Merge pull request #61 from MobleyLab/samar_edits

Samar edits and figure additions

Author: davidlmobley

paper/basic_training.pdf

@@ -1051,13 +1051,17 @@ \subsubsection{Motivation}
 
 \subsubsection{ Ewald Summation}
 
-One way of handling the aforementioned issues in an efficient manner is to use the Ewald summation technique (ref, Ewald 1921). To understand this technique, lets represent the relation between the charge distriution and the coulombic potential  in the differntial form (Poisson equation) :
+One way of handling the aforementioned issues in an efficient manner is to use the Ewald summation technique (ref, Ewald 1921). To understand this technique, lets represent the relation between the charge distribution and the coulombic potential  in the differential form (Poisson equation) :
 
 \[
 \Delta \phi(\boldsymbol{x}) = - \frac{1}{\epsilon} \rho(\boldsymbol{x}) 
 \] 
 
-where,  $ \boldsymbol{x} \epsilon R^3 $ ,  $\phi(\boldsymbol{x})$ is the potential at point $\boldsymbol{x}$, $\rho(\boldsymbol{x})$ is the charge at point $\boldsymbol{x}$ and $\epsilon$ is the permissivity of the medium. This equation is an elliptical partial differential equation(pde) of the second order. The standard way to determine the potential from this equation is a two step method - discretization of the equation followed by solution. These techniques however depend on the smoothness of the functions - $\rho$ and $\phi$ - involved. However, in the case of charge distribution in our simulation system, $\rho$ is a set of delta functions which are clearly not smooth! As $\rho$ is not smooth, $\phi$ is not smooth either. \\
+where,   $\phi(\boldsymbol{x})$ is the potential at point $\boldsymbol{x}$, $\rho(\boldsymbol{x})$ is the charge density at point $\boldsymbol{x}$ and $\epsilon$ is the permissivity of the medium. 
+This equation is an elliptical partial differential equation(pde) of the second order. 
+The standard way to determine the potential from this equation is a two step method - discretization of the equation followed by solution. These techniques however depend on the smoothness of the functions - $\rho$ and $\phi$ - involved. 
+However, in the case of charge distribution in our simulation system, $\rho$ is a set of delta functions which are clearly not smooth! 
+As $\rho$ is not smooth, $\phi$ is not smooth either. \\
 
 
 Ewald method is based on replacing the point charge distributions by smooth charge distributions in order to use the fast solvation techniques of the pde. The most common smooth function used in Ewald method is the gaussian distribution although other distributions have been used as well. Thus,
@@ -1071,8 +1075,8 @@ \subsubsection{ Ewald Summation}
 
 \begin{figure}[h]
 \centering
-\includegraphics[width=\linewidth]{charges_ewald.pdf}
-    \caption{Point charges can be split into Direct space and reciprocal space charges. Direct space charge consists of the original charges and gaussian-distributed screening charge. Reciprocal space charge is only the gaussian-distributed charge.}
+\includegraphics[width=\linewidth]{ewald.pdf}
+    \caption{Screening charge distribution. (top) Original charge distribution. (bottom)Point charges can be split into Direct space(blue) and Reciprocal space charges(red). Direct space charge consists of the original charges and gaussian-distributed screening charge. Reciprocal space charge is only the gaussian-distributed charge. }
 \label{charges_ewald}
 \end{figure}
 
@@ -1083,6 +1087,14 @@ \subsubsection{ Ewald Summation}
 
 Unlike the potential due to the original charge, the potential due to direct space charge decays rapidly as shown in the figure. This is due to erfc function which decays very fast. Infact, it decays even faster than the Van der Waals term $r^{-6}$ and hence the cutoff used for Van der Waals can be used for direct space coulombic potential calculation as well.
 
+\begin{figure}[h]
+\centering
+\includegraphics[width=\linewidth]{decay_comparison.pdf}
+    \caption{Comparison of decay of original $r^{-1}$ term(blue,*), erfc(r) in direct space(black,-) and $r^{-6}$ in van der waals term (red, -.).  }
+\label{charges_ewald}
+\end{figure}
+
+
 Potential due to the long range charge however doesn't decay rapidly and if calculated in the direct space would require summation over the infinite images. However, as we discussed earlier, the smoothness of the  charge $\rho^{lr}$ (and hence potential ($\phi^{lr}$) allows the use of fast pde solvers. Fourier based solvers use the important result that differentiation operation in direct space corresponds to multiplication by (ik) in reciprocal space!
 
 \[