2\rangle = 2d D t$,
in a cubic simulation box with side length $L=20l_{\rm c}$.
Deviations from the analytical results calculated with the molecular-chaos
assumption are seen for small $\Delta t^*$.

\section{Dissipative Particle Dynamics (DPD)} \label{sec:dpd}

\subsection{Simulation Method}

The DPD thermostat is a modified Langevin thermostat,
where friction and noise forces are applied to the relative velocities
of pairs of neighboring particles \cite{hoog92,groo97,espa98}.
The equation of motion for the $i$-th particle with mass $m$ is given by
\begin{equation}
\label{eq:dpd+a}
m \frac{d {\bf v}_{i}}{dt} =
 - \frac{\partial U}{\partial {\bf r}_i}
+ \sum_{j\not=i} \left\{-w_{ij}{\bf v}_{ij}
   \cdot{\bf \hat{r}}_{ij} +
      \sqrt{w_{ij}}{\xi}_{ij}(t)\right\}{\bf \hat{r}}_{ij},
\end{equation}
where ${\bf v}_{ij}={\bf v}_{i}-{\bf v}_{j}$,
${\bf r}_{ij}= {\bf r}_{i}-{\bf r}_{j}$, and
${\bf \hat{r}}_{ij}={\bf r}_{ij}/{r}_{ij}$,
with weight $w_{ij}=w(r_{ij})$.
The Gaussian white noise ${\xi}_{ij}(t)$
obeys the fluctuation-dissipation theorem, with
$\langle \xi_{ij}(t) \rangle  = 0$ and
$\langle \xi_{ij}(t) \xi_{i'j'}(t')\rangle  =
         2 k_{\rm B}T (\delta _{ii'}\delta _{jj'}+\delta _{ij'}\delta _{ij'})
\delta(t-t')$.
This thermostat is applied only in the direction ${\bf \hat{r}}_{ij}$
to conserve the angular momentum. We denote this original method
here DPD$+a$.

In DPD, a linear weight function
$w_{ij}=w_{\rm 1}(r_{ij})=\gamma(1-r_{ij}/r_{\rm {cut}})$
is typically employed, which vanishes beyond the cutoff distance
$r_{ij}=r_{\rm {cut}}$.
Furthermore, DPD is usually combined with a soft repulsive
potential $U$; however, we only consider the ideal-gas equation state (with
potential $U=0$) in this paper.

The DPD equation~(\ref{eq:dpd+a}) is discretized by
the Shardlow's S1 splitting algorithm~\cite{shar03}, where
each thermostat of the $ij$ pair is integrated separately,
\begin{eqnarray}\label{eq:split+a}
{\bf v}_{i}^{\rm {new}} &=&
  {\bf v}_{i} + \{-a_{\rm {dp}}(r_{ij}){\bf v}_{ij} \cdot{\bf \hat{r}}_{ij}
    + b_{\rm {dp}}(r_{ij}){\xi}_{ij,n}\}{\bf \hat{r}}_{ij}, \nonumber \\
{\bf v}_{j}^{\rm {new}} &=&
   {\bf v}_{j} - \{-a_{\rm {dp}}(r_{ij}){\bf v}_{ij} \cdot{\bf \hat{r}}_{ij}
    + b_{\rm {dp}}(r_{ij}){\xi}_{ij,n}\}{\bf \hat{r}}_{ij}, \nonumber \\
\end{eqnarray}
with
\begin{equation}\label{eq:splitab}
a_{\rm {dp}}(r_{ij}) = \frac{w_{ij} \Delta t/m}{1+w_{ij} \Delta t/m},\ \
b_{\rm {dp}}(r_{ij})= \frac{\sqrt{w_{ij}\Delta t}/m}{1+ w_{ij}\Delta t/m}.
\end{equation}
The discretized Gaussian noise ${\xi}_{ij,n}$ is determined by the variance
$\langle \xi_{ij,n} \xi_{i'j',n'}\rangle  =
2 k_{\rm B}T (\delta _{ii'}\delta _{jj'}
                    + \delta _{ij'}\delta _{ij'}) \delta_{nn'}$.
This splitting algorithm belongs to the class of generalized
Lowe-Anderson thermostats~\cite{pete04},
because the factors $a_{\rm {dp}}(r_{ij})$ and $b_{\rm {dp}}(r_{ij})$
satisfy the relation $b_{\rm {dp}}=\sqrt{a_{\rm {dp}}(1-a_{\rm {dp}})/m}$
\cite{nogu07}.

DPD can be modified to remove angular-momentum conservation. We
denoted this technique here DPD$-a$. It has been introduced in
Ref.~\cite{nogu07} to explore the similarities and differences between
DPD and MPC methods.
In this case, the equation of motion reads \cite{nogu07}
\begin{equation}
\label{eq:dpd-a}
m \frac{d {\bf v}_{i}}{dt} =
 - \frac{\partial U}{\partial {\bf r}_i}
+ \sum_{j\not=i} \Big\{-w_{ij}{\bf v}_{ij}
 + \sqrt{w_{ij}}{\mbox{\boldmath $\xi$}}_{ij}(t)\Big\}.
\end{equation}
The splitting algorithm can also be applied to DPD$-a$ as
${\bf v}_{i}^{\rm {new}} = {\bf v}_{i} -a_{\rm {dp}}(r_{ij}){\bf v}_{ij}
+ b_{\rm {dp}}(r_{ij}){\mbox{\boldmath $\xi$}}_{ij,n}$.

The combination of DPD$+a$ and DPD$-a$, denoted `transverse DPD',
with an equation of motion determined by the difference of the
right-hand sides of Eqs.~(\ref{eq:dpd-a}) and (\ref{eq:dpd+a}),
has been suggested very recently~\cite{jung08}.
A similar anisotropic friction has been used in
the standard Langevin equation
to treat polymer entanglement implicitly in polymer melts~\cite{bird87}
and dilute polymer solutions~\cite{nogu00}.

The DPD thermostat can be generalized into a multibody thermostat
(denoted DPD-MT$-a$)~\cite{nogu07}, which is defined by the equation
of motion
\begin{eqnarray}\label{eq:dpdmt}
m \frac{d {\bf v}_{i}}{dt} &=& - \frac{\partial U}{\partial {\bf r}_i}
- w_i^{\rm 0} ({\bf v}_{i}-{\bf v}_{i}^{\rm G} )
 + \sqrt{w_i^{\rm 0}}{\mbox{\boldmath $\xi$}}_{i}(t) \\ \nonumber
&&+ \sum_{j\not=i} w_{ij}\Bigg\{({\bf v}_{j}-{\bf v}_{j}^{\rm G} )
- \frac{{\mbox{\boldmath $\xi$}}_{j}(t)}{\sqrt{w_j^{\rm 0}}}\Bigg\},
\end{eqnarray}
where $w_i^{\rm 0}=\sum_{j\not=i} w_{ij}$, and
${\bf v}_{i}^{\rm G}= \sum_{j\not=i} w_{ij}{\bf v}_{j}/w_i^{\rm 0}$ is
the weighted mean velocity.
The second term on the right-hand side of Eq.~(\ref{eq:dpdmt}) is the
friction term between the $i$-th particle and its neighbors,
and $N_{\rm {nb}}/2$ thermostats in Eq.~(\ref{eq:dpd-a})
are unified into a single thermostat,
where $N_{\rm {nb}}$ is the average number of the neighbors with
$r_{ij}<r_{\rm {cut}}$.
The third and fourth terms on the right-hand side of Eq.~(\ref{eq:dpdmt})
are needed to conserve the translational momentum.

Angular momentum can be conserved in DPD-MT,
when the thermostat for the $i$-th particle is applied only in the direction
${\bf r}_{i,{\rm G}}={\bf r}_i-{\bf r}_i^{\rm G}$,
where the weighted center of mass is
${\bf r}_{i}^{\rm G}= \sum_{j\not=i} w_{ij}{\bf r}_{j}/w_i^{\rm 0}$.
The equation of motion of DPD-MT$+a$ is thus given by
\begin{eqnarray}\label{eq:dpdmt+a}
m \frac{d {\bf v}_{i}}{dt} &=& - \frac{\partial U}{\partial {\bf r}_i}
+\Big\{- w_i^{\rm 0} ({\bf v}_{i}-{\bf v}_{i}^{\rm G} )
                               \cdot{\bf \hat{r}}_{i{\rm G}}
  + \sqrt{w_i^{\rm 0}}{\xi}_{i}(t)\Big\}{\bf \hat{r}}_{i{\rm G}}  \nonumber \\
&& + \sum_{j\not=i} w_{ij}\Bigg\{({\bf v}_{j}-{\bf v}_{j}^{\rm G} )
                               \cdot{\bf \hat{r}}_{j{\rm G}}
  - \frac{{\xi}_{j}(t)}{\sqrt{w_j^{\rm 0}}}\Bigg\}{\bf \hat{r}}_{j{\rm G}} .
\end{eqnarray}

Shardlow's S1 splitting algorithm~\cite{shar03} can
be applied to both DPD-MT$-a$ and DPD-MT$+a$.
Eq.~(\ref{eq:dpdmt}) of DPD-MT$-a$ is discretized such that
each thermostat of the $i,i^{\rm G}$ pair is integrated separately,
\begin{eqnarray}\label{eq:splitmt}
{\bf v}_{i}^{\rm {new}} &=& {\bf v}_{i} -a^{{\rm mt}}_i
  ({\bf v}_{i}-{\bf v}_{i}^{\rm G} ) + b^{{\rm mt}}_i{\mbox{\boldmath $\xi$}}_{i,n},\\
{\bf v}_{j}^{\rm {new}} &=& {\bf v}_{j} + \frac{w_{ij}}{w_i^{\rm 0}}
\Big\{ a^{{\rm mt}}_i
   ({\bf v}_{i}-{\bf v}_{i}^{\rm G} ) - b^{{\rm mt}}_i{\mbox{\boldmath $\xi$}}_{i,n} \Big\}.
\nonumber
\end{eqnarray}
The factors $a^{\rm {mt}}_i$ and $b^{\rm {mt}}_i$ are given by
\begin{equation}\label{eq:splitmtab}
a^{{\rm mt}}_i = \frac{w_i^{\rm 0} \Delta t/m}
                        {1+\nu_i w_i^{\rm 0}\Delta t/2m}, \
b^{{\rm mt}}_i = \frac{\sqrt{w_i^{\rm 0}\Delta t}/m}
                        {1+ \nu_i w_i^{\rm 0}\Delta t/2m},
\end{equation}
where $\nu_i= 1+ \sum_{j\not=i} {w_{ij}}^2/(w_i^{\rm 0})^2$.

\subsection{Transport Coefficients}

We now derive analytical expressions for the viscosity $\eta$ and
self-diffusion constant $D$ of DPD$-a$ and DPD-MT$\pm a$ with ideal-gas
equation of state (with potential $U=0$).
The corresponding derivations for DPD$+a$~\cite{nogu07a}
can be straightforwardly carried over to this case.

The correlations of DPD$\pm a$ results from a multitude of pairwise
collisions, so that
$1-s_{\rm m}=\langle\Pi_j s_{ij}\rangle$ and $1-c_{\rm m}=
\langle\Pi_j c_{ij}\rangle$.
Eq.~(\ref{eq:split+a}) together with a molecular-chaos assumption implies
$s_{ij}= 1- a_{\rm {dp}}{\hat{x}_{ij}}^2$,
$c_{ij}= 1- a_{\rm {dp}}({\hat{x}_{ij}}^2+{\hat{y}_{ij}}^2)
+ 4{a_{\rm {dp}}}^2{\hat{x}_{ij}}^2{\hat{y}_{ij}}^2$
for DPD$+a$, and
$s_{ij}= 1- a_{\rm {dp}}$,
$c_{ij}= 1- 2a_{\rm {dp}} + 2{a_{\rm {dp}}}^2$
for DPD$-a$.
For an ideal gas, the number of particles $k$ per volume $\Delta V$
is given by the Poisson distribution, $P(k)= e^{-n\Delta V}(n\Delta V)^k/k!$,
so that $\langle c^k\rangle=\exp\{(-1+c)n\Delta V\}$ for some constant $c$.
This implies $1-s_{\rm m}=\exp(-1+\sum_j \langle s_{ij}\rangle)$.

The collisional stress $\sigma_{xy}^{\rm {col}}$ is the momentum flux due
to DPD collisions crossing a plane at $y=y_0$.
After interchange of the order of integration,
$\sigma_{xy}^{\rm {col}}$ is given by
\begin{equation}\label{eq:cst+dp}
\sigma_{xy}^{\rm {col}} = -\frac{mn^2}{2\Delta t} \int d{\bf r}_{ij}\ (v_{i,x}^{\rm {new}}-v_{i,x})y_{ij},
\end{equation}
where Eq.~(\ref{eq:split+a}) and $\langle  v_{ij,x}\rangle=\dot\gamma y_{ij}$
have been used.
Thus, the diffusion constant and viscosity of DPD$+a$ are
given by Eq.~(\ref{eq:kvis0}) with \cite{nogu07a}
\begin{eqnarray}\label{eq:c+dp}
D &=& \frac{k_{\rm B}T\Delta t}{m} \label{eq:dif+dp}
      \left( \frac{1}{1-\exp(-n[a_{\rm {dp}}(r)]_g/d)} - \frac{1}{2} \right), \\
c_{\rm m} &=& 1 -
     \exp\left\{n \left[-\frac{2a_{\rm {dp}}(r)}{d} +
        \frac{4a_{\rm {dp}}(r)^2}{d(d+2)} \right]_g\right\}, \\
\label{eq:cvis+dp}
\eta_{\rm {col}}
 &=& \frac{n^2}{2d(d+2)} \left[\frac{w r^2}{1+w\Delta t/m} \right]_g,\\
&& [w]_g \equiv \int g(r)w(r) dV.
\end{eqnarray}
Similarly, for DPD$-a$, the viscosity and diffusion constant are found to be
\begin{eqnarray}\label{eq:c-dp}
D &=& \frac{k_{\rm B}T\Delta t}{m}
           \left( \frac{1}{1-\exp(-n[a_{\rm {dp}}(r)]_g)} - \frac{1}{2} \right),\\
c_{\rm m} &=& 1- \exp\left\{2n \left[-a_{\rm {dp}}(r)+ a_{\rm {dp}}(r)^2
                            \right]_g\right\}, \\
\eta_{\rm {col}}
 &=& \frac{n^2}{2d}\left[\frac{w r^2}{1+w\Delta t/m} \right]_g.
\end{eqnarray}
The only differences between the expressions for $D$, $c_m$ and
$\eta_{\rm {col}}$ in DPD$+a$ and DPD$-a$ are prefactors containing
$d$ and $d+2$.

To simplify the equations of DPD-MT,
the factors $a^{\rm {mt}}_i$ and $\nu_i$ are pre-averaged as
\begin{equation}\label{eq:splitmtam}
a_{\rm {m}} = \frac{n[w]_g \Delta t/m}{ 1+\nu_{\rm m} n[w]_g\Delta t/2m}, \ \
\nu_{\rm m}= 1+\frac{[w^2]_g}{n[w]_g^2}.
\end{equation}
Then $D$ and $\eta_{\rm {col}}$ of DPD-MT$+a$ are given by
\begin{eqnarray}\label{eq:c-mt+a}
D &=& \frac{k_{\rm B}T\Delta t}{m}
           \left\{ \frac{1}{1-\exp(- \nu_{\rm m}a_{\rm m}/d)} - \frac{1}{2} \right\},\\
c_{\rm m} &=& 1- \exp\left\{- \frac{2a_{\rm m}\nu_{\rm m}}{d} +
\frac{2{a_{\rm m}}^2{\nu_{\rm m}}^2}{d(d+2)} \right\}, \\
\eta_{\rm {col}}
 &=& \frac{n[w^2r^2]_g \ \ }{d(d+2)[w]_g (1+ \nu_{\rm m} n[w]_g\Delta t/2m  )}.
\end{eqnarray}
Finally, for DPD-MT$-a$, we find
\begin{eqnarray}\label{eq:c-mt-a}
D &=& \frac{k_{\rm B}T\Delta t}{m}
           \left\{ \frac{1}{1-\exp(- \nu_{\rm m} a_{\rm m})} - \frac{1}{2} \right\},\\
c_{\rm m} &=& 1- \exp(-2a_{\rm m}\nu_{\rm m} +  {a_{\rm m}}^2{\nu_{\rm m}}^2 ), \\
\eta_{\rm {col}}
 &=& \frac{n[w^2r^2]_g}{d[w]_g (1+\nu_{\rm m} n[w]_g\Delta t/2m  )}.
\end{eqnarray}

\begin{figure}
\includegraphics{fig9.eps}
\caption{\label{fig:visdpd}
(Color online)
Dependence of the viscosity $\eta$ on $\Delta t^*$ of DPD$\pm a$ and
DPD-MT$\pm a$ in three-dimensional space
for $n r_{\rm {cut}}^3=3$ and $\gamma\tau_{\rm 0}/m=9$.
Symbols represent the numerical data of DPD$+a$ ($\circ$),
DPD$-a$ ($\times$, $\Box$), DPD-MT$+a$ ($\triangle$), and DPD$-a$ ($\diamond$).
Dashed and solid lines represent analytical results for DPD$-a$ and other
DPD methods, respectively.
Error bars are smaller than the size of symbols.
}
\end{figure}

\subsection{Numerical Results}

Fig.~\ref{fig:visdpd} shows the viscosity of various DPD fluids
with an ideal-gas equation of state and the linear weight
$w_{\rm 1}(r_{ij})$.  The viscosity and time step are normalized
by $\eta_{\rm 0}=\sqrt{mk_{\rm B}T}/{r_{\rm {cut}}}^{d-1}$ and
$\tau_{\rm 0}=r_{\rm {cut}}\sqrt{m/k_{\rm B}T}$, respectively.
The dimensionless time step is $\Delta t^*=\Delta t/\tau_{\rm 0}$,
as before.
There is in general good agreement between analytical and numerical results.
However, small deviations are visible.
One reason for these deviations is that the molecular-chaos assumption
is not perfectly valid~\cite{mast99}.
In the case of DPD-MT$\pm a$, another reason is the pre-averaging procedure
used in the derivation of the analytical expressions, which neglects
some correlations.

The kinetic (collisional) viscosities of DPD$+a$ and DPD-MT$+a$ are larger
(smaller) than those of the `$-a$' versions, since
angular-momentum conservation reduces the momentum transfer in DPD collisions.
A similar behavior has also been found for MPC$\pm a$ in
Sec.~\ref{sec:mpc}.

\section{Thermostating Mesoscale Fluids under Flow}
\label{sec:thermo}

In experiments, systems are usually thermostated on their boundaries.
 However, in simulations,
thermostats typically act on all fluid particles in order to avoid
temperature gradients.  In flows, the temperature is defined under
the assumption of local equilibrium.
In the MPC and DPD families,
the length scales which define this ``local'' environment are
$l_{\rm c}$ and $r_{\rm {cut}}$, respectively.
On these scales, the thermal fluctuations should be separated from
the macroscopic flow, and the thermostats should act on the local
kinetic energy to fix the temperature.

The conditions on the shear rate $\dot\gamma$ for this local equilibrium
to hold are obtained as follows.
All of thermostats of the MPC family are profile-unbiased thermostats
(PUT)~\cite{evan86}.
Thus, the condition for a maximum shear rate of PUT~\cite{loos92} also
apply to MPC.
In simple shear flow with low Reynolds number,
the particle velocities are characterized by
$\langle{\bf v}_i({\bf r}_i)\rangle = \dot\gamma y_i {\bf e}_x$ and
$\langle {\bf v}_i({\bf r}_i)^2 \rangle = d k_{\rm B}T/m
                                         + \dot\gamma^2 {y_i}^2$.
In MPC, the particle velocity ${\bf v}_{i,{\rm c}}$ relative to the
center-of-mass velocity of a MPC collision cell is employed to calculate
the kinetic energy in the local rest frame,
\begin{equation}\label{eq:mpcT}
 \frac{1}{N_{\rm c}-1} \sum_{i \in \rm {cell}}
          \langle {{\bf v}_{i,{\rm c}}}^2 \rangle=
  \frac{d k_{\rm B} T}{m} + \frac{\dot\gamma^2 {l_{\rm c}}^2}{12},
\end{equation}
where the average is taken over all particles in a cell.
For $\dot\gamma l_{\rm c} \ll \sqrt{k_{\rm B} T/m}$,
the second term in Eq.~(\ref{eq:mpcT}) is negligible, and
the thermal fluctuations and shear are well separated. On the other
hand, for $\dot\gamma l_{\rm c} \gtrsim \sqrt{k_{\rm B} T/m}$,
the thermostats couple with the macroscopic flow and
may modify the flow behavior.

In DPD$+a$, the relative velocity ${\bf v}_{ij}$ of neighboring particles
is employed instead,
\begin{equation}\label{eq:dpdT}
\langle ({\bf v}_{ij}\cdot{\bf \hat{r}}_{ij})^2 \rangle=
  \frac{2k_{\rm B} T}{m} +  \frac{ [(\dot\gamma r\hat{x}\hat{y})^2 w]_g}{[w]_g}.
\end{equation}
For the linear weight $w_1(r_{ij})$ and uniform radial distribution
function $g(r)$, the second term in Eq.~(\ref{eq:dpdT}) is
$\{(d+1)/(d+2)^2(d+3)\}(\dot\gamma r_{\rm {cut}})^2$.
Thus, the condition for thermostats to provide local equilibrium conditions
is $\dot\gamma r_{\rm {cut}} \ll \sqrt{k_{\rm B} T/m}$.

To study the hydrodynamic behavior of complex fluids,
the parameter ranges of simulations should of course also
match physical conditions of experiments.
Thus, the simulation parameters have to be chosen such as to adjust
dimensionless hydrodynamic quantities, like
the Reynolds number, the Schmidt number, and the Knudsen number.

\section{Summary}

MPC and DPD are very versatile simulation techniques for mesoscale
hydrodynamics. By employing different types of collision rules and
thermostats, it is possible to construct a variety of algorithms with
different properies. One of the important properties is whether an
algorithm does or does not conserve angular momentum.
The angular momentum conservation can
be switched on or off in each variant of MPC and DPD.

In addition to MPC algorithms suggested previously, we have introduced
here an angular-momentum conserving version of the widely used
stochastic-rotation-dynamics algorithm of MPC. This algorithm has to
be used with some caution, because compared to other MPC$+a$ techniques,
it does not give a uniform radial distribution function. However, the
deviations are small for sufficiently large particle numbers per cell
and not too small time step.

We have derived analytical expressions for the viscosity $\eta$ and the
self-diffusion constant $D$ of various MPC and DPD methods.
The theoretical results show very good agreement with numerical results.
Many similarities between MPC and DPD are seen in the derivation of
$\eta$ and $D$ and the relation between the `$-a$' and `$+a$' versions,
We believe that these similarities apply generally for particle-based
hydrodynamics methods.

\begin{acknowledgments}
We thank T.~Ihle (North Dakota State University) and I.O.~G\"{o}tze for
helpful discussions.
Support of this work by the DFG through the SFB TR6, ``Physics of
Colloidal Dispersions in External Fields'', is gratefully acknowledged.
\end{acknowledgments}

\bibliographystyle{apsrev}

redpajama_set_name: RedPajamaArXiv


It can't end like that, can it? Yury Bykov's impressively bleak drama of bureaucracy, expedience, and the expendability of the poor seems to surge toward a finale that is both inevitable and impossible. The story, gutsy and wrenching, concerns a Russian plumber who susses out that a housing project in which some 800 people live will almost certainly collapse in a day or two. The film, Bykov's third and best, is set in the parody of capitalism that is the new Russia. Since it's in a grim realist mode, dwelling on domestic violence and the ravages of poverty, you'll likely expect that it can't possibly end well — that the forces that first stacked the poor up in that crumbling cinderblock won't profit from their being sprung from it.
But everything's complicated by that title card proclaiming that The Fool was completed with the aid of the Ministry of Culture of the Russian Federation. Everything we know about the land of Putin tells us that The Fool must end in catastrophe, with it being more convenient for the poor to be killed than for government officials' corruption to be exposed. But everything else we know about Putin's Russia tells us that its Ministry of Culture would never let that ending out into the world. As in many great films from Iran or China, the tension here is twofold: Will the system in which the film was made compromise the truth of that story?
Will the system in which the film was made compromise the truth of its story?
Here's a spoiler for this brave and extraordinary work of fabulistic social realism: The answer to that last question is no. If anything, The Fool goes further in its indictment than audiences might be willing to follow. It's easy to cheer an attack on a system that punishes the poor and impoverishes them more, but Bykov is unsparing in his depictions of that system's victims. The men of that housing project tend to be fists-first bruisers, the women helpless and embittered — and, with some despair, the film finds those traits in most of its characters, flush or broke.
Much of the film transpires in and on top of the building. Dima (Artem Bystrov), the regular-guy plumber, dashes through the hallways, showing officials the damage but trying not to upset the residents, a sorry population of drunks and addicts wary of anyone who seems to have some power. Bykov makes a telling contrast between the fleet, you-are-there photography of Dima's trips through the complex and an extended sequence at a party of the local swells and bureaucrats responsible for the looming tragedy. In a banquet hall of white tablecloths, thumping bass, and drunken profiteers, Dima beseeches Nina Galaganova (Natalya Surkova) — the mayor of this small, unnamed town — to take action to save these lives. After much skepticism, Nina agrees, only to then confront the truth of what that action might cost her: not just potential exposure and criminal proceedings, but the chance that she will no longer be one of Russia's haves — that she might be reduced to living like the people whose existence now depends on her. Surkova's performance is a marvel of anguish and realpolitik, and it's the heart of the film. Bykov's moral tale is clear-eyed and callused over, worrying not over individual lives but over a nation's soul. Its chief question, though, should resonate with anyone enjoying anything more than subsistence-level living: How much suffering will you accept to keep you from being one of them?


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Community Relations Overview
Automated Phone Calls and Emails
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Annual Hazardous Materials Notification for Parents and Staff
Career and Technical Education Non-Discrimination Policy
Student Discrimination, Harassment or Bullying Reporting and Investigation Flow Chart
Student Discrimination, Harassment or Bullying Reporting Form
EC Children Records Destruction Notice
Non-Discrimination Statement and Title IX
Requests for Proposals (RFP)
Student Directory Information Declination Form
CHCCS About Community Relations
The Public Information Office serves as the communication link between the district and the families whose children attend our schools.
The staff issues regular press releases to local media about district events, publishes news articles, designs special publications and calendars, schedules Blackboard Connect calls to remind parents of key district activities, maintains the content on the district website and our social media sites, and sends regular news updates to employees and the community through its mailing lists.
The staff also helps to coordinate crisis communication during inclement weather or school safety situations.
Contact Public Relations
Andy Jenks
ajenks@chccs.k12.nc.us
Robin Nucilli
rnucilli@chccs.k12.nc.us
sphillips@chccs.k12.nc.us
CHCCS Publications
Points of Pride (PoP)
CHCCS PoP 2019_20.pdf 2.37 MB (Last Modified on July 7, 2021)
District Headlines
East Chapel Hill's Brian Link Named 2021-22 Regional Teacher of the Year
Brian Link, Social Justice Civics and Economics teacher at East Chapel Hill High School, and 2021-22 Chapel Hill-Carrboro City Schools Teacher of the Year, has been named the Burroughs-Wellcome Fund North Central Teacher of the Year winner, one of only nine North Carolina Regional Teachers of the Year.
Focusing on Staff Wellness at FPGB in the "Zen Den"
"Zen Den". The words conjure all sorts of possibilities. If you close your eyes, imagine the calm and quiet, then you've begun to envision the newly reconfigured space at Frank Porter Graham Bilingüe Elementary School. The FPGB "Zen Den" is now open all day, every day for teachers and staff to seize a few minutes of respite and serenity.
National Merit Scholar Semifinalists Announced for 2022
On September 15, the National Merit Scholarship Corporation announced approximately 16,000 Semifinalists in the National Merit Scholarship Program. Twenty-eight Chapel Hill-Carrboro City Schools students earned Semifinalist awards, representing Carrboro High School, Chapel Hill High School and East Chapel Hill High School.
For the 15th Year, the Public School Foundation and East Rotary Provide School Supplies to Teachers
For 15 years, the Chapel Hill-Carrboro Public School Foundation and East Chapel Hill Rotary Club have partnered to kick off the school year by hosting a supply store with much needed school supplies for teachers. The past two years have brought many challenges as the event has been reconfigured as COVID-safe. But on the morning of August 19, a team of PSF and Rotary members and community volunteers carried out the annual event of giving school supplies to grateful teachers across the district.
Chapel Hill HS "Proconian" Publishes PSF-Funded Series
A 2019 Public School Foundation grant awarded to Chapel Hill High School journalism teacher, Stephen Head, serves as a perfect example of the flexibility required for so many projects impacted by the pandemic. Head wrote the grant application to bring in a professional journalist as a guest teacher, with the goal of writing a series of "Proconian" stories about the lives of the workers constructing the new buildings at CHHS. More than 18 months after the original project was funded, the series, "Covid-19 Stories," became one about how the coronavirus pandemic affected various individuals.
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text:
Fit India Movement is a nation-wide movement in India to encourage people to remain healthy and fit by including physical activities and sports in their daily lives. Fit India Movement was launched by Prime Minister of India Narendra Modi at Indira Gandhi Stadium in New Delhi on 29 August 2019 (National Sports Day). Fit India was founded by Shri Suparno Satpathy in 1993

The campaign has a "Fitness Pledge" that reads, I promise to myself that I will devote time for physical activity and sports every day and I will encourage my family members and neighbours to be physically fit and make India a fit nation.

Execution

Government

A committee has been formed to advice government on this campaign. It is composed of various government officials, members of Indian Olympic Association, national sports federations, private bodies and fitness promoters.

As per union HRD department directive states and union territories can request for funds as per guidelines to procure listed fitness items for children studying in government schools. The equipment purchased from the grants are to be maintained in workable condition by the authorities concerned. It is also mandated to keep record of working, repairable and damaged beyond repair equipment. The schools are also allowed to include their traditional and regional games.

Private schools
In November 2019, the PM announced Fit India grading for schools that will rank them in order of the fitness. Eligible schools can apply for ranking and once confirmed they will be allowed to use the Fit India logo and flag.

Institutions & Individuals

There are representatives of private bodies such as Confederation of Indian Industries (CII), Sequoia Fitness and Sports Technology Fitness365 Reliance Foundation, JSW Cement and JSW Paints, SE TransStadia Pvt. Ltd., Tata Trust, ASSOCHAM India, Federation of Indian Chambers of Commerce & Industry (FICCI),government institutions like Ministry Of Defence, Sainik School Society along with Jaiganesh actors Shilpa Shetty and Milind Soman etc. as well as Fitness Apps like the SS app.Nitin is god

On 6 September 2019, Lok Sabha speaker Shree Omprakash Krishna Birla conducted a Fit India session in the parliament complex and invited upon all MPs to pursue the movement in their constituencies.

References

Health campaigns
Health education in India
Modi administration initiatives

redpajama_set_name: RedPajamaWikipedia


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Exhibit G for Ordinance 6064
26.88.208 - Wind Energy Special Use Standards
Purpose. This section establishes standards for the siting and operation of wind energy systems and facilities. This section is intended to implement the requirements of Government Code section 65892.13, while protecting the scenic and natural resources of the county and the health, safety and welfare of its residents to the extent permitted by law.
Applicability. These standards apply to all wind energy systems and facilities as allowed by the base zoning district as shown in Table 1 subject to the General Development standards for Renewable Energy Facilities in Section 26.88.200.
Exempt Accessory Wind Energy Systems. In any zoning district, accessory wind energy systems that are attached to a wall, roof or structural member of a legally established building are exempt from the development standards set forth herein, subject to the height and setback restrictions of the underlying zoning district.
Limitations on Location and Use. Wind energy systems and facilities shall not be located on a site that is:
Within a scenic corridor identified by the open space element of the General Plan;
Within a special studies zone established in compliance with the Alquist-Priolo Earthquake Fault Zoning Act;
Subject to a conservation easement established in compliance with Civil Code Section 815 et seq., that prohibits wind energy systems or facilities;
Subject to an open space easement established in compliance with Government Code Section 51070 et seq., that prohibits wind energy systems or facilities;
Subject to an agricultural conservation easement established in compliance with Government Code Section 10200 et seq., that prohibits wind energy systems or facilities;
Subject to a Williamson Act contract established in compliance with Government Code Section 51200 et seq., that prohibits small wind energy systems or facilities; or
Listed in the National Register of Historic Places, or the California Register of Historic Resources, in compliance with Public Resources Code Section 5024.1, or contains a structure that is so listed.
Development Standards. The following standards shall apply to wind energy systems and facilities, in addition to the General Development Standards for Renewable Energy set forth in 26.88.200, unless otherwise exempt:
The system's tower shall be set back a minimum distance equal to the height of the tower from all parcel lines, and a minimum distance of ten (10 ′) feet from any other structure on the parcel on which the system is located. On parcels of ten (10) acres or more, the parcel line setback may be reduced if the applicant demonstrates that:
Because of topography, strict adherence to the setback requirement would result in greater visibility of the system's tower than a reduced setback, and
The system's tower is set back a minimum distance equal to the height of the tower from any structure on adjoining parcels;
The system's tower and supporting structures shall comply with any applicable fire setback requirements in the fire safe standards (Chapter 13, Article V of this code);
The system's tower shall not exceed a maximum height of forty (40 ′) feet on a parcel of less than one (1) acre, a maximum of sixty-five (65′) feet on a parcel of one (1) to less than five (5) acres, and maximum height of eighty (80′) feet on a parcel of five (5) acres or more, unless a use permit is obtained;
The system's tower shall be set back from and not project above the top of any visually prominent ridgeline;
The system's tower shall not significantly impair a scenic vista from a county- designated or state-designated scenic corridor;
The system's tower shall be located and screened by landforms, natural vegetation or other means to minimize visual impacts on neighboring residences and public roads, public trails and other public areas;
The system's tower and supporting structures shall be painted a single, neutral, non-reflective, non-glossy (for example, earth-tones, gray, black) that, to the extent possible, visually blends the system with the surrounding natural and built environments.
The system's turbine shall be approved by the California Energy Commission as qualifying under the Emerging Renewables Fund of the commission's Renewables Investment Plan or certified by a national program recognized and approved by the commission;
The system shall be designed and constructed in compliance with the Uniform Building Code and National Electric Code. The safety of the design and construction shall be certified by a California-licensed mechanical, structural or civil engineer;
The system shall comply with all applicable Federal Aviation Administration requirements, including Subpart B (commencing with Section 77.11) of Part 77 of Title 14 of the Code of Federal Regulations regarding installations close to airports, and the State Aeronautics Act (Part 1 (commencing with Section 21001) of Division 9 of the Public Utilities Code);
The system shall be equipped with manual and automatic over speed controls. The conformance of rotor and over speed control design and fabrication to good engineering practices shall be certified by a California-licensed mechanical, structural or civil engineer;
The system's tower-climbing apparatus and blade tips shall be no closer than fifteen (15) feet from ground level unless the system is enclosed by a six (6′)-foot high fence;
The system's utility lines shall be underground where economically practical;
Where vegetation is removed in the construction of the system or an access road to the system, landscaping shall be planted to minimize visual impacts, avoid erosion and maintain stability of soils;
The system shall be operated such that no electro-magnetic interference is caused;
No more than one (1) accessory system shall be allowed on a parcel;
Decibel levels generated by the system shall not exceed the maximum noise levels applied pursuant to the noise element of the general plan, except during short-term events including utility outages and severe wind storms;
Brand names or advertising associated with the system or the system's installation shall not be visible from any public place;
Signs warning of high voltage electricity shall be posted on stationary portions of the system's tower and any supporting structures, and at gated entry points to the site at a height of five (5) feet above the ground;
Upon abandonment or termination of the system's use, the entire facility, including the system's tower, turbine, supporting structures and all equipment, shall be removed and the site shall be restored to its pre-construction condition or other authorized use.
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text:
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	public function getComments()
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	}

	public function getUserPointActionDescription()
	{
		return $this->getComments();
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}

redpajama_set_name: RedPajamaGithub


In Hobby Lobby and Conestoga, the Supreme Court must uphold the Religious Freedom Restoration Act.
These are the words of then-Representative Charles Schumer as he championed his bill, the Religious Freedom Restoration Act (RFRA), on the floor of the House in 1993.
#ad#This month, the Supreme Court has a second shot at rectifying its decision in Smith — this time, with the aid of RFRA — when it considers two challenges to the HHS mandate, from Hobby Lobby and Conestoga Wood.
The greatness of our Constitution lies in its design to protect a multiplicity of opinions from being silenced in favor of the agendas of the reigning governmental power. By passing RFRA, Congress ensured that the government could not limit the exercise of religious beliefs without clearing a significant hurdle: the burden of proving that the measure serves a compelling government interest and cannot be met through a less-restrictive method.
Freedom of conscience extends well beyond where a person worships. It encompasses a person's whole being — who he is, how he acts, and the daily decisions he makes. It can include nothing less than the way a person lives all aspects of her life, from her private life to her public life and commercial conduct.
The First Amendment guarantees that we have freedom to live our lives according to our religious beliefs and moral convictions, free from government coercion. An individual does not abandon this freedom simply because he enters the stream of commerce. And yet, that is exactly what the administration is trying to force on businesses like Hobby Lobby and Conestoga under the HHS mandate.
The importance of protecting this standard is demonstrated by the more than 50 amicus briefs — nearly three times as many as filed by the opposition — in favor of Hobby Lobby and Conestoga Wood, from sources including 107 members of Congress, Democrats and Republicans alike, who have urged the Supreme Court to strike down the HHS mandate as unconstitutional because the government cannot meet its required burden of proof under RFRA.
— John Boozman is the junior U.S. senator from Arkansas, and J. Randy Forbes represents Virginia's fourth congressional district in the House of Representatives.


The Americana Troupe (AT) plays and sings a wide variety of gospel, old-time, bluegrass and country-flavored songs for Trinity worship services, area senior centers and other organizations in the greater Warren and Morris County area. The AT enjoys arranging favorite hymns for listeners to sing and tap their toes to. Acoustic, stringed instruments and an accordion are among the instruments played by the troupe.


text:
Adaptive Insights Closes $75 Million Investment to Power New Era of Corporate Performance Management for Modern Finance Leaders
PALO ALTO, C.A.—Adaptive Insights, the worldwide leader in cloud corporate performance management (CPM)for the biggest brands and the hottest companies, today announced that the company has secured $75 million in funding to accelerate product roadmap innovation and global expansion. With its cloud-based finance solution, Adaptive provides market-changing opportunities both to Excel users and companies that rely on legacy on-premises solutions. With 2,700 customers, more than three times the customers of all other cloud CPM vendors combined, Adaptive helps high-growth companies scale through IPO and beyond, while also enabling blue chip enterprises to reinvent core processes to outpace the competition.
New investor JMI Equity joins the round along with existing investors Norwest Venture Partners, ONSET Ventures, Bessemer Venture Partners, Cardinal Venture Capital, Monitor Ventures, and Information Venture Partners. JMI Equity is focused on high-growth enterprise software companies and brings a track record of success in scaling cloud market leaders, including Eloqua and ServiceNow.
"Financial and operational management is the cornerstone for every successful company. Adaptive leads this market through tremendous product innovation, strong executive leadership, and a rapidly growing, fanatical customer base. These characteristics define high performing SaaS companies," said Peter Arrowsmith, general partner at JMI Equity. Arrowsmith will be joining Adaptive as a Board Observer.
CEO Tom Bogan, experienced enterprise software executive and chairman of the board at Citrix Systems (CTXS), is at the helm of Adaptive and is applying his 25-year success record in building and growing large, publicly traded enterprise and cloud software companies.
"CFOs are increasingly taking center stage in the strategy and direction of their organizations and looking for the right technology to provide real-time data and powerful insights to gain a 360-degree view of their business," commented Bogan. "Adaptive invented the cloud CPM market and is in an unparalleled position to help finance leaders take their planning and analysis to the next level, particularly as they continue to innovate, drive company growth, and manage business transformation."
The recent funding is further validation of Adaptive's success trajectory, which includes:
Growing over 50% year-over-year growth in total annual recurring revenue (ARR) bookings, driven by new midsize and enterprise customers and expanded deal sizes globally.
Expanding international growth of 75% YoY in Q1.
Largest ecosystems of 200+ partners, with partners contributing to half of Adaptive's new ARR across 16 different countries.
Enhanced cloud-based Adaptive product suite focused on performance, scalability and a feature set targeting enterprise customers that includes rich integration capabilities. And recently announced Adaptive Solutions is a new approach that leverages a combination of model templates, packaged content and best practice professional services gained through thousands of successful customer implementations. Adaptive Revenue, the first Adaptive Solution, differentiates by being the only industry solution that brings together sales, finance, and service stakeholders on a unified cloud CPM and business intelligence (BI) platform for planning, reporting, and analysis of the entire revenue lifecycle.
Industry validation with the Highest Ability to Execute score of any vendor in the Visionaries Quadrant in Gartner's 2015 CPM Magic Quadrant.
Adaptive is seeing tremendous adoption among large enterprises across multiple sectors, including American Family Insurance, Coca-Cola, Kendall-Jackson Family Wines, Philips, Southwest Gas, and University of Arizona. Adaptive also recently surpassed the 500-customer mark for high-growth technology companies, including FireEye, LinkedIn, and ZenDesk.
About Adaptive Insights
Adaptive Insights is the leader in cloud corporate performance management (CPM). Via its software as a service (SaaS) platform, the company offers capabilities for budgeting, forecasting, reporting, consolidation, dashboards, and business intelligence that empower finance, sales, and other business leaders with insight to drive true competitive advantage. The Adaptive Suite is sold direct or is available through Adaptive's robust cloud CPM channel ecosystem of 200+ partners, including Accenture, Armanino, BDO, Cohn Reznick, Intacct, KPMG, McGladrey, and Plex Systems. NetSuite also offers Adaptive Insights as its NetSuite Financial Planning Module.
More than 2,700 companies in 85 countries use Adaptive Insights. These range from mid-sized companies and nonprofits to large corporations, including AAA, Boston Scientific, CORT, Konica Minolta, NetSuite, Philips, and Siemens. Adaptive Insights is headquartered in Palo Alto, Calif. For more information, visit www.AdaptiveInsights.com, the Adaptive Insights Blog, and follow Adaptive on LinkedIn, Twitter, Facebook, and YouTube.
JMI Equity is a growth equity firm focused on investing in leading software and services companies. Founded in 1992, JMI has invested in over 115 businesses in its target markets and has raised more than $3.1 billion of committed capital. Combining domain expertise and active partnership, JMI works with management teams to build market-leading businesses and create long-term shareholder value. Realized investments from JMI funds include Autotask, Axeda, BigMachines, Eloqua, Gemcom, Halogen and ServiceNow. www.jmi.com
For Adaptive Insights:
Emily Borders
Highwire PR
PH: 415-963-4174 ext. 1
For JMI:
Allyson Vento
Abernathy MacGregor Group