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Revision 2853 by tim, Sun Jun 11 02:23:00 2006 UTC vs.
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1  
2 < \chapter{\label{chapt:methodology}LANGEVIN DYNAMICS for RIGID BODIES of ARBITRARY SHAPE}
2 > \chapter{\label{chapt:methodology}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE}
3  
4   \section{Introduction}
5  
# Line 98 | Line 98 | sophisticated rigid body dynamics.
98   estimation of friction tensor from hydrodynamics theory into the
99   sophisticated rigid body dynamics.
100  
101 < \section{Method{\label{methodSec}}}
101 > \section{Computational methods{\label{methodSec}}}
102  
103   \subsection{\label{introSection:frictionTensor}Friction Tensor}
104   Theoretically, the friction kernel can be determined using velocity
# Line 360 | Line 360 | joining center of resistance $R$ and origin $O$.
360   where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
361   joining center of resistance $R$ and origin $O$.
362  
363 < \subsection{Langevin dynamics for rigid particles of arbitrary shape}
363 > \subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}}
364  
365   Consider a Langevin equation of motions in generalized coordinates
366   \begin{equation}
# Line 369 | Line 369 | $V_i = V_i(v_i,\omega _i)$. The right side of Eq.
369   \end{equation}
370   where $M_i$ is a $6\times6$ generalized diagonal mass (include mass
371   and moment of inertial) matrix and $V_i$ is a generalized velocity,
372 < $V_i = V_i(v_i,\omega _i)$. The right side of Eq.
372 > $V_i = V_i(v_i,\omega _i)$. The right side of Eq.~
373   (\ref{LDGeneralizedForm}) consists of three generalized forces in
374   lab-fixed frame, systematic force $F_{s,i}$, dissipative force
375   $F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the
# Line 403 | Line 403 | with zero mean and variance
403   \begin{equation}
404   \left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle  =
405   \left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle  =
406 < 2k_B T\Xi _R \delta (t - t').
406 > 2k_B T\Xi _R \delta (t - t'). \label{randomForce}
407   \end{equation}
408  
409   The equation of motion for $v_i$ can be written as
# Line 516 | Line 516 | be advanced to the same time value.
516   \end{align*}
517  
518   \section{Results and discussion}
519 +
520 + The Langevin algorithm described in previous section has been
521 + implemented in {\sc oopse}\cite{Meineke2005} and applied to the
522 + studies of kinetics and thermodynamic properties in several systems.
523 +
524 + \subsection{Temperature control}
525 +
526 + As shown in Eq.~\ref{randomForce}, random collisions associated with
527 + the solvent's thermal motions is controlled by the external
528 + temperature. The capability to maintain the temperature of the whole
529 + system was usually used to measure the stability and efficiency of
530 + the algorithm. In order to verify the stability of this new
531 + algorithm, a series of simulations are performed on system
532 + consisiting of 256 SSD water molecules with different viscosities.
533 + Initial configuration for the simulations is taken from a 1ns NVT
534 + simulation with a cubic box of 19.7166~\AA. All simulation are
535 + carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns
536 + with reference temperature at 300~K. Average temperature as a
537 + function of $\eta$ is shown in Table \ref{langevin:viscosity} where
538 + the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 -
539 + 1$ poise. The better temperature control at higher viscosity can be
540 + explained by the finite size effect and relative slow relaxation
541 + rate at lower viscosity regime.
542 + \begin{table}
543 + \caption{Average temperatures from Langevin dynamics simulations of
544 + SSD water molecules with reference temperature at 300~K.}
545 + \label{langevin:viscosity}
546 + \begin{center}
547 + \begin{tabular}{|l|l|l|}
548 +  \hline
549 +  \eta & ${\rm{T}}_{{\rm{avg}}}$ & ${\rm{T}}_{{\rm{rms}}}$ \\
550 +  1    & 300.47 & 10.99 \\
551 +  0.1  & 301.19 & 11.136 \\
552 +  0.01 & 303.04 & 11.796 \\
553 +  \hline
554 + \end{tabular}
555 + \end{center}
556 + \end{table}
557  
558 + Another set of calculation were performed to study the efficiency of
559 + temperature control using different temperature coupling schemes.
560 + The starting configuration is cooled to 173~K and evolved using NVE,
561 + NVT, and Langevin dynamic with time step of 2 fs.
562 + Fig.~\ref{langevin:temperature} shows the heating curve obtained as
563 + the systems reach equilibrium. The orange curve in
564 + Fig.~\ref{langevin:temperature} represents the simulation using
565 + Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps
566 + which gives reasonable tight coupling, while the blue one from
567 + Langevin dynamics with viscosity of 0.1 poise demonstrates a faster
568 + scaling to the desire temperature. In extremely lower friction
569 + regime (when $ \eta  \approx 0$), Langevin dynamics becomes normal
570 + NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses
571 + the temperature control ability.
572 +
573 +
574 + \begin{figure}
575 + \centering
576 + \includegraphics[width=\linewidth]{temperature.eps}
577 + \caption[Plot of Temperature Fluctuation Versus Time]{Plot of
578 + temperature fluctuation versus time.} \label{langevin:temperature}
579 + \end{figure}
580 +
581 + \subsection{Langevin dynamics of banana-shaped molecule}
582 +
583 + \begin{figure}
584 + \centering
585 + \includegraphics[width=\linewidth]{one_banana.eps}
586 + \caption[]{.} \label{langevin:banana}
587 + \end{figure}
588 +
589 + \begin{figure}
590 + \centering
591 + \includegraphics[width=\linewidth]{roughShell.eps}
592 + \caption[Rough Shell]{Rough Shell.} \label{langevin:roughShell}
593 + \end{figure}
594 +
595 + \begin{figure}
596 + \centering
597 + \includegraphics[width=\linewidth]{twoBanana.eps}
598 + \caption[Two Banana Shaped Molecules]{Two Banana Shaped Molecules.}
599 + \label{langevin:twoBanana}
600 + \end{figure}
601 +
602   \section{Conclusions}

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