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1		\chapter{\label{chapt:introduction}INTRODUCTION AND THEORETICAL BACKGROUND}
2
3	–	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
4	–
5	–	As a special discipline of molecular modeling, Molecular dynamics
6	–	has proven to be a powerful tool for studying the functions of
7	–	biological systems, providing structural, thermodynamic and
8	–	dynamical information.
9	–
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
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
25	+	\begin{equation}
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
34	+	\begin{equation}
35	+	F_{ij} = -F_{ji}.
36	+	\label{introEquation:newtonThirdLaw}
37	+	\end{equation}
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
41	+	conservation theorem concerns the angular momentum of a particle.
42	+	The angular momentum $L$ of a particle with respect to an origin
43	+	from which $r$ is measured is defined to be
44	+	\begin{equation}
45	+	L \equiv r \times p \label{introEquation:angularMomentumDefinition}
46	+	\end{equation}
47	+	The torque $\tau$ with respect to the same origin is defined to be
48	+	\begin{equation}
49	+	\tau \equiv r \times F \label{introEquation:torqueDefinition}
50	+	\end{equation}
51	+	Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
52	+	\[
53	+	\dot L = \frac{d}{{dt}}(r \times p) = (\dot r \times p) + (r \times
54	+	\dot p)
55	+	\]
56	+	since
57	+	\[
58	+	\dot r \times p = \dot r \times mv = m\dot r \times \dot r \equiv 0
59	+	\]
60	+	thus,
61	+	\begin{equation}
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 is conserved,
67	+	\begin{equation}E = T + V. \label{introEquation:energyConservation}
68	+	\end{equation}
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
35	<	interaction. In order to overcome some of the practical difficulties
36	<	which arise in attempts to apply Newton's equation to complex
37	<	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	<	\subsection{\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
49	<	the kinetic, $K$, and potential energies, $U$.
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}
54	–
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
59	<	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	<	\subsection{\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 85 | Line 126 | independent of generalized velocities, the generalized
126		Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
127		introduced by William Rowan Hamilton in 1833 as a re-formulation of
128		classical mechanics. If the potential energy of a system is
129	<	independent of generalized velocities, the generalized momenta can
89	<	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 96 | 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}
99	–
139		With the help of the generalized momenta, we may now define a new
140		quantity $H$ by the equation
141		\begin{equation}
#	Line 104 | Line 143 | $L$ is the Lagrangian function for the system.
143		\label{introEquation:hamiltonianDefByLagrangian}
144		\end{equation}
145		where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
146	<	$L$ is the Lagrangian function for the system.
147	<
109	<	Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
110	<	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 138 | Line 175 | t}}
175		t}}
176		\label{introEquation:motionHamiltonianTime}
177		\end{equation}
178	<
142	<	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.
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
192	<	equations.
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
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}. \label{introEquation:conserveHalmitonian}
203	>	\end{equation}
204
158	–	\subsection{\label{introSection:poissonBrackets}Poisson Brackets}
159	–
160	–	\subsection{\label{introSection:canonicalTransformation}Canonical
161	–	Transformation}
162	–
205		\section{\label{introSection:statisticalMechanics}Statistical
206		Mechanics}
207
208	<	The thermodynamic behaviors and properties  of Molecular Dynamics
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}
214	>	\subsection{\label{introSection:ensemble}Phase Space and Ensemble}
215	>
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	+	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	+	\rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
243	+	\label{introEquation:densityDistribution}
244	+	\end{equation}
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	+	There are several different types of ensembles with different
287	+	statistical characteristics. As a function of macroscopic
288	+	parameters, such as temperature \textit{etc}, the partition function
289	+	can be used to describe the statistical properties of a system in
290	+	thermodynamic equilibrium. As an ensemble of systems, each of which
291	+	is known to be thermally isolated and conserve energy, the
292	+	Microcanonical ensemble (NVE) has a partition function like,
293	+	\begin{equation}
294	+	\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition}
295	+	\end{equation}
296	+	A canonical ensemble (NVT) is an ensemble of systems, each of which
297	+	can share its energy with a large heat reservoir. The distribution
298	+	of the total energy amongst the possible dynamical states is given
299	+	by the partition function,
300	+	\begin{equation}
301	+	\Omega (N,V,T) = e^{ - \beta A}.
302	+	\label{introEquation:NVTPartition}
303	+	\end{equation}
304	+	Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
305	+	TS$. Since most experiments are carried out under constant pressure
306	+	condition, the isothermal-isobaric ensemble (NPT) plays a very
307	+	important role in molecular simulations. The isothermal-isobaric
308	+	ensemble allow the system to exchange energy with a heat bath of
309	+	temperature $T$ and to change the volume as well. Its partition
310	+	function is given as
311	+	\begin{equation}
312	+	\Delta (N,P,T) =  - e^{\beta G}.
313	+	\label{introEquation:NPTPartition}
314	+	\end{equation}
315	+	Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy.
316	+
317	+	\subsection{\label{introSection:liouville}Liouville's theorem}
318	+
319	+	Liouville's theorem is the foundation on which statistical mechanics
320	+	rests. It describes the time evolution of the phase space
321	+	distribution function. In order to calculate the rate of change of
322	+	$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider
323	+	the two faces perpendicular to the $q_1$ axis, which are located at
324	+	$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the
325	+	opposite face is given by the expression,
326	+	\begin{equation}
327	+	\left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
328	+	\right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
329	+	}}\delta q_1 } \right)\delta q_2  \ldots \delta q_f \delta p_1
330	+	\ldots \delta p_f .
331	+	\end{equation}
332	+	Summing all over the phase space, we obtain
333	+	\begin{equation}
334	+	\frac{{d(\delta N)}}{{dt}} =  - \sum\limits_{i = 1}^f {\left[ {\rho
335	+	\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} +
336	+	\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left(
337	+	{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  + \frac{{\partial
338	+	\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1
339	+	\ldots \delta q_f \delta p_1  \ldots \delta p_f .
340	+	\end{equation}
341	+	Differentiating the equations of motion in Hamiltonian formalism
342	+	(\ref{introEquation:motionHamiltonianCoordinate},
343	+	\ref{introEquation:motionHamiltonianMomentum}), we can show,
344	+	\begin{equation}
345	+	\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }}
346	+	+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
347	+	\end{equation}
348	+	which cancels the first terms of the right hand side. Furthermore,
349	+	dividing $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
350	+	p_f $ in both sides, we can write out Liouville's theorem in a
351	+	simple form,
352	+	\begin{equation}
353	+	\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f
354	+	{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i  +
355	+	\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
356	+	\label{introEquation:liouvilleTheorem}
357	+	\end{equation}
358	+	Liouville's theorem states that the distribution function is
359	+	constant along any trajectory in phase space. In classical
360	+	statistical mechanics, since the number of system copies in an
361	+	ensemble is huge and constant, we can assume the local density has
362	+	no reason (other than classical mechanics) to change,
363	+	\begin{equation}
364	+	\frac{{\partial \rho }}{{\partial t}} = 0.
365	+	\label{introEquation:stationary}
366	+	\end{equation}
367	+	In such stationary system, the density of distribution $\rho$ can be
368	+	connected to the Hamiltonian $H$ through Maxwell-Boltzmann
369	+	distribution,
370	+	\begin{equation}
371	+	\rho  \propto e^{ - \beta H}
372	+	\label{introEquation:densityAndHamiltonian}
373	+	\end{equation}
374	+
375	+	\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}}
376	+	Lets consider a region in the phase space,
377	+	\begin{equation}
378	+	\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
379	+	\end{equation}
380	+	If this region is small enough, the density $\rho$ can be regarded
381	+	as uniform over the whole integral. Thus, the number of phase points
382	+	inside this region is given by,
383	+	\begin{equation}
384	+	\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
385	+	dp_1 } ..dp_f.
386	+	\end{equation}
387	+
388	+	\begin{equation}
389	+	\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho
390	+	\frac{d}{{dt}}(\delta v) = 0.
391	+	\end{equation}
392	+	With the help of the stationary assumption
393	+	(Eq.~\ref{introEquation:stationary}), we obtain the principle of
394	+	\emph{conservation of volume in phase space},
395	+	\begin{equation}
396	+	\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
397	+	...dq_f dp_1 } ..dp_f  = 0.
398	+	\label{introEquation:volumePreserving}
399	+	\end{equation}
400	+
401	+	\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}}
402	+
403	+	Liouville's theorem can be expressed in a variety of different forms
404	+	which are convenient within different contexts. For any two function
405	+	$F$ and $G$ of the coordinates and momenta of a system, the Poisson
406	+	bracket ${F, G}$ is defined as
407	+	\begin{equation}
408	+	\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial
409	+	F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} -
410	+	\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial
411	+	q_i }}} \right)}.
412	+	\label{introEquation:poissonBracket}
413	+	\end{equation}
414	+	Substituting equations of motion in Hamiltonian formalism
415	+	(Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
416	+	Eq.~\ref{introEquation:motionHamiltonianMomentum}) into
417	+	(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite
418	+	Liouville's theorem using Poisson bracket notion,
419	+	\begin{equation}
420	+	\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
421	+	{\rho ,H} \right\}.
422	+	\label{introEquation:liouvilleTheromInPoissin}
423	+	\end{equation}
424	+	Moreover, the Liouville operator is defined as
425	+	\begin{equation}
426	+	iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial
427	+	p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial
428	+	H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)}
429	+	\label{introEquation:liouvilleOperator}
430	+	\end{equation}
431	+	In terms of Liouville operator, Liouville's equation can also be
432	+	expressed as
433	+	\begin{equation}
434	+	\left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - iL\rho
435	+	\label{introEquation:liouvilleTheoremInOperator}
436	+	\end{equation}
437	+	which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$.
438		\subsection{\label{introSection:ergodic}The Ergodic Hypothesis}
439	+
440	+	Various thermodynamic properties can be calculated from Molecular
441	+	Dynamics simulation. By comparing experimental values with the
442	+	calculated properties, one can determine the accuracy of the
443	+	simulation and the quality of the underlying model. However, both
444	+	experiments and computer simulations are usually performed during a
445	+	certain time interval and the measurements are averaged over a
446	+	period of time which is different from the average behavior of
447	+	many-body system in Statistical Mechanics. Fortunately, the Ergodic
448	+	Hypothesis makes a connection between time average and the ensemble
449	+	average. It states that the time average and average over the
450	+	statistical ensemble are identical \cite{Frenkel1996, Leach2001}:
451	+	\begin{equation}
452	+	\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
453	+	\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
454	+	{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp
455	+	\end{equation}
456	+	where $\langle  A(q , p) \rangle_t$ is an equilibrium value of a
457	+	physical quantity and $\rho (p(t), q(t))$ is the equilibrium
458	+	distribution function. If an observation is averaged over a
459	+	sufficiently long time (longer than the relaxation time), all
460	+	accessible microstates in phase space are assumed to be equally
461	+	probed, giving a properly weighted statistical average. This allows
462	+	the researcher freedom of choice when deciding how best to measure a
463	+	given observable. In case an ensemble averaged approach sounds most
464	+	reasonable, the Monte Carlo methods\cite{Metropolis1949} can be
465	+	utilized. Or if the system lends itself to a time averaging
466	+	approach, the Molecular Dynamics techniques in
467	+	Sec.~\ref{introSection:molecularDynamics} will be the best
468	+	choice\cite{Frenkel1996}.
469	+
470	+	\section{\label{introSection:geometricIntegratos}Geometric Integrators}
471	+	A variety of numerical integrators have been proposed to simulate
472	+	the motions of atoms in MD simulation. They usually begin with
473	+	initial conditionals and move the objects in the direction governed
474	+	by the differential equations. However, most of them ignore the
475	+	hidden physical laws contained within the equations. Since 1990,
476	+	geometric integrators, which preserve various phase-flow invariants
477	+	such as symplectic structure, volume and time reversal symmetry,
478	+	were developed to address this issue\cite{Dullweber1997,
479	+	McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which
480	+	happens to be a simple example of symplectic integrator, continues
481	+	to gain popularity in the molecular dynamics community. This fact
482	+	can be partly explained by its geometric nature.
483	+
484	+	\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds}
485	+	A \emph{manifold} is an abstract mathematical space. It looks
486	+	locally like Euclidean space, but when viewed globally, it may have
487	+	more complicated structure. A good example of manifold is the
488	+	surface of Earth. It seems to be flat locally, but it is round if
489	+	viewed as a whole. A \emph{differentiable manifold} (also known as
490	+	\emph{smooth manifold}) is a manifold on which it is possible to
491	+	apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is
492	+	defined as a pair $(M, \omega)$ which consists of a
493	+	\emph{differentiable manifold} $M$ and a close, non-degenerated,
494	+	bilinear symplectic form, $\omega$. A symplectic form on a vector
495	+	space $V$ is a function $\omega(x, y)$ which satisfies
496	+	$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
497	+	\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
498	+	$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in
499	+	vector field is an example of symplectic form. One of the
500	+	motivations to study \emph{symplectic manifolds} in Hamiltonian
501	+	Mechanics is that a symplectic manifold can represent all possible
502	+	configurations of the system and the phase space of the system can
503	+	be described by it's cotangent bundle\cite{Jost2002}. Every
504	+	symplectic manifold is even dimensional. For instance, in Hamilton
505	+	equations, coordinate and momentum always appear in pairs.
506	+
507	+	\subsection{\label{introSection:ODE}Ordinary Differential Equations}
508	+
509	+	For an ordinary differential system defined as
510	+	\begin{equation}
511	+	\dot x = f(x)
512	+	\end{equation}
513	+	where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if
514	+	$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian
515	+	function and $J$ is the skew-symmetric matrix
516	+	\begin{equation}
517	+	J = \left( {\begin{array}{*{20}c}
518	+	0 & I  \\
519	+	{ - I} & 0  \\
520	+	\end{array}} \right)
521	+	\label{introEquation:canonicalMatrix}
522	+	\end{equation}
523	+	where $I$ is an identity matrix. Using this notation, Hamiltonian
524	+	system can be rewritten as,
525	+	\begin{equation}
526	+	\frac{d}{{dt}}x = J\nabla _x H(x).
527	+	\label{introEquation:compactHamiltonian}
528	+	\end{equation}In this case, $f$ is
529	+	called a \emph{Hamiltonian vector field}. Another generalization of
530	+	Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986},
531	+	\begin{equation}
532	+	\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
533	+	\end{equation}
534	+	The most obvious change being that matrix $J$ now depends on $x$.
535	+
536	+	\subsection{\label{introSection:exactFlow}Exact Propagator}
537	+
538	+	Let $x(t)$ be the exact solution of the ODE
539	+	system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can
540	+	define its exact propagator(solution) $\varphi_\tau$
541	+	\[ x(t+\tau)
542	+	=\varphi_\tau(x(t))
543	+	\]
544	+	where $\tau$ is a fixed time step and $\varphi$ is a map from phase
545	+	space to itself. The propagator has the continuous group property,
546	+	\begin{equation}
547	+	\varphi _{\tau _1 }  \circ \varphi _{\tau _2 }  = \varphi _{\tau _1
548	+	+ \tau _2 } .
549	+	\end{equation}
550	+	In particular,
551	+	\begin{equation}
552	+	\varphi _\tau   \circ \varphi _{ - \tau }  = I
553	+	\end{equation}
554	+	Therefore, the exact propagator is self-adjoint,
555	+	\begin{equation}
556	+	\varphi _\tau   = \varphi _{ - \tau }^{ - 1}.
557	+	\end{equation}
558	+	The exact propagator can also be written in terms of operator,
559	+	\begin{equation}
560	+	\varphi _\tau  (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial
561	+	}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
562	+	\label{introEquation:exponentialOperator}
563	+	\end{equation}
564	+	In most cases, it is not easy to find the exact propagator
565	+	$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$,
566	+	which is usually called an integrator. The order of an integrator
567	+	$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to
568	+	order $p$,
569	+	\begin{equation}
570	+	\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1})
571	+	\end{equation}
572	+
573	+	\subsection{\label{introSection:geometricProperties}Geometric Properties}
574	+
575	+	The hidden geometric properties\cite{Budd1999, Marsden1998} of an
576	+	ODE and its propagator play important roles in numerical studies.
577	+	Many of them can be found in systems which occur naturally in
578	+	applications. Let $\varphi$ be the propagator of Hamiltonian vector
579	+	field, $\varphi$ is a \emph{symplectic} propagator if it satisfies,
580	+	\begin{equation}
581	+	{\varphi '}^T J \varphi ' = J.
582	+	\end{equation}
583	+	According to Liouville's theorem, the symplectic volume is invariant
584	+	under a Hamiltonian propagator, which is the basis for classical
585	+	statistical mechanics. Furthermore, the propagator of a Hamiltonian
586	+	vector field on a symplectic manifold can be shown to be a
587	+	symplectomorphism. As to the Poisson system,
588	+	\begin{equation}
589	+	{\varphi '}^T J \varphi ' = J \circ \varphi
590	+	\end{equation}
591	+	is the property that must be preserved by the integrator. It is
592	+	possible to construct a \emph{volume-preserving} propagator for a
593	+	source free ODE ($ \nabla \cdot f = 0 $), if the propagator
594	+	satisfies $ \det d\varphi  = 1$. One can show easily that a
595	+	symplectic propagator will be volume-preserving. Changing the
596	+	variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will
597	+	result in a new system,
598	+	\[
599	+	\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
600	+	\]
601	+	The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
602	+	In other words, the propagator of this vector field is reversible if
603	+	and only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
604	+	conserved quantity of a general differential function is a function
605	+	$ G:R^{2d}  \to R^d $ which is constant for all solutions of the ODE
606	+	$\frac{{dx}}{{dt}} = f(x)$ ,
607	+	\[
608	+	\frac{{dG(x(t))}}{{dt}} = 0.
609	+	\]
610	+	Using the chain rule, one may obtain,
611	+	\[
612	+	\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G,
613	+	\]
614	+	which is the condition for conserved quantities. For a canonical
615	+	Hamiltonian system, the time evolution of an arbitrary smooth
616	+	function $G$ is given by,
617	+	\begin{eqnarray}
618	+	\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\
619	+	& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)).
620	+	\label{introEquation:firstIntegral1}
621	+	\end{eqnarray}
622	+	Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1}
623	+	can be rewritten as
624	+	\[
625	+	\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
626	+	\]
627	+	Therefore, the sufficient condition for $G$ to be a conserved
628	+	quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As
629	+	is well known, the Hamiltonian (or energy) H of a Hamiltonian system
630	+	is a conserved quantity, which is due to the fact $\{ H,H\}  = 0$.
631	+	When designing any numerical methods, one should always try to
632	+	preserve the structural properties of the original ODE and its
633	+	propagator.
634	+
635	+	\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
636	+	A lot of well established and very effective numerical methods have
637	+	been successful precisely because of their symplectic nature even
638	+	though this fact was not recognized when they were first
639	+	constructed. The most famous example is the Verlet-leapfrog method
640	+	in molecular dynamics. In general, symplectic integrators can be
641	+	constructed using one of four different methods.
642	+	\begin{enumerate}
643	+	\item Generating functions
644	+	\item Variational methods
645	+	\item Runge-Kutta methods
646	+	\item Splitting methods
647	+	\end{enumerate}
648	+	Generating functions\cite{Channell1990} tend to lead to methods
649	+	which are cumbersome and difficult to use. In dissipative systems,
650	+	variational methods can capture the decay of energy
651	+	accurately\cite{Kane2000}. Since they are geometrically unstable
652	+	against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
653	+	methods are not suitable for Hamiltonian system. Recently, various
654	+	high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003}
655	+	have been developed to overcome this instability. However, due to
656	+	computational penalty involved in implementing the Runge-Kutta
657	+	methods, they have not attracted much attention from the Molecular
658	+	Dynamics community. Instead, splitting methods have been widely
659	+	accepted since they exploit natural decompositions of the
660	+	system\cite{Tuckerman1992, McLachlan1998}.
661	+
662	+	\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
663	+
664	+	The main idea behind splitting methods is to decompose the discrete
665	+	$\varphi_h$ as a composition of simpler propagators,
666	+	\begin{equation}
667	+	\varphi _h  = \varphi _{h_1 }  \circ \varphi _{h_2 }  \ldots  \circ
668	+	\varphi _{h_n }
669	+	\label{introEquation:FlowDecomposition}
670	+	\end{equation}
671	+	where each of the sub-propagator is chosen such that each represent
672	+	a simpler integration of the system. Suppose that a Hamiltonian
673	+	system takes the form,
674	+	\[
675	+	H = H_1 + H_2.
676	+	\]
677	+	Here, $H_1$ and $H_2$ may represent different physical processes of
678	+	the system. For instance, they may relate to kinetic and potential
679	+	energy respectively, which is a natural decomposition of the
680	+	problem. If $H_1$ and $H_2$ can be integrated using exact
681	+	propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a
682	+	simple first order expression is then given by the Lie-Trotter
683	+	formula
684	+	\begin{equation}
685	+	\varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
686	+	\label{introEquation:firstOrderSplitting}
687	+	\end{equation}
688	+	where $\varphi _h$ is the result of applying the corresponding
689	+	continuous $\varphi _i$ over a time $h$. By definition, as
690	+	$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it
691	+	must follow that each operator $\varphi_i(t)$ is a symplectic map.
692	+	It is easy to show that any composition of symplectic propagators
693	+	yields a symplectic map,
694	+	\begin{equation}
695	+	(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi
696	+	'\phi ' = \phi '^T J\phi ' = J,
697	+	\label{introEquation:SymplecticFlowComposition}
698	+	\end{equation}
699	+	where $\phi$ and $\psi$ both are symplectic maps. Thus operator
700	+	splitting in this context automatically generates a symplectic map.
701	+	The Lie-Trotter
702	+	splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces
703	+	local errors proportional to $h^2$, while the Strang splitting gives
704	+	a second-order decomposition,
705	+	\begin{equation}
706	+	\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
707	+	_{1,h/2} , \label{introEquation:secondOrderSplitting}
708	+	\end{equation}
709	+	which has a local error proportional to $h^3$. The Strang
710	+	splitting's popularity in molecular simulation community attribute
711	+	to its symmetric property,
712	+	\begin{equation}
713	+	\varphi _h^{ - 1} = \varphi _{ - h}.
714	+	\label{introEquation:timeReversible}
715	+	\end{equation}
716	+
717	+	\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}}
718	+	The classical equation for a system consisting of interacting
719	+	particles can be written in Hamiltonian form,
720	+	\[
721	+	H = T + V
722	+	\]
723	+	where $T$ is the kinetic energy and $V$ is the potential energy.
724	+	Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one
725	+	obtains the following:
726	+	\begin{align}
727	+	q(\Delta t) &= q(0) + \dot{q}(0)\Delta t +
728	+	\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, %
729	+	\label{introEquation:Lp10a} \\%
730	+	%
731	+	\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m}
732	+	\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. %
733	+	\label{introEquation:Lp10b}
734	+	\end{align}
735	+	where $F(t)$ is the force at time $t$. This integration scheme is
736	+	known as \emph{velocity verlet} which is
737	+	symplectic(\ref{introEquation:SymplecticFlowComposition}),
738	+	time-reversible(\ref{introEquation:timeReversible}) and
739	+	volume-preserving (\ref{introEquation:volumePreserving}). These
740	+	geometric properties attribute to its long-time stability and its
741	+	popularity in the community. However, the most commonly used
742	+	velocity verlet integration scheme is written as below,
743	+	\begin{align}
744	+	\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &=
745	+	\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\%
746	+	%
747	+	q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),%
748	+	\label{introEquation:Lp9b}\\%
749	+	%
750	+	\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) +
751	+	\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c}
752	+	\end{align}
753	+	From the preceding splitting, one can see that the integration of
754	+	the equations of motion would follow:
755	+	\begin{enumerate}
756	+	\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position.
757	+
758	+	\item Use the half step velocities to move positions one whole step, $\Delta t$.
759	+
760	+	\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move.
761	+
762	+	\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
763	+	\end{enumerate}
764	+	By simply switching the order of the propagators in the splitting
765	+	and composing a new integrator, the \emph{position verlet}
766	+	integrator, can be generated,
767	+	\begin{align}
768	+	\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) +
769	+	\frac{{\Delta t}}{{2m}}\dot q(0)} \right], %
770	+	\label{introEquation:positionVerlet1} \\%
771	+	%
772	+	q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
773	+	q(\Delta t)} \right]. %
774	+	\label{introEquation:positionVerlet2}
775	+	\end{align}
776	+
777	+	\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
778	+
779	+	The Baker-Campbell-Hausdorff formula can be used to determine the
780	+	local error of a splitting method in terms of the commutator of the
781	+	operators(\ref{introEquation:exponentialOperator}) associated with
782	+	the sub-propagator. For operators $hX$ and $hY$ which are associated
783	+	with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have
784	+	\begin{equation}
785	+	\exp (hX + hY) = \exp (hZ)
786	+	\end{equation}
787	+	where
788	+	\begin{equation}
789	+	hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left(
790	+	{[X,[X,Y]] + [Y,[Y,X]]} \right) +  \ldots .
791	+	\end{equation}
792	+	Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by
793	+	\[
794	+	[X,Y] = XY - YX .
795	+	\]
796	+	Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974}
797	+	to the Strang splitting, we can obtain
798	+	\begin{eqnarray*}
799	+	\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 \\
800	+	&   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
801	+	&   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots
802	+	).
803	+	\end{eqnarray*}
804	+	Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local
805	+	error of Strang splitting is proportional to $h^3$. The same
806	+	procedure can be applied to a general splitting of the form
807	+	\begin{equation}
808	+	\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
809	+	1} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
810	+	\end{equation}
811	+	A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher
812	+	order methods. Yoshida proposed an elegant way to compose higher
813	+	order methods based on symmetric splitting\cite{Yoshida1990}. Given
814	+	a symmetric second order base method $ \varphi _h^{(2)} $, a
815	+	fourth-order symmetric method can be constructed by composing,
816	+	\[
817	+	\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
818	+	h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
819	+	\]
820	+	where $ \alpha  =  - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta
821	+	= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric
822	+	integrator $ \varphi _h^{(2n + 2)}$ can be composed by
823	+	\begin{equation}
824	+	\varphi _h^{(2n + 2)}  = \varphi _{\alpha h}^{(2n)}  \circ \varphi
825	+	_{\beta h}^{(2n)}  \circ \varphi _{\alpha h}^{(2n)},
826	+	\end{equation}
827	+	if the weights are chosen as
828	+	\[
829	+	\alpha  =  - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta =
830	+	\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} .
831	+	\]
832	+
833	+	\section{\label{introSection:molecularDynamics}Molecular Dynamics}
834	+
835	+	As one of the principal tools of molecular modeling, Molecular
836	+	dynamics has proven to be a powerful tool for studying the functions
837	+	of biological systems, providing structural, thermodynamic and
838	+	dynamical information. The basic idea of molecular dynamics is that
839	+	macroscopic properties are related to microscopic behavior and
840	+	microscopic behavior can be calculated from the trajectories in
841	+	simulations. For instance, instantaneous temperature of a
842	+	Hamiltonian system of $N$ particles can be measured by
843	+	\[
844	+	T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
845	+	\]
846	+	where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
847	+	respectively, $f$ is the number of degrees of freedom, and $k_B$ is
848	+	the Boltzman constant.
849
850	+	A typical molecular dynamics run consists of three essential steps:
851	+	\begin{enumerate}
852	+	\item Initialization
853	+	\begin{enumerate}
854	+	\item Preliminary preparation
855	+	\item Minimization
856	+	\item Heating
857	+	\item Equilibration
858	+	\end{enumerate}
859	+	\item Production
860	+	\item Analysis
861	+	\end{enumerate}
862	+	These three individual steps will be covered in the following
863	+	sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
864	+	initialization of a simulation. Sec.~\ref{introSection:production}
865	+	will discuss issues of production runs.
866	+	Sec.~\ref{introSection:Analysis} provides the theoretical tools for
867	+	analysis of trajectories.
868	+
869	+	\subsection{\label{introSec:initialSystemSettings}Initialization}
870	+
871	+	\subsubsection{\textbf{Preliminary preparation}}
872	+
873	+	When selecting the starting structure of a molecule for molecular
874	+	simulation, one may retrieve its Cartesian coordinates from public
875	+	databases, such as RCSB Protein Data Bank \textit{etc}. Although
876	+	thousands of crystal structures of molecules are discovered every
877	+	year, many more remain unknown due to the difficulties of
878	+	purification and crystallization. Even for molecules with known
879	+	structures, some important information is missing. For example, a
880	+	missing hydrogen atom which acts as donor in hydrogen bonding must
881	+	be added. Moreover, in order to include electrostatic interactions,
882	+	one may need to specify the partial charges for individual atoms.
883	+	Under some circumstances, we may even need to prepare the system in
884	+	a special configuration. For instance, when studying transport
885	+	phenomenon in membrane systems, we may prepare the lipids in a
886	+	bilayer structure instead of placing lipids randomly in solvent,
887	+	since we are not interested in the slow self-aggregation process.
888	+
889	+	\subsubsection{\textbf{Minimization}}
890	+
891	+	It is quite possible that some of molecules in the system from
892	+	preliminary preparation may be overlapping with each other. This
893	+	close proximity leads to high initial potential energy which
894	+	consequently jeopardizes any molecular dynamics simulations. To
895	+	remove these steric overlaps, one typically performs energy
896	+	minimization to find a more reasonable conformation. Several energy
897	+	minimization methods have been developed to exploit the energy
898	+	surface and to locate the local minimum. While converging slowly
899	+	near the minimum, steepest descent method is extremely robust when
900	+	systems are strongly anharmonic. Thus, it is often used to refine
901	+	structures from crystallographic data. Relying on the Hessian,
902	+	advanced methods like Newton-Raphson converge rapidly to a local
903	+	minimum, but become unstable if the energy surface is far from
904	+	quadratic. Another factor that must be taken into account, when
905	+	choosing energy minimization method, is the size of the system.
906	+	Steepest descent and conjugate gradient can deal with models of any
907	+	size. Because of the limits on computer memory to store the hessian
908	+	matrix and the computing power needed to diagonalize these matrices,
909	+	most Newton-Raphson methods can not be used with very large systems.
910	+
911	+	\subsubsection{\textbf{Heating}}
912	+
913	+	Typically, heating is performed by assigning random velocities
914	+	according to a Maxwell-Boltzman distribution for a desired
915	+	temperature. Beginning at a lower temperature and gradually
916	+	increasing the temperature by assigning larger random velocities, we
917	+	end up setting the temperature of the system to a final temperature
918	+	at which the simulation will be conducted. In heating phase, we
919	+	should also keep the system from drifting or rotating as a whole. To
920	+	do this, the net linear momentum and angular momentum of the system
921	+	is shifted to zero after each resampling from the Maxwell -Boltzman
922	+	distribution.
923	+
924	+	\subsubsection{\textbf{Equilibration}}
925	+
926	+	The purpose of equilibration is to allow the system to evolve
927	+	spontaneously for a period of time and reach equilibrium. The
928	+	procedure is continued until various statistical properties, such as
929	+	temperature, pressure, energy, volume and other structural
930	+	properties \textit{etc}, become independent of time. Strictly
931	+	speaking, minimization and heating are not necessary, provided the
932	+	equilibration process is long enough. However, these steps can serve
933	+	as a means to arrive at an equilibrated structure in an effective
934	+	way.
935	+
936	+	\subsection{\label{introSection:production}Production}
937	+
938	+	The production run is the most important step of the simulation, in
939	+	which the equilibrated structure is used as a starting point and the
940	+	motions of the molecules are collected for later analysis. In order
941	+	to capture the macroscopic properties of the system, the molecular
942	+	dynamics simulation must be performed by sampling correctly and
943	+	efficiently from the relevant thermodynamic ensemble.
944	+
945	+	The most expensive part of a molecular dynamics simulation is the
946	+	calculation of non-bonded forces, such as van der Waals force and
947	+	Coulombic forces \textit{etc}. For a system of $N$ particles, the
948	+	complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
949	+	which makes large simulations prohibitive in the absence of any
950	+	algorithmic tricks. A natural approach to avoid system size issues
951	+	is to represent the bulk behavior by a finite number of the
952	+	particles. However, this approach will suffer from surface effects
953	+	at the edges of the simulation. To offset this, \textit{Periodic
954	+	boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to
955	+	simulate bulk properties with a relatively small number of
956	+	particles. In this method, the simulation box is replicated
957	+	throughout space to form an infinite lattice. During the simulation,
958	+	when a particle moves in the primary cell, its image in other cells
959	+	move in exactly the same direction with exactly the same
960	+	orientation. Thus, as a particle leaves the primary cell, one of its
961	+	images will enter through the opposite face.
962	+	\begin{figure}
963	+	\centering
964	+	\includegraphics[width=\linewidth]{pbc.eps}
965	+	\caption[An illustration of periodic boundary conditions]{A 2-D
966	+	illustration of periodic boundary conditions. As one particle leaves
967	+	the left of the simulation box, an image of it enters the right.}
968	+	\label{introFig:pbc}
969	+	\end{figure}
970	+
971	+	%cutoff and minimum image convention
972	+	Another important technique to improve the efficiency of force
973	+	evaluation is to apply spherical cutoffs where particles farther
974	+	than a predetermined distance are not included in the calculation
975	+	\cite{Frenkel1996}. The use of a cutoff radius will cause a
976	+	discontinuity in the potential energy curve. Fortunately, one can
977	+	shift a simple radial potential to ensure the potential curve go
978	+	smoothly to zero at the cutoff radius. The cutoff strategy works
979	+	well for Lennard-Jones interaction because of its short range
980	+	nature. However, simply truncating the electrostatic interaction
981	+	with the use of cutoffs has been shown to lead to severe artifacts
982	+	in simulations. The Ewald summation, in which the slowly decaying
983	+	Coulomb potential is transformed into direct and reciprocal sums
984	+	with rapid and absolute convergence, has proved to minimize the
985	+	periodicity artifacts in liquid simulations. Taking the advantages
986	+	of the fast Fourier transform (FFT) for calculating discrete Fourier
987	+	transforms, the particle mesh-based
988	+	methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
989	+	$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the
990	+	\emph{fast multipole method}\cite{Greengard1987, Greengard1994},
991	+	which treats Coulombic interactions exactly at short range, and
992	+	approximate the potential at long range through multipolar
993	+	expansion. In spite of their wide acceptance at the molecular
994	+	simulation community, these two methods are difficult to implement
995	+	correctly and efficiently. Instead, we use a damped and
996	+	charge-neutralized Coulomb potential method developed by Wolf and
997	+	his coworkers\cite{Wolf1999}. The shifted Coulomb potential for
998	+	particle $i$ and particle $j$ at distance $r_{rj}$ is given by:
999	+	\begin{equation}
1000	+	V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
1001	+	r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
1002	+	R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
1003	+	r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb}
1004	+	\end{equation}
1005	+	where $\alpha$ is the convergence parameter. Due to the lack of
1006	+	inherent periodicity and rapid convergence,this method is extremely
1007	+	efficient and easy to implement.
1008	+	\begin{figure}
1009	+	\centering
1010	+	\includegraphics[width=\linewidth]{shifted_coulomb.eps}
1011	+	\caption[An illustration of shifted Coulomb potential]{An
1012	+	illustration of shifted Coulomb potential.}
1013	+	\label{introFigure:shiftedCoulomb}
1014	+	\end{figure}
1015	+
1016	+	%multiple time step
1017	+
1018	+	\subsection{\label{introSection:Analysis} Analysis}
1019	+
1020	+	Recently, advanced visualization technique have become applied to
1021	+	monitor the motions of molecules. Although the dynamics of the
1022	+	system can be described qualitatively from animation, quantitative
1023	+	trajectory analysis is more useful. According to the principles of
1024	+	Statistical Mechanics in
1025	+	Sec.~\ref{introSection:statisticalMechanics}, one can compute
1026	+	thermodynamic properties, analyze fluctuations of structural
1027	+	parameters, and investigate time-dependent processes of the molecule
1028	+	from the trajectories.
1029	+
1030	+	\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}
1031	+
1032	+	Thermodynamic properties, which can be expressed in terms of some
1033	+	function of the coordinates and momenta of all particles in the
1034	+	system, can be directly computed from molecular dynamics. The usual
1035	+	way to measure the pressure is based on virial theorem of Clausius
1036	+	which states that the virial is equal to $-3Nk_BT$. For a system
1037	+	with forces between particles, the total virial, $W$, contains the
1038	+	contribution from external pressure and interaction between the
1039	+	particles:
1040	+	\[
1041	+	W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
1042	+	f_{ij} } } \right\rangle
1043	+	\]
1044	+	where $f_{ij}$ is the force between particle $i$ and $j$ at a
1045	+	distance $r_{ij}$. Thus, the expression for the pressure is given
1046	+	by:
1047	+	\begin{equation}
1048	+	P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
1049	+	< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
1050	+	\end{equation}
1051	+
1052	+	\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}
1053	+
1054	+	Structural Properties of a simple fluid can be described by a set of
1055	+	distribution functions. Among these functions,the \emph{pair
1056	+	distribution function}, also known as \emph{radial distribution
1057	+	function}, is of most fundamental importance to liquid theory.
1058	+	Experimentally, pair distribution functions can be gathered by
1059	+	Fourier transforming raw data from a series of neutron diffraction
1060	+	experiments and integrating over the surface factor
1061	+	\cite{Powles1973}. The experimental results can serve as a criterion
1062	+	to justify the correctness of a liquid model. Moreover, various
1063	+	equilibrium thermodynamic and structural properties can also be
1064	+	expressed in terms of the radial distribution function
1065	+	\cite{Allen1987}. The pair distribution functions $g(r)$ gives the
1066	+	probability that a particle $i$ will be located at a distance $r$
1067	+	from a another particle $j$ in the system
1068	+	\begin{equation}
1069	+	g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
1070	+	\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
1071	+	(r)}{\rho}.
1072	+	\end{equation}
1073	+	Note that the delta function can be replaced by a histogram in
1074	+	computer simulation. Peaks in $g(r)$ represent solvent shells, and
1075	+	the height of these peaks gradually decreases to 1 as the liquid of
1076	+	large distance approaches the bulk density.
1077	+
1078	+
1079	+	\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
1080	+	Properties}}
1081	+
1082	+	Time-dependent properties are usually calculated using \emph{time
1083	+	correlation functions}, which correlate random variables $A$ and $B$
1084	+	at two different times,
1085	+	\begin{equation}
1086	+	C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
1087	+	\label{introEquation:timeCorrelationFunction}
1088	+	\end{equation}
1089	+	If $A$ and $B$ refer to same variable, this kind of correlation
1090	+	function is called an \emph{autocorrelation function}. One example
1091	+	of an auto correlation function is the velocity auto-correlation
1092	+	function which is directly related to transport properties of
1093	+	molecular liquids:
1094	+	\[
1095	+	D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
1096	+	\right\rangle } dt
1097	+	\]
1098	+	where $D$ is diffusion constant. Unlike the velocity autocorrelation
1099	+	function, which is averaged over time origins and over all the
1100	+	atoms, the dipole autocorrelation functions is calculated for the
1101	+	entire system. The dipole autocorrelation function is given by:
1102	+	\[
1103	+	c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
1104	+	\right\rangle
1105	+	\]
1106	+	Here $u_{tot}$ is the net dipole of the entire system and is given
1107	+	by
1108	+	\[
1109	+	u_{tot} (t) = \sum\limits_i {u_i (t)}.
1110	+	\]
1111	+	In principle, many time correlation functions can be related to
1112	+	Fourier transforms of the infrared, Raman, and inelastic neutron
1113	+	scattering spectra of molecular liquids. In practice, one can
1114	+	extract the IR spectrum from the intensity of the molecular dipole
1115	+	fluctuation at each frequency using the following relationship:
1116	+	\[
1117	+	\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
1118	+	i2\pi vt} dt}.
1119	+	\]
1120	+
1121		\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
1122
1123	<	\section{\label{introSection:correlationFunctions}Correlation Functions}
1123	>	Rigid bodies are frequently involved in the modeling of different
1124	>	areas, from engineering, physics, to chemistry. For example,
1125	>	missiles and vehicles are usually modeled by rigid bodies.  The
1126	>	movement of the objects in 3D gaming engines or other physics
1127	>	simulators is governed by rigid body dynamics. In molecular
1128	>	simulations, rigid bodies are used to simplify protein-protein
1129	>	docking studies\cite{Gray2003}.
1130
1131	<	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
1131	>	It is very important to develop stable and efficient methods to
1132	>	integrate the equations of motion for orientational degrees of
1133	>	freedom. Euler angles are the natural choice to describe the
1134	>	rotational degrees of freedom. However, due to $\frac {1}{sin
1135	>	\theta}$ singularities, the numerical integration of corresponding
1136	>	equations of these motion is very inefficient and inaccurate.
1137	>	Although an alternative integrator using multiple sets of Euler
1138	>	angles can overcome this difficulty\cite{Barojas1973}, the
1139	>	computational penalty and the loss of angular momentum conservation
1140	>	still remain. A singularity-free representation utilizing
1141	>	quaternions was developed by Evans in 1977\cite{Evans1977}.
1142	>	Unfortunately, this approach uses a nonseparable Hamiltonian
1143	>	resulting from the quaternion representation, which prevents the
1144	>	symplectic algorithm from being utilized. Another different approach
1145	>	is to apply holonomic constraints to the atoms belonging to the
1146	>	rigid body. Each atom moves independently under the normal forces
1147	>	deriving from potential energy and constraint forces which are used
1148	>	to guarantee the rigidness. However, due to their iterative nature,
1149	>	the SHAKE and Rattle algorithms also converge very slowly when the
1150	>	number of constraints increases\cite{Ryckaert1977, Andersen1983}.
1151
1152	<	\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
1152	>	A break-through in geometric literature suggests that, in order to
1153	>	develop a long-term integration scheme, one should preserve the
1154	>	symplectic structure of the propagator. By introducing a conjugate
1155	>	momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's
1156	>	equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was
1157	>	proposed to evolve the Hamiltonian system in a constraint manifold
1158	>	by iteratively satisfying the orthogonality constraint $Q^T Q = 1$.
1159	>	An alternative method using the quaternion representation was
1160	>	developed by Omelyan\cite{Omelyan1998}. However, both of these
1161	>	methods are iterative and inefficient. In this section, we descibe a
1162	>	symplectic Lie-Poisson integrator for rigid bodies developed by
1163	>	Dullweber and his coworkers\cite{Dullweber1997} in depth.
1164
1165	<	\subsection{\label{introSection:hydroynamics}Hydrodynamics}
1165	>	\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies}
1166	>	The motion of a rigid body is Hamiltonian with the Hamiltonian
1167	>	function
1168	>	\begin{equation}
1169	>	H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
1170	>	V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
1171	>	\label{introEquation:RBHamiltonian}
1172	>	\end{equation}
1173	>	Here, $q$ and $Q$  are the position and rotation matrix for the
1174	>	rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ , and
1175	>	$J$, a diagonal matrix, is defined by
1176	>	\[
1177	>	I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
1178	>	\]
1179	>	where $I_{ii}$ is the diagonal element of the inertia tensor. This
1180	>	constrained Hamiltonian equation is subjected to a holonomic
1181	>	constraint,
1182	>	\begin{equation}
1183	>	Q^T Q = 1, \label{introEquation:orthogonalConstraint}
1184	>	\end{equation}
1185	>	which is used to ensure rotation matrix's unitarity. Differentiating
1186	>	Eq.~\ref{introEquation:orthogonalConstraint} and using
1187	>	Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain,
1188	>	\begin{equation}
1189	>	Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1190	>	\label{introEquation:RBFirstOrderConstraint}
1191	>	\end{equation}
1192	>	Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
1193	>	\ref{introEquation:motionHamiltonianMomentum}), one can write down
1194	>	the equations of motion,
1195	>	\begin{eqnarray}
1196	>	\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\
1197	>	\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\
1198	>	\frac{{dQ}}{{dt}} & = & PJ^{ - 1},  \label{introEquation:RBMotionRotation}\\
1199	>	\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
1200	>	\end{eqnarray}
1201	>	In general, there are two ways to satisfy the holonomic constraints.
1202	>	We can use a constraint force provided by a Lagrange multiplier on
1203	>	the normal manifold to keep the motion on constraint space. Or we
1204	>	can simply evolve the system on the constraint manifold. These two
1205	>	methods have been proved to be equivalent. The holonomic constraint
1206	>	and equations of motions define a constraint manifold for rigid
1207	>	bodies
1208	>	\[
1209	>	M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
1210	>	\right\}.
1211	>	\]
1212	>	Unfortunately, this constraint manifold is not the cotangent bundle
1213	>	$T^* SO(3)$ which can be consider as a symplectic manifold on Lie
1214	>	rotation group $SO(3)$. However, it turns out that under symplectic
1215	>	transformation, the cotangent space and the phase space are
1216	>	diffeomorphic. By introducing
1217	>	\[
1218	>	\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
1219	>	\]
1220	>	the mechanical system subject to a holonomic constraint manifold $M$
1221	>	can be re-formulated as a Hamiltonian system on the cotangent space
1222	>	\[
1223	>	T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
1224	>	1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
1225	>	\]
1226	>	For a body fixed vector $X_i$ with respect to the center of mass of
1227	>	the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
1228	>	given as
1229	>	\begin{equation}
1230	>	X_i^{lab} = Q X_i + q.
1231	>	\end{equation}
1232	>	Therefore, potential energy $V(q,Q)$ is defined by
1233	>	\[
1234	>	V(q,Q) = V(Q X_0 + q).
1235	>	\]
1236	>	Hence, the force and torque are given by
1237	>	\[
1238	>	\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
1239	>	\]
1240	>	and
1241	>	\[
1242	>	\nabla _Q V(q,Q) = F(q,Q)X_i^t
1243	>	\]
1244	>	respectively. As a common choice to describe the rotation dynamics
1245	>	of the rigid body, the angular momentum on the body fixed frame $\Pi
1246	>	= Q^t P$ is introduced to rewrite the equations of motion,
1247	>	\begin{equation}
1248	>	\begin{array}{l}
1249	>	\dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
1250	>	\dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
1251	>	\end{array}
1252	>	\label{introEqaution:RBMotionPI}
1253	>	\end{equation}
1254	>	as well as holonomic constraints $\Pi J^{ - 1}  + J^{ - 1} \Pi ^t  =
1255	>	0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a
1256	>	matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism,
1257	>	\begin{equation}
1258	>	v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
1259	>	{\begin{array}{*{20}c}
1260	>	0 & { - v_3 } & {v_2 }  \\
1261	>	{v_3 } & 0 & { - v_1 }  \\
1262	>	{ - v_2 } & {v_1 } & 0  \\
1263	>	\end{array}} \right),
1264	>	\label{introEquation:hatmapIsomorphism}
1265	>	\end{equation}
1266	>	will let us associate the matrix products with traditional vector
1267	>	operations
1268	>	\[
1269	>	\hat vu = v \times u.
1270	>	\]
1271	>	Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
1272	>	matrix,
1273	>	\begin{eqnarray}
1274	>	(\dot \Pi  - \dot \Pi ^T )&= &(\Pi  - \Pi ^T )(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
1275	>	& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  -
1276	>	(\Lambda  - \Lambda ^T ). \label{introEquation:skewMatrixPI}
1277	>	\end{eqnarray}
1278	>	Since $\Lambda$ is symmetric, the last term of
1279	>	Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
1280	>	Lagrange multiplier $\Lambda$ is absent from the equations of
1281	>	motion. This unique property eliminates the requirement of
1282	>	iterations which can not be avoided in other methods\cite{Kol1997,
1283	>	Omelyan1998}. Applying the hat-map isomorphism, we obtain the
1284	>	equation of motion for angular momentum on body frame
1285	>	\begin{equation}
1286	>	\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
1287	>	F_i (r,Q)} \right) \times X_i }.
1288	>	\label{introEquation:bodyAngularMotion}
1289	>	\end{equation}
1290	>	In the same manner, the equation of motion for rotation matrix is
1291	>	given by
1292	>	\[
1293	>	\dot Q = Qskew(I^{ - 1} \pi ).
1294	>	\]
1295	>
1296	>	\subsection{\label{introSection:SymplecticFreeRB}Symplectic
1297	>	Lie-Poisson Integrator for Free Rigid Body}
1298	>
1299	>	If there are no external forces exerted on the rigid body, the only
1300	>	contribution to the rotational motion is from the kinetic energy
1301	>	(the first term of \ref{introEquation:bodyAngularMotion}). The free
1302	>	rigid body is an example of a Lie-Poisson system with Hamiltonian
1303	>	function
1304	>	\begin{equation}
1305	>	T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
1306	>	\label{introEquation:rotationalKineticRB}
1307	>	\end{equation}
1308	>	where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
1309	>	Lie-Poisson structure matrix,
1310	>	\begin{equation}
1311	>	J(\pi ) = \left( {\begin{array}{*{20}c}
1312	>	0 & {\pi _3 } & { - \pi _2 }  \\
1313	>	{ - \pi _3 } & 0 & {\pi _1 }  \\
1314	>	{\pi _2 } & { - \pi _1 } & 0  \\
1315	>	\end{array}} \right).
1316	>	\end{equation}
1317	>	Thus, the dynamics of free rigid body is governed by
1318	>	\begin{equation}
1319	>	\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
1320	>	\end{equation}
1321	>	One may notice that each $T_i^r$ in
1322	>	Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly.
1323	>	For instance, the equations of motion due to $T_1^r$ are given by
1324	>	\begin{equation}
1325	>	\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
1326	>	\label{introEqaution:RBMotionSingleTerm}
1327	>	\end{equation}
1328	>	with
1329	>	\[ R_1  = \left( {\begin{array}{*{20}c}
1330	>	0 & 0 & 0  \\
1331	>	0 & 0 & {\pi _1 }  \\
1332	>	0 & { - \pi _1 } & 0  \\
1333	>	\end{array}} \right).
1334	>	\]
1335	>	The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is
1336	>	\[
1337	>	\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
1338	>	Q(0)e^{\Delta tR_1 }
1339	>	\]
1340	>	with
1341	>	\[
1342	>	e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
1343	>	0 & 0 & 0  \\
1344	>	0 & {\cos \theta _1 } & {\sin \theta _1 }  \\
1345	>	0 & { - \sin \theta _1 } & {\cos \theta _1 }  \\
1346	>	\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
1347	>	\]
1348	>	To reduce the cost of computing expensive functions in $e^{\Delta
1349	>	tR_1 }$, we can use Cayley transformation to obtain a single-aixs
1350	>	propagator,
1351	>	\[
1352	>	e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
1353	>	).
1354	>	\]
1355	>	The propagator maps for $T_2^r$ and $T_3^r$ can be found in the same
1356	>	manner. In order to construct a second-order symplectic method, we
1357	>	split the angular kinetic Hamiltonian function into five terms
1358	>	\[
1359	>	T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
1360	>	) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
1361	>	(\pi _1 ).
1362	>	\]
1363	>	By concatenating the propagators corresponding to these five terms,
1364	>	we can obtain an symplectic integrator,
1365	>	\[
1366	>	\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
1367	>	\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
1368	>	\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1369	>	_1 }.
1370	>	\]
1371	>	The non-canonical Lie-Poisson bracket ${F, G}$ of two function
1372	>	$F(\pi )$ and $G(\pi )$ is defined by
1373	>	\[
1374	>	\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
1375	>	).
1376	>	\]
1377	>	If the Poisson bracket of a function $F$ with an arbitrary smooth
1378	>	function $G$ is zero, $F$ is a \emph{Casimir}, which is the
1379	>	conserved quantity in Poisson system. We can easily verify that the
1380	>	norm of the angular momentum, $\parallel \pi
1381	>	\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel
1382	>	\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
1383	>	then by the chain rule
1384	>	\[
1385	>	\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
1386	>	}}{2})\pi.
1387	>	\]
1388	>	Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
1389	>	\pi
1390	>	\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
1391	>	Lie-Poisson integrator is found to be both extremely efficient and
1392	>	stable. These properties can be explained by the fact the small
1393	>	angle approximation is used and the norm of the angular momentum is
1394	>	conserved.
1395	>
1396	>	\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
1397	>	Splitting for Rigid Body}
1398	>
1399	>	The Hamiltonian of rigid body can be separated in terms of kinetic
1400	>	energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations
1401	>	of motion corresponding to potential energy and kinetic energy are
1402	>	listed in the below table,
1403	>	\begin{table}
1404	>	\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
1405	>	\begin{center}
1406	>	\begin{tabular}{|l|l|}
1407	>	\hline
1408	>	% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
1409	>	Potential & Kinetic \\
1410	>	$\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
1411	>	$\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
1412	>	$\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
1413	>	$ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
1414	>	\hline
1415	>	\end{tabular}
1416	>	\end{center}
1417	>	\end{table}
1418	>	A second-order symplectic method is now obtained by the composition
1419	>	of the position and velocity propagators,
1420	>	\[
1421	>	\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
1422	>	_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
1423	>	\]
1424	>	Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
1425	>	sub-propagators which corresponding to force and torque
1426	>	respectively,
1427	>	\[
1428	>	\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
1429	>	_{\Delta t/2,\tau }.
1430	>	\]
1431	>	Since the associated operators of $\varphi _{\Delta t/2,F} $ and
1432	>	$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order
1433	>	inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the
1434	>	kinetic energy can be separated to translational kinetic term, $T^t
1435	>	(p)$, and rotational kinetic term, $T^r (\pi )$,
1436	>	\begin{equation}
1437	>	T(p,\pi ) =T^t (p) + T^r (\pi ).
1438	>	\end{equation}
1439	>	where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
1440	>	defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore,
1441	>	the corresponding propagators are given by
1442	>	\[
1443	>	\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
1444	>	_{\Delta t,T^r }.
1445	>	\]
1446	>	Finally, we obtain the overall symplectic propagators for freely
1447	>	moving rigid bodies
1448	>	\begin{eqnarray}
1449	>	\varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \notag\\
1450	>	& & \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\\
1451	>	& & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
1452	>	\label{introEquation:overallRBFlowMaps}
1453	>	\end{eqnarray}
1454	>
1455	>	\section{\label{introSection:langevinDynamics}Langevin Dynamics}
1456	>	As an alternative to newtonian dynamics, Langevin dynamics, which
1457	>	mimics a simple heat bath with stochastic and dissipative forces,
1458	>	has been applied in a variety of studies. This section will review
1459	>	the theory of Langevin dynamics. A brief derivation of generalized
1460	>	Langevin equation will be given first. Following that, we will
1461	>	discuss the physical meaning of the terms appearing in the equation
1462	>	as well as the calculation of friction tensor from hydrodynamics
1463	>	theory.
1464	>
1465	>	\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
1466	>
1467	>	A harmonic bath model, in which an effective set of harmonic
1468	>	oscillators are used to mimic the effect of a linearly responding
1469	>	environment, has been widely used in quantum chemistry and
1470	>	statistical mechanics. One of the successful applications of
1471	>	Harmonic bath model is the derivation of the Generalized Langevin
1472	>	Dynamics (GLE). Lets consider a system, in which the degree of
1473	>	freedom $x$ is assumed to couple to the bath linearly, giving a
1474	>	Hamiltonian of the form
1475	>	\begin{equation}
1476	>	H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
1477	>	\label{introEquation:bathGLE}.
1478	>	\end{equation}
1479	>	Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated
1480	>	with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian,
1481	>	\[
1482	>	H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
1483	>	}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  \omega _\alpha ^2 }
1484	>	\right\}}
1485	>	\]
1486	>	where the index $\alpha$ runs over all the bath degrees of freedom,
1487	>	$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
1488	>	the harmonic bath masses, and $\Delta U$ is a bilinear system-bath
1489	>	coupling,
1490	>	\[
1491	>	\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
1492	>	\]
1493	>	where $g_\alpha$ are the coupling constants between the bath
1494	>	coordinates ($x_ \alpha$) and the system coordinate ($x$).
1495	>	Introducing
1496	>	\[
1497	>	W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
1498	>	}}{{2m_\alpha  w_\alpha ^2 }}} x^2
1499	>	\]
1500	>	and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
1501	>	\[
1502	>	H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
1503	>	{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
1504	>	w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
1505	>	w_\alpha ^2 }}x} \right)^2 } \right\}}.
1506	>	\]
1507	>	Since the first two terms of the new Hamiltonian depend only on the
1508	>	system coordinates, we can get the equations of motion for
1509	>	Generalized Langevin Dynamics by Hamilton's equations,
1510	>	\begin{equation}
1511	>	m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
1512	>	\sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   -
1513	>	\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)},
1514	>	\label{introEquation:coorMotionGLE}
1515	>	\end{equation}
1516	>	and
1517	>	\begin{equation}
1518	>	m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x_\alpha   -
1519	>	\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
1520	>	\label{introEquation:bathMotionGLE}
1521	>	\end{equation}
1522	>	In order to derive an equation for $x$, the dynamics of the bath
1523	>	variables $x_\alpha$ must be solved exactly first. As an integral
1524	>	transform which is particularly useful in solving linear ordinary
1525	>	differential equations,the Laplace transform is the appropriate tool
1526	>	to solve this problem. The basic idea is to transform the difficult
1527	>	differential equations into simple algebra problems which can be
1528	>	solved easily. Then, by applying the inverse Laplace transform, also
1529	>	known as the Bromwich integral, we can retrieve the solutions of the
1530	>	original problems. Let $f(t)$ be a function defined on $ [0,\infty )
1531	>	$, the Laplace transform of $f(t)$ is a new function defined as
1532	>	\[
1533	>	L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
1534	>	\]
1535	>	where  $p$ is real and  $L$ is called the Laplace Transform
1536	>	Operator. Below are some important properties of Laplace transform
1537	>	\begin{eqnarray*}
1538	>	L(x + y)  & = & L(x) + L(y) \\
1539	>	L(ax)     & = & aL(x) \\
1540	>	L(\dot x) & = & pL(x) - px(0) \\
1541	>	L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
1542	>	L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
1543	>	\end{eqnarray*}
1544	>	Applying the Laplace transform to the bath coordinates, we obtain
1545	>	\begin{eqnarray*}
1546	>	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), \\
1547	>	L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}. \\
1548	>	\end{eqnarray*}
1549	>	By the same way, the system coordinates become
1550	>	\begin{eqnarray*}
1551	>	mL(\ddot x) & = &
1552	>	- \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\}}  \\
1553	>	& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}.
1554	>	\end{eqnarray*}
1555	>	With the help of some relatively important inverse Laplace
1556	>	transformations:
1557	>	\[
1558	>	\begin{array}{c}
1559	>	L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
1560	>	L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
1561	>	L(1) = \frac{1}{p} \\
1562	>	\end{array}
1563	>	\]
1564	>	we obtain
1565	>	\begin{eqnarray*}
1566	>	m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
1567	>	\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
1568	>	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
1569	>	_\alpha  t)\dot x(t - \tau )d\tau } } \right\}}  \\
1570	>	& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
1571	>	x_\alpha (0) - \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}}
1572	>	\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
1573	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
1574	>	\end{eqnarray*}
1575	>	\begin{eqnarray*}
1576	>	m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
1577	>	{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
1578	>	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
1579	>	t)\dot x(t - \tau )d} \tau }  \\
1580	>	& & + \sum\limits_{\alpha  = 1}^N {\left\{ {\left[ {g_\alpha
1581	>	x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha  }}}
1582	>	\right]\cos (\omega _\alpha  t) + \frac{{g_\alpha  \dot x_\alpha
1583	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)} \right\}}
1584	>	\end{eqnarray*}
1585	>	Introducing a \emph{dynamic friction kernel}
1586	>	\begin{equation}
1587	>	\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
1588	>	}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
1589	>	\label{introEquation:dynamicFrictionKernelDefinition}
1590	>	\end{equation}
1591	>	and \emph{a random force}
1592	>	\begin{equation}
1593	>	R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
1594	>	- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
1595	>	\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
1596	>	(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
1597	>	\label{introEquation:randomForceDefinition}
1598	>	\end{equation}
1599	>	the equation of motion can be rewritten as
1600	>	\begin{equation}
1601	>	m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
1602	>	(t)\dot x(t - \tau )d\tau }  + R(t)
1603	>	\label{introEuqation:GeneralizedLangevinDynamics}
1604	>	\end{equation}
1605	>	which is known as the \emph{generalized Langevin equation}.
1606	>
1607	>	\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
1608	>
1609	>	One may notice that $R(t)$ depends only on initial conditions, which
1610	>	implies it is completely deterministic within the context of a
1611	>	harmonic bath. However, it is easy to verify that $R(t)$ is totally
1612	>	uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)}
1613	>	\right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
1614	>	0.$ This property is what we expect from a truly random process. As
1615	>	long as the model chosen for $R(t)$ was a gaussian distribution in
1616	>	general, the stochastic nature of the GLE still remains.
1617	>	%dynamic friction kernel
1618	>	The convolution integral
1619	>	\[
1620	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
1621	>	\]
1622	>	depends on the entire history of the evolution of $x$, which implies
1623	>	that the bath retains memory of previous motions. In other words,
1624	>	the bath requires a finite time to respond to change in the motion
1625	>	of the system. For a sluggish bath which responds slowly to changes
1626	>	in the system coordinate, we may regard $\xi(t)$ as a constant
1627	>	$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
1628	>	\[
1629	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
1630	>	\]
1631	>	and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1632	>	\[
1633	>	m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
1634	>	\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
1635	>	\]
1636	>	which can be used to describe the effect of dynamic caging in
1637	>	viscous solvents. The other extreme is the bath that responds
1638	>	infinitely quickly to motions in the system. Thus, $\xi (t)$ can be
1639	>	taken as a $delta$ function in time:
1640	>	\[
1641	>	\xi (t) = 2\xi _0 \delta (t)
1642	>	\]
1643	>	Hence, the convolution integral becomes
1644	>	\[
1645	>	\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
1646	>	{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
1647	>	\]
1648	>	and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1649	>	\begin{equation}
1650	>	m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
1651	>	x(t) + R(t) \label{introEquation:LangevinEquation}
1652	>	\end{equation}
1653	>	which is known as the Langevin equation. The static friction
1654	>	coefficient $\xi _0$ can either be calculated from spectral density
1655	>	or be determined by Stokes' law for regular shaped particles. A
1656	>	briefly review on calculating friction tensor for arbitrary shaped
1657	>	particles is given in Sec.~\ref{introSection:frictionTensor}.
1658	>
1659	>	\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
1660	>
1661	>	Defining a new set of coordinates
1662	>	\[
1663	>	q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
1664	>	^2 }}x(0),
1665	>	\]
1666	>	we can rewrite $R(T)$ as
1667	>	\[
1668	>	R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
1669	>	\]
1670	>	And since the $q$ coordinates are harmonic oscillators,
1671	>	\begin{eqnarray*}
1672	>	\left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
1673	>	\left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
1674	>	\left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
1675	>	\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 } }  \\
1676	>	& = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
1677	>	& = &kT\xi (t) \\
1678	>	\end{eqnarray*}
1679	>	Thus, we recover the \emph{second fluctuation dissipation theorem}
1680	>	\begin{equation}
1681	>	\xi (t) = \left\langle {R(t)R(0)} \right\rangle
1682	>	\label{introEquation:secondFluctuationDissipation},
1683	>	\end{equation}
1684	>	which acts as a constraint on the possible ways in which one can
1685	>	model the random force and friction kernel.

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