186 lines
8.0 KiB
TeX
186 lines
8.0 KiB
TeX
\documentclass[../main.tex]{subfiles}
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\resources % search for graphics in ../res/imgs/
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\begin{document}
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\section{Methods}
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For a multi-particles system, we need to modify $\vb{F}$ to account for the
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force of other particles in the system acting upon each other. To do that, we
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add another term to $\vb{F}$
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\begin{equation}
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\vb{F}_i(t, \vb{v}_i, \vb{r}_i) = q_i \vb{E}(t, \vb{r}_i) + q_i \vb{v}_i \cross \vb{B} - \vb{E}_p(t, \vb{r}_i),
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\end{equation}
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where $i$ and $j$ are particle indices and
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\begin{equation}
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\vb{E}_p(t, \vb{r}_i) = q_i k_e \sum_{j \neq i}
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q_j \frac{\vb{r_i} - \vb{r_j}}{\left| \vb{r_i} - \vb{r_j} \right|^3}.
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\label{eq:}
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\end{equation}
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Newton's second law for a particle $i$ is then
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\begin{equation}
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\frac{d^2\vb{r}_i}{dt^2} = \frac{\vb{F}\left(t, \frac{d\vb{r}_i}{dt}, \vb{r_i}\right)}{m_i},
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\label{eq:newtonlaw2}
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\end{equation}
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We can then rewrite the second order ODE from equation~\ref{eq:newtonlaw2}
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into a set of coupled first order ODEs.
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We now rewrite Newton's second law of motion as
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\begin{equation}
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\begin{split}
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\frac{d\vb{r}_i}{dt} &= \vb{v}_i \\
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\frac{d\vb{v}_i}{dt} &= \frac{\vb{F}(t, \vb{v}_i, \vb{r}_i)}{m_i}.
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\end{split}
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\label{eq:coupled}
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\end{equation}
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\subsection{Forward Euler}
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For a single particle, the forward Euler method for a coupled system is
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expressed as
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\begin{equation}
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\begin{split}
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\vb{r}_{i+1} &= \vb{r}_i + h \cdot \frac{d\vb{r}_i}{dt} = \vb{r}_i + h \cdot \vb{v}_i \\
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\vb{v}_{i+1} &= \vb{v}_i + h \cdot \frac{\vb{v}_i}{dt} = \vb{v}_i + h \cdot \frac{\vb{F}\left( t_i, \vb{v}_i, \vb{r}_i \right)}{m},
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\end{split}
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\end{equation}
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where $i$ is the current time step of the particle $m$ is the mass of the particle, and $h$
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is the time step length.
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When dealing with a multi-particle system, we need to ensure that we do not
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update the position of any particles until every particle has calculated their
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next step. An easy way of doing this is to create a copy of all the particles,
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then update the copy, and when all the particles have calculated their next
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step, simply replace the particles with the copies.
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Algorithm~\ref{algo:forwardeuler} provides an overview on how that can be achieved. % Make this better
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\begin{figure}[H]
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\begin{algorithm}[H]
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\caption{Forward Euler method}
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\label{algo:forwardeuler}
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\begin{algorithmic}
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\Procedure{Evolve forward Euler}{$particles, dt$}
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\State $N \leftarrow \text{Number of particles in } particles$
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\State $a \leftarrow \text{Calculate } \frac{\vb{F_i}}{m_i} \text{ for each particle in } particles$
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\For{ $i = 1, 2, \ldots , N$ }
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\State $particles_i.\vb{r} \leftarrow particles_i.\vb{r} + dt \cdot particles_i.\vb{v}$
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\State $particles_i.\vb{v} \leftarrow particles_i.\vb{v} + dt \cdot a_i$
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\EndFor
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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\end{figure}
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\subsection{4th order Runge-Kutta}
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For a single particle, we can express the 4th order Runge-Kutta (RK4) method as
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\begin{equation}
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\begin{split}
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\vb{v}_{i+1} &= \vb{v}_i + \frac{h}{6} \left( \vb{k}_{\vb{v},1}
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+ 2\vb{k}_{\vb{v},2} + 2\vb{k}_{\vb{v},3} + \vb{k}_{\vb{v},4}
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\right) \\
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\vb{r}_{i+1} &= \vb{r}_i + \frac{h}{6} \left( \vb{k}_{\vb{r},1}
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+ 2\vb{k}_{\vb{r},2} + 2\vb{k}_{\vb{r},3} + \vb{k}_{\vb{r},4}
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\right),
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\end{split}
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\end{equation}
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where
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\begin{equation}
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\begin{split}
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\vb{k}_{\vb{v},1} &= \frac{\vb{F}(t_i, \vb{v}_i, \vb{r}_i)}{m} \\
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\vb{k}_{\vb{r},1} &= \vb{v}_i \\
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\vb{k}_{\vb{v},2} &= \frac{\vb{F}(t_i+\frac{h}{2}, \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2}, \vb{r}_i + h \cdot \frac{\vb{k}_{\vb{r},1}}{2})}{m} \\
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\vb{k}_{\vb{r},2} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2} \\
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\vb{k}_{\vb{v},3} &= \frac{\vb{F}(t_i+\frac{h}{2}, \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},2}}{2}, \vb{r}_i + h \frac{\cdot \vb{k}_{\vb{r},2}}{2})}{m} \\
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\vb{k}_{\vb{r},3} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},2}}{2} \\
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\vb{k}_{\vb{v},4} &= \frac{\vb{F}(t_i+h, \vb{v}_i + h \cdot \vb{k}_{\vb{v},3}, \vb{r}_i + h \cdot \vb{k}_{\vb{r},3})}{m} \\
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\vb{k}_{\vb{r},4} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2}.
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\end{split}
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\end{equation}
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In order to find each $\vb{k}_{\vb{r},i}$ and $\vb{k}_{\vb{v},i}$,
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we need to first compute all $\vb{k}_{\vb{r},i}$ and $\vb{k}_{\vb{v},i}$
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for all particles, then we can update the particles in order to compute
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$\vb{k}_{\vb{r},i+1}$ and $\vb{k}_{\vb{v},i+1}$. In order for the algorithm
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to work, we need to save a copy of each particle before starting so that we
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can update the particles correctly for each step.
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This approach would require 8 loops to be able to complete the calculation since
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we cannot update the particles until after all $\vb{k}$ values have been
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computed, however if we create a temporary array that holds particles, we can
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put the updated particles in there, and then use that array in the next loop.
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around 8 separate loops to calculate all $\vb{k}$ values
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\newpage
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\begin{figure}
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\begin{algorithm}[H]
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\caption{RK4 method}
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\label{algo:rk4}
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\begin{algorithmic}
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\Procedure{Evolve RK4}{$dt$}
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\State $N \leftarrow \text{Number of particles inside the Penning trap}$
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\State $orig\_p \leftarrow \text{Copy of particles}$ \Comment{Keep the original particles}
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\State $tmp\_p \leftarrow \text{Array of particles of size }N$
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\State $\vb{k}_{\vb{r}} \leftarrow \text{2D array of vectors of size } 4 \cross N$
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\State $\vb{k}_{\vb{v}} \leftarrow \text{2D array of vectors of size } 4 \cross N$
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\For{ $i = 1, 2, \ldots, N$ }
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\State $\vb{k}_{\vb{r},1,i} \leftarrow \vb{v}$
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\State $\vb{k}_{\vb{v},1,i} \leftarrow \frac{\vb{F}}{m}$
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\State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{2} \cdot \vb{k}_{\vb{r},1,i}$
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\State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{2} \cdot \vb{k}_{\vb{v},1,i}$
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\EndFor
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\State $particles \leftarrow tmp\_p$ \Comment{Update particles}
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\For{ $i = 1, 2, \ldots, N$ }
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\State $\vb{k}_{\vb{r},2,i} \leftarrow \vb{v}$
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\State $\vb{k}_{\vb{v},2,i} \leftarrow \frac{\vb{F}}{m}$
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\State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{2} \cdot \vb{k}_{\vb{r},2,i}$
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\State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{2} \cdot \vb{k}_{\vb{v},2,i}$
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\EndFor
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\State $particles \leftarrow tmp\_p$ \Comment{Update particles}
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\For{ $i = 1, 2, \ldots, N$ }
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\State $\vb{k}_{\vb{r},3,i} \leftarrow \vb{v}$
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\State $\vb{k}_{\vb{v},3,i} \leftarrow \frac{\vb{F}}{m}$
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\State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + dt \cdot \vb{k}_{\vb{r},3,i}$
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\State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + dt \cdot \vb{k}_{\vb{v},3,i}$
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\EndFor
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\State $particles \leftarrow tmp\_p$ \Comment{Update particles}
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\For{ $i = 1, 2, \ldots, N$ }
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\State $\vb{k}_{\vb{r},4,i} \leftarrow \vb{v}$
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\State $\vb{k}_{\vb{v},4,i} \leftarrow \frac{\vb{F}}{m}$
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\State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{6}
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\cdot \left( \vb{k}_{\vb{r},1,i} + \vb{k}_{\vb{r},2,i}
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+ \vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} \right)$
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\State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{6}
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\cdot \left( \vb{k}_{\vb{v},1,i} + \vb{k}_{\vb{v},2,i}
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+ \vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} \right)$
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\EndFor
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\State $particles \leftarrow tmp\_p$ \Comment{Final update}
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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$\vb{F}$ in the algorithm does not take any arguments as it uses the
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velocities and positions of the particles inside the array $particles$ to
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calculate the total force acting on particle $i$.
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\end{figure}
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\biblio
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\end{document}
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