trunk/tengDissertation/Introduction.tex

\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}

\section{\label{introSection:classicalMechanics}Classical
Mechanics}

Using equations of motion derived from Classical Mechanics,
Molecular Dynamics simulations are carried out by integrating the
equations of motion for a given system of particles. There are three
fundamental ideas behind classical mechanics. Firstly, one can
determine the state of a mechanical system at any time of interest;
Secondly, all the mechanical properties of the system at that time
can be determined by combining the knowledge of the properties of
the system with the specification of this state; Finally, the
specification of the state when further combined with the laws of
mechanics will also be sufficient to predict the future behavior of
the system.

\subsection{\label{introSection:newtonian}Newtonian Mechanics}
The discovery of Newton's three laws of mechanics which govern the
motion of particles is the foundation of the classical mechanics.
Newton's first law defines a class of inertial frames. Inertial
frames are reference frames where a particle not interacting with
other bodies will move with constant speed in the same direction.
With respect to inertial frames, Newton's second law has the form
\begin{equation}
F = \frac {dp}{dt} = \frac {mdv}{dt}
\label{introEquation:newtonSecondLaw}
\end{equation}
A point mass interacting with other bodies moves with the
acceleration along the direction of the force acting on it. Let
$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and
$F_{ji}$ be the force that particle $j$ exerts on particle $i$.
Newton's third law states that
\begin{equation}
F_{ij} = -F_{ji}.
 \label{introEquation:newtonThirdLaw}
\end{equation}
Conservation laws of Newtonian Mechanics play very important roles
in solving mechanics problems. The linear momentum of a particle is
conserved if it is free or it experiences no force. The second
conservation theorem concerns the angular momentum of a particle.
The angular momentum $L$ of a particle with respect to an origin
from which $r$ is measured is defined to be
\begin{equation}
L \equiv r \times p \label{introEquation:angularMomentumDefinition}
\end{equation}
The torque $\tau$ with respect to the same origin is defined to be
\begin{equation}
\tau \equiv r \times F \label{introEquation:torqueDefinition}
\end{equation}
Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
\[
\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times
\dot p)
\]
since
\[
\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0
\]
thus,
\begin{equation}
\dot L = r \times \dot p = \tau
\end{equation}
If there are no external torques acting on a body, the angular
momentum of it is conserved. The last conservation theorem state
that if all forces are conservative, energy is conserved,
\begin{equation}E = T + V. \label{introEquation:energyConservation}
\end{equation}
All of these conserved quantities are important factors to determine
the quality of numerical integration schemes for rigid
bodies.\cite{Dullweber1997}

\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}

Newtonian Mechanics suffers from an important limitation: motion can
only be described in cartesian coordinate systems which make it
impossible to predict analytically the properties of the system even
if we know all of the details of the interaction. In order to
overcome some of the practical difficulties which arise in attempts
to apply Newton's equation to complex systems, approximate numerical
procedures may be developed.

\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's
Principle}}

Hamilton introduced the dynamical principle upon which it is
possible to base all of mechanics and most of classical physics.
Hamilton's Principle may be stated as follows: the trajectory, along
which a dynamical system may move from one point to another within a
specified time, is derived by finding the path which minimizes the
time integral of the difference between the kinetic $K$, and
potential energies $U$,
\begin{equation}
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}.
\label{introEquation:halmitonianPrinciple1}
\end{equation}
For simple mechanical systems, where the forces acting on the
different parts are derivable from a potential, the Lagrangian
function $L$ can be defined as the difference between the kinetic
energy of the system and its potential energy,
\begin{equation}
L \equiv K - U = L(q_i ,\dot q_i ).
\label{introEquation:lagrangianDef}
\end{equation}
Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
\begin{equation}
\delta \int_{t_1 }^{t_2 } {L dt = 0} .
\label{introEquation:halmitonianPrinciple2}
\end{equation}

\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The
Equations of Motion in Lagrangian Mechanics}}

For a system of $f$ degrees of freedom, the equations of motion in
the Lagrangian form is
\begin{equation}
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
\label{introEquation:eqMotionLagrangian}
\end{equation}
where $q_{i}$ is generalized coordinate and $\dot{q_{i}}$ is
generalized velocity.

\subsection{\label{introSection:hamiltonian}Hamiltonian Mechanics}

Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
introduced by William Rowan Hamilton in 1833 as a re-formulation of
classical mechanics. If the potential energy of a system is
independent of velocities, the momenta can be defined as
\begin{equation}
p_i = \frac{\partial L}{\partial \dot q_i}
\label{introEquation:generalizedMomenta}
\end{equation}
The Lagrange equations of motion are then expressed by
\begin{equation}
p_i  = \frac{{\partial L}}{{\partial q_i }}
\label{introEquation:generalizedMomentaDot}
\end{equation}
With the help of the generalized momenta, we may now define a new
quantity $H$ by the equation
\begin{equation}
H = \sum\limits_k {p_k \dot q_k }  - L ,
\label{introEquation:hamiltonianDefByLagrangian}
\end{equation}
where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
$L$ is the Lagrangian function for the system. Differentiating
Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain
\begin{equation}
dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
\frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1}
\end{equation}
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second
and fourth terms in the parentheses cancel. Therefore,
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
\begin{equation}
dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
\right)}  - \frac{{\partial L}}{{\partial t}}dt .
\label{introEquation:diffHamiltonian2}
\end{equation}
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
find
\begin{equation}
\frac{{\partial H}}{{\partial p_k }} = \dot {q_k}
\label{introEquation:motionHamiltonianCoordinate}
\end{equation}
\begin{equation}
\frac{{\partial H}}{{\partial q_k }} =  - \dot {p_k}
\label{introEquation:motionHamiltonianMomentum}
\end{equation}
and
\begin{equation}
\frac{{\partial H}}{{\partial t}} =  - \frac{{\partial L}}{{\partial
t}}
\label{introEquation:motionHamiltonianTime}
\end{equation}
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
equation of motion. Due to their symmetrical formula, they are also
known as the canonical equations of motions.\cite{Goldstein2001}

An important difference between Lagrangian approach and the
Hamiltonian approach is that the Lagrangian is considered to be a
function of the generalized velocities $\dot q_i$ and coordinates
$q_i$, while the Hamiltonian is considered to be a function of the
generalized momenta $p_i$ and the conjugate coordinates $q_i$.
Hamiltonian Mechanics is more appropriate for application to
statistical mechanics and quantum mechanics, since it treats the
coordinate and its time derivative as independent variables and it
only works with 1st-order differential equations.\cite{Marion1990}
In Newtonian Mechanics, a system described by conservative forces
conserves the total energy
(Eq.~\ref{introEquation:energyConservation}). It follows that
Hamilton's equations of motion conserve the total Hamiltonian
\begin{equation}
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
}}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian}
\end{equation}

\section{\label{introSection:statisticalMechanics}Statistical
Mechanics}

The thermodynamic behaviors and properties of Molecular Dynamics
simulation are governed by the principle of Statistical Mechanics.
The following section will give a brief introduction to some of the
Statistical Mechanics concepts and theorems presented in this
dissertation.

\subsection{\label{introSection:ensemble}Phase Space and Ensemble}

Mathematically, phase space is the space which represents all
possible states of a system. Each possible state of the system
corresponds to one unique point in the phase space. For mechanical
systems, the phase space usually consists of all possible values of
position and momentum variables. Consider a dynamic system of $f$
particles in a cartesian space, where each of the $6f$ coordinates
and momenta is assigned to one of $6f$ mutually orthogonal axes, the
phase space of this system is a $6f$ dimensional space. A point, $x
=
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over q} _1 , \ldots
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over q} _f
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over p} _1  \ldots
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over p} _f )$ , with a unique set of values of $6f$ coordinates and
momenta is a phase space vector.
%%%fix me

In statistical mechanics, the condition of an ensemble at any time
can be regarded as appropriately specified by the density $\rho$
with which representative points are distributed over the phase
space. The density distribution for an ensemble with $f$ degrees of
freedom is defined as,
\begin{equation}
\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
\label{introEquation:densityDistribution}
\end{equation}
Governed by the principles of mechanics, the phase points change
their locations which changes the density at any time at phase
space. Hence, the density distribution is also to be taken as a
function of the time. The number of systems $\delta N$ at time $t$
can be determined by,
\begin{equation}
\delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
\label{introEquation:deltaN}
\end{equation}
Assuming enough copies of the systems, we can sufficiently
approximate $\delta N$ without introducing discontinuity when we go
from one region in the phase space to another. By integrating over
the whole phase space,
\begin{equation}
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
\label{introEquation:totalNumberSystem}
\end{equation}
gives us an expression for the total number of copies. Hence, the
probability per unit volume in the phase space can be obtained by,
\begin{equation}
\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
\label{introEquation:unitProbability}
\end{equation}
With the help of Eq.~\ref{introEquation:unitProbability} and the
knowledge of the system, it is possible to calculate the average
value of any desired quantity which depends on the coordinates and
momenta of the system. Even when the dynamics of the real system are
complex, or stochastic, or even discontinuous, the average
properties of the ensemble of possibilities as a whole remain well
defined. For a classical system in thermal equilibrium with its
environment, the ensemble average of a mechanical quantity, $\langle
A(q , p) \rangle_t$, takes the form of an integral over the phase
space of the system,
\begin{equation}
\langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
\label{introEquation:ensembelAverage}
\end{equation}

\subsection{\label{introSection:liouville}Liouville's theorem}

Liouville's theorem is the foundation on which statistical mechanics
rests. It describes the time evolution of the phase space
distribution function. In order to calculate the rate of change of
$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider
the two faces perpendicular to the $q_1$ axis, which are located at
$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the
opposite face is given by the expression,
\begin{equation}
\left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
\right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
}}\delta q_1 } \right)\delta q_2  \ldots \delta q_f \delta p_1
\ldots \delta p_f .
\end{equation}
Summing all over the phase space, we obtain
\begin{equation}
\frac{{d(\delta N)}}{{dt}} =  - \sum\limits_{i = 1}^f {\left[ {\rho
\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} +
\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left(
{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  + \frac{{\partial
\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1
\ldots \delta q_f \delta p_1  \ldots \delta p_f .
\end{equation}
Differentiating the equations of motion in Hamiltonian formalism
(\ref{introEquation:motionHamiltonianCoordinate},
\ref{introEquation:motionHamiltonianMomentum}), we can show,
\begin{equation}
\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }}
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
\end{equation}
which cancels the first terms of the right hand side. Furthermore,
dividing $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
p_f $ in both sides, we can write out Liouville's theorem in a
simple form,
\begin{equation}
\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f
{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  +
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
\label{introEquation:liouvilleTheorem}
\end{equation}
Liouville's theorem states that the distribution function is
constant along any trajectory in phase space. In classical
statistical mechanics, since the number of system copies in an
ensemble is huge and constant, we can assume the local density has
no reason (other than classical mechanics) to change,
\begin{equation}
\frac{{\partial \rho }}{{\partial t}} = 0.
\label{introEquation:stationary}
\end{equation}
In such stationary system, the density of distribution $\rho$ can be
connected to the Hamiltonian $H$ through Maxwell-Boltzmann
distribution,
\begin{equation}
\rho  \propto e^{ - \beta H}
\label{introEquation:densityAndHamiltonian}
\end{equation}

\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}}
Lets consider a region in the phase space,
\begin{equation}
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
\end{equation}
If this region is small enough, the density $\rho$ can be regarded
as uniform over the whole integral. Thus, the number of phase points
inside this region is given by,
\begin{equation}
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
dp_1 } ..dp_f.
\end{equation}

\begin{equation}
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho
\frac{d}{{dt}}(\delta v) = 0.
\end{equation}
With the help of the stationary assumption
(Eq.~\ref{introEquation:stationary}), we obtain the principle of
\emph{conservation of volume in phase space},
\begin{equation}
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
...dq_f dp_1 } ..dp_f  = 0.
\label{introEquation:volumePreserving}
\end{equation}

\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}}

Liouville's theorem can be expressed in a variety of different forms
which are convenient within different contexts. For any two function
$F$ and $G$ of the coordinates and momenta of a system, the Poisson
bracket $\{F,G\}$ is defined as
\begin{equation}
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} -
\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial
q_i }}} \right)}.
\label{introEquation:poissonBracket}
\end{equation}
Substituting equations of motion in Hamiltonian formalism
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite
Liouville's theorem using Poisson bracket notion,
\begin{equation}
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
{\rho ,H} \right\}.
\label{introEquation:liouvilleTheromInPoissin}
\end{equation}
Moreover, the Liouville operator is defined as
\begin{equation}
iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)}
\label{introEquation:liouvilleOperator}
\end{equation}
In terms of Liouville operator, Liouville's equation can also be
expressed as
\begin{equation}
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - iL\rho
\label{introEquation:liouvilleTheoremInOperator}
\end{equation}
which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$.
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}

Various thermodynamic properties can be calculated from Molecular
Dynamics simulation. By comparing experimental values with the
calculated properties, one can determine the accuracy of the
simulation and the quality of the underlying model. However, both
experiments and computer simulations are usually performed during a
certain time interval and the measurements are averaged over a
period of time which is different from the average behavior of
many-body system in Statistical Mechanics. Fortunately, the Ergodic
Hypothesis makes a connection between time average and the ensemble
average. It states that the time average and average over the
statistical ensemble are identical:\cite{Frenkel1996, Leach2001}
\begin{equation}
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp
\end{equation}
where $\langle  A(q , p) \rangle_t$ is an equilibrium value of a
physical quantity and $\rho (p(t), q(t))$ is the equilibrium
distribution function. If an observation is averaged over a
sufficiently long time (longer than the relaxation time), all
accessible microstates in phase space are assumed to be equally
probed, giving a properly weighted statistical average. This allows
the researcher freedom of choice when deciding how best to measure a
given observable. In case an ensemble averaged approach sounds most
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be
utilized. Or if the system lends itself to a time averaging
approach, the Molecular Dynamics techniques in
Sec.~\ref{introSection:molecularDynamics} will be the best
choice.\cite{Frenkel1996}

\section{\label{introSection:geometricIntegratos}Geometric Integrators}
A variety of numerical integrators have been proposed to simulate
the motions of atoms in MD simulation. They usually begin with
initial conditions and move the objects in the direction governed by
the differential equations. However, most of them ignore the hidden
physical laws contained within the equations. Since 1990, geometric
integrators, which preserve various phase-flow invariants such as
symplectic structure, volume and time reversal symmetry, were
developed to address this issue.\cite{Dullweber1997, McLachlan1998,
Leimkuhler1999} The velocity Verlet method, which happens to be a
simple example of symplectic integrator, continues to gain
popularity in the molecular dynamics community. This fact can be
partly explained by its geometric nature.

\subsection{\label{introSection:symplecticManifold}Manifolds and Bundles}
A \emph{manifold} is an abstract mathematical space. It looks
locally like Euclidean space, but when viewed globally, it may have
more complicated structure. A good example of manifold is the
surface of Earth. It seems to be flat locally, but it is round if
viewed as a whole. A \emph{differentiable manifold} (also known as
\emph{smooth manifold}) is a manifold on which it is possible to
apply calculus.\cite{Hirsch1997} A \emph{symplectic manifold} is
defined as a pair $(M, \omega)$ which consists of a
\emph{differentiable manifold} $M$ and a close, non-degenerate,
bilinear symplectic form, $\omega$. A symplectic form on a vector
space $V$ is a function $\omega(x, y)$ which satisfies
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in
vector field is an example of symplectic form.
Given vector spaces $V$ and $W$ over same field $F$, $f: V \to W$ is a linear transformation if 
\begin{eqnarray*}
f(x+y) & = & f(x) + f(y) \\
f(ax) & = & af(x)       
\end{eqnarray*}
are always satisfied for any two vectors $x$ and $y$ in $V$ and any scalar $a$ in $F$. One can define the dual vector space $V^*$ of $V$ if any two built-in linear transformations $\phi$ and $\psi$ in $V^*$ satisfy the following definition of addition and scalar multiplication:
\begin{eqnarray*}
(\phi+\psi)(x) & = & \phi(x)+\psi(x) \\
(a\phi)(x) & = & a \phi(x)
\end{eqnarray*}
for all $a$ in $F$ and $x$ in $V$. For a manifold $M$, one can define a tangent vector of a tangent space $TM_q$ at every point $q$
\begin{equation}
\dot q = \mathop {\lim }\limits_{t \to 0} \frac{{\phi (t) - \phi (0)}}{t}       
\end{equation}
where $\phi(0)=q$ and $\phi(t) \in M$. One may also define a cotangent space $T^*M_q$ as the dual space of the tangent space $TM_q$. The tangent space and the cotangent space are isomorphic to each other, since they are both real vector spaces with same dimension.
The union of tangent spaces at every point of $M$ is called the tangent bundle of $M$ and is denoted by $TM$, while cotangent bundle $T^*M$ is defined as the union of the cotangent spaces to $M$.\cite{Jost2002} For a Hamiltonian system with configuration manifold $V$, the $(q,\dot q)$ phase space is the tangent bundle of the configuration manifold $V$, while the cotangent bundle is represented by $(q,p)$. 

\subsection{\label{introSection:ODE}Ordinary Differential Equations}

For an ordinary differential system defined as
\begin{equation}
\dot x = f(x)
\end{equation}
where $x = x(q,p)$, this system is a canonical Hamiltonian, if
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian
function and $J$ is the skew-symmetric matrix
\begin{equation}
J = \left( {\begin{array}{*{20}c}
   0 & I  \\
   { - I} & 0  \\
\end{array}} \right)
\label{introEquation:canonicalMatrix}
\end{equation}
where $I$ is an identity matrix. Using this notation, Hamiltonian
system can be rewritten as,
\begin{equation}
\frac{d}{{dt}}x = J\nabla _x H(x).
\label{introEquation:compactHamiltonian}
\end{equation}In this case, $f$ is
called a \emph{Hamiltonian vector field}. Another generalization of
Hamiltonian dynamics is Poisson Dynamics,\cite{Olver1986}
\begin{equation}
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
\end{equation}
where the most obvious change being that matrix $J$ now depends on
$x$.

\subsection{\label{introSection:exactFlow}Exact Propagator}

Let $x(t)$ be the exact solution of the ODE
system,
\begin{equation}
\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE}
\end{equation} we can
define its exact propagator $\varphi_\tau$:
\[ x(t+\tau)
=\varphi_\tau(x(t))
\]
where $\tau$ is a fixed time step and $\varphi$ is a map from phase
space to itself. The propagator has the continuous group property,
\begin{equation}
\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
+ \tau _2 } .
\end{equation}
In particular,
\begin{equation}
\varphi _\tau   \circ \varphi _{ - \tau }  = I
\end{equation}
Therefore, the exact propagator is self-adjoint,
\begin{equation}
\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
\end{equation}
In most cases, it is not easy to find the exact propagator
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$,
which is usually called an integrator. The order of an integrator
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
order $p$,
\begin{equation}
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1})
\end{equation}

\subsection{\label{introSection:geometricProperties}Geometric Properties}

The hidden geometric properties\cite{Budd1999, Marsden1998} of an
ODE and its propagator play important roles in numerical studies.
Many of them can be found in systems which occur naturally in
applications. Let $\varphi$ be the propagator of Hamiltonian vector
field, $\varphi$ is a \emph{symplectic} propagator if it satisfies,
\begin{equation}
{\varphi '}^T J \varphi ' = J.
\end{equation}
According to Liouville's theorem, the symplectic volume is invariant
under a Hamiltonian propagator, which is the basis for classical
statistical mechanics. Furthermore, the propagator of a Hamiltonian
vector field on a symplectic manifold can be shown to be a
symplectomorphism. As to the Poisson system,
\begin{equation}
{\varphi '}^T J \varphi ' = J \circ \varphi
\end{equation}
is the property that must be preserved by the integrator. It is
possible to construct a \emph{volume-preserving} propagator for a
source free ODE ($ \nabla \cdot f = 0 $), if the propagator
satisfies $ \det d\varphi  = 1$. One can show easily that a
symplectic propagator will be volume-preserving. Changing the
variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will
result in a new system,
\[
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
\]
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
In other words, the propagator of this vector field is reversible if
and only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
conserved quantity of a general differential function is a function
$ G:R^{2d}  \to R^d $ which is constant for all solutions of the ODE
$\frac{{dx}}{{dt}} = f(x)$ ,
\[
\frac{{dG(x(t))}}{{dt}} = 0.
\]
Using the chain rule, one may obtain,
\[
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G,
\]
which is the condition for conserved quantities. For a canonical
Hamiltonian system, the time evolution of an arbitrary smooth
function $G$ is given by,
\begin{eqnarray}
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\
                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)).
\label{introEquation:firstIntegral1}
\end{eqnarray}
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1}
can be rewritten as
\[
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
\]
Therefore, the sufficient condition for $G$ to be a conserved
quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As
is well known, the Hamiltonian (or energy) H of a Hamiltonian system
is a conserved quantity, which is due to the fact $\{ H,H\}  = 0$.
When designing any numerical methods, one should always try to
preserve the structural properties of the original ODE and its
propagator.

\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
A lot of well established and very effective numerical methods have
been successful precisely because of their symplectic nature even
though this fact was not recognized when they were first
constructed. The most famous example is the Verlet-leapfrog method
in molecular dynamics. In general, symplectic integrators can be
constructed using one of four different methods.
\begin{enumerate}
\item Generating functions
\item Variational methods
\item Runge-Kutta methods
\item Splitting methods
\end{enumerate}
Generating functions\cite{Channell1990} tend to lead to methods
which are cumbersome and difficult to use. In dissipative systems,
variational methods can capture the decay of energy
accurately.\cite{Kane2000} Since they are geometrically unstable
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
methods are not suitable for Hamiltonian
system.\cite{Cartwright1992} Recently, various high-order explicit
Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to
overcome this instability. However, due to computational penalty
involved in implementing the Runge-Kutta methods, they have not
attracted much attention from the Molecular Dynamics community.
Instead, splitting methods have been widely accepted since they
exploit natural decompositions of the system.\cite{McLachlan1998,
Tuckerman1992}

\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}

The main idea behind splitting methods is to decompose the discrete
$\varphi_h$ as a composition of simpler propagators,
\begin{equation}
\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
\varphi _{h_n }
\label{introEquation:FlowDecomposition}
\end{equation}
where each of the sub-propagator is chosen such that each represent
a simpler integration of the system. Suppose that a Hamiltonian
system takes the form,
\[
H = H_1 + H_2.
\]
Here, $H_1$ and $H_2$ may represent different physical processes of
the system. For instance, they may relate to kinetic and potential
energy respectively, which is a natural decomposition of the
problem. If $H_1$ and $H_2$ can be integrated using exact
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a
simple first order expression is then given by the Lie-Trotter
formula\cite{Trotter1959}
\begin{equation}
\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
\label{introEquation:firstOrderSplitting}
\end{equation}
where $\varphi _h$ is the result of applying the corresponding
continuous $\varphi _i$ over a time $h$. By definition, as
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
must follow that each operator $\varphi_i(t)$ is a symplectic map.
It is easy to show that any composition of symplectic propagators
yields a symplectic map,
\begin{equation}
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
'\phi ' = \phi '^T J\phi ' = J,
 \label{introEquation:SymplecticFlowComposition}
\end{equation}
where $\phi$ and $\psi$ both are symplectic maps. Thus operator
splitting in this context automatically generates a symplectic map.
The Lie-Trotter
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces
local errors proportional to $h^2$, while the Strang splitting gives
a second-order decomposition,\cite{Strang1968}
\begin{equation}
\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
_{1,h/2} , \label{introEquation:secondOrderSplitting}
\end{equation}
which has a local error proportional to $h^3$. The Strang
splitting's popularity in molecular simulation community attribute
to its symmetric property,
\begin{equation}
\varphi _h^{ - 1} = \varphi _{ - h}.
\label{introEquation:timeReversible}
\end{equation}

\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}}
The classical equation for a system consisting of interacting
particles can be written in Hamiltonian form,
\[
H = T + V
\]
where $T$ is the kinetic energy and $V$ is the potential energy.
Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one
obtains the following:
\begin{align}
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
    \frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
\label{introEquation:Lp10a} \\%
%
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
    \biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
\label{introEquation:Lp10b}
\end{align}
where $F(t)$ is the force at time $t$. This integration scheme is
known as \emph{velocity verlet} which is
symplectic(Eq.~\ref{introEquation:SymplecticFlowComposition}),
time-reversible(Eq.~\ref{introEquation:timeReversible}) and
volume-preserving (Eq.~\ref{introEquation:volumePreserving}). These
geometric properties attribute to its long-time stability and its
popularity in the community. However, the most commonly used
velocity verlet integration scheme is written as below,
\begin{align}
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
    \dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
%
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
    \label{introEquation:Lp9b}\\%
%
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
    \frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c}
\end{align}
From the preceding splitting, one can see that the integration of
the equations of motion would follow:
\begin{enumerate}
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.

\item Use the half step velocities to move positions one whole step, $\Delta t$.

\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move.

\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
\end{enumerate}
By simply switching the order of the propagators in the splitting
and composing a new integrator, the \emph{position verlet}
integrator, can be generated,
\begin{align}
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) +
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], %
\label{introEquation:positionVerlet1} \\%
%
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
q(\Delta t)} \right]. %
 \label{introEquation:positionVerlet2}
\end{align}

\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}

The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used
to determine the local error of a splitting method in terms of the
commutator of the
operators(Eq.~\ref{introEquation:exponentialOperator}) associated
with the sub-propagator. For operators $hX$ and $hY$ which are
associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we
have
\begin{equation}
\exp (hX + hY) = \exp (hZ)
\end{equation}
where
\begin{equation}
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left(
{[X,[X,Y]] + [Y,[Y,X]]} \right) +  \ldots .
\end{equation}
Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by
\[
[X,Y] = XY - YX .
\]
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
to the Strang splitting, we can obtain
\begin{eqnarray*}
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\
                                   &   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
                                   &   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots
                                   ).
\end{eqnarray*}
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local
error of Strang splitting is proportional to $h^3$. The same
procedure can be applied to a general splitting of the form
\begin{equation}
\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
1} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
\end{equation}
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher
order methods. Yoshida proposed an elegant way to compose higher
order methods based on symmetric splitting.\cite{Yoshida1990} Given
a symmetric second order base method $ \varphi _h^{(2)} $, a
fourth-order symmetric method can be constructed by composing,
\[
\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
\]
where $ \alpha  =  - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric
integrator $ \varphi _h^{(2n + 2)}$ can be composed by
\begin{equation}
\varphi _h^{(2n + 2)}  = \varphi _{\alpha h}^{(2n)}  \circ \varphi
_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)},
\end{equation}
if the weights are chosen as
\[
\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
\]

\section{\label{introSection:molecularDynamics}Molecular Dynamics}

As one of the principal tools of molecular modeling, Molecular
dynamics has proven to be a powerful tool for studying the functions
of biological systems, providing structural, thermodynamic and
dynamical information. The basic idea of molecular dynamics is that
macroscopic properties are related to microscopic behavior and
microscopic behavior can be calculated from the trajectories in
simulations. For instance, instantaneous temperature of a
Hamiltonian system of $N$ particles can be measured by
\[
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
\]
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
respectively, $f$ is the number of degrees of freedom, and $k_B$ is
the Boltzman constant.

A typical molecular dynamics run consists of three essential steps:
\begin{enumerate}
  \item Initialization
    \begin{enumerate}
    \item Preliminary preparation
    \item Minimization
    \item Heating
    \item Equilibration
    \end{enumerate}
  \item Production
  \item Analysis
\end{enumerate}
These three individual steps will be covered in the following
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
initialization of a simulation. Sec.~\ref{introSection:production}
 discusses issues of production runs.
Sec.~\ref{introSection:Analysis} provides the theoretical tools for
analysis of trajectories.

\subsection{\label{introSec:initialSystemSettings}Initialization}

\subsubsection{\textbf{Preliminary preparation}}

When selecting the starting structure of a molecule for molecular
simulation, one may retrieve its Cartesian coordinates from public
databases, such as RCSB Protein Data Bank \textit{etc}. Although
thousands of crystal structures of molecules are discovered every
year, many more remain unknown due to the difficulties of
purification and crystallization. Even for molecules with known
structures, some important information is missing. For example, a
missing hydrogen atom which acts as donor in hydrogen bonding must
be added. Moreover, in order to include electrostatic interactions,
one may need to specify the partial charges for individual atoms.
Under some circumstances, we may even need to prepare the system in
a special configuration. For instance, when studying transport
phenomenon in membrane systems, we may prepare the lipids in a
bilayer structure instead of placing lipids randomly in solvent,
since we are not interested in the slow self-aggregation process.

\subsubsection{\textbf{Minimization}}

It is quite possible that some of molecules in the system from
preliminary preparation may be overlapping with each other. This
close proximity leads to high initial potential energy which
consequently jeopardizes any molecular dynamics simulations. To
remove these steric overlaps, one typically performs energy
minimization to find a more reasonable conformation. Several energy
minimization methods have been developed to exploit the energy
surface and to locate the local minimum. While converging slowly
near the minimum, the steepest descent method is extremely robust when
systems are strongly anharmonic. Thus, it is often used to refine
structures from crystallographic data. Relying on the Hessian,
advanced methods like Newton-Raphson converge rapidly to a local
minimum, but become unstable if the energy surface is far from
quadratic. Another factor that must be taken into account, when
choosing energy minimization method, is the size of the system.
Steepest descent and conjugate gradient can deal with models of any
size. Because of the limits on computer memory to store the hessian
matrix and the computing power needed to diagonalize these matrices,
most Newton-Raphson methods can not be used with very large systems.

\subsubsection{\textbf{Heating}}

Typically, heating is performed by assigning random velocities
according to a Maxwell-Boltzman distribution for a desired
temperature. Beginning at a lower temperature and gradually
increasing the temperature by assigning larger random velocities, we
end up setting the temperature of the system to a final temperature
at which the simulation will be conducted. In the heating phase, we
should also keep the system from drifting or rotating as a whole. To
do this, the net linear momentum and angular momentum of the system
is shifted to zero after each resampling from the Maxwell -Boltzman
distribution.

\subsubsection{\textbf{Equilibration}}

The purpose of equilibration is to allow the system to evolve
spontaneously for a period of time and reach equilibrium. The
procedure is continued until various statistical properties, such as
temperature, pressure, energy, volume and other structural
properties \textit{etc}, become independent of time. Strictly
speaking, minimization and heating are not necessary, provided the
equilibration process is long enough. However, these steps can serve
as a mean to arrive at an equilibrated structure in an effective
way.

\subsection{\label{introSection:production}Production}

The production run is the most important step of the simulation, in
which the equilibrated structure is used as a starting point and the
motions of the molecules are collected for later analysis. In order
to capture the macroscopic properties of the system, the molecular
dynamics simulation must be performed by sampling correctly and
efficiently from the relevant thermodynamic ensemble.

The most expensive part of a molecular dynamics simulation is the
calculation of non-bonded forces, such as van der Waals force and
Coulombic forces \textit{etc}. For a system of $N$ particles, the
complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
which makes large simulations prohibitive in the absence of any
algorithmic tricks. A natural approach to avoid system size issues
is to represent the bulk behavior by a finite number of the
particles. However, this approach will suffer from surface effects
at the edges of the simulation. To offset this, \textit{Periodic
boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to
simulate bulk properties with a relatively small number of
particles. In this method, the simulation box is replicated
throughout space to form an infinite lattice. During the simulation,
when a particle moves in the primary cell, its image in other cells
move in exactly the same direction with exactly the same
orientation. Thus, as a particle leaves the primary cell, one of its
images will enter through the opposite face.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{pbc.eps}
\caption[An illustration of periodic boundary conditions]{A 2-D
illustration of periodic boundary conditions. As one particle leaves
the left of the simulation box, an image of it enters the right.}
\label{introFig:pbc}
\end{figure}

%cutoff and minimum image convention
Another important technique to improve the efficiency of force
evaluation is to apply spherical cutoffs where particles farther
than a predetermined distance are not included in the
calculation.\cite{Frenkel1996} The use of a cutoff radius will cause
a discontinuity in the potential energy curve. Fortunately, one can
shift a simple radial potential to ensure the potential curve go
smoothly to zero at the cutoff radius. The cutoff strategy works
well for Lennard-Jones interaction because of its short range
nature. However, simply truncating the electrostatic interaction
with the use of cutoffs has been shown to lead to severe artifacts
in simulations. The Ewald summation, in which the slowly decaying
Coulomb potential is transformed into direct and reciprocal sums
with rapid and absolute convergence, has proved to minimize the
periodicity artifacts in liquid simulations. Taking advantage of
fast Fourier transform (FFT) techniques for calculating discrete
Fourier transforms, the particle mesh-based
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the
\emph{fast multipole method}\cite{Greengard1987, Greengard1994},
which treats Coulombic interactions exactly at short range, and
approximate the potential at long range through multipolar
expansion. In spite of their wide acceptance at the molecular
simulation community, these two methods are difficult to implement
correctly and efficiently. Instead, we use a damped and
charge-neutralized Coulomb potential method developed by Wolf and
his coworkers.\cite{Wolf1999} The shifted Coulomb potential for
particle $i$ and particle $j$ at distance $r_{rj}$ is given by:
\begin{equation}
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
r_{ij})}{r_{ij}}\right\}, \label{introEquation:shiftedCoulomb}
\end{equation}
where $\alpha$ is the convergence parameter. Due to the lack of
inherent periodicity and rapid convergence,this method is extremely
efficient and easy to implement.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{shifted_coulomb.eps}
\caption[An illustration of shifted Coulomb potential]{An
illustration of shifted Coulomb potential.}
\label{introFigure:shiftedCoulomb}
\end{figure}

%multiple time step

\subsection{\label{introSection:Analysis} Analysis}

Recently, advanced visualization techniques have been applied to
monitor the motions of molecules. Although the dynamics of the
system can be described qualitatively from animation, quantitative
trajectory analysis is more useful. According to the principles of
Statistical Mechanics in
Sec.~\ref{introSection:statisticalMechanics}, one can compute
thermodynamic properties, analyze fluctuations of structural
parameters, and investigate time-dependent processes of the molecule
from the trajectories.

\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}

Thermodynamic properties, which can be expressed in terms of some
function of the coordinates and momenta of all particles in the
system, can be directly computed from molecular dynamics. The usual
way to measure the pressure is based on virial theorem of Clausius
which states that the virial is equal to $-3Nk_BT$. For a system
with forces between particles, the total virial, $W$, contains the
contribution from external pressure and interaction between the
particles:
\[
W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
f_{ij} } } \right\rangle
\]
where $f_{ij}$ is the force between particle $i$ and $j$ at a
distance $r_{ij}$. Thus, the expression for the pressure is given
by:
\begin{equation}
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
\end{equation}

\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}

Structural Properties of a simple fluid can be described by a set of
distribution functions. Among these functions,the \emph{pair
distribution function}, also known as \emph{radial distribution
function}, is of most fundamental importance to liquid theory.
Experimentally, pair distribution functions can be gathered by
Fourier transforming raw data from a series of neutron diffraction
experiments and integrating over the surface
factor.\cite{Powles1973} The experimental results can serve as a
criterion to justify the correctness of a liquid model. Moreover,
various equilibrium thermodynamic and structural properties can also
be expressed in terms of the radial distribution
function.\cite{Allen1987} The pair distribution functions $g(r)$
gives the probability that a particle $i$ will be located at a
distance $r$ from a another particle $j$ in the system
\begin{equation}
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
(r)}{\rho}.
\end{equation}
Note that the delta function can be replaced by a histogram in
computer simulation. Peaks in $g(r)$ represent solvent shells, and
the height of these peaks gradually decreases to 1 as the liquid of
large distance approaches the bulk density.


\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
Properties}}

Time-dependent properties are usually calculated using \emph{time
correlation functions}, which correlate random variables $A$ and $B$
at two different times,
\begin{equation}
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
\label{introEquation:timeCorrelationFunction}
\end{equation}
If $A$ and $B$ refer to same variable, this kind of correlation
functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation
function which is directly related to transport properties of
molecular liquids:
\begin{equation}
D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
\right\rangle } dt
\end{equation}
where $D$ is diffusion constant. Unlike the velocity autocorrelation
function, which is averaged over time origins and over all the
atoms, the dipole autocorrelation functions is calculated for the
entire system. The dipole autocorrelation function is given by:
\begin{equation}
c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
\right\rangle
\end{equation}
Here $u_{tot}$ is the net dipole of the entire system and is given
by
\begin{equation}
u_{tot} (t) = \sum\limits_i {u_i (t)}.
\end{equation}
In principle, many time correlation functions can be related to
Fourier transforms of the infrared, Raman, and inelastic neutron
scattering spectra of molecular liquids. In practice, one can
extract the IR spectrum from the intensity of the molecular dipole
fluctuation at each frequency using the following relationship:
\begin{equation}
\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
i2\pi vt} dt}.
\end{equation}

\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}

Rigid bodies are frequently involved in the modeling of different
areas, including engineering, physics and chemistry. For example,
missiles and vehicles are usually modeled by rigid bodies.  The
movement of the objects in 3D gaming engines or other physics
simulators is governed by rigid body dynamics. In molecular
simulations, rigid bodies are used to simplify protein-protein
docking studies.\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.

\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
The Hamiltonian of a rigid body is given by
\begin{equation}
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
\label{introEquation:RBHamiltonian}
\end{equation}
Here, $q$ and $Q$  are the position vector and rotation matrix for
the rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ ,
and $J$, a diagonal matrix, is defined by
\[
I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
\]
where $I_{ii}$ is the diagonal element of the inertia tensor. This
constrained Hamiltonian equation is subjected to a holonomic
constraint,
\begin{equation}
Q^T Q = 1, \label{introEquation:orthogonalConstraint}
\end{equation}
which is used to ensure the rotation matrix's unitarity. Using
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~
\ref{introEquation:motionHamiltonianMomentum}, one can write down
the equations of motion,
\begin{eqnarray}
 \frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\
 \frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\
 \frac{{dQ}}{{dt}} & = & PJ^{ - 1},  \label{introEquation:RBMotionRotation}\\
 \frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
\end{eqnarray}
Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and
using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain,
\begin{equation}
Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
\label{introEquation:RBFirstOrderConstraint}
\end{equation}
In general, there are two ways to satisfy the holonomic constraints.
We can use a constraint force provided by a Lagrange multiplier on
the normal manifold to keep the motion on the constraint space. Or
we can simply evolve the system on the constraint manifold. These
two methods have been proved to be equivalent. The holonomic
constraint and equations of motions define a constraint manifold for
rigid bodies
\[
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
\right\}.
\]
Unfortunately, this constraint manifold is not $T^* SO(3)$ which is
a symplectic manifold on Lie rotation group $SO(3)$. However, it
turns out that under symplectic transformation, the cotangent space
and the phase space are diffeomorphic. By introducing
\[
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
\]
the mechanical system subjected to a holonomic constraint manifold $M$
can be re-formulated as a Hamiltonian system on the cotangent space
\[
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
\]
For a body fixed vector $X_i$ with respect to the center of mass of
the rigid body, its corresponding lab fixed vector $X_i^{lab}$  is
given as
\begin{equation}
X_i^{lab} = Q X_i + q.
\end{equation}
Therefore, potential energy $V(q,Q)$ is defined by
\[
V(q,Q) = V(Q X_0 + q).
\]
Hence, the force and torque are given by
\[
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
\]
and
\[
\nabla _Q V(q,Q) = F(q,Q)X_i^t
\]
respectively. As a common choice to describe the rotation dynamics
of the rigid body, the angular momentum on the body fixed frame $\Pi
= Q^t P$ is introduced to rewrite the equations of motion,
\begin{equation}
 \begin{array}{l}
 \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
 \dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
 \end{array}
 \label{introEqaution:RBMotionPI}
\end{equation}
as well as holonomic constraints $\Pi J^{ - 1}  + J^{ - 1} \Pi ^t  =
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism,
\begin{equation}
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
{\begin{array}{*{20}c}
   0 & { - v_3 } & {v_2 }  \\
   {v_3 } & 0 & { - v_1 }  \\
   { - v_2 } & {v_1 } & 0  \\
\end{array}} \right),
\label{introEquation:hatmapIsomorphism}
\end{equation}
will let us associate the matrix products with traditional vector
operations
\[
\hat vu = v \times u.
\]
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
matrix,
\begin{eqnarray}
(\dot \Pi  - \dot \Pi ^T )&= &(\Pi  - \Pi ^T )(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  -
(\Lambda  - \Lambda ^T ). \label{introEquation:skewMatrixPI}
\end{eqnarray}
Since $\Lambda$ is symmetric, the last term of
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
Lagrange multiplier $\Lambda$ is absent from the equations of
motion. This unique property eliminates the requirement of
iterations which can not be avoided in other methods.\cite{Kol1997,
Omelyan1998} Applying the hat-map isomorphism, we obtain the
equation of motion for angular momentum in the body frame
\begin{equation}
\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
F_i (r,Q)} \right) \times X_i }.
\label{introEquation:bodyAngularMotion}
\end{equation}
In the same manner, the equation of motion for rotation matrix is
given by
\[
\dot Q = Qskew(I^{ - 1} \pi ).
\]

\subsection{\label{introSection:SymplecticFreeRB}Symplectic
Lie-Poisson Integrator for Free Rigid Bodies}

If there are no external forces exerted on the rigid body, the only
contribution to the rotational motion is from the kinetic energy
(the first term of \ref{introEquation:bodyAngularMotion}). The free
rigid body is an example of a Lie-Poisson system with Hamiltonian
function
\begin{equation}
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
\label{introEquation:rotationalKineticRB}
\end{equation}
where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
Lie-Poisson structure matrix,
\begin{equation}
J(\pi ) = \left( {\begin{array}{*{20}c}
   0 & {\pi _3 } & { - \pi _2 }  \\
   { - \pi _3 } & 0 & {\pi _1 }  \\
   {\pi _2 } & { - \pi _1 } & 0  \\
\end{array}} \right).
\end{equation}
Thus, the dynamics of free rigid body is governed by
\begin{equation}
\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
\end{equation}
One may notice that each $T_i^r$ in
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly.
For instance, the equations of motion due to $T_1^r$ are given by
\begin{equation}
\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
\label{introEqaution:RBMotionSingleTerm}
\end{equation}
with
\[ R_1  = \left( {\begin{array}{*{20}c}
   0 & 0 & 0  \\
   0 & 0 & {\pi _1 }  \\
   0 & { - \pi _1 } & 0  \\
\end{array}} \right).
\]
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is
\[
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
Q(0)e^{\Delta tR_1 }
\]
with
\[
e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
   0 & 0 & 0  \\
   0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
   0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
\]
To reduce the cost of computing expensive functions in $e^{\Delta
tR_1 }$, we can use the Cayley transformation to obtain a
single-aixs propagator,
\begin{eqnarray*}
e^{\Delta tR_1 }  & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta
tR_1 ) \\
%
& \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{eqnarray*}
The propagators for $T_2^r$ and $T_3^r$ can be found in the same
manner. In order to construct a second-order symplectic method, we
split the angular kinetic Hamiltonian function into five terms
\[
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
(\pi _1 ).
\]
By concatenating the propagators corresponding to these five terms,
we can obtain an symplectic integrator,
\[
\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
_1 }.
\]
The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by
\[
\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
).
\]
If the Poisson bracket of a function $F$ with an arbitrary smooth
function $G$ is zero, $F$ is a \emph{Casimir}, which is the
conserved quantity in Poisson system. We can easily verify that the
norm of the angular momentum, $\parallel \pi
\parallel$, is a \emph{Casimir}.\cite{McLachlan1993} Let $F(\pi ) = S(\frac{{\parallel
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
then by the chain rule
\[
\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
}}{2})\pi.
\]
Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
\pi
\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
Lie-Poisson integrator is found to be both extremely efficient and
stable. These properties can be explained by the fact the small
angle approximation is used and the norm of the angular momentum is
conserved.

\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
Splitting for Rigid Body}

The Hamiltonian of rigid body can be separated in terms of kinetic
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations
of motion corresponding to potential energy and kinetic energy are
listed in Table~\ref{introTable:rbEquations}.
\begin{table}
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
\label{introTable:rbEquations}
\begin{center}
\begin{tabular}{|l|l|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  Potential & Kinetic \\
  $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
  $\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
  $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
  $ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
  \hline
\end{tabular}
\end{center}
\end{table}
A second-order symplectic method is now obtained by the composition
of the position and velocity propagators,
\[
\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
\]
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
sub-propagators which corresponding to force and torque
respectively,
\[
\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
_{\Delta t/2,\tau }.
\]
Since the associated operators of $\varphi _{\Delta t/2,F} $ and
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the
kinetic energy can be separated to translational kinetic term, $T^t
(p)$, and rotational kinetic term, $T^r (\pi )$,
\begin{equation}
T(p,\pi ) =T^t (p) + T^r (\pi ).
\end{equation}
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore,
the corresponding propagators are given by
\[
\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
_{\Delta t,T^r }.
\]
Finally, we obtain the overall symplectic propagators for freely
moving rigid bodies
\begin{eqnarray}
 \varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \notag\\
  & & \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \notag\\
  & & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .
\label{introEquation:overallRBFlowMaps}
\end{eqnarray}

\section{\label{introSection:langevinDynamics}Langevin Dynamics}
As an 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. This section will review
the theory of Langevin dynamics. A brief derivation of the generalized
Langevin equation will be given first. Following that, we will
discuss the physical meaning of the terms appearing in the equation.

\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}

A harmonic bath model, in which an effective set of harmonic
oscillators are used to mimic the effect of a linearly responding
environment, has been widely used in quantum chemistry and
statistical mechanics. One of the successful applications of
Harmonic bath model is the derivation of the Generalized Langevin
Dynamics (GLE). Consider a system, in which the degree of
freedom $x$ is assumed to couple to the bath linearly, giving a
Hamiltonian of the form
\begin{equation}
H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
\label{introEquation:bathGLE}.
\end{equation}
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian,
\[
H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  x_\alpha ^2 }
\right\}}
\]
where the index $\alpha$ runs over all the bath degrees of freedom,
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath
coupling,
\[
\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
\]
where $g_\alpha$ are the coupling constants between the bath
coordinates ($x_ \alpha$) and the system coordinate ($x$).
Introducing
\[
W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
}}{{2m_\alpha  w_\alpha ^2 }}} x^2
\]
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
\[
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
w_\alpha ^2 }}x} \right)^2 } \right\}}.
\]
Since the first two terms of the new Hamiltonian depend only on the
system coordinates, we can get the equations of motion for
Generalized Langevin Dynamics by Hamilton's equations,
\begin{equation}
m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
\sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   -
\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)},
\label{introEquation:coorMotionGLE}
\end{equation}
and
\begin{equation}
m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x_\alpha   -
\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
\label{introEquation:bathMotionGLE}
\end{equation}
In order to derive an equation for $x$, the dynamics of the bath
variables $x_\alpha$ must be solved exactly first. As an integral
transform which is particularly useful in solving linear ordinary
differential equations,the Laplace transform is the appropriate tool
to solve this problem. The basic idea is to transform the difficult
differential equations into simple algebra problems which can be
solved easily. Then, by applying the inverse Laplace transform, we
can retrieve the solutions of the original problems. Let $f(t)$ be a
function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$
is a new function defined as
\[
L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
\]
where  $p$ is real and  $L$ is called the Laplace Transform
Operator. Below are some important properties of the Laplace transform
\begin{eqnarray*}
 L(x + y)  & = & L(x) + L(y) \\
 L(ax)     & = & aL(x) \\
 L(\dot x) & = & pL(x) - px(0) \\
 L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
 L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
 \end{eqnarray*}
Applying the Laplace transform to the bath coordinates, we obtain
\begin{eqnarray*}
p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) & = & - \omega _\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha  }}L(x), \\
L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}. \\
\end{eqnarray*}
In the same way, the system coordinates become
\begin{eqnarray*}
 mL(\ddot x) & = &
  - \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0) - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}  \\
  & & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}.
\end{eqnarray*}
With the help of some relatively important inverse Laplace
transformations:
\[
\begin{array}{c}
 L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
 L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
 L(1) = \frac{1}{p} \\
 \end{array}
\]
we obtain
\begin{eqnarray*}
m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
x_\alpha (0) - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}}
\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}\\
%
& = & -
\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha
= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha
^2 }}} \right)\cos (\omega _\alpha
t)\dot x(t - \tau )d} \tau }  \\
& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha  }}}
\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
\end{eqnarray*}
Introducing a \emph{dynamic friction kernel}
\begin{equation}
\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
\label{introEquation:dynamicFrictionKernelDefinition}
\end{equation}
and \emph{a random force}
\begin{equation}
R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
\label{introEquation:randomForceDefinition}
\end{equation}
the equation of motion can be rewritten as
\begin{equation}
m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
(t)\dot x(t - \tau )d\tau }  + R(t)
\label{introEuqation:GeneralizedLangevinDynamics}
\end{equation}
which is known as the \emph{generalized Langevin equation} (GLE).

\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}

One may notice that $R(t)$ depends only on initial conditions, which
implies it is completely deterministic within the context of a
harmonic bath. However, it is easy to verify that $R(t)$ is totally
uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)}
\right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
0.$ This property is what we expect from a truly random process. As
long as the model chosen for $R(t)$ was a gaussian distribution in
general, the stochastic nature of the GLE still remains.
%dynamic friction kernel
The convolution integral
\[
\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
\]
depends on the entire history of the evolution of $x$, which implies
that the bath retains memory of previous motions. In other words,
the bath requires a finite time to respond to change in the motion
of the system. For a sluggish bath which responds slowly to changes
in the system coordinate, we may regard $\xi(t)$ as a constant
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
\[
\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
\]
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
\[
m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
\]
which can be used to describe the effect of dynamic caging in
viscous solvents. The other extreme is the bath that responds
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be
taken as a $delta$ function in time:
\[
\xi (t) = 2\xi _0 \delta (t).
\]
Hence, the convolution integral becomes
\[
\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
\]
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
\begin{equation}
m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
x(t) + R(t) \label{introEquation:LangevinEquation}
\end{equation}
which is known as the Langevin equation. The static friction
coefficient $\xi _0$ can either be calculated from spectral density
or be determined by Stokes' law for regular shaped particles. A
brief review on calculating friction tensors for arbitrary shaped
particles is given in Sec.~\ref{introSection:frictionTensor}.

\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}

Defining a new set of coordinates
\[
q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
^2 }}x(0),
\]
we can rewrite $R(t)$ as
\[
R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
\]
And since the $q$ coordinates are harmonic oscillators,
\begin{eqnarray*}
 \left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
 \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
 \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
 \left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha  {\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle } }  \\
  & = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
  & = &kT\xi (t)
\end{eqnarray*}
Thus, we recover the \emph{second fluctuation dissipation theorem}
\begin{equation}
\xi (t) = \left\langle {R(t)R(0)} \right\rangle
\label{introEquation:secondFluctuationDissipation},
\end{equation}
which acts as a constraint on the possible ways in which one can
model the random force and friction kernel.
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