Changed the variable name for the potential's Green function from \phi to \varphi.

theory/Multipoles/fmm_summary.tex

@@ -6,7 +6,7 @@ evaluate for each particle in a system the potential and associated
 forces generated by all the other particles. Mathematically, this means
 evaluate
 \begin{equation}
-  \phi(\mathbf{x}_a) = \sum_{b \neq a} G m_b\phi(\mathbf{x}_a -
+  \phi(\mathbf{x}_a) = \sum_{b \neq a} G m_b\varphi(\mathbf{x}_a -
   \mathbf{x}_b)\qquad \forall~a\in N
   \label{eq:fmm:n_body}
 \end{equation}
@@ -23,10 +23,11 @@ arising from nearby particles. \\
 
 In what follows, we use the compact multi-index notation of
 \cite{Dehnen2014} (repeated in appendix \ref{sec:multi_index_notation}
-for completeness) to simplify expressions. $\mathbf{k}$, $\mathbf{m}$
-and $\mathbf{n}$ are multi-indices and $\mathbf{r}$, $\mathbf{R}$,
-$\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$ are vectors, whilst $a$
-and $b$ are particle indices.\\
+for completeness) to simplify expressions and ease
+comparisons. $\mathbf{k}$, $\mathbf{m}$ and $\mathbf{n}$ are
+multi-indices and $\mathbf{r}$, $\mathbf{R}$, $\mathbf{x}$,
+$\mathbf{y}$ and $\mathbf{z}$ are vectors, whilst $a$ and $b$ are
+particle indices.\\
 
 \begin{figure}
 \includegraphics[width=\columnwidth]{cells.pdf}
@@ -46,34 +47,34 @@ with centres of mass $\mathbf{z}_A$ and  $\mathbf{z}_B$
 respectively, as shown on Fig.~\ref{fig:fmm:cells}, the potential
 generated by $b$ at the location of $a$ can be rewritten as
 \begin{align}
-  \phi(\mathbf{x}_a - \mathbf{x}_b)
-  &= \phi\left(\mathbf{x}_a - \mathbf{z}_A - \mathbf{x}_b +
+  \varphi(\mathbf{x}_a - \mathbf{x}_b)
+  &= \varphi\left(\mathbf{x}_a - \mathbf{z}_A - \mathbf{x}_b +
   \mathbf{z}_B + \mathbf{z}_A - \mathbf{z}_B\right)  \nonumber \\
-  &= \phi\left(\mathbf{r}_a - \mathbf{r}_b + \mathbf{R}\right)
+  &= \varphi\left(\mathbf{r}_a - \mathbf{r}_b + \mathbf{R}\right)
   \nonumber \\
   &= \sum_\mathbf{k} \frac{1}{\mathbf{k}!} \left(\mathbf{r}_a -
-  \mathbf{r}_b\right)^{\mathbf{k}} \nabla^{\mathbf{k}}\phi(\mathbf{R})
+  \mathbf{r}_b\right)^{\mathbf{k}} \nabla^{\mathbf{k}}\varphi(\mathbf{R})
   \nonumber \\
   &= \sum_\mathbf{k} \frac{1}{\mathbf{k}!} \sum_{\mathbf{n} <
     \mathbf{k}} \binom{\mathbf{k}}{\mathbf{n}} \mathbf{r}_a^{\mathbf{n}}
   \left(-\mathbf{r}_b\right)^{\mathbf{k} - \mathbf{n}}
-  \nabla^{\mathbf{k}}\phi(\mathbf{R})\nonumber \\
+  \nabla^{\mathbf{k}}\varphi(\mathbf{R})\nonumber \\
   &= \sum_\mathbf{n} \frac{1}{\mathbf{n}!} \mathbf{r}_a^{\mathbf{n}}
   \sum_\mathbf{m} \frac{1}{\mathbf{m}!}
-  \left(-\mathbf{r}_b\right)^\mathbf{m} \nabla^{\mathbf{n}+\mathbf{m}} \phi(\mathbf{R}),
+  \left(-\mathbf{r}_b\right)^\mathbf{m} \nabla^{\mathbf{n}+\mathbf{m}} \varphi(\mathbf{R}),
 \end{align}
-where we used the Taylor expansion of $\phi$ around $\mathbf{R} \equiv
+where we used the Taylor expansion of $\varphi$ around $\mathbf{R} \equiv
 \mathbf{z}_A - \mathbf{z}_B$ on the third line, used $\mathbf{r}_a
 \equiv \mathbf{x}_a - \mathbf{z}_A$, $\mathbf{r}_b \equiv \mathbf{x}_b
 - \mathbf{z}_B$ throughout and defined $\mathbf{m} \equiv
 \mathbf{k}-\mathbf{n}$ on the last line. Expanding the series only up
 to order $p$, we get
 \begin{equation}
-  \phi(\mathbf{x}_a - \mathbf{x}_b) \approx \sum_{\mathbf{n}}^{p}
+  \varphi(\mathbf{x}_a - \mathbf{x}_b) \approx \sum_{\mathbf{n}}^{p}
   \frac{1}{\mathbf{n}!} \mathbf{r}_a^{\mathbf{n}} \sum_{\mathbf{m}}^{p
     -|\mathbf{n}|} 
   \frac{1}{\mathbf{m}!} \left(-\mathbf{r}_b\right)^\mathbf{m}
-  \nabla^{\mathbf{n}+\mathbf{m}} \phi(\mathbf{R}),
+  \nabla^{\mathbf{n}+\mathbf{m}} \varphi(\mathbf{R}),
   \label{eq:fmm:fmm_one_part}
 \end{equation}
 with the approximation converging as $p\rightarrow\infty$ towards the
@@ -82,13 +83,13 @@ correct value provided $|\mathbf{R}|<|\mathbf{r}_a +
 combine their contributions to the potential at location
 $\mathbf{x}_a$ in cell $A$, we get
 \begin{align}
-  \phi_{BA}(\mathbf{x}_a) &= \sum_{b\in B} m_b\phi(\mathbf{x}_a -
+  \phi_{BA}(\mathbf{x}_a) &= \sum_{b\in B}G m_b\varphi(\mathbf{x}_a -
   \mathbf{x}_b)  \label{eq:fmm:fmm_one_cell}  \\
-  &\approx \sum_{\mathbf{n}}^{p}
+  &\approx G\sum_{\mathbf{n}}^{p}
   \frac{1}{\mathbf{n}!} \mathbf{r}_a^{\mathbf{n}} \sum_{\mathbf{m}}
     ^{p -|\mathbf{n}|}
   \frac{1}{\mathbf{m}!} \sum_{b\in B} m_b\left(-\mathbf{r}_b\right)^\mathbf{m}
-  \nabla^{\mathbf{n}+\mathbf{m}} \phi(\mathbf{R}) \nonumber. 
+  \nabla^{\mathbf{n}+\mathbf{m}} \varphi(\mathbf{R}) \nonumber. 
 \end{align}
 This last equation forms the basis of the FMM. The algorithm
 decomposes the equation into three separated sums evaluated at
@@ -106,12 +107,12 @@ compute the second kernel, \textsc{M2L} (multipole to local
 expansion), which corresponds to the interaction of a cell with
 another one:
 \begin{equation}
-  \mathsf{F}_{\mathbf{n}}(\mathbf{z}_A) = \sum_{\mathbf{m}}^{p -|\mathbf{n}|}
+  \mathsf{F}_{\mathbf{n}}(\mathbf{z}_A) = G\sum_{\mathbf{m}}^{p -|\mathbf{n}|}
   \mathsf{M}_{\mathbf{m}}(\mathbf{z}_B)
   \mathsf{D}_{\mathbf{n}+\mathbf{m}}(\mathbf{R}), \label{eq:fmm:M2L} 
 \end{equation}
 where $\mathsf{D}_{\mathbf{n}+\mathbf{m}}(\mathbf{R}) \equiv
-\nabla^{\mathbf{n}+\mathbf{m}} \phi(\mathbf{R})$ is an order $n+m$
+\nabla^{\mathbf{n}+\mathbf{m}} \varphi(\mathbf{R})$ is an order $n+m$
 derivative of the potential. This is the computationally expensive
 step of the FMM algorithm as the number of operations in a naive
 implementation using cartesian coordinates scales as
@@ -165,8 +166,8 @@ multiplications (provided $\mathsf{D}$ can be evaluated quickly, see
 Sec.~\ref{ssec:grav_derivatives}), which are extremely efficient
 instructions on modern architectures. However, the fully expanded sums
 can lead to rather large and prone to typo expressions. To avoid any
-misshap, we use a \texttt{python} to generate C code in which all the
-sums are unrolled and correct by construction. In \swift, we
+mishaps, we use a \texttt{python} script to generate C code in which
+all the sums are unrolled and correct by construction. In \swift, we
 implemented the kernels up to order $p=5$, as it proved to be accurate
 enough for our purpose, but this could be extended to higher order
 easily. This implies storing $56$ numbers per cell for each

theory/Multipoles/gravity_derivatives.tex

@@ -2,36 +2,36 @@
 \label{ssec:grav_derivatives}
 
 The calculation of all the
-$\mathsf{D}_\mathbf{n}(x,y,z) \equiv \nabla^{\mathbf{n}}\phi(x,y,z)$ terms up
+$\mathsf{D}_\mathbf{n}(x,y,z) \equiv \nabla^{\mathbf{n}}\varphi(x,y,z)$ terms up
 to the relevent order can be quite tedious and it is beneficial to
 automatize the whole setup. Ideally, one would like to have an
 expression for each of these terms that is only made of multiplications
 and additions of each of the coordinates and the inverse distance. We
-achieve this by writing $\phi$ as a composition of functions
-$\phi(u(x,y,z))$ and apply the \textit{Fa\`a di Bruno}
+achieve this by writing $\varphi$ as a composition of functions
+$\varphi(u(x,y,z))$ and apply the \textit{Fa\`a di Bruno}
 formula \citep[i.e. the ``chain rule'' for higher order derivatives,
  see e.g.][]{Hardy2006} to construct our terms:
 \begin{equation}
 \label{eq:faa_di_bruno}
-\frac{\partial^n}{\partial x_1 \cdots \partial x_n} \phi(u)
-= \sum_{A} \phi^{(|A|)}(u) \prod_{B \in
+\frac{\partial^n}{\partial x_1 \cdots \partial x_n} \varphi(u)
+= \sum_{A} \varphi^{(|A|)}(u) \prod_{B \in
 A} \frac{\partial^{|B|}}{\prod_{c\in B}\partial x_c} u(x,y,z),
 \end{equation}
 where $A$ is the set of all partitions of $\lbrace1,\cdots, n\rbrace$,
 $B$ is a block of a partition in the set $A$ and $|\cdot|$ denotes the
-cardinality of a set. For generic functions $\phi$ and $u$ this
+cardinality of a set. For generic functions $\varphi$ and $u$ this
 formula yields an untracktable number of terms; an 8th-order
 derivative will have $4140$ (!)  terms in the sum\footnote{The number
   of terms in the sum is given by the Bell number of the same
   order.}. \\ For the un-softened gravitational potential, we choose to write
 \begin{align}
-   \phi(x,y,z) &= 1 / \sqrt{u(x,y,z)}, \\
+   \varphi(x,y,z) &= 1 / \sqrt{u(x,y,z)}, \\
    u(x,y,z) &= x^2 + y^2 + z^2.
 \end{align}
-This choice allows to have derivatives of any order of $\phi(u)$ that
+This choice allows to have derivatives of any order of $\varphi(u)$ that
 can be easily expressed and only depend on powers of $u$:
 \begin{equation}
-\phi^{(n)}(u) = (-1)^n\cdot\frac{(2n-1)!!}{2^n}\cdot\frac{1}{u^{n+\frac{1}{2}}},
+\varphi^{(n)}(u) = (-1)^n\cdot\frac{(2n-1)!!}{2^n}\cdot\frac{1}{u^{n+\frac{1}{2}}},
 \end{equation}
 where $!!$ denotes the semi-factorial. More importantly, this
 choice of decomposition allows us to have non-zero derivatives of $u$
@@ -39,7 +39,7 @@ only up to second order in $x$, $y$ or $z$. The number of non-zero
 terms in eq. \ref{eq:faa_di_bruno} is hence drastically reduced. For
 instance, when computing $\mathsf{D}_{(4,1,3)}(\mathbf{r}) \equiv
 \frac{\partial^8}{\partial x^4 \partial y \partial z^3}
-\phi(u(x,y,z))$, $4100$ of the $4140$ terms will involve at least one
+\varphi(u(x,y,z))$, $4100$ of the $4140$ terms will involve at least one
 zero-valued derivative (e.g. $\partial^3/\partial x^3$ or
 $\partial^2/\partial x\partial y$) of $u$. Furthermore, among the 40
 remaining terms, many will involve the same combination of derivatives

theory/Multipoles/plot_potential.py

@@ -141,7 +141,7 @@ plot([epsilon, epsilon], [-10, 10], 'k-', alpha=0.5, lw=0.5)
 plot([epsilon/plummer_equivalent_factor, epsilon/plummer_equivalent_factor], [0, 10], 'k-', alpha=0.5, lw=0.5)
 
 ylim(0, 2.3)
-ylabel("$\\phi(r)$", labelpad=1)
+ylabel("$\\varphi(r)$", labelpad=1)
 #yticks([0., 0.5, 1., 1.5, 2., 2.5], ["$%.1f$"%(0.*epsilon), "$%.1f$"%(0.5*epsilon), "$%.1f$"%(1.*epsilon), "$%.1f$"%(1.5*epsilon), "$%.1f$"%(2.*epsilon)])
 
 xlim(0,r_max_plot)
@@ -163,6 +163,6 @@ xticks([0., 0.5, 1., 1.5, 2., 2.5], ["$%.1f$"%(0./epsilon), "", "$%.1f$"%(1./eps
 xlabel("$r/H$", labelpad=-7)
 
 ylim(0, 0.95)
-ylabel("$|\\overrightarrow{\\nabla}\\phi(r)|$", labelpad=0)
+ylabel("$|\\overrightarrow{\\nabla}\\varphi(r)|$", labelpad=0)
 
 savefig("potential.pdf")

theory/Multipoles/potential_derivatives.tex

@@ -7,7 +7,7 @@ the notation $\mathbf{r}=(r_x, r_y, r_z)$, $r = |\mathbf{r}|$ and
 $u=r/H$. Starting from the potential (Eq. \ref{eq:fmm:potential},
 reproduced here for clarity), 
 \begin{align}
-\mathsf{D}_{000}(\mathbf{r}) = \phi (\mathbf{r},H) = 
+\mathsf{D}_{000}(\mathbf{r}) = \varphi (\mathbf{r},H) = 
 \left\lbrace\begin{array}{rcl}
 \frac{1}{H} \left(-3u^7 + 15u^6 - 28u^5 + 21u^4 - 7u^2 + 3\right) & \mbox{if} & u < 1,\\
 \frac{1}{r} & \mbox{if} & u \geq 1, 
@@ -19,7 +19,7 @@ we can construct the higher order terms by successively applying the
 relevant ones here split by order.
 
 \begin{align}
-\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \phi (\mathbf{r},H) = 
+\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r},H) = 
 \left\lbrace\begin{array}{rcl}
 -\frac{r_x}{H^3} \left(21u^5 - 90u^4 + 140u^3 - 84u^2 + 14\right) & \mbox{if} & u < 1,\\
 -\frac{r_x}{r^3} & \mbox{if} & u \geq 1, 
@@ -30,7 +30,7 @@ relevant ones here split by order.
 \noindent\rule{6cm}{0.4pt}
 
 \begin{align}
-\mathsf{D}_{200}(\mathbf{r}) = \frac{\partial^2}{\partial r_x^2} \phi (\mathbf{r},H) = 
+\mathsf{D}_{200}(\mathbf{r}) = \frac{\partial^2}{\partial r_x^2} \varphi (\mathbf{r},H) = 
 \left\lbrace\begin{array}{rcl}
 \frac{r_x^2}{H^5}\left(-105u^3+360u^2-420u+168\right) -
 \frac{1}{H^3} \left(21u^5 - 90u^4 + 140u^3 - 84u^2 + 14\right) & \mbox{if} & u < 1,\\
@@ -40,7 +40,7 @@ relevant ones here split by order.
 \end{align}
 
 \begin{align}
-\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \phi (\mathbf{r},H) = 
+\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \varphi (\mathbf{r},H) = 
 \left\lbrace\begin{array}{rcl}
 \frac{r_xr_y}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
 3\frac{r_xr_y}{r^5} & \mbox{if} & u \geq 1, 
@@ -51,7 +51,7 @@ relevant ones here split by order.
 \noindent\rule{6cm}{0.4pt}
 
 \begin{align}
-\mathsf{D}_{300}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \phi (\mathbf{r},H) = 
+\mathsf{D}_{300}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r},H) = 
 \left\lbrace\begin{array}{rcl}
 -\frac{r_x^3}{H^7} \left(315u - 720 + 420u^{-1}\right) +
 \frac{3r_x}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
@@ -61,7 +61,7 @@ relevant ones here split by order.
 \end{align}
 
 \begin{align}
-\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \phi (\mathbf{r},H) = 
+\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r},H) = 
 \left\lbrace\begin{array}{rcl}
 -\frac{r_x^2r_y}{H^7} \left(315u - 720 + 420u^{-1}\right) +
 \frac{r_y}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
@@ -72,7 +72,7 @@ relevant ones here split by order.
 
 
 \begin{align}
-\mathsf{D}_{111}(\mathbf{r}) = \frac{\partial^3}{\partial r_x\partial r_y\partial r_z} \phi (\mathbf{r},H) = 
+\mathsf{D}_{111}(\mathbf{r}) = \frac{\partial^3}{\partial r_x\partial r_y\partial r_z} \varphi (\mathbf{r},H) = 
 \left\lbrace\begin{array}{rcl}
 -\frac{r_xr_yr_z}{H^7} \left(315u - 720 + 420u^{-1}\right) & \mbox{if} & u < 1,\\
 -15\frac{r_xr_yr_z}{r^7} & \mbox{if} & u \geq 1, 

theory/Multipoles/potential_softening.tex

@@ -10,9 +10,10 @@ $\epsilon$. Instead of the commonly used spline kernel of
 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 division. We set
-$\tilde\delta(\mathbf{x}) = \rho(|\mathbf{x}|) = W(|\mathbf{x}|,
-3\epsilon_{\rm Plummer})$, with $W(r, H)$ given by
+architectures; it also does not require any divisions to evaluate the
+softened forces. We set $\tilde\delta(\mathbf{x}) =
+\rho(|\mathbf{x}|) = W(|\mathbf{x}|, 3\epsilon_{\rm Plummer})$, with
+$W(r, H)$ given by
 
 \begin{align}
 W(r,H) &= \frac{21}{2\pi H^3} \times \nonumber \\
@@ -22,9 +23,9 @@ W(r,H) &= \frac{21}{2\pi H^3} \times \nonumber \\
 \end{array}
 \right.
 \end{align}
-and $u = r/H$. The potential $\phi(r,H)$ corresponding to this density distribution reads
+and $u = r/H$. The potential $\varphi(r,H)$ corresponding to this density distribution reads
 \begin{align}
-\phi = 
+\varphi = 
 \left\lbrace\begin{array}{rcl}
 \frac{1}{H} (-3u^7 + 15u^6 - 28u^5 + 21u^4 - 7u^2 + 3) & \mbox{if} & u < 1,\\
 \frac{1}{r} & \mbox{if} & u \geq 1.