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#	Line 3 | Line 3 | Mechanics}
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 | The torque $\tau$ with respect to the same origin is d
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 bodies
71	>	\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 | classical mechanics. If the potential energy of a syst
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 | where $ \dot q_1  \ldots \dot q_f $ are generalized ve
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 theorem 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 conditionals and move the objects in the direction governed
443	>	by the differential equations. However, most of them ignore the
444	>	hidden physical laws contained within the equations. Since 1990,
445	>	geometric integrators, which preserve various phase-flow invariants
446	>	such as symplectic structure, volume and time reversal symmetry,
447	>	were developed to address this issue\cite{Dullweber1997,
448	>	McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which
449	>	happens to be a simple example of symplectic integrator, continues
450	>	to gain popularity in the molecular dynamics community. This fact
451	>	can be 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
303	<	\emph{differentiable manifold}. A \emph{symplectic manifold} is
304	<	defined as a pair $(M, \omega)$ which consisting of a
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-degenerated,
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 $x = x(q,p)^T$, 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	>	The most obvious change being that matrix $J$ now depends on $x$.
504
505	<	\subsection{\label{introSection:exactFlow}The Exact Flow of ODE}
505	>	\subsection{\label{introSection:exactFlow}Exact Propagator}
506
507	<	\subsection{\label{introSection:hamiltonianSplitting}Hamiltonian Splitting}
507	>	Let $x(t)$ be the exact solution of the ODE
508	>	system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can
509	>	define its exact propagator(solution) $\varphi_\tau$
510	>	\[ x(t+\tau)
511	>	=\varphi_\tau(x(t))
512	>	\]
513	>	where $\tau$ is a fixed time step and $\varphi$ is a map from phase
514	>	space to itself. The propagator has the continuous group property,
515	>	\begin{equation}
516	>	\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
517	>	+ \tau _2 } .
518	>	\end{equation}
519	>	In particular,
520	>	\begin{equation}
521	>	\varphi _\tau   \circ \varphi _{ - \tau }  = I
522	>	\end{equation}
523	>	Therefore, the exact propagator is self-adjoint,
524	>	\begin{equation}
525	>	\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
526	>	\end{equation}
527	>	The exact propagator can also be written in terms of operator,
528	>	\begin{equation}
529	>	\varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
530	>	}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
531	>	\label{introEquation:exponentialOperator}
532	>	\end{equation}
533	>	In most cases, it is not easy to find the exact propagator
534	>	$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$,
535	>	which is usually called an integrator. The order of an integrator
536	>	$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
537	>	order $p$,
538	>	\begin{equation}
539	>	\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1})
540	>	\end{equation}
541
542	<	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
542	>	\subsection{\label{introSection:geometricProperties}Geometric Properties}
543
544	<	As a special discipline of molecular modeling, Molecular dynamics
545	<	has proven to be a powerful tool for studying the functions of
546	<	biological systems, providing structural, thermodynamic and
547	<	dynamical information.
548	<
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	<
544	>	The hidden geometric properties\cite{Budd1999, Marsden1998} of an
545	>	ODE and its propagator play important roles in numerical studies.
546	>	Many of them can be found in systems which occur naturally in
547	>	applications. Let $\varphi$ be the propagator of Hamiltonian vector
548	>	field, $\varphi$ is a \emph{symplectic} propagator if it satisfies,
549		\begin{equation}
550	<	H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
374	<	\label{introEquation:bathGLE}
550	>	{\varphi '}^T J \varphi ' = J.
551		\end{equation}
552	<	where $H_B$ is harmonic bath Hamiltonian,
552	>	According to Liouville's theorem, the symplectic volume is invariant
553	>	under a Hamiltonian propagator, which is the basis for classical
554	>	statistical mechanics. Furthermore, the propagator of a Hamiltonian
555	>	vector field on a symplectic manifold can be shown to be a
556	>	symplectomorphism. As to the Poisson system,
557	>	\begin{equation}
558	>	{\varphi '}^T J \varphi ' = J \circ \varphi
559	>	\end{equation}
560	>	is the property that must be preserved by the integrator. It is
561	>	possible to construct a \emph{volume-preserving} propagator for a
562	>	source free ODE ($ \nabla \cdot f = 0 $), if the propagator
563	>	satisfies $ \det d\varphi  = 1$. One can show easily that a
564	>	symplectic propagator will be volume-preserving. Changing the
565	>	variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will
566	>	result in a new system,
567		\[
568	<	H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
379	<	}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
568	>	\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
569		\]
570	<	and $\Delta U$ is bilinear system-bath coupling,
570	>	The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
571	>	In other words, the propagator of this vector field is reversible if
572	>	and only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
573	>	conserved quantity of a general differential function is a function
574	>	$ G:R^{2d}  \to R^d $ which is constant for all solutions of the ODE
575	>	$\frac{{dx}}{{dt}} = f(x)$ ,
576		\[
577	<	\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
577	>	\frac{{dG(x(t))}}{{dt}} = 0.
578		\]
579	<	Completing the square,
579	>	Using the chain rule, one may obtain,
580		\[
581	<	H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
388	<	{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
389	<	w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
390	<	w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
391	<	1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
581	>	\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G,
582		\]
583	<	and putting it back into Eq.~\ref{introEquation:bathGLE},
583	>	which is the condition for conserved quantities. For a canonical
584	>	Hamiltonian system, the time evolution of an arbitrary smooth
585	>	function $G$ is given by,
586	>	\begin{eqnarray}
587	>	\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\
588	>	& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)).
589	>	\label{introEquation:firstIntegral1}
590	>	\end{eqnarray}
591	>	Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1}
592	>	can be rewritten as
593		\[
594	<	H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
396	<	{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
397	<	w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
398	<	w_\alpha ^2 }}x} \right)^2 } \right\}}
594	>	\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
595		\]
596	<	where
596	>	Therefore, the sufficient condition for $G$ to be a conserved
597	>	quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As
598	>	is well known, the Hamiltonian (or energy) H of a Hamiltonian system
599	>	is a conserved quantity, which is due to the fact $\{ H,H\}  = 0$.
600	>	When designing any numerical methods, one should always try to
601	>	preserve the structural properties of the original ODE and its
602	>	propagator.
603	>
604	>	\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
605	>	A lot of well established and very effective numerical methods have
606	>	been successful precisely because of their symplectic nature even
607	>	though this fact was not recognized when they were first
608	>	constructed. The most famous example is the Verlet-leapfrog method
609	>	in molecular dynamics. In general, symplectic integrators can be
610	>	constructed using one of four different methods.
611	>	\begin{enumerate}
612	>	\item Generating functions
613	>	\item Variational methods
614	>	\item Runge-Kutta methods
615	>	\item Splitting methods
616	>	\end{enumerate}
617	>	Generating functions\cite{Channell1990} tend to lead to methods
618	>	which are cumbersome and difficult to use. In dissipative systems,
619	>	variational methods can capture the decay of energy
620	>	accurately\cite{Kane2000}. Since they are geometrically unstable
621	>	against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
622	>	methods are not suitable for Hamiltonian system. Recently, various
623	>	high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003}
624	>	have been developed to overcome this instability. However, due to
625	>	computational penalty involved in implementing the Runge-Kutta
626	>	methods, they have not attracted much attention from the Molecular
627	>	Dynamics community. Instead, splitting methods have been widely
628	>	accepted since they exploit natural decompositions of the
629	>	system\cite{Tuckerman1992, McLachlan1998}.
630	>
631	>	\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
632	>
633	>	The main idea behind splitting methods is to decompose the discrete
634	>	$\varphi_h$ as a composition of simpler propagators,
635	>	\begin{equation}
636	>	\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
637	>	\varphi _{h_n }
638	>	\label{introEquation:FlowDecomposition}
639	>	\end{equation}
640	>	where each of the sub-propagator is chosen such that each represent
641	>	a simpler integration of the system. Suppose that a Hamiltonian
642	>	system takes the form,
643		\[
644	<	W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
403	<	}}{{2m_\alpha  w_\alpha ^2 }}} x^2
644	>	H = H_1 + H_2.
645		\]
646	<	Since the first two terms of the new Hamiltonian depend only on the
647	<	system coordinates, we can get the equations of motion for
648	<	Generalized Langevin Dynamics by Hamilton's equations
649	<	\ref{introEquation:motionHamiltonianCoordinate,
650	<	introEquation:motionHamiltonianMomentum},
646	>	Here, $H_1$ and $H_2$ may represent different physical processes of
647	>	the system. For instance, they may relate to kinetic and potential
648	>	energy respectively, which is a natural decomposition of the
649	>	problem. If $H_1$ and $H_2$ can be integrated using exact
650	>	propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a
651	>	simple first order expression is then given by the Lie-Trotter
652	>	formula
653	>	\begin{equation}
654	>	\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
655	>	\label{introEquation:firstOrderSplitting}
656	>	\end{equation}
657	>	where $\varphi _h$ is the result of applying the corresponding
658	>	continuous $\varphi _i$ over a time $h$. By definition, as
659	>	$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
660	>	must follow that each operator $\varphi_i(t)$ is a symplectic map.
661	>	It is easy to show that any composition of symplectic propagators
662	>	yields a symplectic map,
663	>	\begin{equation}
664	>	(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
665	>	'\phi ' = \phi '^T J\phi ' = J,
666	>	\label{introEquation:SymplecticFlowComposition}
667	>	\end{equation}
668	>	where $\phi$ and $\psi$ both are symplectic maps. Thus operator
669	>	splitting in this context automatically generates a symplectic map.
670	>	The Lie-Trotter
671	>	splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces
672	>	local errors proportional to $h^2$, while the Strang splitting gives
673	>	a second-order decomposition,
674	>	\begin{equation}
675	>	\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
676	>	_{1,h/2} , \label{introEquation:secondOrderSplitting}
677	>	\end{equation}
678	>	which has a local error proportional to $h^3$. The Strang
679	>	splitting's popularity in molecular simulation community attribute
680	>	to its symmetric property,
681	>	\begin{equation}
682	>	\varphi _h^{ - 1} = \varphi _{ - h}.
683	>	\label{introEquation:timeReversible}
684	>	\end{equation}
685	>
686	>	\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}}
687	>	The classical equation for a system consisting of interacting
688	>	particles can be written in Hamiltonian form,
689	>	\[
690	>	H = T + V
691	>	\]
692	>	where $T$ is the kinetic energy and $V$ is the potential energy.
693	>	Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one
694	>	obtains the following:
695		\begin{align}
696	<	\dot p &=  - \frac{{\partial H}}{{\partial x}}
697	<	&= m\ddot x
698	<	&= - \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)}
699	<	\label{introEq:Lp5}
696	>	q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
697	>	\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
698	>	\label{introEquation:Lp10a} \\%
699	>	%
700	>	\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
701	>	\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
702	>	\label{introEquation:Lp10b}
703		\end{align}
704	<	, and
704	>	where $F(t)$ is the force at time $t$. This integration scheme is
705	>	known as \emph{velocity verlet} which is
706	>	symplectic(\ref{introEquation:SymplecticFlowComposition}),
707	>	time-reversible(\ref{introEquation:timeReversible}) and
708	>	volume-preserving (\ref{introEquation:volumePreserving}). These
709	>	geometric properties attribute to its long-time stability and its
710	>	popularity in the community. However, the most commonly used
711	>	velocity verlet integration scheme is written as below,
712		\begin{align}
713	<	\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
714	<	&= m\ddot x_\alpha
715	<	&= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
713	>	\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
714	>	\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
715	>	%
716	>	q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
717	>	\label{introEquation:Lp9b}\\%
718	>	%
719	>	\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
720	>	\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c}
721		\end{align}
722	+	From the preceding splitting, one can see that the integration of
723	+	the equations of motion would follow:
724	+	\begin{enumerate}
725	+	\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
726
727	<	\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
727	>	\item Use the half step velocities to move positions one whole step, $\Delta t$.
728
729	<	\[
426	<	L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
427	<	\]
729	>	\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move.
730
731	<	\[
732	<	L(x + y) = L(x) + L(y)
733	<	\]
731	>	\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
732	>	\end{enumerate}
733	>	By simply switching the order of the propagators in the splitting
734	>	and composing a new integrator, the \emph{position verlet}
735	>	integrator, can be generated,
736	>	\begin{align}
737	>	\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) +
738	>	\frac{{\Delta t}}{{2m}}\dot q(0)} \right], %
739	>	\label{introEquation:positionVerlet1} \\%
740	>	%
741	>	q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
742	>	q(\Delta t)} \right]. %
743	>	\label{introEquation:positionVerlet2}
744	>	\end{align}
745
746	<	\[
434	<	L(ax) = aL(x)
435	<	\]
746	>	\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
747
748	+	The Baker-Campbell-Hausdorff formula can be used to determine the
749	+	local error of a splitting method in terms of the commutator of the
750	+	operators(\ref{introEquation:exponentialOperator}) associated with
751	+	the sub-propagator. For operators $hX$ and $hY$ which are associated
752	+	with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have
753	+	\begin{equation}
754	+	\exp (hX + hY) = \exp (hZ)
755	+	\end{equation}
756	+	where
757	+	\begin{equation}
758	+	hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left(
759	+	{[X,[X,Y]] + [Y,[Y,X]]} \right) +  \ldots .
760	+	\end{equation}
761	+	Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by
762		\[
763	<	L(\dot x) = pL(x) - px(0)
763	>	[X,Y] = XY - YX .
764		\]
765	<
765	>	Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
766	>	to the Strang splitting, we can obtain
767	>	\begin{eqnarray*}
768	>	\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 \\
769	>	&   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
770	>	&   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots
771	>	).
772	>	\end{eqnarray*}
773	>	Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local
774	>	error of Strang splitting is proportional to $h^3$. The same
775	>	procedure can be applied to a general splitting of the form
776	>	\begin{equation}
777	>	\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
778	>	1} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
779	>	\end{equation}
780	>	A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher
781	>	order methods. Yoshida proposed an elegant way to compose higher
782	>	order methods based on symmetric splitting\cite{Yoshida1990}. Given
783	>	a symmetric second order base method $ \varphi _h^{(2)} $, a
784	>	fourth-order symmetric method can be constructed by composing,
785		\[
786	<	L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
786	>	\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
787	>	h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
788		\]
789	<
789	>	where $ \alpha  =  - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta
790	>	= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric
791	>	integrator $ \varphi _h^{(2n + 2)}$ can be composed by
792	>	\begin{equation}
793	>	\varphi _h^{(2n + 2)}  = \varphi _{\alpha h}^{(2n)}  \circ \varphi
794	>	_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)},
795	>	\end{equation}
796	>	if the weights are chosen as
797		\[
798	<	L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
798	>	\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
799	>	\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
800		\]
801
802	<	Some relatively important transformation,
450	<	\[
451	<	L(\cos at) = \frac{p}{{p^2  + a^2 }}
452	<	\]
802	>	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
803
804	+	As one of the principal tools of molecular modeling, Molecular
805	+	dynamics has proven to be a powerful tool for studying the functions
806	+	of biological systems, providing structural, thermodynamic and
807	+	dynamical information. The basic idea of molecular dynamics is that
808	+	macroscopic properties are related to microscopic behavior and
809	+	microscopic behavior can be calculated from the trajectories in
810	+	simulations. For instance, instantaneous temperature of a
811	+	Hamiltonian system of $N$ particles can be measured by
812		\[
813	<	L(\sin at) = \frac{a}{{p^2  + a^2 }}
813	>	T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
814		\]
815	+	where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
816	+	respectively, $f$ is the number of degrees of freedom, and $k_B$ is
817	+	the Boltzman constant.
818
819	<	\[
820	<	L(1) = \frac{1}{p}
821	<	\]
822	<
823	<	First, the bath coordinates,
824	<	\[
825	<	p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
826	<	_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
827	<	}}L(x)
828	<	\]
829	<	\[
830	<	L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
831	<	px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
832	<	\]
833	<	Then, the system coordinates,
834	<	\begin{align}
835	<	mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
836	<	\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,
819	>	A typical molecular dynamics run consists of three essential steps:
820	>	\begin{enumerate}
821	>	\item Initialization
822	>	\begin{enumerate}
823	>	\item Preliminary preparation
824	>	\item Minimization
825	>	\item Heating
826	>	\item Equilibration
827	>	\end{enumerate}
828	>	\item Production
829	>	\item Analysis
830	>	\end{enumerate}
831	>	These three individual steps will be covered in the following
832	>	sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
833	>	initialization of a simulation. Sec.~\ref{introSection:production}
834	>	will discuss issues of production runs.
835	>	Sec.~\ref{introSection:Analysis} provides the theoretical tools for
836	>	analysis of trajectories.
837
838	<	\begin{align}
488	<	m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
489	<	\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
490	<	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
491	<	_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
492	<	- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
493	<	(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
494	<	_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
495	<	%
496	<	&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
497	<	{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
498	<	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
499	<	t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
500	<	{\left[ {g_\alpha  x_\alpha  (0) - \frac{{g_\alpha  }}{{m_\alpha
501	<	\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
502	<	\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
503	<	(\omega _\alpha  t)} \right\}}
504	<	\end{align}
838	>	\subsection{\label{introSec:initialSystemSettings}Initialization}
839
840	+	\subsubsection{\textbf{Preliminary preparation}}
841	+
842	+	When selecting the starting structure of a molecule for molecular
843	+	simulation, one may retrieve its Cartesian coordinates from public
844	+	databases, such as RCSB Protein Data Bank \textit{etc}. Although
845	+	thousands of crystal structures of molecules are discovered every
846	+	year, many more remain unknown due to the difficulties of
847	+	purification and crystallization. Even for molecules with known
848	+	structures, some important information is missing. For example, a
849	+	missing hydrogen atom which acts as donor in hydrogen bonding must
850	+	be added. Moreover, in order to include electrostatic interactions,
851	+	one may need to specify the partial charges for individual atoms.
852	+	Under some circumstances, we may even need to prepare the system in
853	+	a special configuration. For instance, when studying transport
854	+	phenomenon in membrane systems, we may prepare the lipids in a
855	+	bilayer structure instead of placing lipids randomly in solvent,
856	+	since we are not interested in the slow self-aggregation process.
857	+
858	+	\subsubsection{\textbf{Minimization}}
859	+
860	+	It is quite possible that some of molecules in the system from
861	+	preliminary preparation may be overlapping with each other. This
862	+	close proximity leads to high initial potential energy which
863	+	consequently jeopardizes any molecular dynamics simulations. To
864	+	remove these steric overlaps, one typically performs energy
865	+	minimization to find a more reasonable conformation. Several energy
866	+	minimization methods have been developed to exploit the energy
867	+	surface and to locate the local minimum. While converging slowly
868	+	near the minimum, steepest descent method is extremely robust when
869	+	systems are strongly anharmonic. Thus, it is often used to refine
870	+	structures from crystallographic data. Relying on the Hessian,
871	+	advanced methods like Newton-Raphson converge rapidly to a local
872	+	minimum, but become unstable if the energy surface is far from
873	+	quadratic. Another factor that must be taken into account, when
874	+	choosing energy minimization method, is the size of the system.
875	+	Steepest descent and conjugate gradient can deal with models of any
876	+	size. Because of the limits on computer memory to store the hessian
877	+	matrix and the computing power needed to diagonalize these matrices,
878	+	most Newton-Raphson methods can not be used with very large systems.
879	+
880	+	\subsubsection{\textbf{Heating}}
881	+
882	+	Typically, heating is performed by assigning random velocities
883	+	according to a Maxwell-Boltzman distribution for a desired
884	+	temperature. Beginning at a lower temperature and gradually
885	+	increasing the temperature by assigning larger random velocities, we
886	+	end up setting the temperature of the system to a final temperature
887	+	at which the simulation will be conducted. In heating phase, we
888	+	should also keep the system from drifting or rotating as a whole. To
889	+	do this, the net linear momentum and angular momentum of the system
890	+	is shifted to zero after each resampling from the Maxwell -Boltzman
891	+	distribution.
892	+
893	+	\subsubsection{\textbf{Equilibration}}
894	+
895	+	The purpose of equilibration is to allow the system to evolve
896	+	spontaneously for a period of time and reach equilibrium. The
897	+	procedure is continued until various statistical properties, such as
898	+	temperature, pressure, energy, volume and other structural
899	+	properties \textit{etc}, become independent of time. Strictly
900	+	speaking, minimization and heating are not necessary, provided the
901	+	equilibration process is long enough. However, these steps can serve
902	+	as a means to arrive at an equilibrated structure in an effective
903	+	way.
904	+
905	+	\subsection{\label{introSection:production}Production}
906	+
907	+	The production run is the most important step of the simulation, in
908	+	which the equilibrated structure is used as a starting point and the
909	+	motions of the molecules are collected for later analysis. In order
910	+	to capture the macroscopic properties of the system, the molecular
911	+	dynamics simulation must be performed by sampling correctly and
912	+	efficiently from the relevant thermodynamic ensemble.
913	+
914	+	The most expensive part of a molecular dynamics simulation is the
915	+	calculation of non-bonded forces, such as van der Waals force and
916	+	Coulombic forces \textit{etc}. For a system of $N$ particles, the
917	+	complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
918	+	which makes large simulations prohibitive in the absence of any
919	+	algorithmic tricks. A natural approach to avoid system size issues
920	+	is to represent the bulk behavior by a finite number of the
921	+	particles. However, this approach will suffer from surface effects
922	+	at the edges of the simulation. To offset this, \textit{Periodic
923	+	boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to
924	+	simulate bulk properties with a relatively small number of
925	+	particles. In this method, the simulation box is replicated
926	+	throughout space to form an infinite lattice. During the simulation,
927	+	when a particle moves in the primary cell, its image in other cells
928	+	move in exactly the same direction with exactly the same
929	+	orientation. Thus, as a particle leaves the primary cell, one of its
930	+	images will enter through the opposite face.
931	+	\begin{figure}
932	+	\centering
933	+	\includegraphics[width=\linewidth]{pbc.eps}
934	+	\caption[An illustration of periodic boundary conditions]{A 2-D
935	+	illustration of periodic boundary conditions. As one particle leaves
936	+	the left of the simulation box, an image of it enters the right.}
937	+	\label{introFig:pbc}
938	+	\end{figure}
939	+
940	+	%cutoff and minimum image convention
941	+	Another important technique to improve the efficiency of force
942	+	evaluation is to apply spherical cutoffs where particles farther
943	+	than a predetermined distance are not included in the calculation
944	+	\cite{Frenkel1996}. The use of a cutoff radius will cause a
945	+	discontinuity in the potential energy curve. Fortunately, one can
946	+	shift a simple radial potential to ensure the potential curve go
947	+	smoothly to zero at the cutoff radius. The cutoff strategy works
948	+	well for Lennard-Jones interaction because of its short range
949	+	nature. However, simply truncating the electrostatic interaction
950	+	with the use of cutoffs has been shown to lead to severe artifacts
951	+	in simulations. The Ewald summation, in which the slowly decaying
952	+	Coulomb potential is transformed into direct and reciprocal sums
953	+	with rapid and absolute convergence, has proved to minimize the
954	+	periodicity artifacts in liquid simulations. Taking the advantages
955	+	of the fast Fourier transform (FFT) for calculating discrete Fourier
956	+	transforms, the particle mesh-based
957	+	methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
958	+	$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the
959	+	\emph{fast multipole method}\cite{Greengard1987, Greengard1994},
960	+	which treats Coulombic interactions exactly at short range, and
961	+	approximate the potential at long range through multipolar
962	+	expansion. In spite of their wide acceptance at the molecular
963	+	simulation community, these two methods are difficult to implement
964	+	correctly and efficiently. Instead, we use a damped and
965	+	charge-neutralized Coulomb potential method developed by Wolf and
966	+	his coworkers\cite{Wolf1999}. The shifted Coulomb potential for
967	+	particle $i$ and particle $j$ at distance $r_{rj}$ is given by:
968		\begin{equation}
969	<	m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
970	<	(t)\dot x(t - \tau )d\tau }  + R(t)
971	<	\label{introEuqation:GeneralizedLangevinDynamics}
969	>	V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
970	>	r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
971	>	R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
972	>	r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb}
973		\end{equation}
974	<	%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
975	<	%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
974	>	where $\alpha$ is the convergence parameter. Due to the lack of
975	>	inherent periodicity and rapid convergence,this method is extremely
976	>	efficient and easy to implement.
977	>	\begin{figure}
978	>	\centering
979	>	\includegraphics[width=\linewidth]{shifted_coulomb.eps}
980	>	\caption[An illustration of shifted Coulomb potential]{An
981	>	illustration of shifted Coulomb potential.}
982	>	\label{introFigure:shiftedCoulomb}
983	>	\end{figure}
984	>
985	>	%multiple time step
986	>
987	>	\subsection{\label{introSection:Analysis} Analysis}
988	>
989	>	Recently, advanced visualization technique have become applied to
990	>	monitor the motions of molecules. Although the dynamics of the
991	>	system can be described qualitatively from animation, quantitative
992	>	trajectory analysis is more useful. According to the principles of
993	>	Statistical Mechanics in
994	>	Sec.~\ref{introSection:statisticalMechanics}, one can compute
995	>	thermodynamic properties, analyze fluctuations of structural
996	>	parameters, and investigate time-dependent processes of the molecule
997	>	from the trajectories.
998	>
999	>	\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}
1000	>
1001	>	Thermodynamic properties, which can be expressed in terms of some
1002	>	function of the coordinates and momenta of all particles in the
1003	>	system, can be directly computed from molecular dynamics. The usual
1004	>	way to measure the pressure is based on virial theorem of Clausius
1005	>	which states that the virial is equal to $-3Nk_BT$. For a system
1006	>	with forces between particles, the total virial, $W$, contains the
1007	>	contribution from external pressure and interaction between the
1008	>	particles:
1009		\[
1010	<	\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
1011	<	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
1010	>	W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
1011	>	f_{ij} } } \right\rangle
1012		\]
1013	<	For an infinite harmonic bath, we can use the spectral density and
1014	<	an integral over frequencies.
1013	>	where $f_{ij}$ is the force between particle $i$ and $j$ at a
1014	>	distance $r_{ij}$. Thus, the expression for the pressure is given
1015	>	by:
1016	>	\begin{equation}
1017	>	P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
1018	>	< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
1019	>	\end{equation}
1020
1021	+	\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}
1022	+
1023	+	Structural Properties of a simple fluid can be described by a set of
1024	+	distribution functions. Among these functions,the \emph{pair
1025	+	distribution function}, also known as \emph{radial distribution
1026	+	function}, is of most fundamental importance to liquid theory.
1027	+	Experimentally, pair distribution functions can be gathered by
1028	+	Fourier transforming raw data from a series of neutron diffraction
1029	+	experiments and integrating over the surface factor
1030	+	\cite{Powles1973}. The experimental results can serve as a criterion
1031	+	to justify the correctness of a liquid model. Moreover, various
1032	+	equilibrium thermodynamic and structural properties can also be
1033	+	expressed in terms of the radial distribution function
1034	+	\cite{Allen1987}. The pair distribution functions $g(r)$ gives the
1035	+	probability that a particle $i$ will be located at a distance $r$
1036	+	from a another particle $j$ in the system
1037	+	\begin{equation}
1038	+	g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
1039	+	\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
1040	+	(r)}{\rho}.
1041	+	\end{equation}
1042	+	Note that the delta function can be replaced by a histogram in
1043	+	computer simulation. Peaks in $g(r)$ represent solvent shells, and
1044	+	the height of these peaks gradually decreases to 1 as the liquid of
1045	+	large distance approaches the bulk density.
1046	+
1047	+
1048	+	\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
1049	+	Properties}}
1050	+
1051	+	Time-dependent properties are usually calculated using \emph{time
1052	+	correlation functions}, which correlate random variables $A$ and $B$
1053	+	at two different times,
1054	+	\begin{equation}
1055	+	C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
1056	+	\label{introEquation:timeCorrelationFunction}
1057	+	\end{equation}
1058	+	If $A$ and $B$ refer to same variable, this kind of correlation
1059	+	function is called an \emph{autocorrelation function}. One example
1060	+	of an auto correlation function is the velocity auto-correlation
1061	+	function which is directly related to transport properties of
1062	+	molecular liquids:
1063		\[
1064	<	R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
1065	<	- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
523	<	\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
524	<	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
1064	>	D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
1065	>	\right\rangle } dt
1066		\]
1067	<	The random forces depend only on initial conditions.
1068	<
1069	<	\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
1070	<	So we can define a new set of coordinates,
1067	>	where $D$ is diffusion constant. Unlike the velocity autocorrelation
1068	>	function, which is averaged over time origins and over all the
1069	>	atoms, the dipole autocorrelation functions is calculated for the
1070	>	entire system. The dipole autocorrelation function is given by:
1071		\[
1072	<	q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
1073	<	^2 }}x(0)
1072	>	c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
1073	>	\right\rangle
1074		\]
1075	<	This makes
1075	>	Here $u_{tot}$ is the net dipole of the entire system and is given
1076	>	by
1077		\[
1078	<	R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
1078	>	u_{tot} (t) = \sum\limits_i {u_i (t)}.
1079		\]
1080	<	And since the $q$ coordinates are harmonic oscillators,
1080	>	In principle, many time correlation functions can be related to
1081	>	Fourier transforms of the infrared, Raman, and inelastic neutron
1082	>	scattering spectra of molecular liquids. In practice, one can
1083	>	extract the IR spectrum from the intensity of the molecular dipole
1084	>	fluctuation at each frequency using the following relationship:
1085		\[
1086	<	\begin{array}{l}
1087	<	\left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
542	<	\left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
543	<	\end{array}
1086	>	\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
1087	>	i2\pi vt} dt}.
1088		\]
1089
1090	<	\begin{align}
547	<	\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
548	<	{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
549	<	(t)q_\beta  (0)} \right\rangle } }
550	<	%
551	<	&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
552	<	\right\rangle \cos (\omega _\alpha  t)}
553	<	%
554	<	&= kT\xi (t)
555	<	\end{align}
1090	>	\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
1091
1092	<	\begin{equation}
1093	<	\xi (t) = \left\langle {R(t)R(0)} \right\rangle
1094	<	\label{introEquation:secondFluctuationDissipation}
1095	<	\end{equation}
1092	>	Rigid bodies are frequently involved in the modeling of different
1093	>	areas, from engineering, physics, to chemistry. For example,
1094	>	missiles and vehicles are usually modeled by rigid bodies.  The
1095	>	movement of the objects in 3D gaming engines or other physics
1096	>	simulators is governed by rigid body dynamics. In molecular
1097	>	simulations, rigid bodies are used to simplify protein-protein
1098	>	docking studies\cite{Gray2003}.
1099
1100	<	\section{\label{introSection:hydroynamics}Hydrodynamics}
1100	>	It is very important to develop stable and efficient methods to
1101	>	integrate the equations of motion for orientational degrees of
1102	>	freedom. Euler angles are the natural choice to describe the
1103	>	rotational degrees of freedom. However, due to $\frac {1}{sin
1104	>	\theta}$ singularities, the numerical integration of corresponding
1105	>	equations of these motion is very inefficient and inaccurate.
1106	>	Although an alternative integrator using multiple sets of Euler
1107	>	angles can overcome this difficulty\cite{Barojas1973}, the
1108	>	computational penalty and the loss of angular momentum conservation
1109	>	still remain. A singularity-free representation utilizing
1110	>	quaternions was developed by Evans in 1977\cite{Evans1977}.
1111	>	Unfortunately, this approach uses a nonseparable Hamiltonian
1112	>	resulting from the quaternion representation, which prevents the
1113	>	symplectic algorithm from being utilized. Another different approach
1114	>	is to apply holonomic constraints to the atoms belonging to the
1115	>	rigid body. Each atom moves independently under the normal forces
1116	>	deriving from potential energy and constraint forces which are used
1117	>	to guarantee the rigidness. However, due to their iterative nature,
1118	>	the SHAKE and Rattle algorithms also converge very slowly when the
1119	>	number of constraints increases\cite{Ryckaert1977, Andersen1983}.
1120
1121	<	\subsection{\label{introSection:frictionTensor} Friction Tensor}
1122	<	\subsection{\label{introSection:analyticalApproach}Analytical
1123	<	Approach}
1121	>	A break-through in geometric literature suggests that, in order to
1122	>	develop a long-term integration scheme, one should preserve the
1123	>	symplectic structure of the propagator. By introducing a conjugate
1124	>	momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's
1125	>	equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was
1126	>	proposed to evolve the Hamiltonian system in a constraint manifold
1127	>	by iteratively satisfying the orthogonality constraint $Q^T Q = 1$.
1128	>	An alternative method using the quaternion representation was
1129	>	developed by Omelyan\cite{Omelyan1998}. However, both of these
1130	>	methods are iterative and inefficient. In this section, we descibe a
1131	>	symplectic Lie-Poisson integrator for rigid bodies developed by
1132	>	Dullweber and his coworkers\cite{Dullweber1997} in depth.
1133
1134	<	\subsection{\label{introSection:approximationApproach}Approximation
1135	<	Approach}
1134	>	\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
1135	>	The motion of a rigid body is Hamiltonian with the Hamiltonian
1136	>	function
1137	>	\begin{equation}
1138	>	H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
1139	>	V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
1140	>	\label{introEquation:RBHamiltonian}
1141	>	\end{equation}
1142	>	Here, $q$ and $Q$  are the position vector and rotation matrix for
1143	>	the rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ ,
1144	>	and $J$, a diagonal matrix, is defined by
1145	>	\[
1146	>	I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
1147	>	\]
1148	>	where $I_{ii}$ is the diagonal element of the inertia tensor. This
1149	>	constrained Hamiltonian equation is subjected to a holonomic
1150	>	constraint,
1151	>	\begin{equation}
1152	>	Q^T Q = 1, \label{introEquation:orthogonalConstraint}
1153	>	\end{equation}
1154	>	which is used to ensure the rotation matrix's unitarity. Using
1155	>	Equation (\ref{introEquation:motionHamiltonianCoordinate},
1156	>	\ref{introEquation:motionHamiltonianMomentum}), one can write down
1157	>	the equations of motion,
1158	>	\begin{eqnarray}
1159	>	\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\
1160	>	\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\
1161	>	\frac{{dQ}}{{dt}} & = & PJ^{ - 1},  \label{introEquation:RBMotionRotation}\\
1162	>	\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
1163	>	\end{eqnarray}
1164	>	Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and
1165	>	using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain,
1166	>	\begin{equation}
1167	>	Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1168	>	\label{introEquation:RBFirstOrderConstraint}
1169	>	\end{equation}
1170	>	In general, there are two ways to satisfy the holonomic constraints.
1171	>	We can use a constraint force provided by a Lagrange multiplier on
1172	>	the normal manifold to keep the motion on the constraint space. Or
1173	>	we can simply evolve the system on the constraint manifold. These
1174	>	two methods have been proved to be equivalent. The holonomic
1175	>	constraint and equations of motions define a constraint manifold for
1176	>	rigid bodies
1177	>	\[
1178	>	M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
1179	>	\right\}.
1180	>	\]
1181	>	Unfortunately, this constraint manifold is not $T^* SO(3)$ which is
1182	>	a symplectic manifold on Lie rotation group $SO(3)$. However, it
1183	>	turns out that under symplectic transformation, the cotangent space
1184	>	and the phase space are diffeomorphic. By introducing
1185	>	\[
1186	>	\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
1187	>	\]
1188	>	the mechanical system subject to a holonomic constraint manifold $M$
1189	>	can be re-formulated as a Hamiltonian system on the cotangent space
1190	>	\[
1191	>	T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
1192	>	1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
1193	>	\]
1194	>	For a body fixed vector $X_i$ with respect to the center of mass of
1195	>	the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
1196	>	given as
1197	>	\begin{equation}
1198	>	X_i^{lab} = Q X_i + q.
1199	>	\end{equation}
1200	>	Therefore, potential energy $V(q,Q)$ is defined by
1201	>	\[
1202	>	V(q,Q) = V(Q X_0 + q).
1203	>	\]
1204	>	Hence, the force and torque are given by
1205	>	\[
1206	>	\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
1207	>	\]
1208	>	and
1209	>	\[
1210	>	\nabla _Q V(q,Q) = F(q,Q)X_i^t
1211	>	\]
1212	>	respectively. As a common choice to describe the rotation dynamics
1213	>	of the rigid body, the angular momentum on the body fixed frame $\Pi
1214	>	= Q^t P$ is introduced to rewrite the equations of motion,
1215	>	\begin{equation}
1216	>	\begin{array}{l}
1217	>	\dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
1218	>	\dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
1219	>	\end{array}
1220	>	\label{introEqaution:RBMotionPI}
1221	>	\end{equation}
1222	>	as well as holonomic constraints $\Pi J^{ - 1}  + J^{ - 1} \Pi ^t  =
1223	>	0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a
1224	>	matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism,
1225	>	\begin{equation}
1226	>	v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
1227	>	{\begin{array}{*{20}c}
1228	>	0 & { - v_3 } & {v_2 }  \\
1229	>	{v_3 } & 0 & { - v_1 }  \\
1230	>	{ - v_2 } & {v_1 } & 0  \\
1231	>	\end{array}} \right),
1232	>	\label{introEquation:hatmapIsomorphism}
1233	>	\end{equation}
1234	>	will let us associate the matrix products with traditional vector
1235	>	operations
1236	>	\[
1237	>	\hat vu = v \times u.
1238	>	\]
1239	>	Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
1240	>	matrix,
1241	>	\begin{eqnarray}
1242	>	(\dot \Pi  - \dot \Pi ^T )&= &(\Pi  - \Pi ^T )(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
1243	>	& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  -
1244	>	(\Lambda  - \Lambda ^T ). \label{introEquation:skewMatrixPI}
1245	>	\end{eqnarray}
1246	>	Since $\Lambda$ is symmetric, the last term of
1247	>	Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
1248	>	Lagrange multiplier $\Lambda$ is absent from the equations of
1249	>	motion. This unique property eliminates the requirement of
1250	>	iterations which can not be avoided in other methods\cite{Kol1997,
1251	>	Omelyan1998}. Applying the hat-map isomorphism, we obtain the
1252	>	equation of motion for angular momentum in the body frame
1253	>	\begin{equation}
1254	>	\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
1255	>	F_i (r,Q)} \right) \times X_i }.
1256	>	\label{introEquation:bodyAngularMotion}
1257	>	\end{equation}
1258	>	In the same manner, the equation of motion for rotation matrix is
1259	>	given by
1260	>	\[
1261	>	\dot Q = Qskew(I^{ - 1} \pi ).
1262	>	\]
1263
1264	<	\subsection{\label{introSection:centersRigidBody}Centers of Rigid
1265	<	Body}
1264	>	\subsection{\label{introSection:SymplecticFreeRB}Symplectic
1265	>	Lie-Poisson Integrator for Free Rigid Bodies}
1266	>
1267	>	If there are no external forces exerted on the rigid body, the only
1268	>	contribution to the rotational motion is from the kinetic energy
1269	>	(the first term of \ref{introEquation:bodyAngularMotion}). The free
1270	>	rigid body is an example of a Lie-Poisson system with Hamiltonian
1271	>	function
1272	>	\begin{equation}
1273	>	T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
1274	>	\label{introEquation:rotationalKineticRB}
1275	>	\end{equation}
1276	>	where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
1277	>	Lie-Poisson structure matrix,
1278	>	\begin{equation}
1279	>	J(\pi ) = \left( {\begin{array}{*{20}c}
1280	>	0 & {\pi _3 } & { - \pi _2 }  \\
1281	>	{ - \pi _3 } & 0 & {\pi _1 }  \\
1282	>	{\pi _2 } & { - \pi _1 } & 0  \\
1283	>	\end{array}} \right).
1284	>	\end{equation}
1285	>	Thus, the dynamics of free rigid body is governed by
1286	>	\begin{equation}
1287	>	\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
1288	>	\end{equation}
1289	>	One may notice that each $T_i^r$ in
1290	>	Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly.
1291	>	For instance, the equations of motion due to $T_1^r$ are given by
1292	>	\begin{equation}
1293	>	\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
1294	>	\label{introEqaution:RBMotionSingleTerm}
1295	>	\end{equation}
1296	>	with
1297	>	\[ R_1  = \left( {\begin{array}{*{20}c}
1298	>	0 & 0 & 0  \\
1299	>	0 & 0 & {\pi _1 }  \\
1300	>	0 & { - \pi _1 } & 0  \\
1301	>	\end{array}} \right).
1302	>	\]
1303	>	The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is
1304	>	\[
1305	>	\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
1306	>	Q(0)e^{\Delta tR_1 }
1307	>	\]
1308	>	with
1309	>	\[
1310	>	e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
1311	>	0 & 0 & 0  \\
1312	>	0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
1313	>	0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
1314	>	\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
1315	>	\]
1316	>	To reduce the cost of computing expensive functions in $e^{\Delta
1317	>	tR_1 }$, we can use the Cayley transformation to obtain a
1318	>	single-aixs propagator,
1319	>	\begin{eqnarray*}
1320	>	e^{\Delta tR_1 }  & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta
1321	>	tR_1 ) \\
1322	>	%
1323	>	& \approx & \left( \begin{array}{ccc}
1324	>	1 & 0 & 0 \\
1325	>	0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4}  & -\frac{\theta}{1+
1326	>	\theta^2 / 4} \\
1327	>	0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 +
1328	>	\theta^2 / 4}
1329	>	\end{array}
1330	>	\right).
1331	>	\end{eqnarray*}
1332	>	The propagators for $T_2^r$ and $T_3^r$ can be found in the same
1333	>	manner. In order to construct a second-order symplectic method, we
1334	>	split the angular kinetic Hamiltonian function into five terms
1335	>	\[
1336	>	T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
1337	>	) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
1338	>	(\pi _1 ).
1339	>	\]
1340	>	By concatenating the propagators corresponding to these five terms,
1341	>	we can obtain an symplectic integrator,
1342	>	\[
1343	>	\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
1344	>	\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
1345	>	\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1346	>	_1 }.
1347	>	\]
1348	>	The non-canonical Lie-Poisson bracket ${F, G}$ of two function
1349	>	$F(\pi )$ and $G(\pi )$ is defined by
1350	>	\[
1351	>	\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
1352	>	).
1353	>	\]
1354	>	If the Poisson bracket of a function $F$ with an arbitrary smooth
1355	>	function $G$ is zero, $F$ is a \emph{Casimir}, which is the
1356	>	conserved quantity in Poisson system. We can easily verify that the
1357	>	norm of the angular momentum, $\parallel \pi
1358	>	\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let$ F(\pi ) = S(\frac{{\parallel
1359	>	\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
1360	>	then by the chain rule
1361	>	\[
1362	>	\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
1363	>	}}{2})\pi.
1364	>	\]
1365	>	Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
1366	>	\pi
1367	>	\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
1368	>	Lie-Poisson integrator is found to be both extremely efficient and
1369	>	stable. These properties can be explained by the fact the small
1370	>	angle approximation is used and the norm of the angular momentum is
1371	>	conserved.
1372	>
1373	>	\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
1374	>	Splitting for Rigid Body}
1375	>
1376	>	The Hamiltonian of rigid body can be separated in terms of kinetic
1377	>	energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations
1378	>	of motion corresponding to potential energy and kinetic energy are
1379	>	listed in the below table,
1380	>	\begin{table}
1381	>	\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
1382	>	\begin{center}
1383	>	\begin{tabular}{|l|l|}
1384	>	\hline
1385	>	% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
1386	>	Potential & Kinetic \\
1387	>	$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
1388	>	$\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
1389	>	$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
1390	>	$ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
1391	>	\hline
1392	>	\end{tabular}
1393	>	\end{center}
1394	>	\end{table}
1395	>	A second-order symplectic method is now obtained by the composition
1396	>	of the position and velocity propagators,
1397	>	\[
1398	>	\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
1399	>	_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
1400	>	\]
1401	>	Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
1402	>	sub-propagators which corresponding to force and torque
1403	>	respectively,
1404	>	\[
1405	>	\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
1406	>	_{\Delta t/2,\tau }.
1407	>	\]
1408	>	Since the associated operators of $\varphi _{\Delta t/2,F} $ and
1409	>	$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order
1410	>	inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the
1411	>	kinetic energy can be separated to translational kinetic term, $T^t
1412	>	(p)$, and rotational kinetic term, $T^r (\pi )$,
1413	>	\begin{equation}
1414	>	T(p,\pi ) =T^t (p) + T^r (\pi ).
1415	>	\end{equation}
1416	>	where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
1417	>	defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore,
1418	>	the corresponding propagators are given by
1419	>	\[
1420	>	\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
1421	>	_{\Delta t,T^r }.
1422	>	\]
1423	>	Finally, we obtain the overall symplectic propagators for freely
1424	>	moving rigid bodies
1425	>	\begin{eqnarray}
1426	>	\varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \notag\\
1427	>	& & \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\\
1428	>	& & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .
1429	>	\label{introEquation:overallRBFlowMaps}
1430	>	\end{eqnarray}
1431	>
1432	>	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
1433	>	As an alternative to newtonian dynamics, Langevin dynamics, which
1434	>	mimics a simple heat bath with stochastic and dissipative forces,
1435	>	has been applied in a variety of studies. This section will review
1436	>	the theory of Langevin dynamics. A brief derivation of generalized
1437	>	Langevin equation will be given first. Following that, we will
1438	>	discuss the physical meaning of the terms appearing in the equation
1439	>	as well as the calculation of friction tensor from hydrodynamics
1440	>	theory.
1441	>
1442	>	\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
1443	>
1444	>	A harmonic bath model, in which an effective set of harmonic
1445	>	oscillators are used to mimic the effect of a linearly responding
1446	>	environment, has been widely used in quantum chemistry and
1447	>	statistical mechanics. One of the successful applications of
1448	>	Harmonic bath model is the derivation of the Generalized Langevin
1449	>	Dynamics (GLE). Lets consider a system, in which the degree of
1450	>	freedom $x$ is assumed to couple to the bath linearly, giving a
1451	>	Hamiltonian of the form
1452	>	\begin{equation}
1453	>	H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
1454	>	\label{introEquation:bathGLE}.
1455	>	\end{equation}
1456	>	Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated
1457	>	with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian,
1458	>	\[
1459	>	H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
1460	>	}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  \omega _\alpha ^2 }
1461	>	\right\}}
1462	>	\]
1463	>	where the index $\alpha$ runs over all the bath degrees of freedom,
1464	>	$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
1465	>	the harmonic bath masses, and $\Delta U$ is a bilinear system-bath
1466	>	coupling,
1467	>	\[
1468	>	\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
1469	>	\]
1470	>	where $g_\alpha$ are the coupling constants between the bath
1471	>	coordinates ($x_ \alpha$) and the system coordinate ($x$).
1472	>	Introducing
1473	>	\[
1474	>	W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
1475	>	}}{{2m_\alpha  w_\alpha ^2 }}} x^2
1476	>	\]
1477	>	and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
1478	>	\[
1479	>	H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
1480	>	{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
1481	>	w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
1482	>	w_\alpha ^2 }}x} \right)^2 } \right\}}.
1483	>	\]
1484	>	Since the first two terms of the new Hamiltonian depend only on the
1485	>	system coordinates, we can get the equations of motion for
1486	>	Generalized Langevin Dynamics by Hamilton's equations,
1487	>	\begin{equation}
1488	>	m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
1489	>	\sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   -
1490	>	\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)},
1491	>	\label{introEquation:coorMotionGLE}
1492	>	\end{equation}
1493	>	and
1494	>	\begin{equation}
1495	>	m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x_\alpha   -
1496	>	\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
1497	>	\label{introEquation:bathMotionGLE}
1498	>	\end{equation}
1499	>	In order to derive an equation for $x$, the dynamics of the bath
1500	>	variables $x_\alpha$ must be solved exactly first. As an integral
1501	>	transform which is particularly useful in solving linear ordinary
1502	>	differential equations,the Laplace transform is the appropriate tool
1503	>	to solve this problem. The basic idea is to transform the difficult
1504	>	differential equations into simple algebra problems which can be
1505	>	solved easily. Then, by applying the inverse Laplace transform, we
1506	>	can retrieve the solutions of the original problems. Let $f(t)$ be a
1507	>	function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$
1508	>	is a new function defined as
1509	>	\[
1510	>	L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
1511	>	\]
1512	>	where  $p$ is real and  $L$ is called the Laplace Transform
1513	>	Operator. Below are some important properties of Laplace transform
1514	>	\begin{eqnarray*}
1515	>	L(x + y)  & = & L(x) + L(y) \\
1516	>	L(ax)     & = & aL(x) \\
1517	>	L(\dot x) & = & pL(x) - px(0) \\
1518	>	L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
1519	>	L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
1520	>	\end{eqnarray*}
1521	>	Applying the Laplace transform to the bath coordinates, we obtain
1522	>	\begin{eqnarray*}
1523	>	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), \\
1524	>	L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}. \\
1525	>	\end{eqnarray*}
1526	>	In the same way, the system coordinates become
1527	>	\begin{eqnarray*}
1528	>	mL(\ddot x) & = &
1529	>	- \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\}}  \\
1530	>	& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}.
1531	>	\end{eqnarray*}
1532	>	With the help of some relatively important inverse Laplace
1533	>	transformations:
1534	>	\[
1535	>	\begin{array}{c}
1536	>	L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
1537	>	L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
1538	>	L(1) = \frac{1}{p} \\
1539	>	\end{array}
1540	>	\]
1541	>	we obtain
1542	>	\begin{eqnarray*}
1543	>	m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
1544	>	\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
1545	>	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
1546	>	_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
1547	>	& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
1548	>	x_\alpha (0) - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}}
1549	>	\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
1550	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}\\
1551	>	%
1552	>	& = & -
1553	>	\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha
1554	>	= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha
1555	>	^2 }}} \right)\cos (\omega _\alpha
1556	>	t)\dot x(t - \tau )d} \tau }  \\
1557	>	& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
1558	>	x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha  }}}
1559	>	\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
1560	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
1561	>	\end{eqnarray*}
1562	>	Introducing a \emph{dynamic friction kernel}
1563	>	\begin{equation}
1564	>	\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
1565	>	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
1566	>	\label{introEquation:dynamicFrictionKernelDefinition}
1567	>	\end{equation}
1568	>	and \emph{a random force}
1569	>	\begin{equation}
1570	>	R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
1571	>	- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
1572	>	\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
1573	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
1574	>	\label{introEquation:randomForceDefinition}
1575	>	\end{equation}
1576	>	the equation of motion can be rewritten as
1577	>	\begin{equation}
1578	>	m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
1579	>	(t)\dot x(t - \tau )d\tau }  + R(t)
1580	>	\label{introEuqation:GeneralizedLangevinDynamics}
1581	>	\end{equation}
1582	>	which is known as the \emph{generalized Langevin equation}.
1583	>
1584	>	\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
1585	>
1586	>	One may notice that $R(t)$ depends only on initial conditions, which
1587	>	implies it is completely deterministic within the context of a
1588	>	harmonic bath. However, it is easy to verify that $R(t)$ is totally
1589	>	uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)}
1590	>	\right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
1591	>	0.$ This property is what we expect from a truly random process. As
1592	>	long as the model chosen for $R(t)$ was a gaussian distribution in
1593	>	general, the stochastic nature of the GLE still remains.
1594	>	%dynamic friction kernel
1595	>	The convolution integral
1596	>	\[
1597	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
1598	>	\]
1599	>	depends on the entire history of the evolution of $x$, which implies
1600	>	that the bath retains memory of previous motions. In other words,
1601	>	the bath requires a finite time to respond to change in the motion
1602	>	of the system. For a sluggish bath which responds slowly to changes
1603	>	in the system coordinate, we may regard $\xi(t)$ as a constant
1604	>	$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
1605	>	\[
1606	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
1607	>	\]
1608	>	and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1609	>	\[
1610	>	m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
1611	>	\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
1612	>	\]
1613	>	which can be used to describe the effect of dynamic caging in
1614	>	viscous solvents. The other extreme is the bath that responds
1615	>	infinitely quickly to motions in the system. Thus, $\xi (t)$ can be
1616	>	taken as a $delta$ function in time:
1617	>	\[
1618	>	\xi (t) = 2\xi _0 \delta (t)
1619	>	\]
1620	>	Hence, the convolution integral becomes
1621	>	\[
1622	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
1623	>	{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
1624	>	\]
1625	>	and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1626	>	\begin{equation}
1627	>	m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
1628	>	x(t) + R(t) \label{introEquation:LangevinEquation}
1629	>	\end{equation}
1630	>	which is known as the Langevin equation. The static friction
1631	>	coefficient $\xi _0$ can either be calculated from spectral density
1632	>	or be determined by Stokes' law for regular shaped particles. A
1633	>	brief review on calculating friction tensors for arbitrary shaped
1634	>	particles is given in Sec.~\ref{introSection:frictionTensor}.
1635	>
1636	>	\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
1637	>
1638	>	Defining a new set of coordinates
1639	>	\[
1640	>	q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
1641	>	^2 }}x(0),
1642	>	\]
1643	>	we can rewrite $R(T)$ as
1644	>	\[
1645	>	R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
1646	>	\]
1647	>	And since the $q$ coordinates are harmonic oscillators,
1648	>	\begin{eqnarray*}
1649	>	\left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
1650	>	\left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
1651	>	\left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
1652	>	\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 } }  \\
1653	>	& = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
1654	>	& = &kT\xi (t)
1655	>	\end{eqnarray*}
1656	>	Thus, we recover the \emph{second fluctuation dissipation theorem}
1657	>	\begin{equation}
1658	>	\xi (t) = \left\langle {R(t)R(0)} \right\rangle
1659	>	\label{introEquation:secondFluctuationDissipation},
1660	>	\end{equation}
1661	>	which acts as a constraint on the possible ways in which one can
1662	>	model the random force and friction kernel.

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