Wrote the section of the documentation on the EoS.

theory/SPH/EoS/eos.tex

-aa
 
+In \swift, all transformations between thermodynamical quantities are
+computed using a pre-defined equation of state (EoS) for the gas. This
+allows user to switch EoS without having to modify the equations of
+SPH. The definition of the EoS takes the form of simple relations
+between thermodynamic quantities. All of them must be specified even
+when they default to constants.
+
+%#######################################################################################################
+\subsection{Ideal Gas}
+\label{sec:eos:ideal}
+
+This is the simplest and most common equation of state. It corresponds
+to a gas obeying a pressure law $P = (\gamma-1) \rho u$, where $u$ is
+the gas' internal energy, $\rho$ its density and $\gamma$ the
+(user defined) adiabatic index, often assumed to be $5/3$ or $7/5$. For such a gas,
+we have the following relations between pressure ($P$), internal energy
+($u$), density ($\rho$), sound speed ($c$) and entropic function ($A$):
+
+\begin{align}
+  P &= P_{\rm eos}(\rho, u) \equiv  (\gamma-1) \rho
+  u \label{eq:eos:ideal:P_from_u} \\
+  c &= c_{\rm eos}(\rho, u) \equiv \sqrt{\gamma (\gamma-1)
+    u} \label{eq:eos:ideal:c_from_u} \\
+  A &= A_{\rm eos}(\rho, u) \equiv(\gamma-1) u
+  \rho^{\gamma-1} \label{eq:eos:ideal:A_from_u} \\
+  ~ \nonumber\\
+  P &= P_{\rm eos}(\rho, A) \equiv A
+  \rho^{\gamma} \label{eq:eos:ideal:P_from_A} \\
+  c &= c_{\rm eos}(\rho, A) \equiv \sqrt{\gamma \rho^{\gamma-1}
+    A} \label{eq:eos:ideal:c_from_A} \\
+  u &= u_{\rm eos}(\rho, A) \equiv A \rho^{\gamma-1} /
+  (\gamma-1) \label{eq:eos:ideal:u_from_A}
+\end{align}
+Note that highly optimised functions to compute $x^\gamma$ and other
+useful powers of $\gamma$ have been implemented in \swift for the most
+commonly used values of $\gamma$.
+
+
+%#######################################################################################################
+\subsection{Isothermal Gas}
+\label{sec:eos:isothermal}
+
+An isothermal equation of state can be useful for certain test cases
+or to speed-up the generation of glass files. This EoS corresponds to
+a gas with contant (user-specified) thermal energy $u_{\rm cst}$. For such a gas,
+we have the following (trivial) relations between pressure ($P$), internal energy
+($u$), density ($\rho$), sound speed ($c$) and entropic function ($A$):
+
+\begin{align}
+  P &= P_{\rm eos}(\rho, u) \equiv  (\gamma-1) \rho
+  u_{\rm cst} \label{eq:eos:isothermal:P_from_u} \\
+  c &= c_{\rm eos}(\rho, u) \equiv \sqrt{\gamma (\gamma-1)
+    u_{\rm cst}} = c_{\rm cst} \label{eq:eos:isothermal:c_from_u} \\
+  A &= A_{\rm eos}(\rho, u) \equiv(\gamma-1) u_{\rm cst}
+  \rho^{\gamma-1} \label{eq:eos:isothermal:A_from_u} \\ 
+  ~ \nonumber\\
+  P &= P_{\rm eos}(\rho, A) \equiv (\gamma-1) \rho
+  u_{\rm cst} \label{eq:eos:isothermal:P_from_A} \\
+  c &= c_{\rm eos}(\rho, A) \equiv \sqrt{\gamma (\gamma-1)
+    u_{\rm cst}} = c_{\rm cst} \label{eq:eos:isothermal:c_from_A} \\
+  u &= u_{\rm eos}(\rho, A) \equiv u_{\rm
+    cst} \label{eq:eos:isothermal:u_from_A} 
+\end{align}

theory/SPH/Flavours/sph_flavours.tex

@@ -241,7 +241,7 @@ The prediction step is also identical to the \MinimalSPH case with the
 entropic function replacing the thermal energy.
 
 \begin{align}
-  \vec{x}_i &\rightarrow \vec{x}_i + \vec{v}_i \Delta t  \label{eq:sph:gadget2:drift__x} \\
+  \vec{x}_i &\rightarrow \vec{x}_i + \vec{v}_i \Delta t  \label{eq:sph:gadget2:drift_x} \\
   h_i &\rightarrow h_i \exp\left(\frac{1}{h_i} \frac{dh_i}{dt}
   \Delta t\right), \label{eq:sph:gadget2:drift_h}\\
   \rho_i &\rightarrow \rho_i \exp\left(-\frac{3}{h_i} \frac{dh_i}{dt}

theory/SPH/Kernels/kernel_definitions.tex

@@ -183,11 +183,13 @@ All kernels available in \swift are shown on Fig.~\ref{fig:sph:kernels}.
 \includegraphics[width=\columnwidth]{kernels.pdf}
 \caption{The kernel functions available in \swift for a mean
   inter-particle separation $\langle x\rangle=1.5$ and a resolution
-  $\eta=1.2348$. The corresponding kernel support radii $H$ (shown by
-  arrows) and number of neighours $N_{\rm ngb}$ are indicated on the
-  figure. A Gaussian kernel with the same smoothing length is shown
-  for comparison. Note that all these kernels have the \emph{same
-    resolution} despite having vastly different number of neighbours.}
+  $\eta=1.2348$ shown in linear (top) and log (bottom) units to
+  highlight their differences. The corresponding kernel support radii
+  $H$ (shown by arrows) and number of neighours $N_{\rm ngb}$ are
+  indicated on the figure. A Gaussian kernel with the same smoothing
+  length is shown for comparison. Note that all these kernels have
+  the \emph{same resolution} despite having vastly different number of
+  neighbours.}
 \label{fig:sph:kernels}
 \end{figure}