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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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between the native and denatured states. Because of its stability |
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against noise, Langevin dynamics is very suitable for studying |
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remagnetization processes in various |
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systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For |
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instance, the oscillation power spectrum of nanoparticles from |
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Langevin dynamics simulation has the same peak frequencies for |
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different wave vectors,which recovers the property of magnetic |
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excitations in small finite structures\cite{Berkov2005a}. In an |
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attempt to reduce the computational cost of simulation, multiple |
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time stepping (MTS) methods have been introduced and have been of |
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great interest to macromolecule and protein |
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community\cite{Tuckerman1992}. Relying on the observation that |
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forces between distant atoms generally demonstrate slower |
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fluctuations than forces between close atoms, MTS method are |
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generally implemented by evaluating the slowly fluctuating forces |
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less frequently than the fast ones. Unfortunately, nonlinear |
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instability resulting from increasing timestep in MTS simulation |
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have became a critical obstruction preventing the long time |
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simulation. Due to the coupling to the heat bath, Langevin dynamics |
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has been shown to be able to damp out the resonance artifact more |
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efficiently\cite{Sandu1999}. |
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systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
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oscillation power spectrum of nanoparticles from Langevin dynamics |
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simulation has the same peak frequencies for different wave |
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vectors,which recovers the property of magnetic excitations in small |
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finite structures\cite{Berkov2005a}. In an attempt to reduce the |
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computational cost of simulation, multiple time stepping (MTS) |
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methods have been introduced and have been of great interest to |
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macromolecule and protein community\cite{Tuckerman1992}. Relying on |
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the observation that forces between distant atoms generally |
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demonstrate slower fluctuations than forces between close atoms, MTS |
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method are generally implemented by evaluating the slowly |
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fluctuating forces less frequently than the fast ones. |
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Unfortunately, nonlinear instability resulting from increasing |
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timestep in MTS simulation have became a critical obstruction |
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preventing the long time simulation. Due to the coupling to the heat |
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bath, Langevin dynamics has been shown to be able to damp out the |
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resonance artifact more efficiently\cite{Sandu1999}. |
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%review langevin/browninan dynamics for arbitrarily shaped rigid body |
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Combining Langevin or Brownian dynamics with rigid body dynamics, |
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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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due to the complexity of the elliptic integral, only the ellipsoid |
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with the restriction of two axes having to be equal, \textit{i.e.} |
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prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
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exactly. Introducing an elliptic integral parameter $S$ for prolate, |
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exactly. Introducing an elliptic integral parameter $S$ for prolate |
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: |
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\[ |
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S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
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} }}{b}, |
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\] |
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and oblate, |
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and oblate : |
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\[ |
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S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
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}}{a} |
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tensors |
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\[ |
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\begin{array}{l} |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
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\end{array}, |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
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2a}}, |
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\end{array} |
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\] |
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and |
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\[ |
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\begin{array}{l} |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
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\end{array}. |
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\] |
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\subsection{Langevin Dynamics of Banana Shaped Molecule} |
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In order to compare the |
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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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\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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Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
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made of 256 pentane molecules and two banana shaped molecules at |
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273~K. It has an equivalent implicit solvent system containing only |
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two banana shaped molecules 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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molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{vacf.eps} |
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\caption[Plots of Velocity Auto-correlation functions]{Velocity |
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Auto-correlation function of NVE (blue) and Langevin dynamics |
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(red).} \label{langevin:twoBanana} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{uacf.eps} |
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\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
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256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
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molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
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\end{figure} |
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\section{Conclusions} |
