--- trunk/tengDissertation/Langevin.tex	2006/06/23 21:33:52	2882
+++ trunk/tengDissertation/Langevin.tex	2006/06/26 13:43:22	2890
@@ -28,24 +28,23 @@ systems\cite{Garcia-Palacios1998,Berkov2002,Denisov200
 between the native and denatured states. Because of its stability
 against noise, Langevin dynamics is very suitable for studying
 remagnetization processes in various
-systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For
-instance, the oscillation power spectrum of nanoparticles from
-Langevin dynamics simulation has the same peak frequencies for
-different wave vectors,which recovers the property of magnetic
-excitations in small finite structures\cite{Berkov2005a}. In an
-attempt to reduce the computational cost of simulation, multiple
-time stepping (MTS) methods have been introduced and have been of
-great interest to macromolecule and protein
-community\cite{Tuckerman1992}. Relying on the observation that
-forces between distant atoms generally demonstrate slower
-fluctuations than forces between close atoms, MTS method are
-generally implemented by evaluating the slowly fluctuating forces
-less frequently than the fast ones. Unfortunately, nonlinear
-instability resulting from increasing timestep in MTS simulation
-have became a critical obstruction preventing the long time
-simulation. Due to the coupling to the heat bath, Langevin dynamics
-has been shown to be able to damp out the resonance artifact more
-efficiently\cite{Sandu1999}.
+systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the
+oscillation power spectrum of nanoparticles from Langevin dynamics
+simulation has the same peak frequencies for different wave
+vectors,which recovers the property of magnetic excitations in small
+finite structures\cite{Berkov2005a}. In an attempt to reduce the
+computational cost of simulation, multiple time stepping (MTS)
+methods have been introduced and have been of great interest to
+macromolecule and protein community\cite{Tuckerman1992}. Relying on
+the observation that forces between distant atoms generally
+demonstrate slower fluctuations than forces between close atoms, MTS
+method are generally implemented by evaluating the slowly
+fluctuating forces less frequently than the fast ones.
+Unfortunately, nonlinear instability resulting from increasing
+timestep in MTS simulation have became a critical obstruction
+preventing the long time simulation. Due to the coupling to the heat
+bath, Langevin dynamics has been shown to be able to damp out the
+resonance artifact more efficiently\cite{Sandu1999}.
 
 %review langevin/browninan dynamics for arbitrarily shaped rigid body
 Combining Langevin or Brownian dynamics with rigid body dynamics,
@@ -79,11 +78,11 @@ term\cite{Beard2001}. As a complement to IBD which has
 average acceleration is not always true for cooperative motion which
 is common in protein motion. An inertial Brownian dynamics (IBD) was
 proposed to address this issue by adding an inertial correction
-term\cite{Beard2001}. As a complement to IBD which has a lower bound
+term\cite{Beard2000}. As a complement to IBD which has a lower bound
 in time step because of the inertial relaxation time, long-time-step
 inertial dynamics (LTID) can be used to investigate the inertial
 behavior of the polymer segments in low friction
-regime\cite{Beard2001}. LTID can also deal with the rotational
+regime\cite{Beard2000}. LTID can also deal with the rotational
 dynamics for nonskew bodies without translation-rotation coupling by
 separating the translation and rotation motion and taking advantage
 of the analytical solution of hydrodynamics properties. However,
@@ -172,12 +171,13 @@ exactly. Introducing an elliptic integral parameter $S
 due to the complexity of the elliptic integral, only the ellipsoid
 with the restriction of two axes having to be equal, \textit{i.e.}
 prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved
-exactly. Introducing an elliptic integral parameter $S$ for prolate,
+exactly. Introducing an elliptic integral parameter $S$ for prolate
+:
 \[
 S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
 } }}{b},
 \]
-and oblate,
+and oblate :
 \[
 S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
 }}{a}
@@ -186,14 +186,15 @@ tensors
 tensors
 \[
 \begin{array}{l}
- \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
- \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
- \end{array},
+ \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}}. \\
+ \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S +
+ 2a}},
+ \end{array}
 \]
 and
 \[
 \begin{array}{l}
- \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
+ \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}}, \\
  \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
  \end{array}.
 \]
@@ -541,8 +542,8 @@ rate at lower viscosity regime.
 explained by the finite size effect and relative slow relaxation
 rate at lower viscosity regime.
 \begin{table}
-\caption{Average temperatures from Langevin dynamics simulations of
-SSD water molecules with reference temperature at 300~K.}
+\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF
+SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.}
 \label{langevin:viscosity}
 \begin{center}
 \begin{tabular}{|l|l|l|}
@@ -578,22 +579,19 @@ temperature fluctuation versus time.} \label{langevin:
 temperature fluctuation versus time.} \label{langevin:temperature}
 \end{figure}
 
-\subsection{Langevin Dynamics of Banana Shaped Molecule}
+\subsection{Langevin Dynamics of Banana Shaped Molecules}
 
 In order to verify that Langevin dynamics can mimic the dynamics of
 the systems absent of explicit solvents, we carried out two sets of
 simulations and compare their dynamic properties.
-
-\subsubsection{Simulations Contain One Banana Shaped Molecule}
-
-Fig.~\ref{langevin:oneBanana} shows a snapshot of the simulation
-made of 256 pentane molecules and one banana shaped molecule at
+Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation
+made of 256 pentane molecules and two banana shaped molecules at
 273~K. It has an equivalent implicit solvent system containing only
-one banana shaped molecule with viscosity of 0.289 center poise. To
+two banana shaped molecules with viscosity of 0.289 center poise. To
 calculate the hydrodynamic properties of the banana shaped molecule,
 we create a rough shell model (see Fig.~\ref{langevin:roughShell}),
 in which the banana shaped molecule is represented as a ``shell''
-made of 2266 small identical beads with size of 0.3 $\AA$ on the
+made of 2266 small identical beads with size of 0.3 \AA on the
 surface. Applying the procedure described in
 Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
 identified the center of resistance at $(0, 0.7482, -0.1988)$, as
@@ -608,6 +606,13 @@ -6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-1
 0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\
 \end{array}} \right).
 \]
+Curves of velocity auto-correlation functions in
+Fig.~\ref{langevin:vacf} were shown to match each other very well.
+However, because of the stochastic nature, simulation using Langevin
+dynamics was shown to decay slightly fast. In order to study the
+rotational motion of the molecules, we also calculated the auto-
+correlation function of the principle axis of the second GB
+particle, $u$.
 
 \begin{figure}
 \centering
@@ -616,16 +621,6 @@ model for banana shaped molecule.} \label{langevin:rou
 model for banana shaped molecule.} \label{langevin:roughShell}
 \end{figure}
 
-%\begin{figure}
-%\centering
-%\includegraphics[width=\linewidth]{oneBanana.eps}
-%\caption[Snapshot from Simulation of One Banana Shaped Molecules and
-%256 Pentane Molecules]{Snapshot from simulation of one Banana shaped
-%molecules and 256 pentane molecules.} \label{langevin:oneBanana}
-%\end{figure}
-
-\subsubsection{Simulations Contain Two Banana Shaped Molecules}
-
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{twoBanana.eps}
@@ -634,4 +629,31 @@ molecules and 256 pentane molecules.} \label{langevin:
 molecules and 256 pentane molecules.} \label{langevin:twoBanana}
 \end{figure}
 
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{vacf.eps}
+\caption[Plots of Velocity Auto-correlation Functions]{Velocity
+auto-correlation functions in NVE (blue) and Langevin dynamics
+(red).} \label{langevin:vacf}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{uacf.eps}
+\caption[Auto-correlation functions of the principle axis of the
+middle GB particle]{Auto-correlation functions of the principle axis
+of the middle GB particle in NVE (blue) and Langevin dynamics
+(red).} \label{langevin:twoBanana}
+\end{figure}
+
 \section{Conclusions}
+
+We have presented a new Langevin algorithm by incorporating the
+hydrodynamics properties of arbitrary shaped molecules into an
+advanced symplectic integration scheme. The temperature control
+ability of this algorithm was demonstrated by a set of simulations
+with different viscosities. It was also shown to have significant
+advantage of producing rapid thermal equilibration over
+Nos\'{e}-Hoover method. Further studies in systems involving banana
+shaped molecules illustrated that the dynamic properties could be
+preserved by using this new algorithm as an implicit solvent model.
