small grammar improvement

Author: simongravelle

lammps-tutorials.tex

@@ -3716,17 +3716,17 @@ \subsubsection{Method 1: Free sampling}
 
 \paragraph{Data analysis}
 
-Once the simulation is over, the density profile shows that the density in the center of the box is
+Once the simulation is complete, the density profile shows that the density in the center of the box is
 about two orders of magnitude lower than inside the reservoir (Fig.\,\ref{fig:US-density}).
-Then, let us plot $-R T \ln(\rho/\rho_\mathrm{bulk})$ (i.e.~Eq.\,\eqref{eq:G} where
+Next, we plot $-R T \ln(\rho/\rho_\mathrm{bulk})$ (i.e.~Eq.\,\eqref{eq:G} where
 the pressure ratio $p/p_\mathrm{bulk}$ is replaced by the density ratio
-$\rho/\rho_\mathrm{bulk}$ assuming the system is an ideal gas) and compare it
+$\rho/\rho_\mathrm{bulk}$, assuming the system behaves as an ideal gas) and compare it
 with the imposed potential $U$ from Eq.\,\eqref{eq:U} (Fig.\,\ref{fig:US-FreeSampling}).
-The value for the reference density $\rho_\text{bulk} = 0.0033~\text{\AA{}}^{-3}$
+The reference density, $\rho_\text{bulk} = 0.0033~\text{\AA{}}^{-3}$,
 was estimated by measuring the density of the reservoir from the raw density
 profiles. The agreement between the MD results and the imposed energy profile
-is reasonable, despite some noise in the central part (where fewer data points
-are available due to the repulsive potential).
+is reasonable, despite some noise in the central part, where fewer data points
+are available due to the repulsive potential.
 
 \begin{figure}
 \centering
@@ -3756,12 +3756,13 @@ \subsubsection{Method 1: Free sampling}
 \end{figure}
 
 \paragraph{The limits of free sampling}
-If we increase the value of $U_0$, the average number of atoms in the central
-region will decrease, making it difficult to obtain a good resolution for the free
-energy profile. For instance, when we run the same simulation using $U_0 = 10 \epsilon$,
-which corresponds to a situation where $U_0 \approx 10 k_\text{B} T$, not a single
-atom explores the central part of the simulation box during the 8 ns of simulation.
-In this case, using an enhanced sampling method is preferred; see the next section.
+
+Increasing the value of $U_0$ reduces the average number of atoms in the central
+region, making it challenging to achieve good resolution for
+the free energy profile. For example, running the same simulation with $U_0 = 10 \epsilon$,
+corresponding to $U_0 \approx 10 k_\text{B} T$, results in no atoms exploring
+the central part of the simulation box during the simulation.
+In such case, employing an enhanced sampling method is recommended, as done in the next section.
 
 \subsubsection{Method 2: Umbrella sampling}
 Umbrella sampling is a biased molecular dynamics method in which additional forces