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\begin{document}

\title{An algorithm for performing Langevin dynamics on rigid bodies of arbitrary shape }

\author{Xiuquan Sun, Teng Lin and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail:
gezelter@nd.edu} \\
Department of Chemistry and Biochemistry\\
University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}

\maketitle \doublespacing

\begin{abstract}

\end{abstract}

\newpage

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\section{Introduction}

%applications of langevin dynamics
As alternative to Newtonian dynamics, Langevin dynamics, which
mimics a simple heat bath with stochastic and dissipative forces,
has been applied in a variety of studies. The stochastic treatment
of the solvent enables us to carry out substantially longer time
simulations. Implicit solvent Langevin dynamics simulations of
met-enkephalin not only outperform explicit solvent simulations for
computational efficiency, but also agrees very well with explicit
solvent simulations for dynamical properties.\cite{Shen2002}
Recently, applying Langevin dynamics with the UNRES model, Liow and
his coworkers suggest that protein folding pathways can be possibly
explored within a reasonable amount of time.\cite{Liwo2005} The
stochastic nature of the Langevin dynamics also enhances the
sampling of the system and increases the probability of crossing
energy barriers.\cite{Banerjee2004, Cui2003} Combining Langevin
dynamics with Kramers's theory, Klimov and Thirumalai identified
free-energy barriers by studying the viscosity dependence of the
protein folding rates.\cite{Klimov1997} In order to account for
solvent induced interactions missing from implicit solvent model,
Kaya incorporated desolvation free energy barrier into implicit
coarse-grained solvent model in protein folding/unfolding studies
and discovered a higher free energy barrier 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{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}

%review rigid body dynamics
Rigid bodies are frequently involved in the modeling of different
areas, from engineering, physics, to chemistry. For example,
missiles and vehicle are usually modeled by rigid bodies.  The
movement of the objects in 3D gaming engine or other physics
simulator is governed by the rigid body dynamics. In molecular
simulation, rigid body is used to simplify the model in
protein-protein docking study{\cite{Gray2003}}.

It is very important to develop stable and efficient methods to
integrate the equations of motion for orientational degrees of
freedom. Euler angles are the natural choice to describe the
rotational degrees of freedom. However, due to $\frac {1}{sin
\theta}$ singularities, the numerical integration of corresponding
equations of these motion is very inefficient and inaccurate.
Although an alternative integrator using multiple sets of Euler
angles can overcome this difficulty\cite{Barojas1973}, the
computational penalty and the loss of angular momentum conservation
still remain. A singularity-free representation utilizing
quaternions was developed by Evans in 1977.\cite{Evans1977}
Unfortunately, this approach used a nonseparable Hamiltonian
resulting from the quaternion representation, which prevented the
symplectic algorithm from being utilized. Another different approach
is to apply holonomic constraints to the atoms belonging to the
rigid body. Each atom moves independently under the normal forces
deriving from potential energy and constraint forces which are used
to guarantee the rigidness. However, due to their iterative nature,
the SHAKE and Rattle algorithms also converge very slowly when the
number of constraints increases.\cite{Ryckaert1977, Andersen1983}

A break-through in geometric literature suggests that, in order to
develop a long-term integration scheme, one should preserve the
symplectic structure of the propagator. By introducing a conjugate
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was
proposed to evolve the Hamiltonian system in a constraint manifold
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$.
An alternative method using the quaternion representation was
developed by Omelyan.\cite{Omelyan1998} However, both of these
methods are iterative and inefficient. In this section, we descibe a
symplectic Lie-Poisson integrator for rigid bodies developed by
Dullweber and his coworkers\cite{Dullweber1997} in depth.

%review langevin/browninan dynamics for arbitrarily shaped rigid body
Combining Langevin or Brownian dynamics with rigid body dynamics,
one can study slow processes in biomolecular systems. Modeling DNA
as a chain of rigid beads, which are subject to harmonic potentials
as well as excluded volume potentials, Mielke and his coworkers
discovered rapid superhelical stress generations from the stochastic
simulation of twin supercoiling DNA with response to induced
torques.\cite{Mielke2004} Membrane fusion is another key biological
process which controls a variety of physiological functions, such as
release of neurotransmitters \textit{etc}. A typical fusion event
happens on the time scale of a millisecond, which is impractical to
study using atomistic models with newtonian mechanics. With the help
of coarse-grained rigid body model and stochastic dynamics, the
fusion pathways were explored by many
researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the
difficulty of numerical integration of anisotropic rotation, most of
the rigid body models are simply modeled using spheres, cylinders,
ellipsoids or other regular shapes in stochastic simulations. In an
effort to account for the diffusion anisotropy of arbitrary
particles, Fernandes and de la Torre improved the original Brownian
dynamics simulation algorithm\cite{Ermak1978,Allison1991} by
incorporating a generalized $6\times6$ diffusion tensor and
introducing a simple rotation evolution scheme consisting of three
consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected
errors and biases are introduced into the system due to the
arbitrary order of applying the noncommuting rotation
operators.\cite{Beard2003} Based on the observation the momentum
relaxation time is much less than the time step, one may ignore the
inertia in Brownian dynamics. However, the assumption of zero
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{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{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,
typical nonskew bodies like cylinders and ellipsoids are inadequate
to represent most complex macromolecule assemblies. These intricate
molecules have been represented by a set of beads and their
hydrodynamic properties can be calculated using variants on the
standard hydrodynamic interaction tensors.

The goal of the present work is to develop a Langevin dynamics
algorithm for arbitrary-shaped rigid particles by integrating the
accurate estimation of friction tensor from hydrodynamics theory
into the sophisticated rigid body dynamics algorithms.

\section{Computational Methods{\label{methodSec}}}

\subsection{\label{introSection:frictionTensor}Friction Tensor}
Theoretically, the friction kernel can be determined using the
velocity autocorrelation function. However, this approach becomes
impractical when the system becomes more and more complicated.
Instead, various approaches based on hydrodynamics have been
developed to calculate the friction coefficients. In general, the
friction tensor $\Xi$ is a $6\times 6$ matrix given by
\[
\Xi  = \left( {\begin{array}{*{20}c}
   {\Xi _{}^{tt} } & {\Xi _{}^{rt} }  \\
   {\Xi _{}^{tr} } & {\Xi _{}^{rr} }  \\
\end{array}} \right).
\]
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$
translational friction tensor and rotational resistance (friction)
tensor respectively, while ${\Xi^{tr} }$ is translation-rotation
coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling
tensor. When a particle moves in a fluid, it may experience friction
force or torque along the opposite direction of the velocity or
angular velocity,
\[
\left( \begin{array}{l}
 F_R  \\
 \tau _R  \\
 \end{array} \right) =  - \left( {\begin{array}{*{20}c}
   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
\end{array}} \right)\left( \begin{array}{l}
 v \\
 w \\
 \end{array} \right)
\]
where $F_r$ is the friction force and $\tau _R$ is the friction
torque.

\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}}

For a spherical particle with slip boundary conditions, the
translational and rotational friction constant can be calculated
from Stoke's law,
\[
\Xi ^{tt}  = \left( {\begin{array}{*{20}c}
   {6\pi \eta R} & 0 & 0  \\
   0 & {6\pi \eta R} & 0  \\
   0 & 0 & {6\pi \eta R}  \\
\end{array}} \right)
\]
and
\[
\Xi ^{rr}  = \left( {\begin{array}{*{20}c}
   {8\pi \eta R^3 } & 0 & 0  \\
   0 & {8\pi \eta R^3 } & 0  \\
   0 & 0 & {8\pi \eta R^3 }  \\
\end{array}} \right)
\]
where $\eta$ is the viscosity of the solvent and $R$ is the
hydrodynamic radius.

Other non-spherical shapes, such as cylinders and ellipsoids, are
widely used as references for developing new hydrodynamics theory,
because their properties can be calculated exactly. In 1936, Perrin
extended Stokes's law to general ellipsoids, also called a triaxial
ellipsoid, which is given in Cartesian coordinates
by\cite{Perrin1934, Perrin1936}
\[
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2
}} = 1
\]
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately,
due to the complexity of the elliptic integral, only the ellipsoid
with the restriction of two axes being 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
ellipsoids :
\[
S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
} }}{b},
\]
and oblate ellipsoids:
\[
S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
}}{a},
\]
one can write down the translational and rotational resistance
tensors
\begin{eqnarray*}
 \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{eqnarray*}
and
\begin{eqnarray*}
 \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{eqnarray*}

\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}}

Unlike spherical and other simply shaped molecules, there is no
analytical solution for the friction tensor for arbitrarily shaped
rigid molecules. The ellipsoid of revolution model and general
triaxial ellipsoid model have been used to approximate the
hydrodynamic properties of rigid bodies. However, since the mapping
from all possible ellipsoidal spaces, $r$-space, to all possible
combination of rotational diffusion coefficients, $D$-space, is not
unique\cite{Wegener1979} as well as the intrinsic coupling between
translational and rotational motion of rigid bodies, general
ellipsoids are not always suitable for modeling arbitrarily shaped
rigid molecules. A number of studies have been devoted to
determining the friction tensor for irregularly shaped rigid bodies
using more advanced methods where the molecule of interest was
modeled by a combinations of spheres\cite{Carrasco1999} and the
hydrodynamics properties of the molecule can be calculated using the
hydrodynamic interaction tensor. Let us consider a rigid assembly of
$N$ beads immersed in a continuous medium. Due to hydrodynamic
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different
than its unperturbed velocity $v_i$,
\[
v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
\]
where $F_i$ is the frictional force, and $T_{ij}$ is the
hydrodynamic interaction tensor. The friction force of $i$th bead is
proportional to its ``net'' velocity
\begin{equation}
F_i  = \zeta _i v_i  - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }.
\label{introEquation:tensorExpression}
\end{equation}
This equation is the basis for deriving the hydrodynamic tensor. In
1930, Oseen and Burgers gave a simple solution to
Eq.~\ref{introEquation:tensorExpression}
\begin{equation}
T_{ij}  = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij}
R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor}
\end{equation}
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
A second order expression for element of different size was
introduced by Rotne and Prager\cite{Rotne1969} and improved by
Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977}
\begin{equation}
T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
_i^2  + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} -
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
\label{introEquation:RPTensorNonOverlapped}
\end{equation}
Both of the Eq.~\ref{introEquation:oseenTensor} and
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption
$R_{ij} \ge \sigma _i  + \sigma _j$. An alternative expression for
overlapping beads with the same radius, $\sigma$, is given by
\begin{equation}
T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I +
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
\label{introEquation:RPTensorOverlapped}
\end{equation}
To calculate the resistance tensor at an arbitrary origin $O$, we
construct a $3N \times 3N$ matrix consisting of $N \times N$
$B_{ij}$ blocks
\begin{equation}
B = \left( {\begin{array}{*{20}c}
   {B_{11} } &  \ldots  & {B_{1N} }  \\
    \vdots  &  \ddots  &  \vdots   \\
   {B_{N1} } &  \cdots  & {B_{NN} }  \\
\end{array}} \right),
\end{equation}
where $B_{ij}$ is given by
\[
B_{ij}  = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij}
)T_{ij}
\]
where $\delta _{ij}$ is the Kronecker delta function. Inverting the
$B$ matrix, we obtain
\[
C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
   {C_{11} } &  \ldots  & {C_{1N} }  \\
    \vdots  &  \ddots  &  \vdots   \\
   {C_{N1} } &  \cdots  & {C_{NN} }  \\
\end{array}} \right),
\]
which can be partitioned into $N \times N$ $3 \times 3$ block
$C_{ij}$. With the help of $C_{ij}$ and the skew matrix $U_i$
\[
U_i  = \left( {\begin{array}{*{20}c}
   0 & { - z_i } & {y_i }  \\
   {z_i } & 0 & { - x_i }  \\
   { - y_i } & {x_i } & 0  \\
\end{array}} \right)
\]
where $x_i$, $y_i$, $z_i$ are the components of the vector joining
bead $i$ and origin $O$, the elements of resistance tensor at
arbitrary origin $O$ can be written as
\begin{eqnarray}
 \Xi _{}^{tt}  & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\
 \Xi _{}^{tr}  & = & \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
 \Xi _{}^{rr}  & = &  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\
\label{introEquation:ResistanceTensorArbitraryOrigin}
\end{eqnarray}
The resistance tensor depends on the origin to which they refer. The
proper location for applying the friction force is the center of
resistance (or center of reaction), at which the trace of rotational
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value.
Mathematically, the center of resistance is defined as an unique
point of the rigid body at which the translation-rotation coupling
tensors are symmetric,
\begin{equation}
\Xi^{tr}  = \left( {\Xi^{tr} } \right)^T
\label{introEquation:definitionCR}
\end{equation}
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin},
we can easily derive that the translational resistance tensor is
origin independent, while the rotational resistance tensor and
translation-rotation coupling resistance tensor depend on the
origin. Given the resistance tensor at an arbitrary origin $O$, and
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can
obtain the resistance tensor at $P$ by
\begin{equation}
\begin{array}{l}
 \Xi _P^{tt}  = \Xi _O^{tt}  \\
 \Xi _P^{tr}  = \Xi _P^{rt}  = \Xi _O^{tr}  - U_{OP} \Xi _O^{tt}  \\
 \Xi _P^{rr}  = \Xi _O^{rr}  - U_{OP} \Xi _O^{tt} U_{OP}  + \Xi _O^{tr} U_{OP}  - U_{OP} \Xi _O^{{tr} ^{^T }}  \\
 \end{array}
 \label{introEquation:resistanceTensorTransformation}
\end{equation}
where
\[
U_{OP}  = \left( {\begin{array}{*{20}c}
   0 & { - z_{OP} } & {y_{OP} }  \\
   {z_i } & 0 & { - x_{OP} }  \\
   { - y_{OP} } & {x_{OP} } & 0  \\
\end{array}} \right)
\]
Using Eq.~\ref{introEquation:definitionCR} and
Eq.~\ref{introEquation:resistanceTensorTransformation}, one can
locate the position of center of resistance,
\begin{eqnarray*}
 \left( \begin{array}{l}
 x_{OR}  \\
 y_{OR}  \\
 z_{OR}  \\
 \end{array} \right) & = &\left( {\begin{array}{*{20}c}
   {(\Xi _O^{rr} )_{yy}  + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} }  \\
   { - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz}  + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} }  \\
   { - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx}  + (\Xi _O^{rr} )_{yy} }  \\
\end{array}} \right)^{ - 1}  \\
  & & \left( \begin{array}{l}
 (\Xi _O^{tr} )_{yz}  - (\Xi _O^{tr} )_{zy}  \\
 (\Xi _O^{tr} )_{zx}  - (\Xi _O^{tr} )_{xz}  \\
 (\Xi _O^{tr} )_{xy}  - (\Xi _O^{tr} )_{yx}  \\
 \end{array} \right) \\
\end{eqnarray*}
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
joining center of resistance $R$ and origin $O$.

\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}}

Consider the Langevin equations of motion in generalized coordinates
\begin{equation}
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)}  + F_{r,i} (t)
\label{LDGeneralizedForm}
\end{equation}
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass
and moment of inertial) matrix and $V_i$ is a generalized velocity,
$V_i = V_i(v_i,\omega _i)$. The right side of
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in
lab-fixed frame, systematic force $F_{s,i}$, dissipative force
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the
system in Newtownian mechanics typically refers to lab-fixed frame,
it is also convenient to handle the rotation of rigid body in
body-fixed frame. Thus the friction and random forces are calculated
in body-fixed frame and converted back to lab-fixed frame by:
\[
\begin{array}{l}
 F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\
 F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\
 \end{array}
\]
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to
the body-fixed velocity at center of resistance $v_{R,i}^b$ and
angular velocity $\omega _i$
\begin{equation}
F_{r,i}^b (t) = \left( \begin{array}{l}
 f_{r,i}^b (t) \\
 \tau _{r,i}^b (t) \\
 \end{array} \right) =  - \left( {\begin{array}{*{20}c}
   {\Xi _{R,t} } & {\Xi _{R,c}^T }  \\
   {\Xi _{R,c} } & {\Xi _{R,r} }  \\
\end{array}} \right)\left( \begin{array}{l}
 v_{R,i}^b (t) \\
 \omega _i (t) \\
 \end{array} \right),
\end{equation}
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable
with zero mean and variance
\begin{equation}
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle  =
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle  =
2k_B T\Xi _R \delta (t - t'). \label{randomForce}
\end{equation}
The equation of motion for $v_i$ can be written as
\begin{equation}
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) +
f_{r,i}^l (t)
\end{equation}
Since the frictional force is applied at the center of resistance
which generally does not coincide with the center of mass, an extra
torque is exerted at the center of mass. Thus, the net body-fixed
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is
given by
\begin{equation}
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b
\end{equation}
where $r_{MR}$ is the vector from the center of mass to the center
of the resistance. Instead of integrating the angular velocity in
lab-fixed frame, we consider the equation of angular momentum in
body-fixed frame
\begin{equation}
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t)
+ \tau _{r,i}^b(t)
\end{equation}
Embedding the friction terms into force and torque, one can
integrate the langevin equations of motion for rigid body of
arbitrary shape in a velocity-Verlet style 2-part algorithm, where
$h= \delta t$:

{\tt moveA:}
\begin{align*}
{\bf v}\left(t + h / 2\right)  &\leftarrow  {\bf v}(t)
    + \frac{h}{2} \left( {\bf f}(t) / m \right), \\
%
{\bf r}(t + h) &\leftarrow {\bf r}(t)
    + h  {\bf v}\left(t + h / 2 \right), \\
%
{\bf j}\left(t + h / 2 \right)  &\leftarrow {\bf j}(t)
    + \frac{h}{2} {\bf \tau}^b(t), \\
%
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j}
    (t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right).
\end{align*}
In this context, the $\mathrm{rotate}$ function is the reversible
product of the three body-fixed rotations,
\begin{equation}
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y
/ 2) \cdot \mathsf{G}_x(a_x /2),
\end{equation}
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$,
rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed
axis $\alpha$,
\begin{equation}
\mathsf{G}_\alpha( \theta ) = \left\{
\begin{array}{lcl}
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf
j}(0).
\end{array}
\right.
\end{equation}
$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis
rotation matrix.  For example, in the small-angle limit, the
rotation matrix around the body-fixed x-axis can be approximated as
\begin{equation}
\mathsf{R}_x(\theta) \approx \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4}  & -\frac{\theta}{1+
\theta^2 / 4} \\
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 +
\theta^2 / 4}
\end{array}
\right).
\end{equation}
All other rotations follow in a straightforward manner. After the
first part of the propagation, the forces and body-fixed torques are
calculated at the new positions and orientations

{\tt doForces:}
\begin{align*}
{\bf f}(t + h) &\leftarrow
    - \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\
%
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h)
    \times \frac{\partial V}{\partial {\bf u}}, \\
%
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h)
    \cdot {\bf \tau}^s(t + h).
\end{align*}
Once the forces and torques have been obtained at the new time step,
the velocities can be advanced to the same time value.

{\tt moveB:}
\begin{align*}
{\bf v}\left(t + h \right)  &\leftarrow  {\bf v}\left(t + h / 2
\right)
    + \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\
%
{\bf j}\left(t + h \right)  &\leftarrow {\bf j}\left(t + h / 2
\right)
    + \frac{h}{2} {\bf \tau}^b(t + h) .
\end{align*}

\section{Results}
In order to validate our Langevin integrator for arbitrarily-shaped
rigid bodies, we implemented the algorithm in {\sc
oopse}\cite{Meineke2005} and  compared the results of this algorithm
with the known 
hydrodynamic limiting behavior for a few model systems, and to
microcanonical molecular dynamics simulations for some more
complicated bodies. The model systems and their analytical behavior
(if known) are summarized below. Parameters for the primary particles
comprising our model systems are given in table \ref{tab:parameters},
and a sketch of the arrangement of these primary particles into the
model rigid bodies is shown in figure \ref{fig:models}. In table
\ref{tab:parameters}, $d$ and $l$ are the physical dimensions of
ellipsoidal (Gay-Berne) particles.  For spherical particles, the value
of the Lennard-Jones $\sigma$ parameter is the particle diameter
($d$).  Gay-Berne ellipsoids have an energy scaling parameter,
$\epsilon^s$, which describes the well depth for two identical
ellipsoids in a {\it side-by-side} configuration.  Additionally, a
well depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$,
describes the ratio between the well depths in the {\it end-to-end}
and side-by-side configurations.  For spheres, $\epsilon^r \equiv 1$.
Moments of inertia are also required to describe the motion of primary
particles with orientational degrees of freedom.

\begin{table*}
\begin{minipage}{\linewidth}
\begin{center}
\caption{Parameters for the primary particles in use by the rigid body
models in figure \ref{fig:models}.}
\begin{tabular}{lrcccccccc}
\hline 
 & & & & & & & \multicolumn{3}c{$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$)} \\
 & & $d$ (\AA) & $l$ (\AA) & $\epsilon^s$ (kcal/mol) & $\epsilon^r$ &
$m$ (amu) & $I_{xx}$ & $I_{yy}$ & $I_{zz}$ \\ \hline
Sphere   & & 6.5 & $= d$ & 0.8 & 1 & 190 & 802.75 & 802.75  & 802.75  \\
Ellipsoid & & 4.6 & 13.8  & 0.8 & 0.2 & 200 & 2105 & 2105 & 421 \\
Dumbbell &(2 identical spheres) & 6.5 & $= d$ & 0.8 & 1   & 190 & 802.75 & 802.75 & 802.75 \\
Banana  &(3 identical ellipsoids)& 4.2 & 11.2  & 0.8 & 0.2 & 240 & 10000 & 10000 & 0 \\
Lipid: & Spherical Head & 6.5 & $= d$ & 0.185 & 1 & 196 & & & \\
       & Ellipsoidal Tail & 4.6 & 13.8  & 0.8   & 0.2 & 760 & 45000 & 45000 & 9000 \\
Solvent &  & 4.7 & $= d$ & 0.8 & 1   & 72.06 & & & \\
\hline
\end{tabular}
\label{tab:parameters}
\end{center}
\end{minipage}
\end{table*}

\begin{figure}
\centering
\includegraphics[width=3in]{sketch}
\caption[Sketch of the model systems]{A sketch of the model systems
used in evaluating the behavior of the rigid body Langevin
integrator.} \label{fig:models}
\end{figure}

\subsection{Simulation Methodology}

We performed reference microcanonical simulations with explicit
solvents for each of the different model system.  In each case there
was one solute model and 1929 solvent molecules present in the
simulation box.  All simulations were equilibrated using a
constant-pressure and temperature integrator with target values of 300
K for the temperature and 1 atm for pressure.  Following this stage,
further equilibration and sampling was done in a microcanonical
ensemble.  Since the model bodies are typically quite massive, we were
able to use a time step of 25 fs.  A switching function was applied to
all potentials to smoothly turn off the interactions between a range
of $22$ and $25$ \AA.  The switching function was the standard (cubic)
function,
\begin{equation}
s(r) = 
        \begin{cases}
        1 & \text{if $r \le r_{\text{sw}}$},\\
        \frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2}
        {(r_{\text{cut}} - r_{\text{sw}})^3} 
        & \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\
        0 & \text{if $r > r_{\text{cut}}$.}
        \end{cases}
\label{eq:switchingFunc}
\end{equation}
To measure shear viscosities from our microcanonical simulations, we
used the Einstein form of the pressure correlation function,\cite{hess:209}
\begin{equation}
\eta = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \langle \left(
\int_{t_0}^{t_0 + t} P_{xz}(t') dt' \right)^2 \rangle_{t_0}.
\label{eq:shear}
\end{equation}
A similar form exists for the bulk viscosity
\begin{equation}
\kappa = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \langle \left(
\int_{t_0}^{t_0 + t} 
\left(P\left(t'\right)-\langle P \rangle \right)dt'
\right)^2 \rangle_{t_0}.
\end{equation}
Alternatively, the shear viscosity can also be calculated using a
Green-Kubo formula with the off-diagonal pressure tensor correlation function,
\begin{equation}
\eta = \frac{V}{k_B T} \int_0^{\infty} \langle P_{xz}(t_0) P_{xz}(t_0
+ t) \rangle_{t_0} dt,
\end{equation}
although this method converges extremely slowly and is not practical
for obtaining viscosities from molecular dynamics simulations. 

The Langevin dynamics for the different model systems were performed
at the same temperature as the average temperature of the
microcanonical simulations and with a solvent viscosity taken from
Eq. (\ref{eq:shear}) applied to these simulations.  We used 1024
independent solute simulations to obtain statistics on our Langevin
integrator.

\subsection{Analysis}

The quantities of interest when comparing the Langevin integrator to
analytic hydrodynamic equations and to molecular dynamics simulations
are typically translational diffusion constants and orientational
relaxation times.  Translational diffusion constants for point
particles are computed easily from the long-time slope of the
mean-square displacement,
\begin{equation}
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle,
\end{equation}
of the solute molecules.  For models in which the translational
diffusion tensor (${\bf D}_{tt}$) has non-degenerate eigenvalues
(i.e. any non-spherically-symmetric rigid body), it is possible to
compute the diffusive behavior for motion parallel to each body-fixed
axis by projecting the displacement of the particle onto the
body-fixed reference frame at $t=0$.  With an isotropic solvent, as we
have used in this study, there are differences between the three
diffusion constants, but these must converge to the same value at
longer times.  Translational diffusion constants for the different
shaped models are shown in table \ref{tab:translation}.

In general, the three eigenvalues ($D_1, D_2, D_3$) of the rotational
diffusion tensor (${\bf D}_{rr}$) measure the diffusion of an object
{\it around} a particular body-fixed axis and {\it not} the diffusion
of a vector pointing along the axis.  However, these eigenvalues can
be combined to find 5 characteristic rotational relaxation
times,\cite{PhysRev.119.53,Berne90}
\begin{eqnarray}
1 / \tau_1 & = & 6 D_r + 2 \Delta \\
1 / \tau_2 & = & 6 D_r - 2 \Delta \\
1 / \tau_3 & = & 3 (D_r + D_1)  \\
1 / \tau_4 & = & 3 (D_r + D_2) \\
1 / \tau_5 & = & 3 (D_r + D_3) 
\end{eqnarray}
where 
\begin{equation}
D_r = \frac{1}{3} \left(D_1 + D_2 + D_3 \right)
\end{equation}
and 
\begin{equation}
\Delta = \left( (D_1 - D_2)^2 + (D_3 - D_1 )(D_3 - D_2)\right)^{1/2}
\end{equation}
Each of these characteristic times can be used to predict the decay of
part of the rotational correlation function when $\ell = 2$,
\begin{equation}
C_2(t)  =  \frac{a^2}{N^2} e^{-t/\tau_1} + \frac{b^2}{N^2} e^{-t/\tau_2}.
\end{equation}
This is the same as the $F^2_{0,0}(t)$ correlation function that
appears in Ref. \citen{Berne90}.  The amplitudes of the two decay
terms are expressed in terms of three dimensionless functions of the
eigenvalues: $a = \sqrt{3} (D_1 - D_2)$, $b = (2D_3 - D_1 - D_2 +
2\Delta)$, and $N = 2 \sqrt{\Delta b}$.  Similar expressions can be
obtained for other angular momentum correlation
functions.\cite{PhysRev.119.53,Berne90} In all of the model systems we
studied, only one of the amplitudes of the two decay terms was
non-zero, so it was possible to derive a single relaxation time for
each of the hydrodynamic tensors. In many cases, these characteristic
times are averaged and reported in the literature as a single relaxation
time,\cite{Garcia-de-la-Torre:1997qy}
\begin{equation}
1 / \tau_0 = \frac{1}{5} \sum_{i=1}^5 \tau_{i}^{-1},
\end{equation}
although for the cases reported here, this averaging is not necessary
and only one of the five relaxation times is relevant.

To test the Langevin integrator's behavior for rotational relaxation,
we have compared the analytical orientational relaxation times (if
they are known) with the general result from the diffusion tensor and
with the results from both the explicitly solvated molecular dynamics
and Langevin simulations.  Relaxation times from simulations (both
microcanonical and Langevin), were computed using Legendre polynomial
correlation functions for a unit vector (${\bf u}$) fixed along one or
more of the body-fixed axes of the model.
\begin{equation}
C_{\ell}(t)  =  \langle P_{\ell}\left({\bf u}_{i}(t) \cdot {\bf
u}_{i}(0) \right) 
\end{equation}
For simulations in the high-friction limit, orientational correlation
times can then be obtained from exponential fits of this function, or by
integrating,
\begin{equation}
\tau = \ell (\ell + 1) \int_0^{\infty} C_{\ell}(t) dt.
\end{equation}
In lower-friction solvents, the Legendre correlation functions often
exhibit non-exponential decay, and may not be characterized by a
single decay constant.

In table \ref{tab:rotation} we show the characteristic rotational
relaxation times (based on the diffusion tensor) for each of the model
systems compared with the values obtained via microcanonical and Langevin
simulations.

\subsection{Spherical particles}
Our model system for spherical particles was a Lennard-Jones sphere of
diameter ($\sigma$) 6.5 \AA\ in a sea of smaller spheres ($\sigma$ =
4.7 \AA).  The well depth ($\epsilon$) for both particles was set to
an arbitrary value of 0.8 kcal/mol.  

The Stokes-Einstein behavior of large spherical particles in
hydrodynamic flows is well known, giving translational friction
coefficients of $6 \pi \eta R$ (stick boundary conditions) and
rotational friction coefficients of $8 \pi \eta R^3$.  Recently,
Schmidt and Skinner have computed the behavior of spherical tag
particles in molecular dynamics simulations, and have shown that {\it
slip} boundary conditions ($\Xi_{tt} = 4 \pi \eta R$) may be more
appropriate for molecule-sized spheres embedded in a sea of spherical
qsolvent particles.\cite{Schmidt:2004fj,Schmidt:2003kx}

Our simulation results show similar behavior to the behavior observed
by Schmidt and Skinner.  The diffusion constant obtained from our
microcanonical molecular dynamics simulations lies between the slip
and stick boundary condition results obtained via Stokes-Einstein
behavior.  Since the Langevin integrator assumes Stokes-Einstein stick
boundary conditions in calculating the drag and random forces for
spherical particles, our Langevin routine obtains nearly quantitative
agreement with the hydrodynamic results for spherical particles.  One
avenue for improvement of the method would be to compute elements of
$\Xi_{tt}$ assuming behavior intermediate between the two boundary
conditions.

In these simulations, our spherical particles were structureless, so
there is no way to obtain rotational correlation times for this model
system. 

\subsubsection{Ellipsoids} 
For uniaxial ellipsoids ($a > b = c$) , Perrin's formulae for both
translational and rotational diffusion of each of the body-fixed axes
can be combined to give a single translational diffusion
constant,\cite{Berne90} 
\begin{equation}
D = \frac{k_B T}{6 \pi \eta a} G(\rho),
\label{Dperrin}
\end{equation}
as well as a single rotational diffusion coefficient,
\begin{equation}
\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - \rho^2)
G(\rho) - 1}{1 - \rho^4} \right\}.
\label{ThetaPerrin}
\end{equation}
In these expressions, $G(\rho)$ is a function of the axial ratio
($\rho = b / a$), which for prolate ellipsoids, is
\begin{equation}
G(\rho) = (1- \rho^2)^{-1/2} \ln \left\{ \frac{1 + (1 -
\rho^2)^{1/2}}{\rho} \right\}
\label{GPerrin}
\end{equation}
Again, there is some uncertainty about the correct boundary conditions
to use for molecular-scale ellipsoids in a sea of similarly-sized
solvent particles.  Ravichandran and Bagchi found that {\it slip}
boundary conditions most closely resembled the simulation
results,\cite{Ravichandran:1999fk} in agreement with earlier work of
Tang and Evans.\cite{TANG:1993lr}

Even though there are analytic resistance tensors for ellipsoids, we
constructed a rough-shell model using 2135 beads (each with a diameter
of 0.25 \AA) to approximate the shape of the modle ellipsoid.  We
compared the Langevin dynamics from both the simple ellipsoidal
resistance tensor and the rough shell approximation with
microcanonical simulations and the predictions of Perrin.  As in the
case of our spherical model system, the Langevin integrator reproduces
almost exactly the behavior of the Perrin formulae (which is
unsurprising given that the Perrin formulae were used to derive the
drag and random forces applied to the ellipsoid).  We obtain
translational diffusion constants and rotational correlation times
that are within a few percent of the analytic values for both the
exact treatment of the diffusion tensor as well as the rough-shell
model for the ellipsoid.

The translational diffusion constants from the microcanonical simulations 
agree well with the predictions of the Perrin model, although the rotational
correlation times are a factor of 2 shorter than expected from hydrodynamic 
theory.  One explanation for the slower rotation
of explicitly-solvated ellipsoids is the possibility that solute-solvent 
collisions happen at both ends of the solute whenever the principal 
axis of the ellipsoid is turning. In the upper portion of figure 
\ref{fig:explanation} we sketch a physical picture of this explanation.
Since our Langevin integrator is providing nearly quantitative agreement with 
the Perrin model, it also predicts orientational diffusion for ellipsoids that
exceed explicitly solvated correlation times by a factor of two.

\begin{figure}
\centering
\includegraphics[width=6in]{explanation}
\caption[Explanations of the differences between orientational correlation times for explicitly-solvated models and hydrodynamics predictions]{Explanations of the differences between orientational correlation times for explicitly-solvated models and hydrodynamic predictions.   For the ellipsoids (upper figures), rotation of the principal axis can involve correlated collisions at both sides of the solute.  In the rigid dumbbell model (lower figures), the large size of the explicit solvent spheres prevents them from coming in contact with a substantial fraction of the surface area of the dumbbell.  Therefore, the explicit solvent only provides drag over a substantially reduced surface area of this model, where the hydrodynamic theories utilize the entire surface area for estimating rotational diffusion.
} \label{fig:explanation}
\end{figure}

\subsubsection{Rigid dumbbells}
Perhaps the only {\it composite} rigid body for which analytic
expressions for the hydrodynamic tensor are available is the
two-sphere dumbbell model.  This model consists of two non-overlapping
spheres held by a rigid bond connecting their centers. There are
competing expressions for the 6x6 resistance tensor for this
model. Equation (\ref{introEquation:oseenTensor}) above gives the
original Oseen tensor, while the second order expression introduced by
Rotne and Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la
Torre and Bloomfield,\cite{Torre1977} is given above as
Eq. (\ref{introEquation:RPTensorNonOverlapped}).  In our case, we use
a model dumbbell in which the two spheres are identical Lennard-Jones
particles ($\sigma$ = 6.5 \AA\ , $\epsilon$ = 0.8 kcal / mol) held at
a distance of 6.532 \AA.

The theoretical values for the translational diffusion constant of the
dumbbell are calculated from the work of Stimson and Jeffery, who
studied the motion of this system in a flow parallel to the
inter-sphere axis,\cite{Stimson:1926qy} and Davis, who studied the
motion in a flow {\it perpendicular} to the inter-sphere
axis.\cite{Davis:1969uq} We know of no analytic solutions for the {\it
orientational} correlation times for this model system (other than
those derived from the 6 x 6 tensors mentioned above).

The bead model for this model system comprises the two large spheres
by themselves, while the rough shell approximation used 3368 separate
beads (each with a diameter of 0.25 \AA) to approximate the shape of
the rigid body.  The hydrodynamics tensors computed from both the bead
and rough shell models are remarkably similar.  Computing the initial
hydrodynamic tensor for a rough shell model can be quite expensive (in
this case it requires inverting a 10104 x 10104 matrix), while the
bead model is typically easy to compute (in this case requiring
inversion of a 6 x 6 matrix).  

\begin{figure}
\centering
\includegraphics[width=3in]{RoughShell}
\caption[Model rigid bodies and their rough shell approximations]{The
model rigid bodies (left column) used to test this algorithm and their
rough-shell approximations (right-column) that were used to compute
the hydrodynamic tensors.  The top two models (ellipsoid and dumbbell)
have analytic solutions and were used to test the rough shell
approximation.  The lower two models (banana and lipid) were compared
with explicitly-solvated molecular dynamics simulations. } 
\label{fig:roughShell}
\end{figure}


Once the hydrodynamic tensor has been computed, there is no additional
penalty for carrying out a Langevin simulation with either of the two
different hydrodynamics models.  Our naive expectation is that since
the rigid body's surface is roughened under the various shell models,
the diffusion constants will be even farther from the ``slip''
boundary conditions than observed for the bead model (which uses a
Stokes-Einstein model to arrive at the hydrodynamic tensor).  For the
dumbbell, this prediction is correct although all of the Langevin
diffusion constants are within 6\% of the diffusion constant predicted
from the fully solvated system.

For rotational motion, Langevin integration (and the hydrodynamic tensor) 
yields rotational correlation times that are substantially shorter than those
obtained from explicitly-solvated simulations.  It is likely that this is due
to the large size of the explicit solvent spheres, a feature that prevents 
the solvent from coming in contact with a substantial fraction of the surface 
area of the dumbbell.  Therefore, the explicit solvent only provides drag 
over a substantially reduced surface area of this model, while the 
hydrodynamic theories utilize the entire surface area for estimating 
rotational diffusion.  A sketch of the free volume available in the explicit
solvent simulations is shown in figure \ref{fig:explanation}.

\subsubsection{Ellipsoidal-composite banana-shaped molecules}
Banana-shaped rigid bodies composed of three Gay-Berne ellipsoids 
have been used by Orlandi {\it et al.} to observe
mesophases in coarse-grained models bent-core liquid crystalline
molecules.\cite{Orlandi:2006fk}  We have used the same overlapping
ellipsoids as a way to test the behavior of our algorithm for a
structure of some interest to the materials science community,
although since we are interested in capturing only the hydrodynamic
behavior of this model, we have left out the dipolar interactions of the
original Orlandi model. 
 
A reference system composed of a single banana rigid body embedded in a
sea of 1929 solvent particles was created and run under standard
(microcanonical) molecular dynamics.  The resulting viscosity of this 
mixture was 0.298 centipoise (as estimated using Eq. (\ref{eq:shear})).
To calculate the hydrodynamic properties of the banana rigid body model,
we created a rough shell (see Fig.~\ref{langevin:roughShell}), in which 
the banana is represented as a ``shell'' made of 3321 identical beads 
(0.25 \AA in diameter) distributed on the surface.  Applying the 
procedure described in Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
identified the center of resistance at (0 $\rm{\AA}$, 0.81 $\rm{\AA}$, 0 $\rm{\AA}$), as well as the resistance tensor,
\[
\left( {\begin{array}{*{20}c}
0.9261 & 0 & 0&0&0.08585&0.2057\\
0& 0.9270&-0.007063& 0.08585&0&0\\
0&-0.007063&0.7494&0.2057&0&0\\
0&0.0858&0.2057& 58.64& 0&0\\0.08585&0&0&0&48.30&3.219&\\0.2057&0&0&0&3.219&10.7373\\\end{array}} \right).
\]
where the units for translational, translation-rotation coupling and rotational 
tensors are $\frac{kcal \cdot fs}{mol \cdot \rm{\AA}^2}$, $\frac{kcal \cdot fs}{
mol \cdot \rm{\AA} \cdot rad}$ and $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respe
ctively.

The Langevin rigid-body integrator (and the hydrodynamic diffusion tensor)
are essentially quantitative for translational diffusion of this model.  
Orientational correlation times under the Langevin rigid-body integrator 
are within 11\% of the values obtained from explicit solvent, but these 
models also exhibit some solvent inaccessible surface area in the 
explicitly-solvated case.  

\subsubsection{Composite sphero-ellipsoids}
Spherical heads perched on the ends of Gay-Berne ellipsoids have been
used recently as models for lipid molecules.\cite{SunGezelter08,Ayton01}

The diffusion constants and rotation relaxation times for
different shaped molecules are shown in table \ref{tab:translation}
and \ref{tab:rotation} to compare to the results calculated from NVE
simulations. The theoretical values for sphere is calculated from the
Stokes-Einstein law, the theoretical values for ellipsoid is
calculated from Perrin's fomula,  The exact method is
applied to the langevin dynamics simulations for sphere and ellipsoid,
the bead model is applied to the simulation for dumbbell molecule, and
the rough shell model is applied to ellipsoid, dumbbell, banana and
lipid molecules. The results from all the langevin dynamics
simulations, including exact, bead model and rough shell, match the
theoretical values perfectly for all different shaped molecules. This
indicates that our simulation package for langevin dynamics is working
well. The approxiate methods ( bead model and rough shell model) are
accurate enough for the current simulations. The goal of the langevin
dynamics theory is to replace the explicit solvents by the friction
forces. We compared the dynamic properties of different shaped
molecules in langevin dynamics simulations with that in NVE
simulations. The results are reasonable close. Overall, the
translational diffusion constants calculated from langevin dynamics
simulations are very close to the values from the NVE simulation. For
sphere and lipid molecules, the diffusion constants are a little bit
off from the NVE simulation results. One possible reason is that the
calculation of the viscosity is very difficult to be accurate. Another
possible reason is that although we save very frequently during the
NVE simulations and run pretty long time simulations, there is only
one solute molecule in the system which makes the calculation for the
diffusion constant difficult. The sphere molecule behaves as a free
rotor in the solvent, so there is no rotation relaxation time
calculated from NVE simulations. The rotation relaxation time is not
very close to the NVE simulations results. The banana and lipid
molecules match the NVE simulations results pretty well. The mismatch
between langevin dynamics and NVE simulation for ellipsoid is possibly
caused by the slip boundary condition. For dumbbell, the mismatch is
caused by the size of the solvent molecule is pretty large compared to
dumbbell molecule in NVE simulations.

According to our simulations, the langevin dynamics is a reliable
theory to apply to replace the explicit solvents, especially for the
translation properties. For large molecules, the rotation properties
are also mimiced reasonablly well.

\begin{table*}
\begin{minipage}{\linewidth}
\begin{center}
\caption{Translational diffusion constants (D) for the model systems 
calculated using microcanonical simulations (with explicit solvent), 
theoretical predictions, and Langevin simulations (with implicit solvent).
Analytical solutions for the exactly-solved hydrodynamics models are
from Refs. \citen{Einstein05} (sphere), \citen{Perrin1934} and \citen{Perrin1936}
(ellipsoid), \citen{Stimson:1926qy} and \citen{Davis:1969uq}
(dumbbell). The other model systems have no known analytic solution.
All  diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (=
$10^{-4}$ \AA$^2$  / fs). }
\begin{tabular}{lccccccc}
\hline
 & \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\
\cline{2-3} \cline{5-7}
model & $\eta$ (centipoise)  & D & & Analytical & method & Hydrodynamics & simulation \\ 
\hline
sphere    & 0.348  & 1.64 & & 1.94 & exact       & 1.94 & 1.98 \\
ellipsoid & 0.255  & 2.44 & & 2.34 & exact       & 2.34 & 2.37 \\
          & 0.255  & 2.44 & & 2.34 & rough shell & 2.36 & 2.28 \\
dumbbell  & 0.241  & 2.13 & & 2.09 & bead model  & 2.10 & 2.15 \\
          & 0.241  & 2.13 & & 2.09 & rough shell & 2.03 & 2.01 \\
banana    & 0.298  & 1.53 & &      & rough shell & 1.56 & 1.55 \\
lipid     & 0.349  & 0.96 & &      & rough shell & 1.33 & 1.32 \\
\end{tabular}
\label{tab:translation}
\end{center}
\end{minipage}
\end{table*}

\begin{table*}
\begin{minipage}{\linewidth}
\begin{center}
\caption{Orientational relaxation times ($\tau$) for the model systems using
microcanonical simulation (with explicit solvent), theoretical
predictions, and Langevin simulations (with implicit solvent). All
relaxation times are for the rotational correlation function with
$\ell = 2$ and are reported in units of ps.  The ellipsoidal model has
an exact solution for the orientational correlation time due to
Perrin, but the other model systems have no known analytic solution.}
\begin{tabular}{lccccccc}
\hline
 & \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\
\cline{2-3} \cline{5-7}
model & $\eta$ (centipoise) & $\tau$ & & Perrin & method & Hydrodynamic  & simulation \\ 
\hline
ellipsoid & 0.255  & 46.7 & & 22.0 & exact       & 22.0 & 22.2 \\
          & 0.255  & 46.7 & & 22.0 & rough shell & 22.6 & 22.2 \\
dumbbell  & 0.241  & 14.3 & &      & bead model  & 39.2 & 71.2 \\
          & 0.241  & 14.3 & &      & rough shell & 32.6 & 70.5 \\
banana    & 0.298  & 63.8 & &      & rough shell & 70.9 & 70.9 \\
lipid     & 0.349  & 78.0 & &      & rough shell & 76.9 & 77.9 \\
\hline
\end{tabular}
\label{tab:rotation}
\end{center}
\end{minipage}
\end{table*}

Langevin dynamics simulations are applied to study the formation of
the ripple phase of lipid membranes. The initial configuration is
taken from our molecular dynamics studies on lipid bilayers with
lennard-Jones sphere solvents. The solvent molecules are excluded from
the system, the experimental value of water viscosity is applied to
mimic the heat bath. Fig. XXX is the snapshot of the stable
configuration of the system, the ripple structure stayed stable after
100 ns run. The efficiency of the simulation is increased by one order
of magnitude.

\section{Conclusions}

We have presented a new Langevin algorithm by incorporating the
hydrodynamics properties of arbitrary shaped molecules into an
advanced symplectic integration scheme. 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.


\section{Acknowledgments}
Support for this project was provided by the National Science
Foundation under grant CHE-0134881. T.L. also acknowledges the
financial support from center of applied mathematics at University
of Notre Dame.
\newpage

\bibliographystyle{jcp}
\bibliography{langevin}

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