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\chapter{\label{chapt:methodology}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\chapter{\label{chapt:langevin}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\section{Introduction} |
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average acceleration is not always true for cooperative motion which |
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is common in protein motion. An inertial Brownian dynamics (IBD) was |
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proposed to address this issue by adding an inertial correction |
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term\cite{Beard2001}. As a complement to IBD which has a lower bound |
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term\cite{Beard2000}. As a complement to IBD which has a lower bound |
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in time step because of the inertial relaxation time, long-time-step |
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inertial dynamics (LTID) can be used to investigate the inertial |
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behavior of the polymer segments in low friction |
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regime\cite{Beard2001}. LTID can also deal with the rotational |
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regime\cite{Beard2000}. LTID can also deal with the rotational |
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dynamics for nonskew bodies without translation-rotation coupling by |
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separating the translation and rotation motion and taking advantage |
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of the analytical solution of hydrodynamics properties. However, |
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\subsection{Langevin Dynamics of Banana Shaped Molecule} |
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In order to compare the |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{one_banana.eps} |
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\caption[]{.} \label{langevin:banana} |
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\end{figure} |
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In order to verify that Langevin dynamics can mimic the dynamics of |
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the systems absent of explicit solvents, we carried out two sets of |
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simulations and compare their dynamic properties. |
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\subsubsection{Simulations Contain One Banana Shaped Molecule} |
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Fig.~\ref{langevin:oneBanana} shows a snapshot of the simulation |
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made of 256 pentane molecules and one banana shaped molecule at |
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273~K. It has an equivalent implicit solvent system containing only |
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one banana shaped molecule with viscosity of 0.289 center poise. To |
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calculate the hydrodynamic properties of the banana shaped molecule, |
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we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
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in which the banana shaped molecule is represented as a ``shell'' |
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made of 2266 small identical beads with size of 0.3 $\AA$ on the |
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surface. Applying the procedure described in |
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Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
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identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
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well as the resistance tensor, |
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\[ |
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\left( {\begin{array}{*{20}c} |
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0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
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3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
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-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
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5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
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0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
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0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
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\end{array}} \right). |
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\] |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{roughShell.eps} |
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model for banana shaped molecule.} \label{langevin:roughShell} |
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\end{figure} |
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%\begin{figure} |
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%\centering |
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%\includegraphics[width=\linewidth]{oneBanana.eps} |
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%\caption[Snapshot from Simulation of One Banana Shaped Molecules and |
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%256 Pentane Molecules]{Snapshot from simulation of one Banana shaped |
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%molecules and 256 pentane molecules.} \label{langevin:oneBanana} |
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%\end{figure} |
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\subsubsection{Simulations Contain Two Banana Shaped Molecules} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{twoBanana.eps} |
