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# Line 6 | Line 6 | for a given system of particles. There are three funda
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
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
# Line 17 | Line 17 | motion of particles is the foundation of the classical
17   \subsection{\label{introSection:newtonian}Newtonian Mechanics}
18   The discovery of Newton's three laws of mechanics which govern the
19   motion of particles is the foundation of the classical mechanics.
20 < Newton¡¯s first law defines a class of inertial frames. Inertial
20 > Newton's first law defines a class of inertial frames. Inertial
21   frames are reference frames where a particle not interacting with
22   other bodies will move with constant speed in the same direction.
23 < With respect to inertial frames Newton¡¯s second law has the form
23 > With respect to inertial frames, Newton's second law has the form
24   \begin{equation}
25 < F = \frac {dp}{dt} = \frac {mv}{dt}
25 > F = \frac {dp}{dt} = \frac {mdv}{dt}
26   \label{introEquation:newtonSecondLaw}
27   \end{equation}
28   A point mass interacting with other bodies moves with the
29   acceleration along the direction of the force acting on it. Let
30   $F_{ij}$ be the force that particle $i$ exerts on particle $j$, and
31   $F_{ji}$ be the force that particle $j$ exerts on particle $i$.
32 < Newton¡¯s third law states that
32 > Newton's third law states that
33   \begin{equation}
34   F_{ij} = -F_{ji}
35   \label{introEquation:newtonThirdLaw}
# 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
# Line 68 | Line 68 | important factors to determine the quality of numerica
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}.
71 > schemes for rigid bodies \cite{Dullweber1997}.
72  
73   \subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
74  
75 < Newtonian Mechanics suffers from two important limitations: it
76 < describes their motion in special cartesian coordinate systems.
77 < Another limitation of Newtonian mechanics becomes obvious when we
78 < try to describe systems with large numbers of particles. It becomes
79 < very difficult to predict the properties of the system by carrying
80 < out calculations involving the each individual interaction between
81 < all the particles, even if we know all of the details of the
82 < interaction. In order to overcome some of the practical difficulties
83 < which arise in attempts to apply Newton's equation to complex
84 < system, alternative procedures may be developed.
75 > Newtonian Mechanics suffers from two important limitations: motions
76 > can only be described in cartesian coordinate systems. Moreover, It
77 > become impossible to predict analytically the properties of the
78 > system even if we know all of the details of the interaction. In
79 > order to overcome some of the practical difficulties which arise in
80 > attempts to apply Newton's equation to complex system, approximate
81 > numerical 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,
87 > possible to base all of mechanics and most of classical physics.
88 > Hamilton's Principle may be stated as follows,
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
93 < the kinetic, $K$, and potential energies, $U$ \cite{Tolman1979}.
93 > the kinetic, $K$, and potential energies, $U$.
94   \begin{equation}
95   \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
96   \label{introEquation:halmitonianPrinciple1}
97   \end{equation}
98  
99   For simple mechanical systems, where the forces acting on the
100 < different part are derivable from a potential and the velocities are
101 < small compared with that of light, the Lagrangian function $L$ can
102 < be define as the difference between the kinetic energy of the system
106 < and its potential energy,
100 > different parts are derivable from a potential, the Lagrangian
101 > function $L$ can be defined as the difference between the kinetic
102 > energy of the system and its potential energy,
103   \begin{equation}
104   L \equiv K - U = L(q_i ,\dot q_i ) ,
105   \label{introEquation:lagrangianDef}
# Line 114 | Line 110 | then Eq.~\ref{introEquation:halmitonianPrinciple1} bec
110   \label{introEquation:halmitonianPrinciple2}
111   \end{equation}
112  
113 < \subsubsection{\label{introSection:equationOfMotionLagrangian}The
114 < Equations of Motion in Lagrangian Mechanics}
113 > \subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The
114 > Equations of Motion in Lagrangian Mechanics}}
115  
116   For a holonomic system of $f$ degrees of freedom, the equations of
117   motion in the Lagrangian form is
# Line 132 | Line 128 | classical mechanics. If the potential energy of a syst
128   Arising from Lagrangian Mechanics, Hamiltonian Mechanics was
129   introduced by William Rowan Hamilton in 1833 as a re-formulation of
130   classical mechanics. If the potential energy of a system is
131 < independent of generalized velocities, the generalized momenta can
136 < be defined as
131 > independent of velocities, the momenta can be defined as
132   \begin{equation}
133   p_i = \frac{\partial L}{\partial \dot q_i}
134   \label{introEquation:generalizedMomenta}
# Line 172 | Line 167 | find
167   By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
168   find
169   \begin{equation}
170 < \frac{{\partial H}}{{\partial p_k }} = q_k
170 > \frac{{\partial H}}{{\partial p_k }} = \dot {q_k}
171   \label{introEquation:motionHamiltonianCoordinate}
172   \end{equation}
173   \begin{equation}
174 < \frac{{\partial H}}{{\partial q_k }} =  - p_k
174 > \frac{{\partial H}}{{\partial q_k }} =  - \dot {p_k}
175   \label{introEquation:motionHamiltonianMomentum}
176   \end{equation}
177   and
# Line 193 | Line 188 | Hamiltonian approach is that the Lagrangian is conside
188  
189   An important difference between Lagrangian approach and the
190   Hamiltonian approach is that the Lagrangian is considered to be a
191 < function of the generalized velocities $\dot q_i$ and the
192 < generalized coordinates $q_i$, while the Hamiltonian is considered
193 < to be a function of the generalized momenta $p_i$ and the conjugate
194 < generalized coordinate $q_i$. Hamiltonian Mechanics is more
195 < appropriate for application to statistical mechanics and quantum
196 < mechanics, since it treats the coordinate and its time derivative as
197 < independent variables and it only works with 1st-order differential
203 < equations\cite{Marion1990}.
191 > function of the generalized velocities $\dot q_i$ and coordinates
192 > $q_i$, while the Hamiltonian is considered to be a function of the
193 > generalized momenta $p_i$ and the conjugate coordinates $q_i$.
194 > Hamiltonian Mechanics is more appropriate for application to
195 > statistical mechanics and quantum mechanics, since it treats the
196 > coordinate and its time derivative as independent variables and it
197 > only works with 1st-order differential equations\cite{Marion1990}.
198  
199   In Newtonian Mechanics, a system described by conservative forces
200   conserves the total energy \ref{introEquation:energyConservation}.
# Line 230 | Line 224 | phase space usually consists of all possible values of
224   possible states. Each possible state of the system corresponds to
225   one unique point in the phase space. For mechanical systems, the
226   phase space usually consists of all possible values of position and
227 < momentum variables. Consider a dynamic system in a cartesian space,
228 < where each of the $6f$ coordinates and momenta is assigned to one of
229 < $6f$ mutually orthogonal axes, the phase space of this system is a
230 < $6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 ,
231 < \ldots ,p_f )$, with a unique set of values of $6f$ coordinates and
232 < momenta is a phase space vector.
227 > momentum variables. Consider a dynamic system of $f$ particles in a
228 > cartesian space, where each of the $6f$ coordinates and momenta is
229 > assigned to one of $6f$ mutually orthogonal axes, the phase space of
230 > this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots
231 > ,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$
232 > coordinates and momenta is a phase space vector.
233  
234   A microscopic state or microstate of a classical system is
235   specification of the complete phase space vector of a system at any
# Line 257 | Line 251 | with which representative points are distributed over
251   regions of the phase space. The condition of an ensemble at any time
252   can be regarded as appropriately specified by the density $\rho$
253   with which representative points are distributed over the phase
254 < space. The density of distribution for an ensemble with $f$ degrees
255 < of freedom is defined as,
254 > space. The density distribution for an ensemble with $f$ degrees of
255 > freedom is defined as,
256   \begin{equation}
257   \rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
258   \label{introEquation:densityDistribution}
259   \end{equation}
260   Governed by the principles of mechanics, the phase points change
261 < their value which would change the density at any time at phase
262 < space. Hence, the density of distribution is also to be taken as a
261 > their locations which would change the density at any time at phase
262 > space. Hence, the density distribution is also to be taken as a
263   function of the time.
264  
265   The number of systems $\delta N$ at time $t$ can be determined by,
# Line 273 | Line 267 | The number of systems $\delta N$ at time $t$ can be de
267   \delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
268   \label{introEquation:deltaN}
269   \end{equation}
270 < Assuming a large enough population of systems are exploited, we can
271 < sufficiently approximate $\delta N$ without introducing
272 < discontinuity when we go from one region in the phase space to
273 < another. By integrating over the whole phase space,
270 > Assuming a large enough population of systems, we can sufficiently
271 > approximate $\delta N$ without introducing discontinuity when we go
272 > from one region in the phase space to another. By integrating over
273 > the whole phase space,
274   \begin{equation}
275   N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
276   \label{introEquation:totalNumberSystem}
# Line 293 | Line 287 | complex, or stochastic, or even discontinuous, the ave
287   value of any desired quantity which depends on the coordinates and
288   momenta of the system. Even when the dynamics of the real system is
289   complex, or stochastic, or even discontinuous, the average
290 < properties of the ensemble of possibilities as a whole may still
291 < remain well defined. For a classical system in thermal equilibrium
292 < with its environment, the ensemble average of a mechanical quantity,
293 < $\langle A(q , p) \rangle_t$, takes the form of an integral over the
294 < phase space of the system,
290 > properties of the ensemble of possibilities as a whole remaining
291 > well defined. For a classical system in thermal equilibrium with its
292 > environment, the ensemble average of a mechanical quantity, $\langle
293 > A(q , p) \rangle_t$, takes the form of an integral over the phase
294 > space of the system,
295   \begin{equation}
296   \langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
297   (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
# Line 307 | Line 301 | statistical characteristics. As a function of macrosco
301  
302   There are several different types of ensembles with different
303   statistical characteristics. As a function of macroscopic
304 < parameters, such as temperature \textit{etc}, partition function can
305 < be used to describe the statistical properties of a system in
304 > parameters, such as temperature \textit{etc}, the partition function
305 > can be used to describe the statistical properties of a system in
306   thermodynamic equilibrium.
307  
308   As an ensemble of systems, each of which is known to be thermally
309 < isolated and conserve energy, Microcanonical ensemble(NVE) has a
309 > isolated and conserve energy, the Microcanonical ensemble(NVE) has a
310   partition function like,
311   \begin{equation}
312   \Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
# Line 326 | Line 320 | Here, $A$ is the Helmholtz free energy which is define
320   \label{introEquation:NVTPartition}
321   \end{equation}
322   Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
323 < TS$. Since most experiment are carried out under constant pressure
324 < condition, isothermal-isobaric ensemble(NPT) play a very important
325 < role in molecular simulation. The isothermal-isobaric ensemble allow
326 < the system to exchange energy with a heat bath of temperature $T$
327 < and to change the volume as well. Its partition function is given as
323 > TS$. Since most experiments are carried out under constant pressure
324 > condition, the isothermal-isobaric ensemble(NPT) plays a very
325 > important role in molecular simulations. The isothermal-isobaric
326 > ensemble allow the system to exchange energy with a heat bath of
327 > temperature $T$ and to change the volume as well. Its partition
328 > function is given as
329   \begin{equation}
330   \Delta (N,P,T) =  - e^{\beta G}.
331   \label{introEquation:NPTPartition}
# Line 339 | Line 334 | Here, $G = U - TS + PV$, and $G$ is called Gibbs free
334  
335   \subsection{\label{introSection:liouville}Liouville's theorem}
336  
337 < The Liouville's theorem is the foundation on which statistical
338 < mechanics rests. It describes the time evolution of phase space
337 > Liouville's theorem is the foundation on which statistical mechanics
338 > rests. It describes the time evolution of the phase space
339   distribution function. In order to calculate the rate of change of
340   $\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we
341   consider the two faces perpendicular to the $q_1$ axis, which are
# Line 369 | Line 364 | which cancels the first terms of the right hand side.
364   + \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
365   \end{equation}
366   which cancels the first terms of the right hand side. Furthermore,
367 < divining $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
367 > dividing $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
368   p_f $ in both sides, we can write out Liouville's theorem in a
369   simple form,
370   \begin{equation}
# Line 395 | Line 390 | distribution,
390   \label{introEquation:densityAndHamiltonian}
391   \end{equation}
392  
393 < \subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space}
393 > \subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}}
394   Lets consider a region in the phase space,
395   \begin{equation}
396   \delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
397   \end{equation}
398   If this region is small enough, the density $\rho$ can be regarded
399 < as uniform over the whole phase space. Thus, the number of phase
400 < points inside this region is given by,
399 > as uniform over the whole integral. Thus, the number of phase points
400 > inside this region is given by,
401   \begin{equation}
402   \delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
403   dp_1 } ..dp_f.
# Line 414 | Line 409 | With the help of stationary assumption
409   \end{equation}
410   With the help of stationary assumption
411   (\ref{introEquation:stationary}), we obtain the principle of the
412 < \emph{conservation of extension in phase space},
412 > \emph{conservation of volume in phase space},
413   \begin{equation}
414   \frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
415   ...dq_f dp_1 } ..dp_f  = 0.
416   \label{introEquation:volumePreserving}
417   \end{equation}
418  
419 < \subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms}
419 > \subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}}
420  
421   Liouville's theorem can be expresses in a variety of different forms
422   which are convenient within different contexts. For any two function
# Line 463 | Line 458 | calculated properties, one can determine the accuracy
458   Various thermodynamic properties can be calculated from Molecular
459   Dynamics simulation. By comparing experimental values with the
460   calculated properties, one can determine the accuracy of the
461 < simulation and the quality of the underlying model. However, both of
462 < experiment and computer simulation are usually performed during a
461 > simulation and the quality of the underlying model. However, both
462 > experiments and computer simulations are usually performed during a
463   certain time interval and the measurements are averaged over a
464   period of them which is different from the average behavior of
465 < many-body system in Statistical Mechanics. Fortunately, Ergodic
466 < Hypothesis is proposed to make a connection between time average and
467 < ensemble average. It states that time average and average over the
465 > many-body system in Statistical Mechanics. Fortunately, the Ergodic
466 > Hypothesis makes a connection between time average and the ensemble
467 > average. It states that the time average and average over the
468   statistical ensemble are identical \cite{Frenkel1996, Leach2001}.
469   \begin{equation}
470   \langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
# Line 491 | Line 486 | choice\cite{Frenkel1996}.
486   choice\cite{Frenkel1996}.
487  
488   \section{\label{introSection:geometricIntegratos}Geometric Integrators}
489 < A variety of numerical integrators were proposed to simulate the
490 < motions. They usually begin with an initial conditionals and move
491 < the objects in the direction governed by the differential equations.
492 < However, most of them ignore the hidden physical law contained
493 < within the equations. Since 1990, geometric integrators, which
494 < preserve various phase-flow invariants such as symplectic structure,
495 < volume and time reversal symmetry, are developed to address this
496 < issue\cite{Dullweber1997, McLachlan1998, Leimkuhler1999}. The
497 < velocity verlet method, which happens to be a simple example of
498 < symplectic integrator, continues to gain its popularity in molecular
499 < dynamics community. This fact can be partly explained by its
500 < geometric nature.
489 > A variety of numerical integrators have been proposed to simulate
490 > the motions of atoms in MD simulation. They usually begin with
491 > initial conditionals and move the objects in the direction governed
492 > by the differential equations. However, most of them ignore the
493 > hidden physical laws contained within the equations. Since 1990,
494 > geometric integrators, which preserve various phase-flow invariants
495 > such as symplectic structure, volume and time reversal symmetry, are
496 > developed to address this issue\cite{Dullweber1997, McLachlan1998,
497 > Leimkuhler1999}. The velocity verlet method, which happens to be a
498 > simple example of symplectic integrator, continues to gain
499 > popularity in the molecular dynamics community. This fact can be
500 > partly explained by its geometric nature.
501  
502 < \subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
503 < A \emph{manifold} is an abstract mathematical space. It locally
504 < looks like Euclidean space, but when viewed globally, it may have
505 < more complicate structure. A good example of manifold is the surface
506 < of Earth. It seems to be flat locally, but it is round if viewed as
507 < a whole. A \emph{differentiable manifold} (also known as
508 < \emph{smooth manifold}) is a manifold with an open cover in which
509 < the covering neighborhoods are all smoothly isomorphic to one
510 < another. In other words,it is possible to apply calculus on
516 < \emph{differentiable manifold}. A \emph{symplectic manifold} is
517 < defined as a pair $(M, \omega)$ which consisting of a
502 > \subsection{\label{introSection:symplecticManifold}Symplectic Manifolds}
503 > A \emph{manifold} is an abstract mathematical space. It looks
504 > locally like Euclidean space, but when viewed globally, it may have
505 > more complicated structure. A good example of manifold is the
506 > surface of Earth. It seems to be flat locally, but it is round if
507 > viewed as a whole. A \emph{differentiable manifold} (also known as
508 > \emph{smooth manifold}) is a manifold on which it is possible to
509 > apply calculus on \emph{differentiable manifold}. A \emph{symplectic
510 > manifold} is defined as a pair $(M, \omega)$ which consists of a
511   \emph{differentiable manifold} $M$ and a close, non-degenerated,
512   bilinear symplectic form, $\omega$. A symplectic form on a vector
513   space $V$ is a function $\omega(x, y)$ which satisfies
514   $\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
515   \lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
516 < $\omega(x, x) = 0$. Cross product operation in vector field is an
517 < example of symplectic form.
516 > $\omega(x, x) = 0$. The cross product operation in vector field is
517 > an example of symplectic form.
518  
519 < One of the motivations to study \emph{symplectic manifold} in
519 > One of the motivations to study \emph{symplectic manifolds} in
520   Hamiltonian Mechanics is that a symplectic manifold can represent
521   all possible configurations of the system and the phase space of the
522   system can be described by it's cotangent bundle. Every symplectic
523   manifold is even dimensional. For instance, in Hamilton equations,
524   coordinate and momentum always appear in pairs.
525  
533 – Let  $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map
534 – \[
535 – f : M \rightarrow N
536 – \]
537 – is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and
538 – the \emph{pullback} of $\eta$ under f is equal to $\omega$.
539 – Canonical transformation is an example of symplectomorphism in
540 – classical mechanics.
541 –
526   \subsection{\label{introSection:ODE}Ordinary Differential Equations}
527  
528 < For a ordinary differential system defined as
528 > For an ordinary differential system defined as
529   \begin{equation}
530   \dot x = f(x)
531   \end{equation}
532 < where $x = x(q,p)^T$, this system is canonical Hamiltonian, if
532 > where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if
533   \begin{equation}
534   f(r) = J\nabla _x H(r).
535   \end{equation}
# Line 689 | Line 673 | though this fact was not recognized when they were fir
673   A lot of well established and very effective numerical methods have
674   been successful precisely because of their symplecticities even
675   though this fact was not recognized when they were first
676 < constructed. The most famous example is leapfrog methods in
677 < molecular dynamics. In general, symplectic integrators can be
676 > constructed. The most famous example is the Verlet-leapfrog methods
677 > in molecular dynamics. In general, symplectic integrators can be
678   constructed using one of four different methods.
679   \begin{enumerate}
680   \item Generating functions
# Line 708 | Line 692 | instability. However, due to computational penalty inv
692   high-order explicit Runge-Kutta methods
693   \cite{Owren1992,Chen2003}have been developed to overcome this
694   instability. However, due to computational penalty involved in
695 < implementing the Runge-Kutta methods, they do not attract too much
696 < attention from Molecular Dynamics community. Instead, splitting have
697 < been widely accepted since they exploit natural decompositions of
698 < the system\cite{Tuckerman1992, McLachlan1998}.
695 > implementing the Runge-Kutta methods, they have not attracted much
696 > attention from the Molecular Dynamics community. Instead, splitting
697 > methods have been widely accepted since they exploit natural
698 > decompositions of the system\cite{Tuckerman1992, McLachlan1998}.
699  
700 < \subsubsection{\label{introSection:splittingMethod}Splitting Method}
700 > \subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
701  
702   The main idea behind splitting methods is to decompose the discrete
703   $\varphi_h$ as a composition of simpler flows,
# Line 734 | Line 718 | $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a sim
718   energy respectively, which is a natural decomposition of the
719   problem. If $H_1$ and $H_2$ can be integrated using exact flows
720   $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first
721 < order is then given by the Lie-Trotter formula
721 > order expression is then given by the Lie-Trotter formula
722   \begin{equation}
723   \varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
724   \label{introEquation:firstOrderSplitting}
# Line 760 | Line 744 | _{1,h/2} , \label{introEquation:secondOrderSplitting}
744   \varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
745   _{1,h/2} , \label{introEquation:secondOrderSplitting}
746   \end{equation}
747 < which has a local error proportional to $h^3$. Sprang splitting's
748 < popularity in molecular simulation community attribute to its
749 < symmetric property,
747 > which has a local error proportional to $h^3$. The Sprang
748 > splitting's popularity in molecular simulation community attribute
749 > to its symmetric property,
750   \begin{equation}
751   \varphi _h^{ - 1} = \varphi _{ - h}.
752   \label{introEquation:timeReversible}
753   \end{equation}
754  
755 < \subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method}
755 > \subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Example of Splitting Method}}
756   The classical equation for a system consisting of interacting
757   particles can be written in Hamiltonian form,
758   \[
# Line 828 | Line 812 | q(\Delta t)} \right]. %
812   \label{introEquation:positionVerlet2}
813   \end{align}
814  
815 < \subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
815 > \subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
816  
817   Baker-Campbell-Hausdorff formula can be used to determine the local
818   error of splitting method in terms of commutator of the
# Line 921 | Line 905 | trajectory analysis.
905  
906   \subsection{\label{introSec:initialSystemSettings}Initialization}
907  
908 < \subsubsection{Preliminary preparation}
908 > \subsubsection{\textbf{Preliminary preparation}}
909  
910   When selecting the starting structure of a molecule for molecular
911   simulation, one may retrieve its Cartesian coordinates from public
# Line 939 | Line 923 | interested in self-aggregation and it takes a long tim
923   instead of placing lipids randomly in solvent, since we are not
924   interested in self-aggregation and it takes a long time to happen.
925  
926 < \subsubsection{Minimization}
926 > \subsubsection{\textbf{Minimization}}
927  
928   It is quite possible that some of molecules in the system from
929   preliminary preparation may be overlapped with each other. This
# Line 961 | Line 945 | Newton-Raphson methods can not be used with very large
945   matrix and insufficient storage capacity to store them, most
946   Newton-Raphson methods can not be used with very large models.
947  
948 < \subsubsection{Heating}
948 > \subsubsection{\textbf{Heating}}
949  
950   Typically, Heating is performed by assigning random velocities
951   according to a Gaussian distribution for a temperature. Beginning at
# Line 973 | Line 957 | shifted to zero.
957   net linear momentum and angular momentum of the system should be
958   shifted to zero.
959  
960 < \subsubsection{Equilibration}
960 > \subsubsection{\textbf{Equilibration}}
961  
962   The purpose of equilibration is to allow the system to evolve
963   spontaneously for a period of time and reach equilibrium. The
# Line 1079 | Line 1063 | from the trajectories.
1063   parameters, and investigate time-dependent processes of the molecule
1064   from the trajectories.
1065  
1066 < \subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties}
1066 > \subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamics Properties}}
1067  
1068   Thermodynamics properties, which can be expressed in terms of some
1069   function of the coordinates and momenta of all particles in the
# Line 1101 | Line 1085 | P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\
1085   < j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
1086   \end{equation}
1087  
1088 < \subsubsection{\label{introSection:structuralProperties}Structural Properties}
1088 > \subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}
1089  
1090   Structural Properties of a simple fluid can be described by a set of
1091   distribution functions. Among these functions,\emph{pair
# Line 1141 | Line 1125 | other is essentially zero.
1125   %\label{introFigure:pairDistributionFunction}
1126   %\end{figure}
1127  
1128 < \subsubsection{\label{introSection:timeDependentProperties}Time-dependent
1129 < Properties}
1128 > \subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
1129 > Properties}}
1130  
1131   Time-dependent properties are usually calculated using \emph{time
1132   correlation function}, which correlates random variables $A$ and $B$
# Line 1696 | Line 1680 | which is known as the \emph{generalized Langevin equat
1680   \end{equation}
1681   which is known as the \emph{generalized Langevin equation}.
1682  
1683 < \subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel}
1683 > \subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
1684  
1685   One may notice that $R(t)$ depends only on initial conditions, which
1686   implies it is completely deterministic within the context of a
# Line 1755 | Line 1739 | particles is given in Sec.~\ref{introSection:frictionT
1739   briefly review on calculating friction tensor for arbitrary shaped
1740   particles is given in Sec.~\ref{introSection:frictionTensor}.
1741  
1742 < \subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
1742 > \subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
1743  
1744   Defining a new set of coordinates,
1745   \[
# Line 1821 | Line 1805 | toque.
1805   where $F_r$ is the friction force and $\tau _R$ is the friction
1806   toque.
1807  
1808 < \subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape}
1808 > \subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}}
1809  
1810   For a spherical particle, the translational and rotational friction
1811   constant can be calculated from Stoke's law,
# Line 1883 | Line 1867 | and
1867   \end{array}.
1868   \]
1869  
1870 < \subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape}
1870 > \subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}}
1871  
1872   Unlike spherical and other regular shaped molecules, there is not
1873   analytical solution for friction tensor of any arbitrary shaped

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