| 28 |
|
between the native and denatured states. Because of its stability |
| 29 |
|
against noise, Langevin dynamics is very suitable for studying |
| 30 |
|
remagnetization processes in various |
| 31 |
< |
systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For |
| 32 |
< |
instance, the oscillation power spectrum of nanoparticles from |
| 33 |
< |
Langevin dynamics simulation has the same peak frequencies for |
| 34 |
< |
different wave vectors,which recovers the property of magnetic |
| 35 |
< |
excitations in small finite structures\cite{Berkov2005a}. In an |
| 36 |
< |
attempt to reduce the computational cost of simulation, multiple |
| 37 |
< |
time stepping (MTS) methods have been introduced and have been of |
| 38 |
< |
great interest to macromolecule and protein |
| 39 |
< |
community\cite{Tuckerman1992}. Relying on the observation that |
| 40 |
< |
forces between distant atoms generally demonstrate slower |
| 41 |
< |
fluctuations than forces between close atoms, MTS method are |
| 42 |
< |
generally implemented by evaluating the slowly fluctuating forces |
| 43 |
< |
less frequently than the fast ones. Unfortunately, nonlinear |
| 44 |
< |
instability resulting from increasing timestep in MTS simulation |
| 45 |
< |
have became a critical obstruction preventing the long time |
| 46 |
< |
simulation. Due to the coupling to the heat bath, Langevin dynamics |
| 47 |
< |
has been shown to be able to damp out the resonance artifact more |
| 48 |
< |
efficiently\cite{Sandu1999}. |
| 31 |
> |
systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
| 32 |
> |
oscillation power spectrum of nanoparticles from Langevin dynamics |
| 33 |
> |
simulation has the same peak frequencies for different wave |
| 34 |
> |
vectors,which recovers the property of magnetic excitations in small |
| 35 |
> |
finite structures\cite{Berkov2005a}. In an attempt to reduce the |
| 36 |
> |
computational cost of simulation, multiple time stepping (MTS) |
| 37 |
> |
methods have been introduced and have been of great interest to |
| 38 |
> |
macromolecule and protein community\cite{Tuckerman1992}. Relying on |
| 39 |
> |
the observation that forces between distant atoms generally |
| 40 |
> |
demonstrate slower fluctuations than forces between close atoms, MTS |
| 41 |
> |
method are generally implemented by evaluating the slowly |
| 42 |
> |
fluctuating forces less frequently than the fast ones. |
| 43 |
> |
Unfortunately, nonlinear instability resulting from increasing |
| 44 |
> |
timestep in MTS simulation have became a critical obstruction |
| 45 |
> |
preventing the long time simulation. Due to the coupling to the heat |
| 46 |
> |
bath, Langevin dynamics has been shown to be able to damp out the |
| 47 |
> |
resonance artifact more efficiently\cite{Sandu1999}. |
| 48 |
|
|
| 49 |
|
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
| 50 |
|
Combining Langevin or Brownian dynamics with rigid body dynamics, |
| 171 |
|
due to the complexity of the elliptic integral, only the ellipsoid |
| 172 |
|
with the restriction of two axes having to be equal, \textit{i.e.} |
| 173 |
|
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 174 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 174 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate |
| 175 |
> |
: |
| 176 |
|
\[ |
| 177 |
|
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 178 |
|
} }}{b}, |
| 179 |
|
\] |
| 180 |
< |
and oblate, |
| 180 |
> |
and oblate : |
| 181 |
|
\[ |
| 182 |
|
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 183 |
|
}}{a} |
| 186 |
|
tensors |
| 187 |
|
\[ |
| 188 |
|
\begin{array}{l} |
| 189 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 190 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 191 |
< |
\end{array}, |
| 189 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 190 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 191 |
> |
2a}}, |
| 192 |
> |
\end{array} |
| 193 |
|
\] |
| 194 |
|
and |
| 195 |
|
\[ |
| 196 |
|
\begin{array}{l} |
| 197 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 197 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 198 |
|
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 199 |
|
\end{array}. |
| 200 |
|
\] |
| 542 |
|
explained by the finite size effect and relative slow relaxation |
| 543 |
|
rate at lower viscosity regime. |
| 544 |
|
\begin{table} |
| 545 |
< |
\caption{Average temperatures from Langevin dynamics simulations of |
| 546 |
< |
SSD water molecules with reference temperature at 300~K.} |
| 545 |
> |
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
| 546 |
> |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
| 547 |
|
\label{langevin:viscosity} |
| 548 |
|
\begin{center} |
| 549 |
|
\begin{tabular}{|l|l|l|} |
| 579 |
|
temperature fluctuation versus time.} \label{langevin:temperature} |
| 580 |
|
\end{figure} |
| 581 |
|
|
| 582 |
< |
\subsection{Langevin Dynamics of Banana Shaped Molecule} |
| 582 |
> |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
| 583 |
|
|
| 584 |
|
In order to verify that Langevin dynamics can mimic the dynamics of |
| 585 |
|
the systems absent of explicit solvents, we carried out two sets of |
| 586 |
|
simulations and compare their dynamic properties. |
| 587 |
< |
|
| 588 |
< |
\subsubsection{Simulations Contain One Banana Shaped Molecule} |
| 588 |
< |
|
| 589 |
< |
Fig.~\ref{langevin:oneBanana} shows a snapshot of the simulation |
| 590 |
< |
made of 256 pentane molecules and one banana shaped molecule at |
| 587 |
> |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
| 588 |
> |
made of 256 pentane molecules and two banana shaped molecules at |
| 589 |
|
273~K. It has an equivalent implicit solvent system containing only |
| 590 |
< |
one banana shaped molecule with viscosity of 0.289 center poise. To |
| 590 |
> |
two banana shaped molecules with viscosity of 0.289 center poise. To |
| 591 |
|
calculate the hydrodynamic properties of the banana shaped molecule, |
| 592 |
|
we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 593 |
|
in which the banana shaped molecule is represented as a ``shell'' |
| 594 |
< |
made of 2266 small identical beads with size of 0.3 $\AA$ on the |
| 594 |
> |
made of 2266 small identical beads with size of 0.3 \AA on the |
| 595 |
|
surface. Applying the procedure described in |
| 596 |
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 597 |
|
identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
| 606 |
|
0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
| 607 |
|
\end{array}} \right). |
| 608 |
|
\] |
| 609 |
+ |
Curves of velocity auto-correlation functions in |
| 610 |
+ |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
| 611 |
+ |
However, because of the stochastic nature, simulation using Langevin |
| 612 |
+ |
dynamics was shown to decay slightly fast. In order to study the |
| 613 |
+ |
rotational motion of the molecules, we also calculated the auto- |
| 614 |
+ |
correlation function of the principle axis of the second GB |
| 615 |
+ |
particle, $u$. |
| 616 |
|
|
| 617 |
|
\begin{figure} |
| 618 |
|
\centering |
| 621 |
|
model for banana shaped molecule.} \label{langevin:roughShell} |
| 622 |
|
\end{figure} |
| 623 |
|
|
| 619 |
– |
%\begin{figure} |
| 620 |
– |
%\centering |
| 621 |
– |
%\includegraphics[width=\linewidth]{oneBanana.eps} |
| 622 |
– |
%\caption[Snapshot from Simulation of One Banana Shaped Molecules and |
| 623 |
– |
%256 Pentane Molecules]{Snapshot from simulation of one Banana shaped |
| 624 |
– |
%molecules and 256 pentane molecules.} \label{langevin:oneBanana} |
| 625 |
– |
%\end{figure} |
| 626 |
– |
|
| 627 |
– |
\subsubsection{Simulations Contain Two Banana Shaped Molecules} |
| 628 |
– |
|
| 624 |
|
\begin{figure} |
| 625 |
|
\centering |
| 626 |
|
\includegraphics[width=\linewidth]{twoBanana.eps} |
| 629 |
|
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 630 |
|
\end{figure} |
| 631 |
|
|
| 632 |
+ |
\begin{figure} |
| 633 |
+ |
\centering |
| 634 |
+ |
\includegraphics[width=\linewidth]{vacf.eps} |
| 635 |
+ |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 636 |
+ |
auto-correlation functions in NVE (blue) and Langevin dynamics |
| 637 |
+ |
(red).} \label{langevin:vacf} |
| 638 |
+ |
\end{figure} |
| 639 |
+ |
|
| 640 |
+ |
\begin{figure} |
| 641 |
+ |
\centering |
| 642 |
+ |
\includegraphics[width=\linewidth]{uacf.eps} |
| 643 |
+ |
\caption[Auto-correlation functions of the principle axis of the |
| 644 |
+ |
middle GB particle]{Auto-correlation functions of the principle axis |
| 645 |
+ |
of the middle GB particle in NVE (blue) and Langevin dynamics |
| 646 |
+ |
(red).} \label{langevin:twoBanana} |
| 647 |
+ |
\end{figure} |
| 648 |
+ |
|
| 649 |
|
\section{Conclusions} |
| 650 |
+ |
|
| 651 |
+ |
We have presented a new Langevin algorithm by incorporating the |
| 652 |
+ |
hydrodynamics properties of arbitrary shaped molecules into an |
| 653 |
+ |
advanced symplectic integration scheme. The temperature control |
| 654 |
+ |
ability of this algorithm was demonstrated by a set of simulations |
| 655 |
+ |
with different viscosities. It was also shown to have significant |
| 656 |
+ |
advantage of producing rapid thermal equilibration over |
| 657 |
+ |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
| 658 |
+ |
shaped molecules illustrated that the dynamic properties could be |
| 659 |
+ |
preserved by using this new algorithm as an implicit solvent model. |
