fixed the figure

Author: samarjeet

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,