\subsection{Gravitational softening}
\label{ssec:potential_softening}

To avoid artificial two-body relaxation, the Dirac
$\delta$-distribution corresponding to each particle is convolved with
a softening kernel of a given fixed, but time-variable, scale-length
$H$. Instead of the commonly used spline kernel of
\cite{Monaghan1985} (e.g. in \textsc{Gadget}), we use a C2 kernel
\citep{Wendland1995} which leads to an expression for the force that
is cheaper to compute and has a very similar overall shape. The C2
kernel has the advantage of being branch-free leading to an expression
which is faster to evaluate using vector units available on modern
architectures; it also does not require any divisions to evaluate the
softened forces. We set $\tilde\delta(\mathbf{r}) = \rho(|\mathbf{r}|)
= W(|\mathbf{r}|, 3\epsilon_{\rm Plummer})$, with $W(r, H)$ given by

\begin{align}
W(r,H) &= \frac{21}{2\pi H^3} \times \nonumber \\
&\left\lbrace\begin{array}{rcl}
4u^5 - 15u^4 + 20u^3 - 10u^2 + 1 & \mbox{if} & u < 1,\\
0 & \mbox{if} & u \geq 1,
\end{array}
\right.
\end{align}
and $u = r/H$. The potential $\varphi(r,H)$ corresponding to this density distribution reads
\begin{align}
\varphi(r,H) = 
\left\lbrace\begin{array}{rcl}
f(\frac{r}{H}) \times H^{-1} & \mbox{if} & r < H,\\
r^{-1} & \mbox{if} & r \geq H,
\end{array}
\right.
\label{eq:fmm:potential}
\end{align}
with $f(u) \equiv -3u^7 + 15u^6 - 28u^5 + 21u^4 - 7u^2 + 3$. These
choices lead to a potential at $|\mathbf{x}| = 0$ equal to the central
potential of a Plummer sphere (i.e. $\varphi(0) = 1/\epsilon_{\rm
  Plummer}$)\footnote{Note the factor $3$ in the definition of
  $\rho(|\mathbf{x}|)$ which differs from the factor $2.8$ used for
  the cubic spline kernel as a consequence of the change of the functional
  form of $W$.}. From this expression the softened gravitational force can
be easily obtained:
\begin{align}
\mathbf{\nabla}\varphi(r,H) = \mathbf{r} \cdot
\left\lbrace\begin{array}{rcl}
g(\frac{r}{H}) \times H^{-3} & \mbox{if} & r < H,\\
r^{-3} & \mbox{if} & r \geq H,
\end{array}
\right.
\label{eq:fmm:force}
\end{align}
with $g(u) \equiv f'(u)/u = -21u^5+90u^4-140u^3+84u^2-14$. This last
expression has the advantage of not containing any divisions or
branching (besides the always necessary check for $r<H$), making it
faster to evaluate than the softened force derived from the
\cite{Monaghan1985} spline kernel. Note also, the useful expression
for the norm of the forces:
\begin{align}
|\mathbf{\nabla}\varphi(r,H)| = 
\left\lbrace\begin{array}{rcl}
f'(\frac{r}{H}) \times H^{-2} & \mbox{if} & r < H,\\
r^{-2} & \mbox{if} & r \geq H.
\end{array}
\right.
\label{eq:fmm:force_norm}
\end{align}
The softened density profile, its corresponding potential and
resulting forces are shown on Fig. \ref{fig:fmm:softening} (for more
details about how these are constructed see section 2
of~\cite{Price2007}). Finally, we also compute the change in potential
due to a change in softening length:
\begin{align}
\frac{\partial}{\partial H}\varphi(r,H) = 
\left\lbrace\begin{array}{rcl}
k(\frac{r}{H}) \times H^{-2} & \mbox{if} & r < H,\\
0 & \mbox{if} & r \geq H.
\end{array}
\right.
\label{eq:fmm:potential_h_derivative}
\end{align}
where $k(u)=-24u^7+105u^6-168u^5+105u^4-21u^2$. This term enters the
equation of motion when adaptive softening is used for SPH
\citep[e.g.][]{Price2007}. For comparison purposes, we also implemented the
more traditional spline-kernel softening in \swift.
\begin{figure}
\includegraphics[width=\columnwidth]{potential.pdf}
\caption{The density (top), potential (middle) and forces (bottom)
  generated py a point mass in our softened gravitational scheme.  A
  Plummer-equivalent sphere is shown for comparison. The spline kernel
  of \citet{Monaghan1985} is also depicted but note that it has not
  been normalised to match the Plummer-sphere potential at $r=0$ (as
  is done in simulations) but rather normalised to the Newtonian
  potential at $r=H$ to better highlight the differences in shapes.}
\label{fig:fmm:softening}
\end{figure}
Users specify the value of the Plummer-equivalent softening
$\epsilon_{\rm Plummer}$ in the parameter file.

\subsubsection{Interaction of bodies with different softening lengths}

\textcolor{red}{MORE WORDS HERE.}\\