--- trunk/langevin/langevin.tex	2007/08/03 20:46:41	3205
+++ trunk/langevin/langevin.tex	2008/01/31 22:13:55	3340
@@ -2,7 +2,7 @@
 %\documentclass[aps,prb,preprint]{revtex4}
 \documentclass[11pt]{article}
 \usepackage{endfloat}
-\usepackage{amsmath,bm}
+\usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{times}
 \usepackage{mathptmx}
@@ -17,13 +17,13 @@
 \evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight
 9.0in \textwidth 6.5in \brokenpenalty=10000
 \renewcommand{\baselinestretch}{1.2}
-\renewcommand\citemid{\ } % no comma in optional reference note
+\renewcommand\citemid{\ } % no comma in optional referenc note
 
 \begin{document}
 
 \title{An algorithm for performing Langevin dynamics on rigid bodies of arbitrary shape }
 
-\author{Teng Lin, Xiuquan Sun and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail:
+\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\\
@@ -48,254 +48,265 @@ As alternative to Newtonian dynamics, Langevin dynamic
 \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
+Langevin dynamics, which mimics a simple heat bath with stochastic and
+dissipative forces, has been applied in a variety of situations as an
+alternative to molecular dynamics with explicit solvent molecules.
+The stochastic treatment of the solvent allows the use of simulations
+with substantially longer time and length scales.  In general, the
+dynamic and structural properties obtained from Langevin simulations
+agree quite well with similar properties obtained from explicit
+solvent simulations.
+
+Recent examples of the usefulness of Langevin simulations include a
+study of met-enkephalin in which Langevin simulations predicted
+dynamical properties that were largely in agreement with explicit
+solvent simulations.\cite{Shen2002} By applying Langevin dynamics with
+the UNRES model, Liow and his coworkers suggest that protein folding
+pathways can be explored within a reasonable amount of
+time.\cite{Liwo2005}
+
+The stochastic nature of Langevin dynamics also enhances the sampling
+of the system and increases the probability of crossing energy
+barriers.\cite{Cui2003,Banerjee2004} Combining Langevin dynamics with
+Kramers' 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 the implicit solvent model, Kaya
+incorporated a desolvation free energy barrier into protein
+folding/unfolding studies and discovered a higher free energy barrier
+between the native and denatured states.\cite{HuseyinKaya07012005}
+
+Because of its stability against noise, Langevin dynamics has also
+proven useful for studying remagnetization processes in various
+systems.\cite{Palacios1998,Berkov2002,Denisov2003} [Check: 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}
+Langevin dynamics 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}}.
+In typical LD simulations, the friction and random forces on
+individual atoms are taken from Stokes' law,
+\begin{eqnarray}
+m \dot{v}(t) & = & -\nabla U(x) - \xi m v(t) + R(t) \\
+\langle R(t) \rangle & = & 0 \\
+\langle R(t) R(t') \rangle & = & 2 k_B T \xi m \delta(t - t')
+\end{eqnarray}
+where $\xi \approx 6 \pi \eta a$.  Here $\eta$ is the viscosity of the
+implicit solvent, and $a$ is the hydrodynamic radius of the atom.
 
-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
+The use of rigid substructures,\cite{Chun:2000fj}
+coarse-graining,\cite{Ayton01,Golubkov06,Orlandi:2006fk,SunGezelter08}
+and ellipsoidal representations of protein side chains~\cite{Fogolari:1996lr}
+has made the use of the Stokes-Einstein approximation problematic.  A
+rigid substructure moves as a single unit with orientational as well
+as translational degrees of freedom.  This requires a more general
+treatment of the hydrodynamics than the spherical approximation
+provides.  The atoms involved in a rigid or coarse-grained structure
+should properly have solvent-mediated interactions with each
+other. The theory of interactions {\it between} bodies moving through
+a fluid has been developed over the past century and has been applied
+to simulations of Brownian
+motion.\cite{FIXMAN:1986lr,Ramachandran1996}
+
+In order to account for the diffusion anisotropy of arbitrarily-shaped
+particles, Fernandes and Garc\'{i}a de la Torre improved the original
+Brownian dynamics simulation algorithm~\cite{Ermak1978,Allison1991} by
+incorporating a generalized $6\times6$ diffusion tensor and
+introducing a rotational evolution scheme consisting of three
+consecutive rotations.\cite{Fernandes2002} Unfortunately, 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 proteins. 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 linked polymer segments in a 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 macromolecular
+assemblies.  There is therefore a need for incorporating the
+hydrodynamics of complex (and potentially skew) rigid bodies in the
+library of methods available for performing Langevin simulations.
+
+\subsection{Rigid Body Dynamics}
+Rigid bodies are frequently involved in the modeling of large
+collections of particles that move as a single unit.  In molecular
+simulations, rigid bodies have been used to simplify protein-protein
+docking,\cite{Gray2003} and lipid bilayer
+simulations.\cite{SunGezelter08} Many of the water models in common
+use are also rigid-body
+models,\cite{Jorgensen83,Berendsen81,Berendsen87} although they are
+typically evolved using constraints rather than rigid body equations
+of motion.
+
+Euler angles are a 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 can
+become inaccurate (and inefficient).  Although the use of multiple
+sets of Euler angles can overcome this problem,\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}
+remain.  A singularity-free representation utilizing quaternions was
+developed by Evans in 1977.\cite{Evans1977} The Evans quaternion
+approach uses a nonseparable Hamiltonian, and this has prevented
+symplectic algorithms from being utilized until very
+recently.\cite{Miller2002} 
 
-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.
+Another approach is the application of holonomic constraints to the
+atoms belonging to the rigid body.  Each atom moves independently
+under the normal forces deriving from potential energy and constraints
+are used to guarantee rigidity. However, due to their iterative
+nature, the SHAKE and RATTLE algorithms converge very slowly when the
+number of constraints (and the number of particles that belong to the
+rigid body) increases.\cite{Ryckaert1977,Andersen1983}
 
-%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.
+In order to develop a stable and efficient integration scheme that
+preserves most constants of the motion, symplectic propagators are
+necessary.  By introducing a conjugate momentum to the rotation matrix
+$Q$ and re-formulating Hamilton's equations, a symplectic
+orientational integrator, RSHAKE,\cite{Kol1997} was proposed to evolve
+rigid bodies on 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 suffer from some
+related inefficiencies. A symplectic Lie-Poisson integrator for rigid
+bodies developed by Dullweber {\it et al.}\cite{Dullweber1997} removes
+most of the limitations mentioned above and is therefore the basis for
+our Langevin integrator.
 
 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.
+accurate estimation of friction tensor from hydrodynamics theory into
+a symplectic rigid body dynamics propagator.  In the sections below,
+we review some of the theory of hydrodynamic tensors developed
+primarily for Brownian simulations of multi-particle systems, we then
+present our integration method for a set of generalized Langevin
+equations of motion, and we compare the behavior of the new Langevin
+integrator to dynamical quantities obtained via explicit solvent
+molecular dynamics.
 
-\section{Computational Methods{\label{methodSec}}}
-
-\subsection{\label{introSection:frictionTensor}Friction Tensor}
-Theoretically, the friction kernel can be determined using the
+\subsection{\label{introSection:frictionTensor}The Friction Tensor}
+Theoretically, a complete 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,
-\[
+impractical when the solute becomes complex. Instead, various
+approaches based on hydrodynamics have been developed to calculate the
+friction coefficients. In general, the friction tensor $\Xi$ is a
+$6\times 6$ matrix given by
+\begin{equation}
+\Xi  = \left( \begin{array}{*{20}c}
+   \Xi^{tt} & \Xi^{rt}  \\
+   \Xi^{tr} & \Xi^{rr}  \\
+\end{array} \right).
+\end{equation}
+Here, $\Xi^{tt}$ and $\Xi^{rr}$ are $3 \times 3$ translational and
+rotational resistance (friction) tensors respectively, while
+$\Xi^{tr}$ is translation-rotation coupling tensor and $\Xi^{rt}$ is
+rotation-translation coupling tensor. When a particle moves in a
+fluid, it may experience friction force ($\mathbf{f}_f$) and torque
+($\mathbf{\tau}_f$) in opposition to the directions of the velocity
+($\mathbf{v}$) and body-fixed angular velocity ($\mathbf{\omega}$),
+\begin{equation}
 \left( \begin{array}{l}
- F_R  \\
- \tau _R  \\
- \end{array} \right) =  - \left( {\begin{array}{*{20}c}
-   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
-   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
-\end{array}} \right)\left( \begin{array}{l}
- v \\
- w \\
- \end{array} \right)
-\]
-where $F_r$ is the friction force and $\tau _R$ is the friction
-torque.
+ \mathbf{f}_f  \\
+ \mathbf{\tau}_f  \\
+ \end{array} \right) =  - \left( \begin{array}{*{20}c}
+   \Xi^{tt} & \Xi^{rt}  \\
+   \Xi^{tr} & \Xi^{rr}  \\
+\end{array} \right)\left( \begin{array}{l}
+ \mathbf{v} \\
+ \mathbf{\omega} \\
+ \end{array} \right).
+\end{equation}
 
 \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,
-\[
-\Xi ^{tt}  = \left( {\begin{array}{*{20}c}
+For a spherical particle under ``stick'' boundary conditions, the
+translational and rotational friction tensors can be calculated from
+Stokes' law,
+\begin{equation}
+\Xi^{tt}  = \left( \begin{array}{*{20}c}
    {6\pi \eta R} & 0 & 0  \\
    0 & {6\pi \eta R} & 0  \\
    0 & 0 & {6\pi \eta R}  \\
-\end{array}} \right)
-\]
+\end{array} \right)
+\end{equation}
 and
-\[
-\Xi ^{rr}  = \left( {\begin{array}{*{20}c}
+\begin{equation}
+\Xi^{rr}  = \left( \begin{array}{*{20}c}
    {8\pi \eta R^3 } & 0 & 0  \\
    0 & {8\pi \eta R^3 } & 0  \\
    0 & 0 & {8\pi \eta R^3 }  \\
-\end{array}} \right)
-\]
+\end{array} \right)
+\end{equation}
 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,
+widely used as references for developing new hydrodynamics theories,
 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
+extended Stokes' law to general ellipsoids which are given in
+Cartesian coordinates by~\cite{Perrin1934,Perrin1936}
+\begin{equation}
+\frac{x^2 }{a^2} + \frac{y^2}{b^2} + \frac{z^2 }{c^2} = 1.
+\end{equation}
+Here, the semi-axes are of lengths $a$, $b$, and $c$. Due to the
+complexity of the elliptic integral, only uniaxial ellipsoids, either
+prolate ($a \ge b = c$) or oblate ($a < b = c$), can be solved
+exactly. Introducing an elliptic integral parameter $S$ for prolate,
+\begin{equation}
+S = \frac{2}{\sqrt{a^2  - b^2}} \ln \frac{a + \sqrt{a^2  - b^2}}{b},
+\end{equation}
+and oblate,
+\begin{equation}
+S = \frac{2}{\sqrt {b^2  - a^2 }} \arctan \frac{\sqrt {b^2  - a^2}}{a},
+\end{equation}
+ellipsoids, 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}},
+ \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
+for oblate, 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}}.
+ \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*}
+for prolate ellipsoids. For both spherical and ellipsoidal particles,
+the translation-rotation and rotation-translation coupling tensors are
+zero.
 
 \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
+hydrodynamic properties of rigid bodies. However, the mapping from all
+possible ellipsoidal spaces, $r$-space, to all possible combination of
+rotational diffusion coefficients, $D$-space, is not
+unique.\cite{Wegener1979} Additionally, because there is 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 $v_i$,
-\[
+using more advanced methods where the molecule of interest was modeled
+by a combinations of spheres\cite{Carrasco1999} and the hydrodynamics
+properties of the molecule can be calculated using the hydrodynamic
+interaction tensor. 
+
+Consider a rigid assembly of $N$ beads immersed in a continuous
+medium. Due to hydrodynamic interaction, the ``net'' velocity of $i$th
+bead, $v'_i$ is different than its unperturbed velocity $v_i$,
+\begin{equation}
 v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
-\]
-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
+\end{equation}
+where $F_i$ is the frictional force, and $T_{ij}$ is the hydrodynamic
+interaction tensor. The frictional force on the $i^\mathrm{th}$ 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}
@@ -332,17 +343,17 @@ B = \left( {\begin{array}{*{20}c}
 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} }  \\
+B = \left( \begin{array}{*{20}c}
+   B_{11} &  \ldots  & B_{1N}   \\
     \vdots  &  \ddots  &  \vdots   \\
-   {B_{N1} } &  \cdots  & {B_{NN} }  \\
-\end{array}} \right),
+   B_{N1} &  \cdots  & B_{NN} \\
+\end{array} \right),
 \end{equation}
 where $B_{ij}$ is given by
-\[
+\begin{equation}
 B_{ij}  = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij}
 )T_{ij}
-\]
+\end{equation}
 where $\delta _{ij}$ is the Kronecker delta function. Inverting the
 $B$ matrix, we obtain
 \[
@@ -365,11 +376,26 @@ arbitrary origin $O$ can be written as
 bead $i$ and origin $O$, the elements of resistance tensor at
 arbitrary origin $O$ can be written as
 \begin{eqnarray}
+\label{introEquation:ResistanceTensorArbitraryOrigin}
  \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}
+ \Xi _{}^{rr}  & = &  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } }
+U_j  + 6 \eta V {\bf I}. \notag
 \end{eqnarray}
+The final term in the expression for $\Xi^{rr}$ is correction that
+accounts for errors in the rotational motion of certain kinds of bead
+models. The additive correction uses the solvent viscosity ($\eta$)
+as well as the total volume of the beads that contribute to the
+hydrodynamic model,
+\begin{equation}
+V = \frac{4 \pi}{3} \sum_{i=1}^{N} \sigma_i^3,
+\end{equation}
+where $\sigma_i$ is the radius of bead $i$.  This correction term was
+rigorously tested and compared with the analytical results for
+two-sphere and ellipsoidal systems by Garcia de la Torre and
+Rodes.\cite{Torre:1983lr}
+
+
 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
@@ -400,7 +426,7 @@ U_{OP}  = \left( {\begin{array}{*{20}c}
 \[
 U_{OP}  = \left( {\begin{array}{*{20}c}
    0 & { - z_{OP} } & {y_{OP} }  \\
-   {z_i } & 0 & { - x_{OP} }  \\
+   {z_{OP} } & 0 & { - x_{OP} }  \\
    { - y_{OP} } & {x_{OP} } & 0  \\
 \end{array}} \right)
 \]
@@ -412,90 +438,122 @@ locate the position of center of resistance,
  x_{OR}  \\
  y_{OR}  \\
  z_{OR}  \\
- \end{array} \right) & = &\left( {\begin{array}{*{20}c}
+ \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}  \\
+\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
+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}}
 
+\section{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)
+\mathbf{M} \dot{\mathbf{V}}(t) = \mathbf{F}_{s}(t) +
+\mathbf{F}_{f}(t)  + \mathbf{F}_{r}(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 $\omega _i$
+where $\mathbf{M}$ is a $6 \times 6$ diagonal mass matrix (which
+includes the mass of the rigid body as well as the moments of inertia
+in the body-fixed frame) and $\mathbf{V}$ is a generalized velocity,
+$\mathbf{V} =
+\left\{\mathbf{v},\mathbf{\omega}\right\}$. The right side of
+Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a
+system force $\mathbf{F}_{s}$, a frictional or dissipative force
+$\mathbf{F}_{f}$ and stochastic force $\mathbf{F}_{r}$. While the
+evolution of the system in Newtonian mechanics is typically done in
+the lab-fixed frame, it is convenient to handle the dynamics of rigid
+bodies in the body-fixed frame. Thus the friction and random forces
+are calculated in body-fixed frame and may be converted back to
+lab-fixed frame using the rigid body's rotation matrix ($Q$):
 \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) \\
+\mathbf{F}_{f,r} = 
+\left( \begin{array}{c}
+\mathbf{f}_{f,r} \\
+\mathbf{\tau}_{f,r}
+\end{array} \right)
+=
+\left( \begin{array}{c}
+Q^{T} \mathbf{f}^{b}_{f,r} \\
+Q^{T} \mathbf{\tau}^{b}_{f,r}
+\end{array} \right)
+\end{equation}
+The body-fixed friction force, $\mathbf{F}_{f}^b$, is proportional to
+the velocity at the center of resistance $\mathbf{v}_{R}^b$ (in the
+body-fixed frame) and the angular velocity $\mathbf{\omega}$
+\begin{equation}
+\mathbf{F}_{f}^b (t) = \left( \begin{array}{l}
+ \mathbf{f}_{f}^b (t) \\
+ \mathbf{\tau}_{f}^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}
+ \mathbf{v}_{R}^b (t) \\
+ \mathbf{\omega} (t) \\
  \end{array} \right),
 \end{equation}
-while the random force $F_{r,i}^l$ is a Gaussian stochastic variable
-with zero mean and variance
+while the random force, $\mathbf{F}_{r}$, 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}
+\left\langle {\mathbf{F}_{r}(t) (\mathbf{F}_{r}(t'))^T } \right\rangle  =
+\left\langle {\mathbf{F}_{r}^b (t) (\mathbf{F}_{r}^b (t'))^T } \right\rangle  =
+2 k_B T \Xi_R \delta(t - t'). \label{randomForce}
 \end{equation}
-The equation of motion for $v_i$ can be written as
+$\Xi_R$ is the $6\times6$ resistance tensor at the center of
+resistance.  Once this tensor is known for a given rigid body,
+obtaining a stochastic vector that has the properties in
+Eq. (\ref{eq:randomForce}) can be done efficiently by carrying out a
+one-time Cholesky decomposition to obtain the square root matrix of
+the resistance tensor $\Xi_R = \mathbf{S} \mathbf{S}^{T}$, where
+$\mathbf{S}$ is a lower triangular matrix.\cite{SchlickBook} A vector
+with the statistics required for the random force can then be obtained
+by multiplying $\mathbf{S}$ onto a 6-vector $Z$ which has elements
+chosen from a Gaussian distribution, such that:
 \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)
+\langle Z_i \rangle = 0, \hspace{1in} \langle Z_i \cdot Z_j \rangle = \frac{2 k_B
+T}{\delta t} \delta_{ij}.
 \end{equation}
+The random force, $F_{r}^{b} = \mathbf{S} Z$, can be shown to have the
+correct ohmic
+
+
+Each
+time a random force vector is needed, a gaussian random vector is
+generated and then the square root matrix is multiplied onto this
+vector.
+
+The equation of motion for $\mathbf{v}$ can be written as
+\begin{equation}
+ m \dot{\mathbf{v}} (t) =  \mathbf{f}_{s}^l (t) + \mathbf{f}_{f}^l (t) +
+\mathbf{f}_{r}^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
+frictional torque at the center of mass, $\tau_{f}^b (t)$, is
 given by
 \begin{equation}
-\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b
+\tau_{f}^b \leftarrow \tau_{f}^b + \mathbf{r}_{MR} \times \mathbf{f}_{f}^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)
+\dot j(t) = \tau_{s}^b (t) + \tau_{f}^b (t) + \tau_{r}^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
-$h= \delta t$:
+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*}
@@ -574,148 +632,618 @@ the velocities can be advanced to the same time value.
     + \frac{h}{2} {\bf \tau}^b(t + h) .
 \end{align*}
 
-\section{Results and Discussion}
+\section{Validating the Method\label{sec:validating}}
+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.
 
-The Langevin algorithm described in previous section has been
-implemented in {\sc oopse}\cite{Meineke2005} and applied to studies
-of the static and dynamic properties in several systems.
-
-\subsection{Temperature Control}
-
-As shown in Eq.~\ref{randomForce}, random collisions associated with
-the solvent's thermal motions is controlled by the external
-temperature. The capability to maintain the temperature of the whole
-system was usually used to measure the stability and efficiency of
-the algorithm. In order to verify the stability of this new
-algorithm, a series of simulations are performed on system
-consisiting of 256 SSD water molecules with different viscosities.
-The initial configuration for the simulations is taken from a 1ns
-NVT simulation with a cubic box of 19.7166~\AA. All simulation are
-carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns
-with reference temperature at 300~K. The average temperature as a
-function of $\eta$ is shown in Table \ref{langevin:viscosity} where
-the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 -
-1$ poise. The better temperature control at higher viscosity can be
-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.}
-\label{langevin:viscosity}
+\begin{table*}
+\begin{minipage}{\linewidth}
 \begin{center}
-\begin{tabular}{lll}
-  \hline
-  $\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\
-  \hline
-  1    & 300.47 & 10.99 \\
-  0.1  & 301.19 & 11.136 \\
-  0.01 & 303.04 & 11.796 \\
-  \hline
+\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{table}
+\end{minipage}
+\end{table*}
 
-Another set of calculations were performed to study the efficiency of
-temperature control using different temperature coupling schemes.
-The starting configuration is cooled to 173~K and evolved using NVE,
-NVT, and Langevin dynamic with time step of 2 fs.
-Fig.~\ref{langevin:temperature} shows the heating curve obtained as
-the systems reach equilibrium. The orange curve in
-Fig.~\ref{langevin:temperature} represents the simulation using
-Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps
-which gives reasonable tight coupling, while the blue one from
-Langevin dynamics with viscosity of 0.1 poise demonstrates a faster
-scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal
-NVE (see orange curve in Fig.~\ref{langevin:temperature}) which
-loses the temperature control ability.
-
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{temperature.pdf}
-\caption[Plot of Temperature Fluctuation Versus Time]{Plot of
-temperature fluctuation versus time.} \label{langevin:temperature}
+\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{Langevin Dynamics of Banana Shaped Molecules}
+\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. 
 
-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.
-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
-two banana shaped molecules with viscosity of 0.289 center poise. To
-calculate the hydrodynamic properties of the banana shaped molecule,
-we created 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
-surface. Applying the procedure described in
-Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
-identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$,
--0.1988 $\rm{\AA}$), 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 $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respectively.
-Curves of the 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 faster than MD. 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, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was
-probably due to the reason that we used the experimental viscosity directly instead of calculating bulk viscosity from simulation.
+The model systems studied used both Lennard-Jones spheres as well as
+uniaxial Gay-Berne ellipoids. In its original form, the Gay-Berne
+potential was a single site model for the interactions of rigid
+ellipsoidal molecules.\cite{Gay81} It can be thought of as a
+modification of the Gaussian overlap model originally described by
+Berne and Pechukas.\cite{Berne72} The potential is constructed in the
+familiar form of the Lennard-Jones function using
+orientation-dependent $\sigma$ and $\epsilon$ parameters,
+\begin{equation*}
+V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat
+r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
+{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u
+}_i},
+{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12}
+-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
+{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right]
+\label{eq:gb}
+\end{equation*}
+
+The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
+\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf
+\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters 
+are dependent on the relative orientations of the two ellipsoids (${\bf
+\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the
+inter-ellipsoid separation (${\bf \hat{r}}_{ij}$).  The shape and
+attractiveness of each ellipsoid is governed by a relatively small set
+of parameters: $l$ and $d$ describe the length and width of each
+uniaxial ellipsoid, while $\epsilon^s$, 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.  Details of the potential
+are given elsewhere,\cite{Luckhurst90,Golubkov06,SunGezelter08} and an
+excellent overview of the computational methods that can be used to
+efficiently compute forces and torques for this potential can be found
+in Ref. \citen{Golubkov06}
+
+For the interaction between nonequivalent uniaxial ellipsoids (or
+between spheres and ellipsoids), the spheres are treated as ellipsoids
+with an aspect ratio of 1 ($d = l$) and with an well depth ratio
+($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$).  The form of the
+Gay-Berne potential we are using was generalized by Cleaver {\it et
+al.} and is appropriate for dissimilar uniaxial
+ellipsoids.\cite{Cleaver96}
+
+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} \left\langle \left(
+\int_{t_0}^{t_0 + t} P_{xz}(t') dt' \right)^2 \right\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} \left\langle \left(
+\int_{t_0}^{t_0 + t} 
+\left(P\left(t'\right)-\left\langle P \right\rangle \right)dt'
+\right)^2 \right\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} \left\langle P_{xz}(t_0) P_{xz}(t_0
++ t) \right\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} \left\langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \right\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)  =  \left\langle P_{\ell}\left({\bf u}_{i}(t) \cdot {\bf
+u}_{i}(0) \right) \right\rangle 
+\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
+solvent 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 the explicit solvent simulations, both our solute and solvent
+particles were structureless, exerting no torques upon each other.
+Therefore, there are not rotational correlation times available for
+this model system.
+
+\subsection{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 model 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.
+
+\subsection{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=\linewidth]{roughShell.pdf}
-\caption[Rough shell model for banana shaped molecule]{Rough shell
-model for banana shaped molecule.} \label{langevin:roughShell}
+\includegraphics[width=2in]{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}.
+
+
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{twoBanana.pdf}
-\caption[Snapshot from Simulation of Two Banana Shaped Molecules and
-256 Pentane Molecules]{Snapshot from simulation of two Banana shaped
-molecules and 256 pentane molecules.} \label{langevin:twoBanana}
+\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}
 
+
+
+\subsection{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 for 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{fig: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, ${\bf r} = $(0 \AA, 0.81 \AA, 0 \AA), as
+well as the resistance tensor, 
+\begin{equation*}
+\Xi = 
+\left( {\begin{array}{*{20}c}
+0.9261 & 0 & 0&0&0.08585&0.2057\\
+0& 0.9270&-0.007063& 0.08585&0&0\\
+0&-0.007063&0.7494&0.2057&0&0\\
+0&0.0858&0.2057& 58.64& 0&0\\0.08585&0&0&0&48.30&3.219&\\0.2057&0&0&0&3.219&10.7373\\\end{array}} \right),
+\end{equation*}
+where the units for translational, translation-rotation coupling and
+rotational tensors are (kcal fs / mol \AA$^2$), (kcal fs / mol \AA\ rad),
+and (kcal fs / mol rad$^2$), respectively.
+
+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.  
+
+\subsection{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}
+MORE DETAILS
+
+A reference system composed of a single lipid 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 0.349 centipoise (as estimated using
+Eq. (\ref{eq:shear})).  To calculate the hydrodynamic properties of
+the lipid rigid body model, we created a rough shell (see
+Fig.~\ref{fig:roughShell}), in which the lipid is represented as a
+``shell'' made of 3550 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, ${\bf r} = $(0 \AA, 0 \AA, 1.46
+\AA).
+
+
+\subsection{Summary}
+According to our simulations, the langevin dynamics is a reliable
+theory to apply to replace the explicit solvents, especially for the
+translation properties. For large molecules, the rotation properties
+are also mimiced reasonablly well.
+
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{vacf.pdf}
-\caption[Plots of Velocity Auto-correlation Functions]{Velocity
-auto-correlation functions of NVE (explicit solvent) in blue and
-Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf}
+\includegraphics[width=\linewidth]{graph}
+\caption[Mean squared displacements and orientational
+correlation functions for each of the model rigid bodies.]{The
+mean-squared displacements ($\langle r^2(t) \rangle$) and
+orientational correlation functions ($C_2(t)$) for each of the model
+rigid bodies studied.  The circles are the results for microcanonical
+simulations with explicit solvent molecules, while the other data sets
+are results for Langevin dynamics using the different hydrodynamic
+tensor approximations.  The Perrin model for the ellipsoids is
+considered the ``exact'' hydrodynamic behavior (this can also be said
+for the translational motion of the dumbbell operating under the bead
+model). In most cases, the various hydrodynamics models reproduce
+each other quantitatively.}
+\label{fig:results} 
 \end{figure}
 
+\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    & 0.279  & 3.06 & & 2.42 & exact       & 2.42 & 2.33 \\
+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  & 0.308  & 2.06 & & 1.64 & bead model  & 1.65 & 1.62 \\
+          & 0.308  & 2.06 & & 1.64 & rough shell & 1.59 & 1.62 \\
+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
+sphere    & 0.279  &      & & 9.69 & exact       & 9.69 & 9.64 \\
+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  & 0.308  & 14.1 & &      & bead model  & 50.0 & 50.1 \\
+          & 0.308  & 14.1 & &      & rough shell & 41.5 & 41.3 \\
+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*}
+
+\section{Application: A rigid-body lipid bilayer}
+
+The Langevin dynamics integrator was applied to study the formation of
+corrugated structures emerging from simulations of the coarse grained
+lipid molecular models presented above.  The initial configuration is
+taken from our molecular dynamics studies on lipid bilayers with
+lennard-Jones sphere solvents. The solvent molecules were excluded
+from the system, and the experimental value for the viscosity of water
+at 20C ($\eta = 1.00$ cp) was used to mimic the hydrodynamic effects
+of the solvent.  The absence of explicit solvent molecules and the
+stability of the integrator allowed us to take timesteps of 50 fs.  A
+total simulation run time of 100 ns was sampled.
+Fig. \ref{fig:bilayer} shows the configuration of the system after 100
+ns, and the ripple structure remains stable during the entire
+trajectory.  Compared with using explicit bead-model solvent
+molecules, the efficiency of the simulation has increased by an order
+of magnitude.
+
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{uacf.pdf}
-\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 of NVE (blue) and Langevin dynamics
-(red).} \label{langevin:uacf}
+\includegraphics[width=\linewidth]{bilayer}
+\caption[Snapshot of a bilayer of rigid-body models for lipids]{A
+snapshot of a bilayer composed of rigid-body models for lipid
+molecules evolving using the Langevin integrator described in this
+work.} \label{fig:bilayer}
 \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.
+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}
@@ -725,7 +1253,7 @@ of Notre Dame.
 of Notre Dame.
 \newpage
 
-\bibliographystyle{jcp2}
+\bibliographystyle{jcp}
 \bibliography{langevin}
 
 \end{document}
