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#	Line 3 | Line 3 | Closely related to Classical Mechanics, Molecular Dyna
3		\section{\label{introSection:classicalMechanics}Classical
4		Mechanics}
5
6	<	Closely related to Classical Mechanics, Molecular Dynamics
7	<	simulations are carried out by integrating the equations of motion
8	<	for a given system of particles. There are three fundamental ideas
9	<	behind classical mechanics. Firstly, One can determine the state of
10	<	a mechanical system at any time of interest; Secondly, all the
11	<	mechanical properties of the system at that time can be determined
12	<	by combining the knowledge of the properties of the system with the
13	<	specification of this state; Finally, the specification of the state
14	<	when further combine with the laws of mechanics will also be
15	<	sufficient to predict the future behavior of the system.
6	>	Using equations of motion derived from Classical Mechanics,
7	>	Molecular Dynamics simulations are carried out by integrating the
8	>	equations of motion for a given system of particles. There are three
9	>	fundamental ideas behind classical mechanics. Firstly, one can
10	>	determine the state of a mechanical system at any time of interest;
11	>	Secondly, all the mechanical properties of the system at that time
12	>	can be determined by combining the knowledge of the properties of
13	>	the system with the specification of this state; Finally, the
14	>	specification of the state when further combined with the laws of
15	>	mechanics will also be sufficient to predict the future behavior of
16	>	the system.
17
18		\subsection{\label{introSection:newtonian}Newtonian Mechanics}
19		The discovery of Newton's three laws of mechanics which govern the
20		motion of particles is the foundation of the classical mechanics.
21	<	Newton¡¯s first law defines a class of inertial frames. Inertial
21	>	Newton's first law defines a class of inertial frames. Inertial
22		frames are reference frames where a particle not interacting with
23		other bodies will move with constant speed in the same direction.
24	<	With respect to inertial frames Newton¡¯s second law has the form
24	>	With respect to inertial frames, Newton's second law has the form
25		\begin{equation}
26	<	F = \frac {dp}{dt} = \frac {mv}{dt}
26	>	F = \frac {dp}{dt} = \frac {mdv}{dt}
27		\label{introEquation:newtonSecondLaw}
28		\end{equation}
29		A point mass interacting with other bodies moves with the
30		acceleration along the direction of the force acting on it. Let
31	<	$F_ij$ be the force that particle $i$ exerts on particle $j$, and
32	<	$F_ji$ be the force that particle $j$ exerts on particle $i$.
33	<	Newton¡¯s third law states that
31	>	$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and
32	>	$F_{ji}$ be the force that particle $j$ exerts on particle $i$.
33	>	Newton's third law states that
34		\begin{equation}
35	<	F_ij = -F_ji
35	>	F_{ij} = -F_{ji}.
36		\label{introEquation:newtonThirdLaw}
37		\end{equation}
37	–
38		Conservation laws of Newtonian Mechanics play very important roles
39		in solving mechanics problems. The linear momentum of a particle is
40		conserved if it is free or it experiences no force. The second
#	Line 46 | Line 46 | N \equiv r \times F \label{introEquation:torqueDefinit
46		\end{equation}
47		The torque $\tau$ with respect to the same origin is defined to be
48		\begin{equation}
49	<	N \equiv r \times F \label{introEquation:torqueDefinition}
49	>	\tau \equiv r \times F \label{introEquation:torqueDefinition}
50		\end{equation}
51		Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
52		\[
#	Line 59 | Line 59 | thus,
59		\]
60		thus,
61		\begin{equation}
62	<	\dot L = r \times \dot p = N
62	>	\dot L = r \times \dot p = \tau
63		\end{equation}
64		If there are no external torques acting on a body, the angular
65		momentum of it is conserved. The last conservation theorem state
66	<	that if all forces are conservative, Energy
67	<	\begin{equation}E = T + V \label{introEquation:energyConservation}
66	>	that if all forces are conservative, energy is conserved,
67	>	\begin{equation}E = T + V. \label{introEquation:energyConservation}
68		\end{equation}
69	<	is conserved. All of these conserved quantities are
70	<	important factors to determine the quality of numerical integration
71	<	scheme for rigid body \cite{Dullweber1997}.
69	>	All of these conserved quantities are important factors to determine
70	>	the quality of numerical integration schemes for rigid
71	>	bodies.\cite{Dullweber1997}
72
73		\subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
74
75	<	Newtonian Mechanics suffers from two important limitations: it
76	<	describes their motion in special cartesian coordinate systems.
77	<	Another limitation of Newtonian mechanics becomes obvious when we
78	<	try to describe systems with large numbers of particles. It becomes
79	<	very difficult to predict the properties of the system by carrying
80	<	out calculations involving the each individual interaction between
81	<	all the particles, even if we know all of the details of the
82	<	interaction. In order to overcome some of the practical difficulties
83	<	which arise in attempts to apply Newton's equation to complex
84	<	system, alternative procedures may be developed.
75	>	Newtonian Mechanics suffers from an important limitation: motion can
76	>	only be described in cartesian coordinate systems which make it
77	>	impossible to predict analytically the properties of the system even
78	>	if we know all of the details of the interaction. In order to
79	>	overcome some of the practical difficulties which arise in attempts
80	>	to apply Newton's equation to complex systems, approximate numerical
81	>	procedures may be developed.
82
83	<	\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's
84	<	Principle}
83	>	\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's
84	>	Principle}}
85
86		Hamilton introduced the dynamical principle upon which it is
87	<	possible to base all of mechanics and, indeed, most of classical
88	<	physics. Hamilton's Principle may be stated as follow,
89	<
90	<	The actual trajectory, along which a dynamical system may move from
91	<	one point to another within a specified time, is derived by finding
92	<	the path which minimizes the time integral of the difference between
96	<	the kinetic, $K$, and potential energies, $U$ \cite{tolman79}.
87	>	possible to base all of mechanics and most of classical physics.
88	>	Hamilton's Principle may be stated as follows: the trajectory, along
89	>	which a dynamical system may move from one point to another within a
90	>	specified time, is derived by finding the path which minimizes the
91	>	time integral of the difference between the kinetic $K$, and
92	>	potential energies $U$,
93		\begin{equation}
94	<	\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
94	>	\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}.
95		\label{introEquation:halmitonianPrinciple1}
96		\end{equation}
101	–
97		For simple mechanical systems, where the forces acting on the
98	<	different part are derivable from a potential and the velocities are
99	<	small compared with that of light, the Lagrangian function $L$ can
100	<	be define as the difference between the kinetic energy of the system
106	<	and its potential energy,
98	>	different parts are derivable from a potential, the Lagrangian
99	>	function $L$ can be defined as the difference between the kinetic
100	>	energy of the system and its potential energy,
101		\begin{equation}
102	<	L \equiv K - U = L(q_i ,\dot q_i ) ,
102	>	L \equiv K - U = L(q_i ,\dot q_i ).
103		\label{introEquation:lagrangianDef}
104		\end{equation}
105	<	then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
105	>	Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
106		\begin{equation}
107	<	\delta \int_{t_1 }^{t_2 } {L dt = 0} ,
107	>	\delta \int_{t_1 }^{t_2 } {L dt = 0} .
108		\label{introEquation:halmitonianPrinciple2}
109		\end{equation}
110
111	<	\subsubsection{\label{introSection:equationOfMotionLagrangian}The
112	<	Equations of Motion in Lagrangian Mechanics}
111	>	\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The
112	>	Equations of Motion in Lagrangian Mechanics}}
113
114	<	for a holonomic system of $f$ degrees of freedom, the equations of
115	<	motion in the Lagrangian form is
114	>	For a system of $f$ degrees of freedom, the equations of motion in
115	>	the Lagrangian form is
116		\begin{equation}
117		\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
118		\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
#	Line 132 | Line 126 | independent of generalized velocities, the generalized
126		Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
127		introduced by William Rowan Hamilton in 1833 as a re-formulation of
128		classical mechanics. If the potential energy of a system is
129	<	independent of generalized velocities, the generalized momenta can
136	<	be defined as
129	>	independent of velocities, the momenta can be defined as
130		\begin{equation}
131		p_i = \frac{\partial L}{\partial \dot q_i}
132		\label{introEquation:generalizedMomenta}
#	Line 143 | Line 136 | p_i  = \frac{{\partial L}}{{\partial q_i }}
136		p_i  = \frac{{\partial L}}{{\partial q_i }}
137		\label{introEquation:generalizedMomentaDot}
138		\end{equation}
146	–
139		With the help of the generalized momenta, we may now define a new
140		quantity $H$ by the equation
141		\begin{equation}
#	Line 151 | Line 143 | $L$ is the Lagrangian function for the system.
143		\label{introEquation:hamiltonianDefByLagrangian}
144		\end{equation}
145		where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
146	<	$L$ is the Lagrangian function for the system.
147	<
156	<	Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
157	<	one can obtain
146	>	$L$ is the Lagrangian function for the system. Differentiating
147	>	Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain
148		\begin{equation}
149		dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
150		\frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
151		L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
152	<	L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
152	>	L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1}
153		\end{equation}
154	<	Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
155	<	second and fourth terms in the parentheses cancel. Therefore,
154	>	Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second
155	>	and fourth terms in the parentheses cancel. Therefore,
156		Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
157		\begin{equation}
158		dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
159	<	\right)}  - \frac{{\partial L}}{{\partial t}}dt
159	>	\right)}  - \frac{{\partial L}}{{\partial t}}dt .
160		\label{introEquation:diffHamiltonian2}
161		\end{equation}
162		By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
163		find
164		\begin{equation}
165	<	\frac{{\partial H}}{{\partial p_k }} = q_k
165	>	\frac{{\partial H}}{{\partial p_k }} = \dot {q_k}
166		\label{introEquation:motionHamiltonianCoordinate}
167		\end{equation}
168		\begin{equation}
169	<	\frac{{\partial H}}{{\partial q_k }} =  - p_k
169	>	\frac{{\partial H}}{{\partial q_k }} =  - \dot {p_k}
170		\label{introEquation:motionHamiltonianMomentum}
171		\end{equation}
172		and
#	Line 185 | Line 175 | t}}
175		t}}
176		\label{introEquation:motionHamiltonianTime}
177		\end{equation}
178	<
189	<	Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
178	>	where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
179		Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
180		equation of motion. Due to their symmetrical formula, they are also
181	<	known as the canonical equations of motions \cite{Goldstein01}.
181	>	known as the canonical equations of motions.\cite{Goldstein2001}
182
183		An important difference between Lagrangian approach and the
184		Hamiltonian approach is that the Lagrangian is considered to be a
185	<	function of the generalized velocities $\dot q_i$ and the
186	<	generalized coordinates $q_i$, while the Hamiltonian is considered
187	<	to be a function of the generalized momenta $p_i$ and the conjugate
188	<	generalized coordinate $q_i$. Hamiltonian Mechanics is more
189	<	appropriate for application to statistical mechanics and quantum
190	<	mechanics, since it treats the coordinate and its time derivative as
191	<	independent variables and it only works with 1st-order differential
203	<	equations\cite{Marion90}.
204	<
185	>	function of the generalized velocities $\dot q_i$ and coordinates
186	>	$q_i$, while the Hamiltonian is considered to be a function of the
187	>	generalized momenta $p_i$ and the conjugate coordinates $q_i$.
188	>	Hamiltonian Mechanics is more appropriate for application to
189	>	statistical mechanics and quantum mechanics, since it treats the
190	>	coordinate and its time derivative as independent variables and it
191	>	only works with 1st-order differential equations.\cite{Marion1990}
192		In Newtonian Mechanics, a system described by conservative forces
193	<	conserves the total energy \ref{introEquation:energyConservation}.
194	<	It follows that Hamilton's equations of motion conserve the total
195	<	Hamiltonian.
193	>	conserves the total energy
194	>	(Eq.~\ref{introEquation:energyConservation}). It follows that
195	>	Hamilton's equations of motion conserve the total Hamiltonian
196		\begin{equation}
197		\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
198		H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
199		}}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
200		H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
201		\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
202	<	q_i }}} \right) = 0}
216	<	\label{introEquation:conserveHalmitonian}
202	>	q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian}
203		\end{equation}
204
219	–	When studying Hamiltonian system, it is more convenient to use
220	–	notation
221	–	\begin{equation}
222	–	r = r(q,p)^T
223	–	\end{equation}
224	–	and to introduce a $2n \times 2n$ canonical structure matrix $J$,
225	–	\begin{equation}
226	–	J = \left( {\begin{array}{*{20}c}
227	–	0 & I  \\
228	–	{ - I} & 0  \\
229	–	\end{array}} \right)
230	–	\label{introEquation:canonicalMatrix}
231	–	\end{equation}
232	–	where $I$ is a $n \times n$ identity matrix and $J$ is a
233	–	skew-symmetric matrix ($ J^T  =  - J $). Thus, Hamiltonian system
234	–	can be rewritten as,
235	–	\begin{equation}
236	–	\frac{d}{{dt}}r = J\nabla _r H(r)
237	–	\label{introEquation:compactHamiltonian}
238	–	\end{equation}
239	–
205		\section{\label{introSection:statisticalMechanics}Statistical
206		Mechanics}
207
208		The thermodynamic behaviors and properties of Molecular Dynamics
209		simulation are governed by the principle of Statistical Mechanics.
210		The following section will give a brief introduction to some of the
211	<	Statistical Mechanics concepts presented in this dissertation.
211	>	Statistical Mechanics concepts and theorems presented in this
212	>	dissertation.
213
214	<	\subsection{\label{introSection:ensemble}Ensemble and Phase Space}
214	>	\subsection{\label{introSection:ensemble}Phase Space and Ensemble}
215
216	<	\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
216	>	Mathematically, phase space is the space which represents all
217	>	possible states of a system. Each possible state of the system
218	>	corresponds to one unique point in the phase space. For mechanical
219	>	systems, the phase space usually consists of all possible values of
220	>	position and momentum variables. Consider a dynamic system of $f$
221	>	particles in a cartesian space, where each of the $6f$ coordinates
222	>	and momenta is assigned to one of $6f$ mutually orthogonal axes, the
223	>	phase space of this system is a $6f$ dimensional space. A point, $x
224	>	=
225	>	(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
226	>	\over q} _1 , \ldots
227	>	,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
228	>	\over q} _f
229	>	,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
230	>	\over p} _1  \ldots
231	>	,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
232	>	\over p} _f )$ , with a unique set of values of $6f$ coordinates and
233	>	momenta is a phase space vector.
234	>	%%%fix me
235
236	<	Various thermodynamic properties can be calculated from Molecular
237	<	Dynamics simulation. By comparing experimental values with the
238	<	calculated properties, one can determine the accuracy of the
239	<	simulation and the quality of the underlying model. However, both of
240	<	experiment and computer simulation are usually performed during a
257	<	certain time interval and the measurements are averaged over a
258	<	period of them which is different from the average behavior of
259	<	many-body system in Statistical Mechanics. Fortunately, Ergodic
260	<	Hypothesis is proposed to make a connection between time average and
261	<	ensemble average. It states that time average and average over the
262	<	statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
236	>	In statistical mechanics, the condition of an ensemble at any time
237	>	can be regarded as appropriately specified by the density $\rho$
238	>	with which representative points are distributed over the phase
239	>	space. The density distribution for an ensemble with $f$ degrees of
240	>	freedom is defined as,
241		\begin{equation}
242	<	\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty }
243	<	\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma
266	<	{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq
242	>	\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
243	>	\label{introEquation:densityDistribution}
244		\end{equation}
245	<	where $\langle A \rangle_t$ is an equilibrium value of a physical
246	<	quantity and $\rho (p(t), q(t))$ is the equilibrium distribution
247	<	function. If an observation is averaged over a sufficiently long
248	<	time (longer than relaxation time), all accessible microstates in
249	<	phase space are assumed to be equally probed, giving a properly
250	<	weighted statistical average. This allows the researcher freedom of
251	<	choice when deciding how best to measure a given observable. In case
252	<	an ensemble averaged approach sounds most reasonable, the Monte
253	<	Carlo techniques\cite{metropolis:1949} can be utilized. Or if the
254	<	system lends itself to a time averaging approach, the Molecular
255	<	Dynamics techniques in Sec.~\ref{introSection:molecularDynamics}
256	<	will be the best choice\cite{Frenkel1996}.
245	>	Governed by the principles of mechanics, the phase points change
246	>	their locations which changes the density at any time at phase
247	>	space. Hence, the density distribution is also to be taken as a
248	>	function of the time. The number of systems $\delta N$ at time $t$
249	>	can be determined by,
250	>	\begin{equation}
251	>	\delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
252	>	\label{introEquation:deltaN}
253	>	\end{equation}
254	>	Assuming enough copies of the systems, we can sufficiently
255	>	approximate $\delta N$ without introducing discontinuity when we go
256	>	from one region in the phase space to another. By integrating over
257	>	the whole phase space,
258	>	\begin{equation}
259	>	N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
260	>	\label{introEquation:totalNumberSystem}
261	>	\end{equation}
262	>	gives us an expression for the total number of copies. Hence, the
263	>	probability per unit volume in the phase space can be obtained by,
264	>	\begin{equation}
265	>	\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int
266	>	{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
267	>	\label{introEquation:unitProbability}
268	>	\end{equation}
269	>	With the help of Eq.~\ref{introEquation:unitProbability} and the
270	>	knowledge of the system, it is possible to calculate the average
271	>	value of any desired quantity which depends on the coordinates and
272	>	momenta of the system. Even when the dynamics of the real system are
273	>	complex, or stochastic, or even discontinuous, the average
274	>	properties of the ensemble of possibilities as a whole remain well
275	>	defined. For a classical system in thermal equilibrium with its
276	>	environment, the ensemble average of a mechanical quantity, $\langle
277	>	A(q , p) \rangle_t$, takes the form of an integral over the phase
278	>	space of the system,
279	>	\begin{equation}
280	>	\langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
281	>	(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
282	>	(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
283	>	\label{introEquation:ensembelAverage}
284	>	\end{equation}
285
286	+	\subsection{\label{introSection:liouville}Liouville's theorem}
287	+
288	+	Liouville's theorem is the foundation on which statistical mechanics
289	+	rests. It describes the time evolution of the phase space
290	+	distribution function. In order to calculate the rate of change of
291	+	$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider
292	+	the two faces perpendicular to the $q_1$ axis, which are located at
293	+	$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the
294	+	opposite face is given by the expression,
295	+	\begin{equation}
296	+	\left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
297	+	\right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
298	+	}}\delta q_1 } \right)\delta q_2  \ldots \delta q_f \delta p_1
299	+	\ldots \delta p_f .
300	+	\end{equation}
301	+	Summing all over the phase space, we obtain
302	+	\begin{equation}
303	+	\frac{{d(\delta N)}}{{dt}} =  - \sum\limits_{i = 1}^f {\left[ {\rho
304	+	\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} +
305	+	\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left(
306	+	{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  + \frac{{\partial
307	+	\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1
308	+	\ldots \delta q_f \delta p_1  \ldots \delta p_f .
309	+	\end{equation}
310	+	Differentiating the equations of motion in Hamiltonian formalism
311	+	(\ref{introEquation:motionHamiltonianCoordinate},
312	+	\ref{introEquation:motionHamiltonianMomentum}), we can show,
313	+	\begin{equation}
314	+	\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }}
315	+	+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
316	+	\end{equation}
317	+	which cancels the first terms of the right hand side. Furthermore,
318	+	dividing $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
319	+	p_f $ in both sides, we can write out Liouville's theorem in a
320	+	simple form,
321	+	\begin{equation}
322	+	\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f
323	+	{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  +
324	+	\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
325	+	\label{introEquation:liouvilleTheorem}
326	+	\end{equation}
327	+	Liouville's theorem states that the distribution function is
328	+	constant along any trajectory in phase space. In classical
329	+	statistical mechanics, since the number of system copies in an
330	+	ensemble is huge and constant, we can assume the local density has
331	+	no reason (other than classical mechanics) to change,
332	+	\begin{equation}
333	+	\frac{{\partial \rho }}{{\partial t}} = 0.
334	+	\label{introEquation:stationary}
335	+	\end{equation}
336	+	In such stationary system, the density of distribution $\rho$ can be
337	+	connected to the Hamiltonian $H$ through Maxwell-Boltzmann
338	+	distribution,
339	+	\begin{equation}
340	+	\rho  \propto e^{ - \beta H}
341	+	\label{introEquation:densityAndHamiltonian}
342	+	\end{equation}
343	+
344	+	\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}}
345	+	Lets consider a region in the phase space,
346	+	\begin{equation}
347	+	\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
348	+	\end{equation}
349	+	If this region is small enough, the density $\rho$ can be regarded
350	+	as uniform over the whole integral. Thus, the number of phase points
351	+	inside this region is given by,
352	+	\begin{equation}
353	+	\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
354	+	dp_1 } ..dp_f.
355	+	\end{equation}
356	+
357	+	\begin{equation}
358	+	\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho
359	+	\frac{d}{{dt}}(\delta v) = 0.
360	+	\end{equation}
361	+	With the help of the stationary assumption
362	+	(Eq.~\ref{introEquation:stationary}), we obtain the principle of
363	+	\emph{conservation of volume in phase space},
364	+	\begin{equation}
365	+	\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
366	+	...dq_f dp_1 } ..dp_f  = 0.
367	+	\label{introEquation:volumePreserving}
368	+	\end{equation}
369	+
370	+	\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}}
371	+
372	+	Liouville's theorem can be expressed in a variety of different forms
373	+	which are convenient within different contexts. For any two function
374	+	$F$ and $G$ of the coordinates and momenta of a system, the Poisson
375	+	bracket $\{F,G\}$ is defined as
376	+	\begin{equation}
377	+	\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial
378	+	F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} -
379	+	\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial
380	+	q_i }}} \right)}.
381	+	\label{introEquation:poissonBracket}
382	+	\end{equation}
383	+	Substituting equations of motion in Hamiltonian formalism
384	+	(Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
385	+	Eq.~\ref{introEquation:motionHamiltonianMomentum}) into
386	+	(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite
387	+	Liouville's theorem using Poisson bracket notion,
388	+	\begin{equation}
389	+	\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
390	+	{\rho ,H} \right\}.
391	+	\label{introEquation:liouvilleTheromInPoissin}
392	+	\end{equation}
393	+	Moreover, the Liouville operator is defined as
394	+	\begin{equation}
395	+	iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial
396	+	p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial
397	+	H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)}
398	+	\label{introEquation:liouvilleOperator}
399	+	\end{equation}
400	+	In terms of Liouville operator, Liouville's equation can also be
401	+	expressed as
402	+	\begin{equation}
403	+	\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - iL\rho
404	+	\label{introEquation:liouvilleTheoremInOperator}
405	+	\end{equation}
406	+	which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$.
407	+	\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
408	+
409	+	Various thermodynamic properties can be calculated from Molecular
410	+	Dynamics simulation. By comparing experimental values with the
411	+	calculated properties, one can determine the accuracy of the
412	+	simulation and the quality of the underlying model. However, both
413	+	experiments and computer simulations are usually performed during a
414	+	certain time interval and the measurements are averaged over a
415	+	period of time which is different from the average behavior of
416	+	many-body system in Statistical Mechanics. Fortunately, the Ergodic
417	+	Hypothesis makes a connection between time average and the ensemble
418	+	average. It states that the time average and average over the
419	+	statistical ensemble are identical:\cite{Frenkel1996, Leach2001}
420	+	\begin{equation}
421	+	\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
422	+	\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
423	+	{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp
424	+	\end{equation}
425	+	where $\langle  A(q , p) \rangle_t$ is an equilibrium value of a
426	+	physical quantity and $\rho (p(t), q(t))$ is the equilibrium
427	+	distribution function. If an observation is averaged over a
428	+	sufficiently long time (longer than the relaxation time), all
429	+	accessible microstates in phase space are assumed to be equally
430	+	probed, giving a properly weighted statistical average. This allows
431	+	the researcher freedom of choice when deciding how best to measure a
432	+	given observable. In case an ensemble averaged approach sounds most
433	+	reasonable, the Monte Carlo methods\cite{Metropolis1949} can be
434	+	utilized. Or if the system lends itself to a time averaging
435	+	approach, the Molecular Dynamics techniques in
436	+	Sec.~\ref{introSection:molecularDynamics} will be the best
437	+	choice.\cite{Frenkel1996}
438	+
439		\section{\label{introSection:geometricIntegratos}Geometric Integrators}
440	<	A variety of numerical integrators were proposed to simulate the
441	<	motions. They usually begin with an initial conditionals and move
442	<	the objects in the direction governed by the differential equations.
443	<	However, most of them ignore the hidden physical law contained
444	<	within the equations. Since 1990, geometric integrators, which
445	<	preserve various phase-flow invariants such as symplectic structure,
446	<	volume and time reversal symmetry, are developed to address this
447	<	issue. The velocity verlet method, which happens to be a simple
448	<	example of symplectic integrator, continues to gain its popularity
449	<	in molecular dynamics community. This fact can be partly explained
450	<	by its geometric nature.
440	>	A variety of numerical integrators have been proposed to simulate
441	>	the motions of atoms in MD simulation. They usually begin with
442	>	initial conditions and move the objects in the direction governed by
443	>	the differential equations. However, most of them ignore the hidden
444	>	physical laws contained within the equations. Since 1990, geometric
445	>	integrators, which preserve various phase-flow invariants such as
446	>	symplectic structure, volume and time reversal symmetry, were
447	>	developed to address this issue.\cite{Dullweber1997, McLachlan1998,
448	>	Leimkuhler1999} The velocity Verlet method, which happens to be a
449	>	simple example of symplectic integrator, continues to gain
450	>	popularity in the molecular dynamics community. This fact can be
451	>	partly explained by its geometric nature.
452
453	<	\subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
454	<	A \emph{manifold} is an abstract mathematical space. It locally
455	<	looks like Euclidean space, but when viewed globally, it may have
456	<	more complicate structure. A good example of manifold is the surface
457	<	of Earth. It seems to be flat locally, but it is round if viewed as
458	<	a whole. A \emph{differentiable manifold} (also known as
459	<	\emph{smooth manifold}) is a manifold with an open cover in which
460	<	the covering neighborhoods are all smoothly isomorphic to one
461	<	another. In other words,it is possible to apply calculus on
462	<	\emph{differentiable manifold}. A \emph{symplectic manifold} is
304	<	defined as a pair $(M, \omega)$ which consisting of a
305	<	\emph{differentiable manifold} $M$ and a close, non-degenerated,
453	>	\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds}
454	>	A \emph{manifold} is an abstract mathematical space. It looks
455	>	locally like Euclidean space, but when viewed globally, it may have
456	>	more complicated structure. A good example of manifold is the
457	>	surface of Earth. It seems to be flat locally, but it is round if
458	>	viewed as a whole. A \emph{differentiable manifold} (also known as
459	>	\emph{smooth manifold}) is a manifold on which it is possible to
460	>	apply calculus.\cite{Hirsch1997} A \emph{symplectic manifold} is
461	>	defined as a pair $(M, \omega)$ which consists of a
462	>	\emph{differentiable manifold} $M$ and a close, non-degenerate,
463		bilinear symplectic form, $\omega$. A symplectic form on a vector
464		space $V$ is a function $\omega(x, y)$ which satisfies
465		$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
466		\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
467	<	$\omega(x, x) = 0$. Cross product operation in vector field is an
468	<	example of symplectic form.
467	>	$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in
468	>	vector field is an example of symplectic form. One of the
469	>	motivations to study \emph{symplectic manifolds} in Hamiltonian
470	>	Mechanics is that a symplectic manifold can represent all possible
471	>	configurations of the system and the phase space of the system can
472	>	be described by it's cotangent bundle.\cite{Jost2002} Every
473	>	symplectic manifold is even dimensional. For instance, in Hamilton
474	>	equations, coordinate and momentum always appear in pairs.
475
476	<	One of the motivations to study \emph{symplectic manifold} in
314	<	Hamiltonian Mechanics is that a symplectic manifold can represent
315	<	all possible configurations of the system and the phase space of the
316	<	system can be described by it's cotangent bundle. Every symplectic
317	<	manifold is even dimensional. For instance, in Hamilton equations,
318	<	coordinate and momentum always appear in pairs.
476	>	\subsection{\label{introSection:ODE}Ordinary Differential Equations}
477
478	<	Let  $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map
479	<	\[
480	<	f : M \rightarrow N
481	<	\]
482	<	is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and
483	<	the \emph{pullback} of $\eta$ under f is equal to $\omega$.
484	<	Canonical transformation is an example of symplectomorphism in
485	<	classical mechanics. According to Liouville's theorem, the
486	<	Hamiltonian \emph{flow} or \emph{symplectomorphism} generated by the
487	<	Hamiltonian vector filed preserves the volume form on the phase
488	<	space, which is the basis of classical statistical mechanics.
478	>	For an ordinary differential system defined as
479	>	\begin{equation}
480	>	\dot x = f(x)
481	>	\end{equation}
482	>	where $x = x(q,p)$, this system is a canonical Hamiltonian, if
483	>	$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian
484	>	function and $J$ is the skew-symmetric matrix
485	>	\begin{equation}
486	>	J = \left( {\begin{array}{*{20}c}
487	>	0 & I  \\
488	>	{ - I} & 0  \\
489	>	\end{array}} \right)
490	>	\label{introEquation:canonicalMatrix}
491	>	\end{equation}
492	>	where $I$ is an identity matrix. Using this notation, Hamiltonian
493	>	system can be rewritten as,
494	>	\begin{equation}
495	>	\frac{d}{{dt}}x = J\nabla _x H(x).
496	>	\label{introEquation:compactHamiltonian}
497	>	\end{equation}In this case, $f$ is
498	>	called a \emph{Hamiltonian vector field}. Another generalization of
499	>	Hamiltonian dynamics is Poisson Dynamics,\cite{Olver1986}
500	>	\begin{equation}
501	>	\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
502	>	\end{equation}
503	>	where the most obvious change being that matrix $J$ now depends on
504	>	$x$.
505
506	<	\subsection{\label{introSection:exactFlow}The Exact Flow of ODE}
506	>	\subsection{\label{introSection:exactFlow}Exact Propagator}
507
508	<	\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting}
508	>	Let $x(t)$ be the exact solution of the ODE
509	>	system,
510	>	\begin{equation}
511	>	\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE}
512	>	\end{equation} we can
513	>	define its exact propagator $\varphi_\tau$:
514	>	\[ x(t+\tau)
515	>	=\varphi_\tau(x(t))
516	>	\]
517	>	where $\tau$ is a fixed time step and $\varphi$ is a map from phase
518	>	space to itself. The propagator has the continuous group property,
519	>	\begin{equation}
520	>	\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
521	>	+ \tau _2 } .
522	>	\end{equation}
523	>	In particular,
524	>	\begin{equation}
525	>	\varphi _\tau   \circ \varphi _{ - \tau }  = I
526	>	\end{equation}
527	>	Therefore, the exact propagator is self-adjoint,
528	>	\begin{equation}
529	>	\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
530	>	\end{equation}
531	>	The exact propagator can also be written as an operator,
532	>	\begin{equation}
533	>	\varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
534	>	}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
535	>	\label{introEquation:exponentialOperator}
536	>	\end{equation}
537	>	In most cases, it is not easy to find the exact propagator
538	>	$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$,
539	>	which is usually called an integrator. The order of an integrator
540	>	$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
541	>	order $p$,
542	>	\begin{equation}
543	>	\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1})
544	>	\end{equation}
545
546	<	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
546	>	\subsection{\label{introSection:geometricProperties}Geometric Properties}
547
548	<	As a special discipline of molecular modeling, Molecular dynamics
549	<	has proven to be a powerful tool for studying the functions of
550	<	biological systems, providing structural, thermodynamic and
551	<	dynamical information.
552	<
343	<	\subsection{\label{introSec:mdInit}Initialization}
344	<
345	<	\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
346	<
347	<	\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
348	<
349	<	A rigid body is a body in which the distance between any two given
350	<	points of a rigid body remains constant regardless of external
351	<	forces exerted on it. A rigid body therefore conserves its shape
352	<	during its motion.
353	<
354	<	Applications of dynamics of rigid bodies.
355	<
356	<	\subsection{\label{introSection:lieAlgebra}Lie Algebra}
357	<
358	<	\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
359	<
360	<	\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
361	<
362	<	%\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
363	<
364	<	\section{\label{introSection:correlationFunctions}Correlation Functions}
365	<
366	<	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
367	<
368	<	\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
369	<
370	<	\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
371	<
548	>	The hidden geometric properties\cite{Budd1999, Marsden1998} of an
549	>	ODE and its propagator play important roles in numerical studies.
550	>	Many of them can be found in systems which occur naturally in
551	>	applications. Let $\varphi$ be the propagator of Hamiltonian vector
552	>	field, $\varphi$ is a \emph{symplectic} propagator if it satisfies,
553		\begin{equation}
554	<	H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
374	<	\label{introEquation:bathGLE}
554	>	{\varphi '}^T J \varphi ' = J.
555		\end{equation}
556	<	where $H_B$ is harmonic bath Hamiltonian,
556	>	According to Liouville's theorem, the symplectic volume is invariant
557	>	under a Hamiltonian propagator, which is the basis for classical
558	>	statistical mechanics. Furthermore, the propagator of a Hamiltonian
559	>	vector field on a symplectic manifold can be shown to be a
560	>	symplectomorphism. As to the Poisson system,
561	>	\begin{equation}
562	>	{\varphi '}^T J \varphi ' = J \circ \varphi
563	>	\end{equation}
564	>	is the property that must be preserved by the integrator. It is
565	>	possible to construct a \emph{volume-preserving} propagator for a
566	>	source free ODE ($ \nabla \cdot f = 0 $), if the propagator
567	>	satisfies $ \det d\varphi  = 1$. One can show easily that a
568	>	symplectic propagator will be volume-preserving. Changing the
569	>	variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will
570	>	result in a new system,
571		\[
572	<	H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
379	<	}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
572	>	\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
573		\]
574	<	and $\Delta U$ is bilinear system-bath coupling,
574	>	The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
575	>	In other words, the propagator of this vector field is reversible if
576	>	and only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
577	>	conserved quantity of a general differential function is a function
578	>	$ G:R^{2d}  \to R^d $ which is constant for all solutions of the ODE
579	>	$\frac{{dx}}{{dt}} = f(x)$ ,
580		\[
581	<	\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
581	>	\frac{{dG(x(t))}}{{dt}} = 0.
582		\]
583	<	Completing the square,
583	>	Using the chain rule, one may obtain,
584		\[
585	<	H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
388	<	{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
389	<	w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
390	<	w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
391	<	1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
585	>	\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G,
586		\]
587	<	and putting it back into Eq.~\ref{introEquation:bathGLE},
587	>	which is the condition for conserved quantities. For a canonical
588	>	Hamiltonian system, the time evolution of an arbitrary smooth
589	>	function $G$ is given by,
590	>	\begin{eqnarray}
591	>	\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\
592	>	& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)).
593	>	\label{introEquation:firstIntegral1}
594	>	\end{eqnarray}
595	>	Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1}
596	>	can be rewritten as
597		\[
598	<	H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
396	<	{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
397	<	w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
398	<	w_\alpha ^2 }}x} \right)^2 } \right\}}
598	>	\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
599		\]
600	<	where
600	>	Therefore, the sufficient condition for $G$ to be a conserved
601	>	quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As
602	>	is well known, the Hamiltonian (or energy) H of a Hamiltonian system
603	>	is a conserved quantity, which is due to the fact $\{ H,H\}  = 0$.
604	>	When designing any numerical methods, one should always try to
605	>	preserve the structural properties of the original ODE and its
606	>	propagator.
607	>
608	>	\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
609	>	A lot of well established and very effective numerical methods have
610	>	been successful precisely because of their symplectic nature even
611	>	though this fact was not recognized when they were first
612	>	constructed. The most famous example is the Verlet-leapfrog method
613	>	in molecular dynamics. In general, symplectic integrators can be
614	>	constructed using one of four different methods.
615	>	\begin{enumerate}
616	>	\item Generating functions
617	>	\item Variational methods
618	>	\item Runge-Kutta methods
619	>	\item Splitting methods
620	>	\end{enumerate}
621	>	Generating functions\cite{Channell1990} tend to lead to methods
622	>	which are cumbersome and difficult to use. In dissipative systems,
623	>	variational methods can capture the decay of energy
624	>	accurately.\cite{Kane2000} Since they are geometrically unstable
625	>	against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
626	>	methods are not suitable for Hamiltonian
627	>	system.\cite{Cartwright1992} Recently, various high-order explicit
628	>	Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to
629	>	overcome this instability. However, due to computational penalty
630	>	involved in implementing the Runge-Kutta methods, they have not
631	>	attracted much attention from the Molecular Dynamics community.
632	>	Instead, splitting methods have been widely accepted since they
633	>	exploit natural decompositions of the system.\cite{McLachlan1998,
634	>	Tuckerman1992}
635	>
636	>	\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
637	>
638	>	The main idea behind splitting methods is to decompose the discrete
639	>	$\varphi_h$ as a composition of simpler propagators,
640	>	\begin{equation}
641	>	\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
642	>	\varphi _{h_n }
643	>	\label{introEquation:FlowDecomposition}
644	>	\end{equation}
645	>	where each of the sub-propagator is chosen such that each represent
646	>	a simpler integration of the system. Suppose that a Hamiltonian
647	>	system takes the form,
648		\[
649	<	W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
403	<	}}{{2m_\alpha  w_\alpha ^2 }}} x^2
649	>	H = H_1 + H_2.
650		\]
651	<	Since the first two terms of the new Hamiltonian depend only on the
652	<	system coordinates, we can get the equations of motion for
653	<	Generalized Langevin Dynamics by Hamilton's equations
654	<	\ref{introEquation:motionHamiltonianCoordinate,
655	<	introEquation:motionHamiltonianMomentum},
651	>	Here, $H_1$ and $H_2$ may represent different physical processes of
652	>	the system. For instance, they may relate to kinetic and potential
653	>	energy respectively, which is a natural decomposition of the
654	>	problem. If $H_1$ and $H_2$ can be integrated using exact
655	>	propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a
656	>	simple first order expression is then given by the Lie-Trotter
657	>	formula\cite{Trotter1959}
658	>	\begin{equation}
659	>	\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
660	>	\label{introEquation:firstOrderSplitting}
661	>	\end{equation}
662	>	where $\varphi _h$ is the result of applying the corresponding
663	>	continuous $\varphi _i$ over a time $h$. By definition, as
664	>	$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
665	>	must follow that each operator $\varphi_i(t)$ is a symplectic map.
666	>	It is easy to show that any composition of symplectic propagators
667	>	yields a symplectic map,
668	>	\begin{equation}
669	>	(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
670	>	'\phi ' = \phi '^T J\phi ' = J,
671	>	\label{introEquation:SymplecticFlowComposition}
672	>	\end{equation}
673	>	where $\phi$ and $\psi$ both are symplectic maps. Thus operator
674	>	splitting in this context automatically generates a symplectic map.
675	>	The Lie-Trotter
676	>	splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces
677	>	local errors proportional to $h^2$, while the Strang splitting gives
678	>	a second-order decomposition,\cite{Strang1968}
679	>	\begin{equation}
680	>	\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
681	>	_{1,h/2} , \label{introEquation:secondOrderSplitting}
682	>	\end{equation}
683	>	which has a local error proportional to $h^3$. The Strang
684	>	splitting's popularity in molecular simulation community attribute
685	>	to its symmetric property,
686	>	\begin{equation}
687	>	\varphi _h^{ - 1} = \varphi _{ - h}.
688	>	\label{introEquation:timeReversible}
689	>	\end{equation}
690	>
691	>	\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}}
692	>	The classical equation for a system consisting of interacting
693	>	particles can be written in Hamiltonian form,
694	>	\[
695	>	H = T + V
696	>	\]
697	>	where $T$ is the kinetic energy and $V$ is the potential energy.
698	>	Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one
699	>	obtains the following:
700		\begin{align}
701	<	\dot p &=  - \frac{{\partial H}}{{\partial x}}
702	<	&= m\ddot x
703	<	&= - \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)}
704	<	\label{introEq:Lp5}
701	>	q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
702	>	\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
703	>	\label{introEquation:Lp10a} \\%
704	>	%
705	>	\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
706	>	\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
707	>	\label{introEquation:Lp10b}
708		\end{align}
709	<	, and
709	>	where $F(t)$ is the force at time $t$. This integration scheme is
710	>	known as \emph{velocity verlet} which is
711	>	symplectic(Eq.~\ref{introEquation:SymplecticFlowComposition}),
712	>	time-reversible(Eq.~\ref{introEquation:timeReversible}) and
713	>	volume-preserving (Eq.~\ref{introEquation:volumePreserving}). These
714	>	geometric properties attribute to its long-time stability and its
715	>	popularity in the community. However, the most commonly used
716	>	velocity verlet integration scheme is written as below,
717		\begin{align}
718	<	\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
719	<	&= m\ddot x_\alpha
720	<	&= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
718	>	\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
719	>	\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
720	>	%
721	>	q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
722	>	\label{introEquation:Lp9b}\\%
723	>	%
724	>	\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
725	>	\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c}
726		\end{align}
727	+	From the preceding splitting, one can see that the integration of
728	+	the equations of motion would follow:
729	+	\begin{enumerate}
730	+	\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
731
732	<	\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
732	>	\item Use the half step velocities to move positions one whole step, $\Delta t$.
733
734	<	\[
426	<	L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
427	<	\]
734	>	\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move.
735
736	<	\[
737	<	L(x + y) = L(x) + L(y)
738	<	\]
736	>	\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
737	>	\end{enumerate}
738	>	By simply switching the order of the propagators in the splitting
739	>	and composing a new integrator, the \emph{position verlet}
740	>	integrator, can be generated,
741	>	\begin{align}
742	>	\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) +
743	>	\frac{{\Delta t}}{{2m}}\dot q(0)} \right], %
744	>	\label{introEquation:positionVerlet1} \\%
745	>	%
746	>	q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
747	>	q(\Delta t)} \right]. %
748	>	\label{introEquation:positionVerlet2}
749	>	\end{align}
750
751	<	\[
434	<	L(ax) = aL(x)
435	<	\]
751	>	\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
752
753	+	The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used
754	+	to determine the local error of a splitting method in terms of the
755	+	commutator of the
756	+	operators(Eq.~\ref{introEquation:exponentialOperator}) associated
757	+	with the sub-propagator. For operators $hX$ and $hY$ which are
758	+	associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we
759	+	have
760	+	\begin{equation}
761	+	\exp (hX + hY) = \exp (hZ)
762	+	\end{equation}
763	+	where
764	+	\begin{equation}
765	+	hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left(
766	+	{[X,[X,Y]] + [Y,[Y,X]]} \right) +  \ldots .
767	+	\end{equation}
768	+	Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by
769		\[
770	<	L(\dot x) = pL(x) - px(0)
770	>	[X,Y] = XY - YX .
771		\]
772	<
772	>	Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
773	>	to the Strang splitting, we can obtain
774	>	\begin{eqnarray*}
775	>	\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 \\
776	>	&   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
777	>	&   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots
778	>	).
779	>	\end{eqnarray*}
780	>	Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local
781	>	error of Strang splitting is proportional to $h^3$. The same
782	>	procedure can be applied to a general splitting of the form
783	>	\begin{equation}
784	>	\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
785	>	1} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
786	>	\end{equation}
787	>	A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher
788	>	order methods. Yoshida proposed an elegant way to compose higher
789	>	order methods based on symmetric splitting.\cite{Yoshida1990} Given
790	>	a symmetric second order base method $ \varphi _h^{(2)} $, a
791	>	fourth-order symmetric method can be constructed by composing,
792		\[
793	<	L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
793	>	\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
794	>	h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
795		\]
796	<
796	>	where $ \alpha  =  - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta
797	>	= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric
798	>	integrator $ \varphi _h^{(2n + 2)}$ can be composed by
799	>	\begin{equation}
800	>	\varphi _h^{(2n + 2)}  = \varphi _{\alpha h}^{(2n)}  \circ \varphi
801	>	_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)},
802	>	\end{equation}
803	>	if the weights are chosen as
804		\[
805	<	L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
805	>	\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
806	>	\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
807		\]
808
809	<	Some relatively important transformation,
450	<	\[
451	<	L(\cos at) = \frac{p}{{p^2  + a^2 }}
452	<	\]
809	>	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
810
811	+	As one of the principal tools of molecular modeling, Molecular
812	+	dynamics has proven to be a powerful tool for studying the functions
813	+	of biological systems, providing structural, thermodynamic and
814	+	dynamical information. The basic idea of molecular dynamics is that
815	+	macroscopic properties are related to microscopic behavior and
816	+	microscopic behavior can be calculated from the trajectories in
817	+	simulations. For instance, instantaneous temperature of a
818	+	Hamiltonian system of $N$ particles can be measured by
819		\[
820	<	L(\sin at) = \frac{a}{{p^2  + a^2 }}
820	>	T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
821		\]
822	<
823	<	\[
824	<	L(1) = \frac{1}{p}
460	<	\]
461	<
462	<	First, the bath coordinates,
463	<	\[
464	<	p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
465	<	_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
466	<	}}L(x)
467	<	\]
468	<	\[
469	<	L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
470	<	px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
471	<	\]
472	<	Then, the system coordinates,
473	<	\begin{align}
474	<	mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
475	<	\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{\frac{{g_\alpha
476	<	}}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha
477	<	(0)}}{{p^2  + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha
478	<	}}\omega _\alpha ^2 L(x)} \right\}}
479	<	%
480	<	&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
481	<	\sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x)
482	<	- \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0)
483	<	- \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}
484	<	\end{align}
485	<	Then, the inverse transform,
822	>	where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
823	>	respectively, $f$ is the number of degrees of freedom, and $k_B$ is
824	>	the Boltzman constant.
825
826	<	\begin{align}
827	<	m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
828	<	\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
829	<	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
830	<	_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
831	<	- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
832	<	(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
833	<	_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
834	<	%
835	<	&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
836	<	{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
837	<	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
838	<	t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
839	<	{\left[ {g_\alpha  x_\alpha  (0) - \frac{{g_\alpha  }}{{m_\alpha
840	<	\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
841	<	\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
842	<	(\omega _\alpha  t)} \right\}}
843	<	\end{align}
826	>	A typical molecular dynamics run consists of three essential steps:
827	>	\begin{enumerate}
828	>	\item Initialization
829	>	\begin{enumerate}
830	>	\item Preliminary preparation
831	>	\item Minimization
832	>	\item Heating
833	>	\item Equilibration
834	>	\end{enumerate}
835	>	\item Production
836	>	\item Analysis
837	>	\end{enumerate}
838	>	These three individual steps will be covered in the following
839	>	sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
840	>	initialization of a simulation. Sec.~\ref{introSection:production}
841	>	discusses issues of production runs.
842	>	Sec.~\ref{introSection:Analysis} provides the theoretical tools for
843	>	analysis of trajectories.
844
845	+	\subsection{\label{introSec:initialSystemSettings}Initialization}
846	+
847	+	\subsubsection{\textbf{Preliminary preparation}}
848	+
849	+	When selecting the starting structure of a molecule for molecular
850	+	simulation, one may retrieve its Cartesian coordinates from public
851	+	databases, such as RCSB Protein Data Bank \textit{etc}. Although
852	+	thousands of crystal structures of molecules are discovered every
853	+	year, many more remain unknown due to the difficulties of
854	+	purification and crystallization. Even for molecules with known
855	+	structures, some important information is missing. For example, a
856	+	missing hydrogen atom which acts as donor in hydrogen bonding must
857	+	be added. Moreover, in order to include electrostatic interactions,
858	+	one may need to specify the partial charges for individual atoms.
859	+	Under some circumstances, we may even need to prepare the system in
860	+	a special configuration. For instance, when studying transport
861	+	phenomenon in membrane systems, we may prepare the lipids in a
862	+	bilayer structure instead of placing lipids randomly in solvent,
863	+	since we are not interested in the slow self-aggregation process.
864	+
865	+	\subsubsection{\textbf{Minimization}}
866	+
867	+	It is quite possible that some of molecules in the system from
868	+	preliminary preparation may be overlapping with each other. This
869	+	close proximity leads to high initial potential energy which
870	+	consequently jeopardizes any molecular dynamics simulations. To
871	+	remove these steric overlaps, one typically performs energy
872	+	minimization to find a more reasonable conformation. Several energy
873	+	minimization methods have been developed to exploit the energy
874	+	surface and to locate the local minimum. While converging slowly
875	+	near the minimum, the steepest descent method is extremely robust when
876	+	systems are strongly anharmonic. Thus, it is often used to refine
877	+	structures from crystallographic data. Relying on the Hessian,
878	+	advanced methods like Newton-Raphson converge rapidly to a local
879	+	minimum, but become unstable if the energy surface is far from
880	+	quadratic. Another factor that must be taken into account, when
881	+	choosing energy minimization method, is the size of the system.
882	+	Steepest descent and conjugate gradient can deal with models of any
883	+	size. Because of the limits on computer memory to store the hessian
884	+	matrix and the computing power needed to diagonalize these matrices,
885	+	most Newton-Raphson methods can not be used with very large systems.
886	+
887	+	\subsubsection{\textbf{Heating}}
888	+
889	+	Typically, heating is performed by assigning random velocities
890	+	according to a Maxwell-Boltzman distribution for a desired
891	+	temperature. Beginning at a lower temperature and gradually
892	+	increasing the temperature by assigning larger random velocities, we
893	+	end up setting the temperature of the system to a final temperature
894	+	at which the simulation will be conducted. In the heating phase, we
895	+	should also keep the system from drifting or rotating as a whole. To
896	+	do this, the net linear momentum and angular momentum of the system
897	+	is shifted to zero after each resampling from the Maxwell -Boltzman
898	+	distribution.
899	+
900	+	\subsubsection{\textbf{Equilibration}}
901	+
902	+	The purpose of equilibration is to allow the system to evolve
903	+	spontaneously for a period of time and reach equilibrium. The
904	+	procedure is continued until various statistical properties, such as
905	+	temperature, pressure, energy, volume and other structural
906	+	properties \textit{etc}, become independent of time. Strictly
907	+	speaking, minimization and heating are not necessary, provided the
908	+	equilibration process is long enough. However, these steps can serve
909	+	as a mean to arrive at an equilibrated structure in an effective
910	+	way.
911	+
912	+	\subsection{\label{introSection:production}Production}
913	+
914	+	The production run is the most important step of the simulation, in
915	+	which the equilibrated structure is used as a starting point and the
916	+	motions of the molecules are collected for later analysis. In order
917	+	to capture the macroscopic properties of the system, the molecular
918	+	dynamics simulation must be performed by sampling correctly and
919	+	efficiently from the relevant thermodynamic ensemble.
920	+
921	+	The most expensive part of a molecular dynamics simulation is the
922	+	calculation of non-bonded forces, such as van der Waals force and
923	+	Coulombic forces \textit{etc}. For a system of $N$ particles, the
924	+	complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
925	+	which makes large simulations prohibitive in the absence of any
926	+	algorithmic tricks. A natural approach to avoid system size issues
927	+	is to represent the bulk behavior by a finite number of the
928	+	particles. However, this approach will suffer from surface effects
929	+	at the edges of the simulation. To offset this, \textit{Periodic
930	+	boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to
931	+	simulate bulk properties with a relatively small number of
932	+	particles. In this method, the simulation box is replicated
933	+	throughout space to form an infinite lattice. During the simulation,
934	+	when a particle moves in the primary cell, its image in other cells
935	+	move in exactly the same direction with exactly the same
936	+	orientation. Thus, as a particle leaves the primary cell, one of its
937	+	images will enter through the opposite face.
938	+	\begin{figure}
939	+	\centering
940	+	\includegraphics[width=\linewidth]{pbc.eps}
941	+	\caption[An illustration of periodic boundary conditions]{A 2-D
942	+	illustration of periodic boundary conditions. As one particle leaves
943	+	the left of the simulation box, an image of it enters the right.}
944	+	\label{introFig:pbc}
945	+	\end{figure}
946	+
947	+	%cutoff and minimum image convention
948	+	Another important technique to improve the efficiency of force
949	+	evaluation is to apply spherical cutoffs where particles farther
950	+	than a predetermined distance are not included in the
951	+	calculation.\cite{Frenkel1996} The use of a cutoff radius will cause
952	+	a discontinuity in the potential energy curve. Fortunately, one can
953	+	shift a simple radial potential to ensure the potential curve go
954	+	smoothly to zero at the cutoff radius. The cutoff strategy works
955	+	well for Lennard-Jones interaction because of its short range
956	+	nature. However, simply truncating the electrostatic interaction
957	+	with the use of cutoffs has been shown to lead to severe artifacts
958	+	in simulations. The Ewald summation, in which the slowly decaying
959	+	Coulomb potential is transformed into direct and reciprocal sums
960	+	with rapid and absolute convergence, has proved to minimize the
961	+	periodicity artifacts in liquid simulations. Taking advantage of
962	+	fast Fourier transform (FFT) techniques for calculating discrete
963	+	Fourier transforms, the particle mesh-based
964	+	methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
965	+	$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the
966	+	\emph{fast multipole method}\cite{Greengard1987, Greengard1994},
967	+	which treats Coulombic interactions exactly at short range, and
968	+	approximate the potential at long range through multipolar
969	+	expansion. In spite of their wide acceptance at the molecular
970	+	simulation community, these two methods are difficult to implement
971	+	correctly and efficiently. Instead, we use a damped and
972	+	charge-neutralized Coulomb potential method developed by Wolf and
973	+	his coworkers.\cite{Wolf1999} The shifted Coulomb potential for
974	+	particle $i$ and particle $j$ at distance $r_{rj}$ is given by:
975		\begin{equation}
976	<	m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
977	<	(t)\dot x(t - \tau )d\tau }  + R(t)
978	<	\label{introEuqation:GeneralizedLangevinDynamics}
976	>	V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
977	>	r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
978	>	R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
979	>	r_{ij})}{r_{ij}}\right\}, \label{introEquation:shiftedCoulomb}
980		\end{equation}
981	<	%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
982	<	%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
981	>	where $\alpha$ is the convergence parameter. Due to the lack of
982	>	inherent periodicity and rapid convergence,this method is extremely
983	>	efficient and easy to implement.
984	>	\begin{figure}
985	>	\centering
986	>	\includegraphics[width=\linewidth]{shifted_coulomb.eps}
987	>	\caption[An illustration of shifted Coulomb potential]{An
988	>	illustration of shifted Coulomb potential.}
989	>	\label{introFigure:shiftedCoulomb}
990	>	\end{figure}
991	>
992	>	%multiple time step
993	>
994	>	\subsection{\label{introSection:Analysis} Analysis}
995	>
996	>	Recently, advanced visualization techniques have been applied to
997	>	monitor the motions of molecules. Although the dynamics of the
998	>	system can be described qualitatively from animation, quantitative
999	>	trajectory analysis is more useful. According to the principles of
1000	>	Statistical Mechanics in
1001	>	Sec.~\ref{introSection:statisticalMechanics}, one can compute
1002	>	thermodynamic properties, analyze fluctuations of structural
1003	>	parameters, and investigate time-dependent processes of the molecule
1004	>	from the trajectories.
1005	>
1006	>	\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}
1007	>
1008	>	Thermodynamic properties, which can be expressed in terms of some
1009	>	function of the coordinates and momenta of all particles in the
1010	>	system, can be directly computed from molecular dynamics. The usual
1011	>	way to measure the pressure is based on virial theorem of Clausius
1012	>	which states that the virial is equal to $-3Nk_BT$. For a system
1013	>	with forces between particles, the total virial, $W$, contains the
1014	>	contribution from external pressure and interaction between the
1015	>	particles:
1016		\[
1017	<	\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
1018	<	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
1017	>	W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
1018	>	f_{ij} } } \right\rangle
1019		\]
1020	<	For an infinite harmonic bath, we can use the spectral density and
1021	<	an integral over frequencies.
1020	>	where $f_{ij}$ is the force between particle $i$ and $j$ at a
1021	>	distance $r_{ij}$. Thus, the expression for the pressure is given
1022	>	by:
1023	>	\begin{equation}
1024	>	P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
1025	>	< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
1026	>	\end{equation}
1027
1028	+	\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}
1029	+
1030	+	Structural Properties of a simple fluid can be described by a set of
1031	+	distribution functions. Among these functions,the \emph{pair
1032	+	distribution function}, also known as \emph{radial distribution
1033	+	function}, is of most fundamental importance to liquid theory.
1034	+	Experimentally, pair distribution functions can be gathered by
1035	+	Fourier transforming raw data from a series of neutron diffraction
1036	+	experiments and integrating over the surface
1037	+	factor.\cite{Powles1973} The experimental results can serve as a
1038	+	criterion to justify the correctness of a liquid model. Moreover,
1039	+	various equilibrium thermodynamic and structural properties can also
1040	+	be expressed in terms of the radial distribution
1041	+	function.\cite{Allen1987} The pair distribution functions $g(r)$
1042	+	gives the probability that a particle $i$ will be located at a
1043	+	distance $r$ from a another particle $j$ in the system
1044	+	\begin{equation}
1045	+	g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
1046	+	\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
1047	+	(r)}{\rho}.
1048	+	\end{equation}
1049	+	Note that the delta function can be replaced by a histogram in
1050	+	computer simulation. Peaks in $g(r)$ represent solvent shells, and
1051	+	the height of these peaks gradually decreases to 1 as the liquid of
1052	+	large distance approaches the bulk density.
1053	+
1054	+
1055	+	\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
1056	+	Properties}}
1057	+
1058	+	Time-dependent properties are usually calculated using \emph{time
1059	+	correlation functions}, which correlate random variables $A$ and $B$
1060	+	at two different times,
1061	+	\begin{equation}
1062	+	C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
1063	+	\label{introEquation:timeCorrelationFunction}
1064	+	\end{equation}
1065	+	If $A$ and $B$ refer to same variable, this kind of correlation
1066	+	functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation
1067	+	function which is directly related to transport properties of
1068	+	molecular liquids:
1069	+	\begin{equation}
1070	+	D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
1071	+	\right\rangle } dt
1072	+	\end{equation}
1073	+	where $D$ is diffusion constant. Unlike the velocity autocorrelation
1074	+	function, which is averaged over time origins and over all the
1075	+	atoms, the dipole autocorrelation functions is calculated for the
1076	+	entire system. The dipole autocorrelation function is given by:
1077	+	\begin{equation}
1078	+	c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
1079	+	\right\rangle
1080	+	\end{equation}
1081	+	Here $u_{tot}$ is the net dipole of the entire system and is given
1082	+	by
1083	+	\begin{equation}
1084	+	u_{tot} (t) = \sum\limits_i {u_i (t)}.
1085	+	\end{equation}
1086	+	In principle, many time correlation functions can be related to
1087	+	Fourier transforms of the infrared, Raman, and inelastic neutron
1088	+	scattering spectra of molecular liquids. In practice, one can
1089	+	extract the IR spectrum from the intensity of the molecular dipole
1090	+	fluctuation at each frequency using the following relationship:
1091	+	\begin{equation}
1092	+	\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
1093	+	i2\pi vt} dt}.
1094	+	\end{equation}
1095	+
1096	+	\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
1097	+
1098	+	Rigid bodies are frequently involved in the modeling of different
1099	+	areas, including engineering, physics and chemistry. For example,
1100	+	missiles and vehicles are usually modeled by rigid bodies.  The
1101	+	movement of the objects in 3D gaming engines or other physics
1102	+	simulators is governed by rigid body dynamics. In molecular
1103	+	simulations, rigid bodies are used to simplify protein-protein
1104	+	docking studies.\cite{Gray2003}
1105	+
1106	+	It is very important to develop stable and efficient methods to
1107	+	integrate the equations of motion for orientational degrees of
1108	+	freedom. Euler angles are the natural choice to describe the
1109	+	rotational degrees of freedom. However, due to $\frac {1}{sin
1110	+	\theta}$ singularities, the numerical integration of corresponding
1111	+	equations of these motion is very inefficient and inaccurate.
1112	+	Although an alternative integrator using multiple sets of Euler
1113	+	angles can overcome this difficulty\cite{Barojas1973}, the
1114	+	computational penalty and the loss of angular momentum conservation
1115	+	still remain. A singularity-free representation utilizing
1116	+	quaternions was developed by Evans in 1977.\cite{Evans1977}
1117	+	Unfortunately, this approach used a nonseparable Hamiltonian
1118	+	resulting from the quaternion representation, which prevented the
1119	+	symplectic algorithm from being utilized. Another different approach
1120	+	is to apply holonomic constraints to the atoms belonging to the
1121	+	rigid body. Each atom moves independently under the normal forces
1122	+	deriving from potential energy and constraint forces which are used
1123	+	to guarantee the rigidness. However, due to their iterative nature,
1124	+	the SHAKE and Rattle algorithms also converge very slowly when the
1125	+	number of constraints increases.\cite{Ryckaert1977, Andersen1983}
1126	+
1127	+	A break-through in geometric literature suggests that, in order to
1128	+	develop a long-term integration scheme, one should preserve the
1129	+	symplectic structure of the propagator. By introducing a conjugate
1130	+	momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's
1131	+	equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was
1132	+	proposed to evolve the Hamiltonian system in a constraint manifold
1133	+	by iteratively satisfying the orthogonality constraint $Q^T Q = 1$.
1134	+	An alternative method using the quaternion representation was
1135	+	developed by Omelyan.\cite{Omelyan1998} However, both of these
1136	+	methods are iterative and inefficient. In this section, we descibe a
1137	+	symplectic Lie-Poisson integrator for rigid bodies developed by
1138	+	Dullweber and his coworkers\cite{Dullweber1997} in depth.
1139	+
1140	+	\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
1141	+	The Hamiltonian of a rigid body is given by
1142	+	\begin{equation}
1143	+	H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
1144	+	V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
1145	+	\label{introEquation:RBHamiltonian}
1146	+	\end{equation}
1147	+	Here, $q$ and $Q$  are the position vector and rotation matrix for
1148	+	the rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ ,
1149	+	and $J$, a diagonal matrix, is defined by
1150		\[
1151	<	R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
522	<	- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
523	<	\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
524	<	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
1151	>	I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
1152		\]
1153	<	The random forces depend only on initial conditions.
1154	<
1155	<	\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
1156	<	So we can define a new set of coordinates,
1153	>	where $I_{ii}$ is the diagonal element of the inertia tensor. This
1154	>	constrained Hamiltonian equation is subjected to a holonomic
1155	>	constraint,
1156	>	\begin{equation}
1157	>	Q^T Q = 1, \label{introEquation:orthogonalConstraint}
1158	>	\end{equation}
1159	>	which is used to ensure the rotation matrix's unitarity. Using
1160	>	Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~
1161	>	\ref{introEquation:motionHamiltonianMomentum}, one can write down
1162	>	the equations of motion,
1163	>	\begin{eqnarray}
1164	>	\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\
1165	>	\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\
1166	>	\frac{{dQ}}{{dt}} & = & PJ^{ - 1},  \label{introEquation:RBMotionRotation}\\
1167	>	\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
1168	>	\end{eqnarray}
1169	>	Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and
1170	>	using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain,
1171	>	\begin{equation}
1172	>	Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1173	>	\label{introEquation:RBFirstOrderConstraint}
1174	>	\end{equation}
1175	>	In general, there are two ways to satisfy the holonomic constraints.
1176	>	We can use a constraint force provided by a Lagrange multiplier on
1177	>	the normal manifold to keep the motion on the constraint space. Or
1178	>	we can simply evolve the system on the constraint manifold. These
1179	>	two methods have been proved to be equivalent. The holonomic
1180	>	constraint and equations of motions define a constraint manifold for
1181	>	rigid bodies
1182		\[
1183	<	q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
1184	<	^2 }}x(0)
1183	>	M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
1184	>	\right\}.
1185		\]
1186	<	This makes
1186	>	Unfortunately, this constraint manifold is not $T^* SO(3)$ which is
1187	>	a symplectic manifold on Lie rotation group $SO(3)$. However, it
1188	>	turns out that under symplectic transformation, the cotangent space
1189	>	and the phase space are diffeomorphic. By introducing
1190		\[
1191	<	R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
1191	>	\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
1192		\]
1193	<	And since the $q$ coordinates are harmonic oscillators,
1193	>	the mechanical system subjected to a holonomic constraint manifold $M$
1194	>	can be re-formulated as a Hamiltonian system on the cotangent space
1195		\[
1196	<	\begin{array}{l}
1197	<	\left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
1198	<	\left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
1196	>	T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
1197	>	1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
1198	>	\]
1199	>	For a body fixed vector $X_i$ with respect to the center of mass of
1200	>	the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
1201	>	given as
1202	>	\begin{equation}
1203	>	X_i^{lab} = Q X_i + q.
1204	>	\end{equation}
1205	>	Therefore, potential energy $V(q,Q)$ is defined by
1206	>	\[
1207	>	V(q,Q) = V(Q X_0 + q).
1208	>	\]
1209	>	Hence, the force and torque are given by
1210	>	\[
1211	>	\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
1212	>	\]
1213	>	and
1214	>	\[
1215	>	\nabla _Q V(q,Q) = F(q,Q)X_i^t
1216	>	\]
1217	>	respectively. As a common choice to describe the rotation dynamics
1218	>	of the rigid body, the angular momentum on the body fixed frame $\Pi
1219	>	= Q^t P$ is introduced to rewrite the equations of motion,
1220	>	\begin{equation}
1221	>	\begin{array}{l}
1222	>	\dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
1223	>	\dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
1224		\end{array}
1225	+	\label{introEqaution:RBMotionPI}
1226	+	\end{equation}
1227	+	as well as holonomic constraints $\Pi J^{ - 1}  + J^{ - 1} \Pi ^t  =
1228	+	0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a
1229	+	matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism,
1230	+	\begin{equation}
1231	+	v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
1232	+	{\begin{array}{*{20}c}
1233	+	0 & { - v_3 } & {v_2 }  \\
1234	+	{v_3 } & 0 & { - v_1 }  \\
1235	+	{ - v_2 } & {v_1 } & 0  \\
1236	+	\end{array}} \right),
1237	+	\label{introEquation:hatmapIsomorphism}
1238	+	\end{equation}
1239	+	will let us associate the matrix products with traditional vector
1240	+	operations
1241	+	\[
1242	+	\hat vu = v \times u.
1243		\]
1244	+	Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
1245	+	matrix,
1246	+	\begin{eqnarray}
1247	+	(\dot \Pi  - \dot \Pi ^T )&= &(\Pi  - \Pi ^T )(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
1248	+	& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  -
1249	+	(\Lambda  - \Lambda ^T ). \label{introEquation:skewMatrixPI}
1250	+	\end{eqnarray}
1251	+	Since $\Lambda$ is symmetric, the last term of
1252	+	Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
1253	+	Lagrange multiplier $\Lambda$ is absent from the equations of
1254	+	motion. This unique property eliminates the requirement of
1255	+	iterations which can not be avoided in other methods.\cite{Kol1997,
1256	+	Omelyan1998} Applying the hat-map isomorphism, we obtain the
1257	+	equation of motion for angular momentum in the body frame
1258	+	\begin{equation}
1259	+	\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
1260	+	F_i (r,Q)} \right) \times X_i }.
1261	+	\label{introEquation:bodyAngularMotion}
1262	+	\end{equation}
1263	+	In the same manner, the equation of motion for rotation matrix is
1264	+	given by
1265	+	\[
1266	+	\dot Q = Qskew(I^{ - 1} \pi ).
1267	+	\]
1268
1269	<	\begin{align}
1270	<	\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
1271	<	{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
1272	<	(t)q_\beta  (0)} \right\rangle } }
1269	>	\subsection{\label{introSection:SymplecticFreeRB}Symplectic
1270	>	Lie-Poisson Integrator for Free Rigid Bodies}
1271	>
1272	>	If there are no external forces exerted on the rigid body, the only
1273	>	contribution to the rotational motion is from the kinetic energy
1274	>	(the first term of \ref{introEquation:bodyAngularMotion}). The free
1275	>	rigid body is an example of a Lie-Poisson system with Hamiltonian
1276	>	function
1277	>	\begin{equation}
1278	>	T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
1279	>	\label{introEquation:rotationalKineticRB}
1280	>	\end{equation}
1281	>	where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
1282	>	Lie-Poisson structure matrix,
1283	>	\begin{equation}
1284	>	J(\pi ) = \left( {\begin{array}{*{20}c}
1285	>	0 & {\pi _3 } & { - \pi _2 }  \\
1286	>	{ - \pi _3 } & 0 & {\pi _1 }  \\
1287	>	{\pi _2 } & { - \pi _1 } & 0  \\
1288	>	\end{array}} \right).
1289	>	\end{equation}
1290	>	Thus, the dynamics of free rigid body is governed by
1291	>	\begin{equation}
1292	>	\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
1293	>	\end{equation}
1294	>	One may notice that each $T_i^r$ in
1295	>	Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly.
1296	>	For instance, the equations of motion due to $T_1^r$ are given by
1297	>	\begin{equation}
1298	>	\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
1299	>	\label{introEqaution:RBMotionSingleTerm}
1300	>	\end{equation}
1301	>	with
1302	>	\[ R_1  = \left( {\begin{array}{*{20}c}
1303	>	0 & 0 & 0  \\
1304	>	0 & 0 & {\pi _1 }  \\
1305	>	0 & { - \pi _1 } & 0  \\
1306	>	\end{array}} \right).
1307	>	\]
1308	>	The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is
1309	>	\[
1310	>	\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
1311	>	Q(0)e^{\Delta tR_1 }
1312	>	\]
1313	>	with
1314	>	\[
1315	>	e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
1316	>	0 & 0 & 0  \\
1317	>	0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
1318	>	0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
1319	>	\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
1320	>	\]
1321	>	To reduce the cost of computing expensive functions in $e^{\Delta
1322	>	tR_1 }$, we can use the Cayley transformation to obtain a
1323	>	single-aixs propagator,
1324	>	\begin{eqnarray*}
1325	>	e^{\Delta tR_1 }  & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta
1326	>	tR_1 ) \\
1327		%
1328	<	&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
1329	<	\right\rangle \cos (\omega _\alpha  t)}
1330	<	%
1331	<	&= kT\xi (t)
1332	<	\end{align}
1328	>	& \approx & \left( \begin{array}{ccc}
1329	>	1 & 0 & 0 \\
1330	>	0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4}  & -\frac{\theta}{1+
1331	>	\theta^2 / 4} \\
1332	>	0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 +
1333	>	\theta^2 / 4}
1334	>	\end{array}
1335	>	\right).
1336	>	\end{eqnarray*}
1337	>	The propagators for $T_2^r$ and $T_3^r$ can be found in the same
1338	>	manner. In order to construct a second-order symplectic method, we
1339	>	split the angular kinetic Hamiltonian function into five terms
1340	>	\[
1341	>	T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
1342	>	) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
1343	>	(\pi _1 ).
1344	>	\]
1345	>	By concatenating the propagators corresponding to these five terms,
1346	>	we can obtain an symplectic integrator,
1347	>	\[
1348	>	\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
1349	>	\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
1350	>	\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1351	>	_1 }.
1352	>	\]
1353	>	The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by
1354	>	\[
1355	>	\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
1356	>	).
1357	>	\]
1358	>	If the Poisson bracket of a function $F$ with an arbitrary smooth
1359	>	function $G$ is zero, $F$ is a \emph{Casimir}, which is the
1360	>	conserved quantity in Poisson system. We can easily verify that the
1361	>	norm of the angular momentum, $\parallel \pi
1362	>	\parallel$, is a \emph{Casimir}.\cite{McLachlan1993} Let $F(\pi ) = S(\frac{{\parallel
1363	>	\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
1364	>	then by the chain rule
1365	>	\[
1366	>	\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
1367	>	}}{2})\pi.
1368	>	\]
1369	>	Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
1370	>	\pi
1371	>	\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
1372	>	Lie-Poisson integrator is found to be both extremely efficient and
1373	>	stable. These properties can be explained by the fact the small
1374	>	angle approximation is used and the norm of the angular momentum is
1375	>	conserved.
1376
1377	+	\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
1378	+	Splitting for Rigid Body}
1379	+
1380	+	The Hamiltonian of rigid body can be separated in terms of kinetic
1381	+	energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations
1382	+	of motion corresponding to potential energy and kinetic energy are
1383	+	listed in Table~\ref{introTable:rbEquations}.
1384	+	\begin{table}
1385	+	\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
1386	+	\label{introTable:rbEquations}
1387	+	\begin{center}
1388	+	\begin{tabular}{|l|l|}
1389	+	\hline
1390	+	% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
1391	+	Potential & Kinetic \\
1392	+	$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
1393	+	$\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
1394	+	$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
1395	+	$ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
1396	+	\hline
1397	+	\end{tabular}
1398	+	\end{center}
1399	+	\end{table}
1400	+	A second-order symplectic method is now obtained by the composition
1401	+	of the position and velocity propagators,
1402	+	\[
1403	+	\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
1404	+	_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
1405	+	\]
1406	+	Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
1407	+	sub-propagators which corresponding to force and torque
1408	+	respectively,
1409	+	\[
1410	+	\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
1411	+	_{\Delta t/2,\tau }.
1412	+	\]
1413	+	Since the associated operators of $\varphi _{\Delta t/2,F} $ and
1414	+	$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order
1415	+	inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the
1416	+	kinetic energy can be separated to translational kinetic term, $T^t
1417	+	(p)$, and rotational kinetic term, $T^r (\pi )$,
1418		\begin{equation}
1419	<	\xi (t) = \left\langle {R(t)R(0)} \right\rangle
559	<	\label{introEquation:secondFluctuationDissipation}
1419	>	T(p,\pi ) =T^t (p) + T^r (\pi ).
1420		\end{equation}
1421	+	where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
1422	+	defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore,
1423	+	the corresponding propagators are given by
1424	+	\[
1425	+	\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
1426	+	_{\Delta t,T^r }.
1427	+	\]
1428	+	Finally, we obtain the overall symplectic propagators for freely
1429	+	moving rigid bodies
1430	+	\begin{eqnarray}
1431	+	\varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \notag\\
1432	+	& & \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\\
1433	+	& & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .
1434	+	\label{introEquation:overallRBFlowMaps}
1435	+	\end{eqnarray}
1436
1437	<	\section{\label{introSection:hydroynamics}Hydrodynamics}
1437	>	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
1438	>	As an alternative to newtonian dynamics, Langevin dynamics, which
1439	>	mimics a simple heat bath with stochastic and dissipative forces,
1440	>	has been applied in a variety of studies. This section will review
1441	>	the theory of Langevin dynamics. A brief derivation of the generalized
1442	>	Langevin equation will be given first. Following that, we will
1443	>	discuss the physical meaning of the terms appearing in the equation.
1444
1445	<	\subsection{\label{introSection:frictionTensor} Friction Tensor}
565	<	\subsection{\label{introSection:analyticalApproach}Analytical
566	<	Approach}
1445	>	\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
1446
1447	<	\subsection{\label{introSection:approximationApproach}Approximation
1448	<	Approach}
1447	>	A harmonic bath model, in which an effective set of harmonic
1448	>	oscillators are used to mimic the effect of a linearly responding
1449	>	environment, has been widely used in quantum chemistry and
1450	>	statistical mechanics. One of the successful applications of
1451	>	Harmonic bath model is the derivation of the Generalized Langevin
1452	>	Dynamics (GLE). Consider a system, in which the degree of
1453	>	freedom $x$ is assumed to couple to the bath linearly, giving a
1454	>	Hamiltonian of the form
1455	>	\begin{equation}
1456	>	H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
1457	>	\label{introEquation:bathGLE}.
1458	>	\end{equation}
1459	>	Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated
1460	>	with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian,
1461	>	\[
1462	>	H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
1463	>	}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  x_\alpha ^2 }
1464	>	\right\}}
1465	>	\]
1466	>	where the index $\alpha$ runs over all the bath degrees of freedom,
1467	>	$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
1468	>	the harmonic bath masses, and $\Delta U$ is a bilinear system-bath
1469	>	coupling,
1470	>	\[
1471	>	\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
1472	>	\]
1473	>	where $g_\alpha$ are the coupling constants between the bath
1474	>	coordinates ($x_ \alpha$) and the system coordinate ($x$).
1475	>	Introducing
1476	>	\[
1477	>	W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
1478	>	}}{{2m_\alpha  w_\alpha ^2 }}} x^2
1479	>	\]
1480	>	and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
1481	>	\[
1482	>	H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
1483	>	{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
1484	>	w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
1485	>	w_\alpha ^2 }}x} \right)^2 } \right\}}.
1486	>	\]
1487	>	Since the first two terms of the new Hamiltonian depend only on the
1488	>	system coordinates, we can get the equations of motion for
1489	>	Generalized Langevin Dynamics by Hamilton's equations,
1490	>	\begin{equation}
1491	>	m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
1492	>	\sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   -
1493	>	\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)},
1494	>	\label{introEquation:coorMotionGLE}
1495	>	\end{equation}
1496	>	and
1497	>	\begin{equation}
1498	>	m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x_\alpha   -
1499	>	\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
1500	>	\label{introEquation:bathMotionGLE}
1501	>	\end{equation}
1502	>	In order to derive an equation for $x$, the dynamics of the bath
1503	>	variables $x_\alpha$ must be solved exactly first. As an integral
1504	>	transform which is particularly useful in solving linear ordinary
1505	>	differential equations,the Laplace transform is the appropriate tool
1506	>	to solve this problem. The basic idea is to transform the difficult
1507	>	differential equations into simple algebra problems which can be
1508	>	solved easily. Then, by applying the inverse Laplace transform, we
1509	>	can retrieve the solutions of the original problems. Let $f(t)$ be a
1510	>	function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$
1511	>	is a new function defined as
1512	>	\[
1513	>	L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
1514	>	\]
1515	>	where  $p$ is real and  $L$ is called the Laplace Transform
1516	>	Operator. Below are some important properties of the Laplace transform
1517	>	\begin{eqnarray*}
1518	>	L(x + y)  & = & L(x) + L(y) \\
1519	>	L(ax)     & = & aL(x) \\
1520	>	L(\dot x) & = & pL(x) - px(0) \\
1521	>	L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
1522	>	L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
1523	>	\end{eqnarray*}
1524	>	Applying the Laplace transform to the bath coordinates, we obtain
1525	>	\begin{eqnarray*}
1526	>	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), \\
1527	>	L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}. \\
1528	>	\end{eqnarray*}
1529	>	In the same way, the system coordinates become
1530	>	\begin{eqnarray*}
1531	>	mL(\ddot x) & = &
1532	>	- \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\}}  \\
1533	>	& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}.
1534	>	\end{eqnarray*}
1535	>	With the help of some relatively important inverse Laplace
1536	>	transformations:
1537	>	\[
1538	>	\begin{array}{c}
1539	>	L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
1540	>	L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
1541	>	L(1) = \frac{1}{p} \\
1542	>	\end{array}
1543	>	\]
1544	>	we obtain
1545	>	\begin{eqnarray*}
1546	>	m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
1547	>	\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
1548	>	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
1549	>	_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
1550	>	& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
1551	>	x_\alpha (0) - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}}
1552	>	\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
1553	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}\\
1554	>	%
1555	>	& = & -
1556	>	\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha
1557	>	= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha
1558	>	^2 }}} \right)\cos (\omega _\alpha
1559	>	t)\dot x(t - \tau )d} \tau }  \\
1560	>	& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
1561	>	x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha  }}}
1562	>	\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
1563	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
1564	>	\end{eqnarray*}
1565	>	Introducing a \emph{dynamic friction kernel}
1566	>	\begin{equation}
1567	>	\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
1568	>	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
1569	>	\label{introEquation:dynamicFrictionKernelDefinition}
1570	>	\end{equation}
1571	>	and \emph{a random force}
1572	>	\begin{equation}
1573	>	R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
1574	>	- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
1575	>	\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
1576	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
1577	>	\label{introEquation:randomForceDefinition}
1578	>	\end{equation}
1579	>	the equation of motion can be rewritten as
1580	>	\begin{equation}
1581	>	m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
1582	>	(t)\dot x(t - \tau )d\tau }  + R(t)
1583	>	\label{introEuqation:GeneralizedLangevinDynamics}
1584	>	\end{equation}
1585	>	which is known as the \emph{generalized Langevin equation} (GLE).
1586
1587	<	\subsection{\label{introSection:centersRigidBody}Centers of Rigid
1588	<	Body}
1587	>	\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
1588	>
1589	>	One may notice that $R(t)$ depends only on initial conditions, which
1590	>	implies it is completely deterministic within the context of a
1591	>	harmonic bath. However, it is easy to verify that $R(t)$ is totally
1592	>	uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)}
1593	>	\right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
1594	>	0.$ This property is what we expect from a truly random process. As
1595	>	long as the model chosen for $R(t)$ was a gaussian distribution in
1596	>	general, the stochastic nature of the GLE still remains.
1597	>	%dynamic friction kernel
1598	>	The convolution integral
1599	>	\[
1600	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
1601	>	\]
1602	>	depends on the entire history of the evolution of $x$, which implies
1603	>	that the bath retains memory of previous motions. In other words,
1604	>	the bath requires a finite time to respond to change in the motion
1605	>	of the system. For a sluggish bath which responds slowly to changes
1606	>	in the system coordinate, we may regard $\xi(t)$ as a constant
1607	>	$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
1608	>	\[
1609	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
1610	>	\]
1611	>	and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1612	>	\[
1613	>	m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
1614	>	\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
1615	>	\]
1616	>	which can be used to describe the effect of dynamic caging in
1617	>	viscous solvents. The other extreme is the bath that responds
1618	>	infinitely quickly to motions in the system. Thus, $\xi (t)$ can be
1619	>	taken as a $delta$ function in time:
1620	>	\[
1621	>	\xi (t) = 2\xi _0 \delta (t).
1622	>	\]
1623	>	Hence, the convolution integral becomes
1624	>	\[
1625	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
1626	>	{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
1627	>	\]
1628	>	and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1629	>	\begin{equation}
1630	>	m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
1631	>	x(t) + R(t) \label{introEquation:LangevinEquation}
1632	>	\end{equation}
1633	>	which is known as the Langevin equation. The static friction
1634	>	coefficient $\xi _0$ can either be calculated from spectral density
1635	>	or be determined by Stokes' law for regular shaped particles. A
1636	>	brief review on calculating friction tensors for arbitrary shaped
1637	>	particles is given in Sec.~\ref{introSection:frictionTensor}.
1638	>
1639	>	\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
1640	>
1641	>	Defining a new set of coordinates
1642	>	\[
1643	>	q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
1644	>	^2 }}x(0),
1645	>	\]
1646	>	we can rewrite $R(t)$ as
1647	>	\[
1648	>	R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
1649	>	\]
1650	>	And since the $q$ coordinates are harmonic oscillators,
1651	>	\begin{eqnarray*}
1652	>	\left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
1653	>	\left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
1654	>	\left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
1655	>	\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 } }  \\
1656	>	& = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
1657	>	& = &kT\xi (t)
1658	>	\end{eqnarray*}
1659	>	Thus, we recover the \emph{second fluctuation dissipation theorem}
1660	>	\begin{equation}
1661	>	\xi (t) = \left\langle {R(t)R(0)} \right\rangle
1662	>	\label{introEquation:secondFluctuationDissipation},
1663	>	\end{equation}
1664	>	which acts as a constraint on the possible ways in which one can
1665	>	model the random force and friction kernel.

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