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# Line 31 | Line 31 | F_{ij} = -F_{ji}
31   $F_{ji}$ be the force that particle $j$ exerts on particle $i$.
32   Newton's third law states that
33   \begin{equation}
34 < F_{ij} = -F_{ji}
34 > F_{ij} = -F_{ji}.
35   \label{introEquation:newtonThirdLaw}
36   \end{equation}
37 –
37   Conservation laws of Newtonian Mechanics play very important roles
38   in solving mechanics problems. The linear momentum of a particle is
39   conserved if it is free or it experiences no force. The second
# Line 63 | Line 62 | that if all forces are conservative, Energy
62   \end{equation}
63   If there are no external torques acting on a body, the angular
64   momentum of it is conserved. The last conservation theorem state
65 < that if all forces are conservative, Energy
66 < \begin{equation}E = T + V \label{introEquation:energyConservation}
65 > that if all forces are conservative, energy is conserved,
66 > \begin{equation}E = T + V. \label{introEquation:energyConservation}
67   \end{equation}
68 < is conserved. All of these conserved quantities are
69 < important factors to determine the quality of numerical integration
70 < schemes for rigid bodies \cite{Dullweber1997}.
68 > All of these conserved quantities are important factors to determine
69 > the quality of numerical integration schemes for rigid bodies
70 > \cite{Dullweber1997}.
71  
72   \subsection{\label{introSection:lagrangian}Lagrangian Mechanics}
73  
74 < Newtonian Mechanics suffers from two important limitations: motions
75 < can only be described in cartesian coordinate systems. Moreover, It
76 < become impossible to predict analytically the properties of the
77 < system even if we know all of the details of the interaction. In
78 < order to overcome some of the practical difficulties which arise in
79 < attempts to apply Newton's equation to complex system, approximate
80 < numerical procedures may be developed.
74 > Newtonian Mechanics suffers from a important limitation: motions can
75 > only be described in cartesian coordinate systems which make it
76 > impossible to predict analytically the properties of the system even
77 > if we know all of the details of the interaction. In order to
78 > overcome some of the practical difficulties which arise in attempts
79 > to apply Newton's equation to complex system, approximate numerical
80 > procedures may be developed.
81  
82   \subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's
83   Principle}}
84  
85   Hamilton introduced the dynamical principle upon which it is
86   possible to base all of mechanics and most of classical physics.
87 < Hamilton's Principle may be stated as follows,
88 <
89 < The actual trajectory, along which a dynamical system may move from
90 < one point to another within a specified time, is derived by finding
91 < the path which minimizes the time integral of the difference between
93 < the kinetic, $K$, and potential energies, $U$.
87 > Hamilton's Principle may be stated as follows: the actual
88 > trajectory, along which a dynamical system may move from one point
89 > to another within a specified time, is derived by finding the path
90 > which minimizes the time integral of the difference between the
91 > kinetic $K$, and potential energies $U$,
92   \begin{equation}
93 < \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
93 > \delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}.
94   \label{introEquation:halmitonianPrinciple1}
95   \end{equation}
98 –
96   For simple mechanical systems, where the forces acting on the
97   different parts are derivable from a potential, the Lagrangian
98   function $L$ can be defined as the difference between the kinetic
99   energy of the system and its potential energy,
100   \begin{equation}
101 < L \equiv K - U = L(q_i ,\dot q_i ) ,
101 > L \equiv K - U = L(q_i ,\dot q_i ).
102   \label{introEquation:lagrangianDef}
103   \end{equation}
104 < then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
104 > Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes
105   \begin{equation}
106 < \delta \int_{t_1 }^{t_2 } {L dt = 0} ,
106 > \delta \int_{t_1 }^{t_2 } {L dt = 0} .
107   \label{introEquation:halmitonianPrinciple2}
108   \end{equation}
109  
# Line 138 | Line 135 | p_i  = \frac{{\partial L}}{{\partial q_i }}
135   p_i  = \frac{{\partial L}}{{\partial q_i }}
136   \label{introEquation:generalizedMomentaDot}
137   \end{equation}
141 –
138   With the help of the generalized momenta, we may now define a new
139   quantity $H$ by the equation
140   \begin{equation}
# Line 146 | Line 142 | $L$ is the Lagrangian function for the system.
142   \label{introEquation:hamiltonianDefByLagrangian}
143   \end{equation}
144   where $ \dot q_1  \ldots \dot q_f $ are generalized velocities and
145 < $L$ is the Lagrangian function for the system.
146 <
151 < Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian},
152 < one can obtain
145 > $L$ is the Lagrangian function for the system. Differentiating
146 > Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain
147   \begin{equation}
148   dH = \sum\limits_k {\left( {p_k d\dot q_k  + \dot q_k dp_k  -
149   \frac{{\partial L}}{{\partial q_k }}dq_k  - \frac{{\partial
150   L}}{{\partial \dot q_k }}d\dot q_k } \right)}  - \frac{{\partial
151 < L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1}
151 > L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1}
152   \end{equation}
153 < Making use of  Eq.~\ref{introEquation:generalizedMomenta}, the
154 < second and fourth terms in the parentheses cancel. Therefore,
153 > Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second
154 > and fourth terms in the parentheses cancel. Therefore,
155   Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as
156   \begin{equation}
157   dH = \sum\limits_k {\left( {\dot q_k dp_k  - \dot p_k dq_k }
158 < \right)}  - \frac{{\partial L}}{{\partial t}}dt
158 > \right)}  - \frac{{\partial L}}{{\partial t}}dt .
159   \label{introEquation:diffHamiltonian2}
160   \end{equation}
161   By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can
# Line 180 | Line 174 | t}}
174   t}}
175   \label{introEquation:motionHamiltonianTime}
176   \end{equation}
177 <
184 < Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
177 > where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
178   Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
179   equation of motion. Due to their symmetrical formula, they are also
180   known as the canonical equations of motions \cite{Goldstein2001}.
# Line 195 | Line 188 | only works with 1st-order differential equations\cite{
188   statistical mechanics and quantum mechanics, since it treats the
189   coordinate and its time derivative as independent variables and it
190   only works with 1st-order differential equations\cite{Marion1990}.
198 –
191   In Newtonian Mechanics, a system described by conservative forces
192 < conserves the total energy \ref{introEquation:energyConservation}.
193 < It follows that Hamilton's equations of motion conserve the total
194 < Hamiltonian.
192 > conserves the total energy
193 > (Eq.~\ref{introEquation:energyConservation}). It follows that
194 > Hamilton's equations of motion conserve the total Hamiltonian
195   \begin{equation}
196   \frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial
197   H}}{{\partial q_i }}\dot q_i  + \frac{{\partial H}}{{\partial p_i
198   }}\dot p_i } \right)}  = \sum\limits_i {\left( {\frac{{\partial
199   H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} -
200   \frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial
201 < q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian}
201 > q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian}
202   \end{equation}
203  
204   \section{\label{introSection:statisticalMechanics}Statistical
# Line 227 | Line 219 | this system is a $6f$ dimensional space. A point, $x =
219   momentum variables. Consider a dynamic system of $f$ particles in a
220   cartesian space, where each of the $6f$ coordinates and momenta is
221   assigned to one of $6f$ mutually orthogonal axes, the phase space of
222 < this system is a $6f$ dimensional space. A point, $x = (\rightarrow
223 < q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow
224 < p_f )$, with a unique set of values of $6f$ coordinates and momenta
225 < is a phase space vector.
222 > this system is a $6f$ dimensional space. A point, $x =
223 > (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
224 > \over q} _1 , \ldots
225 > ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
226 > \over q} _f
227 > ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
228 > \over p} _1  \ldots
229 > ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
230 > \over p} _f )$ , with a unique set of values of $6f$ coordinates and
231 > momenta is a phase space vector.
232   %%%fix me
233  
234   In statistical mechanics, the condition of an ensemble at any time
# Line 245 | Line 243 | function of the time.
243   Governed by the principles of mechanics, the phase points change
244   their locations which would change the density at any time at phase
245   space. Hence, the density distribution is also to be taken as a
246 < function of the time.
247 <
250 < The number of systems $\delta N$ at time $t$ can be determined by,
246 > function of the time. The number of systems $\delta N$ at time $t$
247 > can be determined by,
248   \begin{equation}
249   \delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
250   \label{introEquation:deltaN}
# Line 280 | Line 277 | space of the system,
277   \begin{equation}
278   \langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
279   (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
280 < (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}
280 > (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
281   \label{introEquation:ensembelAverage}
282   \end{equation}
283  
# Line 288 | Line 285 | thermodynamic equilibrium.
285   statistical characteristics. As a function of macroscopic
286   parameters, such as temperature \textit{etc}, the partition function
287   can be used to describe the statistical properties of a system in
288 < thermodynamic equilibrium.
289 <
290 < As an ensemble of systems, each of which is known to be thermally
294 < isolated and conserve energy, the Microcanonical ensemble (NVE) has
295 < a partition function like,
288 > thermodynamic equilibrium. As an ensemble of systems, each of which
289 > is known to be thermally isolated and conserve energy, the
290 > Microcanonical ensemble (NVE) has a partition function like,
291   \begin{equation}
292 < \Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
292 > \Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition}
293   \end{equation}
294 < A canonical ensemble (NVT)is an ensemble of systems, each of which
294 > A canonical ensemble (NVT) is an ensemble of systems, each of which
295   can share its energy with a large heat reservoir. The distribution
296   of the total energy amongst the possible dynamical states is given
297   by the partition function,
298   \begin{equation}
299 < \Omega (N,V,T) = e^{ - \beta A}
299 > \Omega (N,V,T) = e^{ - \beta A}.
300   \label{introEquation:NVTPartition}
301   \end{equation}
302   Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
# Line 358 | Line 353 | simple form,
353   \frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)}  = 0 .
354   \label{introEquation:liouvilleTheorem}
355   \end{equation}
361 –
356   Liouville's theorem states that the distribution function is
357   constant along any trajectory in phase space. In classical
358   statistical mechanics, since the number of members in an ensemble is
# Line 415 | Line 409 | Substituting equations of motion in Hamiltonian formal
409   q_i }}} \right)}.
410   \label{introEquation:poissonBracket}
411   \end{equation}
412 < Substituting equations of motion in Hamiltonian formalism(
413 < Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
414 < Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into
412 > Substituting equations of motion in Hamiltonian formalism
413 > (Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
414 > Eq.~\ref{introEquation:motionHamiltonianMomentum}) into
415   (Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite
416   Liouville's theorem using Poisson bracket notion,
417   \begin{equation}
# Line 451 | Line 445 | statistical ensemble are identical \cite{Frenkel1996,
445   many-body system in Statistical Mechanics. Fortunately, the Ergodic
446   Hypothesis makes a connection between time average and the ensemble
447   average. It states that the time average and average over the
448 < statistical ensemble are identical \cite{Frenkel1996, Leach2001}.
448 > statistical ensemble are identical \cite{Frenkel1996, Leach2001}:
449   \begin{equation}
450   \langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
451   \frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
# Line 465 | Line 459 | reasonable, the Monte Carlo techniques\cite{Metropolis
459   a properly weighted statistical average. This allows the researcher
460   freedom of choice when deciding how best to measure a given
461   observable. In case an ensemble averaged approach sounds most
462 < reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be
462 > reasonable, the Monte Carlo methods\cite{Metropolis1949} can be
463   utilized. Or if the system lends itself to a time averaging
464   approach, the Molecular Dynamics techniques in
465   Sec.~\ref{introSection:molecularDynamics} will be the best
# Line 500 | Line 494 | an example of symplectic form.
494   $\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
495   \lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
496   $\omega(x, x) = 0$. The cross product operation in vector field is
497 < an example of symplectic form.
498 <
499 < One of the motivations to study \emph{symplectic manifolds} in
500 < Hamiltonian Mechanics is that a symplectic manifold can represent
501 < all possible configurations of the system and the phase space of the
502 < system can be described by it's cotangent bundle. Every symplectic
503 < manifold is even dimensional. For instance, in Hamilton equations,
510 < coordinate and momentum always appear in pairs.
497 > an example of symplectic form. One of the motivations to study
498 > \emph{symplectic manifolds} in Hamiltonian Mechanics is that a
499 > symplectic manifold can represent all possible configurations of the
500 > system and the phase space of the system can be described by it's
501 > cotangent bundle. Every symplectic manifold is even dimensional. For
502 > instance, in Hamilton equations, coordinate and momentum always
503 > appear in pairs.
504  
505   \subsection{\label{introSection:ODE}Ordinary Differential Equations}
506  
# Line 516 | Line 509 | where $x = x(q,p)^T$, this system is a canonical Hamil
509   \dot x = f(x)
510   \end{equation}
511   where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if
512 + $f(r) = J\nabla _x H(r)$. Here, $H = H (q, p)$ is Hamiltonian
513 + function and $J$ is the skew-symmetric matrix
514   \begin{equation}
520 – f(r) = J\nabla _x H(r).
521 – \end{equation}
522 – $H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric
523 – matrix
524 – \begin{equation}
515   J = \left( {\begin{array}{*{20}c}
516     0 & I  \\
517     { - I} & 0  \\
# Line 531 | Line 521 | system can be rewritten as,
521   where $I$ is an identity matrix. Using this notation, Hamiltonian
522   system can be rewritten as,
523   \begin{equation}
524 < \frac{d}{{dt}}x = J\nabla _x H(x)
524 > \frac{d}{{dt}}x = J\nabla _x H(x).
525   \label{introEquation:compactHamiltonian}
526   \end{equation}In this case, $f$ is
527 < called a \emph{Hamiltonian vector field}.
528 <
539 < Another generalization of Hamiltonian dynamics is Poisson
540 < Dynamics\cite{Olver1986},
527 > called a \emph{Hamiltonian vector field}. Another generalization of
528 > Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986},
529   \begin{equation}
530   \dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
531   \end{equation}
# Line 545 | Line 533 | Let $x(t)$ be the exact solution of the ODE system,
533  
534   \subsection{\label{introSection:exactFlow}Exact Flow}
535  
536 < Let $x(t)$ be the exact solution of the ODE system,
537 < \begin{equation}
538 < \frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}
539 < \end{equation}
540 < The exact flow(solution) $\varphi_\tau$ is defined by
553 < \[
554 < x(t+\tau) =\varphi_\tau(x(t))
536 > Let $x(t)$ be the exact solution of the ODE
537 > system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can
538 > define its exact flow(solution) $\varphi_\tau$
539 > \[ x(t+\tau)
540 > =\varphi_\tau(x(t))
541   \]
542   where $\tau$ is a fixed time step and $\varphi$ is a map from phase
543   space to itself. The flow has the continuous group property,
# Line 573 | Line 559 | The exact flow can also be written in terms of the of
559   }{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x).
560   \label{introEquation:exponentialOperator}
561   \end{equation}
576 –
562   In most cases, it is not easy to find the exact flow $\varphi_\tau$.
563   Instead, we use an approximate map, $\psi_\tau$, which is usually
564   called integrator. The order of an integrator $\psi_\tau$ is $p$, if
# Line 587 | Line 572 | them can be found in systems which occur naturally in
572   The hidden geometric properties\cite{Budd1999, Marsden1998} of an
573   ODE and its flow play important roles in numerical studies. Many of
574   them can be found in systems which occur naturally in applications.
590 –
575   Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is
576   a \emph{symplectic} flow if it satisfies,
577   \begin{equation}
# Line 601 | Line 585 | is the property that must be preserved by the integrat
585   \begin{equation}
586   {\varphi '}^T J \varphi ' = J \circ \varphi
587   \end{equation}
588 < is the property that must be preserved by the integrator.
589 <
590 < It is possible to construct a \emph{volume-preserving} flow for a
591 < source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $
592 < \det d\varphi  = 1$. One can show easily that a symplectic flow will
609 < be volume-preserving.
610 <
611 < Changing the variables $y = h(x)$ in an ODE
588 > is the property that must be preserved by the integrator. It is
589 > possible to construct a \emph{volume-preserving} flow for a source
590 > free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det
591 > d\varphi  = 1$. One can show easily that a symplectic flow will be
592 > volume-preserving. Changing the variables $y = h(x)$ in an ODE
593   (Eq.~\ref{introEquation:ODE}) will result in a new system,
594   \[
595   \dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y).
596   \]
597   The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
598   In other words, the flow of this vector field is reversible if and
599 < only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
600 <
620 < A \emph{first integral}, or conserved quantity of a general
599 > only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $. A
600 > \emph{first integral}, or conserved quantity of a general
601   differential function is a function $ G:R^{2d}  \to R^d $ which is
602   constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
603   \[
# Line 625 | Line 605 | Using chain rule, one may obtain,
605   \]
606   Using chain rule, one may obtain,
607   \[
608 < \sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G,
608 > \sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G,
609   \]
610   which is the condition for conserving \emph{first integral}. For a
611   canonical Hamiltonian system, the time evolution of an arbitrary
612   smooth function $G$ is given by,
633 –
613   \begin{eqnarray}
614 < \frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\
615 <                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
614 > \frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\
615 >                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)).
616   \label{introEquation:firstIntegral1}
617   \end{eqnarray}
618 <
619 <
641 < Using poisson bracket notion, Equation
642 < \ref{introEquation:firstIntegral1} can be rewritten as
618 > Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1}
619 > can be rewritten as
620   \[
621   \frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
622   \]
623   Therefore, the sufficient condition for $G$ to be the \emph{first
624 < integral} of a Hamiltonian system is
648 < \[
649 < \left\{ {G,H} \right\} = 0.
650 < \]
624 > integral} of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$
625   As well known, the Hamiltonian (or energy) H of a Hamiltonian system
626   is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
627 < 0$.
654 <
655 < When designing any numerical methods, one should always try to
627 > 0$. When designing any numerical methods, one should always try to
628   preserve the structural properties of the original ODE and its flow.
629  
630   \subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods}
# Line 668 | Line 640 | constructed using one of four different methods.
640   \item Runge-Kutta methods
641   \item Splitting methods
642   \end{enumerate}
671 –
643   Generating function\cite{Channell1990} tends to lead to methods
644   which are cumbersome and difficult to use. In dissipative systems,
645   variational methods can capture the decay of energy
646   accurately\cite{Kane2000}. Since their geometrically unstable nature
647   against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
648   methods are not suitable for Hamiltonian system. Recently, various
649 < high-order explicit Runge-Kutta methods
650 < \cite{Owren1992,Chen2003}have been developed to overcome this
651 < instability. However, due to computational penalty involved in
652 < implementing the Runge-Kutta methods, they have not attracted much
653 < attention from the Molecular Dynamics community. Instead, splitting
654 < methods have been widely accepted since they exploit natural
655 < decompositions of the system\cite{Tuckerman1992, McLachlan1998}.
649 > high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003}
650 > have been developed to overcome this instability. However, due to
651 > computational penalty involved in implementing the Runge-Kutta
652 > methods, they have not attracted much attention from the Molecular
653 > Dynamics community. Instead, splitting methods have been widely
654 > accepted since they exploit natural decompositions of the
655 > system\cite{Tuckerman1992, McLachlan1998}.
656  
657   \subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
658  
# Line 693 | Line 664 | simpler integration of the system.
664   \label{introEquation:FlowDecomposition}
665   \end{equation}
666   where each of the sub-flow is chosen such that each represent a
667 < simpler integration of the system.
668 <
698 < Suppose that a Hamiltonian system takes the form,
667 > simpler integration of the system. Suppose that a Hamiltonian system
668 > takes the form,
669   \[
670   H = H_1 + H_2.
671   \]
# Line 723 | Line 693 | The Lie-Trotter splitting(\ref{introEquation:firstOrde
693   where $\phi$ and $\psi$ both are symplectic maps. Thus operator
694   splitting in this context automatically generates a symplectic map.
695  
696 < The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting})
697 < introduces local errors proportional to $h^2$, while Strang
698 < splitting gives a second-order decomposition,
696 > The Lie-Trotter
697 > splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces
698 > local errors proportional to $h^2$, while Strang splitting gives a
699 > second-order decomposition,
700   \begin{equation}
701   \varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
702   _{1,h/2} , \label{introEquation:secondOrderSplitting}
# Line 785 | Line 756 | the equations of motion would follow:
756  
757   \item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values.
758   \end{enumerate}
788 –
759   By simply switching the order of the propagators in the splitting
760   and composing a new integrator, the \emph{position verlet}
761   integrator, can be generated,
# Line 823 | Line 793 | to the Sprang splitting, we can obtain
793   \begin{eqnarray*}
794   \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 \\
795                                     &   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
796 <                                   &   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots )
796 >                                   &   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots
797 >                                   ).
798   \end{eqnarray*}
799 < Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0,\] the dominant local
799 > Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local
800   error of Spring splitting is proportional to $h^3$. The same
801 < procedure can be applied to a general splitting,  of the form
801 > procedure can be applied to a general splitting of the form
802   \begin{equation}
803   \varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
804   1} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
# Line 972 | Line 943 | algorithmic tricks.
943   Coulombic forces \textit{etc}. For a system of $N$ particles, the
944   complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
945   which making large simulations prohibitive in the absence of any
946 < algorithmic tricks.
947 <
948 < A natural approach to avoid system size issues is to represent the
949 < bulk behavior by a finite number of the particles. However, this
950 < approach will suffer from the surface effect at the edges of the
951 < simulation. To offset this, \textit{Periodic boundary conditions}
952 < (see Fig.~\ref{introFig:pbc}) is developed to simulate bulk
953 < properties with a relatively small number of particles. In this
954 < method, the simulation box is replicated throughout space to form an
955 < infinite lattice. During the simulation, when a particle moves in
956 < the primary cell, its image in other cells move in exactly the same
957 < direction with exactly the same orientation. Thus, as a particle
987 < leaves the primary cell, one of its images will enter through the
988 < opposite face.
946 > algorithmic tricks. A natural approach to avoid system size issues
947 > is to represent the bulk behavior by a finite number of the
948 > particles. However, this approach will suffer from the surface
949 > effect at the edges of the simulation. To offset this,
950 > \textit{Periodic boundary conditions} (see Fig.~\ref{introFig:pbc})
951 > is developed to simulate bulk properties with a relatively small
952 > number of particles. In this method, the simulation box is
953 > replicated throughout space to form an infinite lattice. During the
954 > simulation, when a particle moves in the primary cell, its image in
955 > other cells move in exactly the same direction with exactly the same
956 > orientation. Thus, as a particle leaves the primary cell, one of its
957 > images will enter through the opposite face.
958   \begin{figure}
959   \centering
960   \includegraphics[width=\linewidth]{pbc.eps}
# Line 1048 | Line 1017 | Statistical Mechanics, Sec.~\ref{introSection:statisti
1017   monitor the motions of molecules. Although the dynamics of the
1018   system can be described qualitatively from animation, quantitative
1019   trajectory analysis are more useful. According to the principles of
1020 < Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics},
1021 < one can compute thermodynamic properties, analyze fluctuations of
1022 < structural parameters, and investigate time-dependent processes of
1023 < the molecule from the trajectories.
1020 > Statistical Mechanics in
1021 > Sec.~\ref{introSection:statisticalMechanics}, one can compute
1022 > thermodynamic properties, analyze fluctuations of structural
1023 > parameters, and investigate time-dependent processes of the molecule
1024 > from the trajectories.
1025  
1026   \subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}}
1027  
# Line 1088 | Line 1058 | expressed in terms of radial distribution function \ci
1058   to justify the correctness of a liquid model. Moreover, various
1059   equilibrium thermodynamic and structural properties can also be
1060   expressed in terms of radial distribution function \cite{Allen1987}.
1091 –
1061   The pair distribution functions $g(r)$ gives the probability that a
1062   particle $i$ will be located at a distance $r$ from a another
1063   particle $j$ in the system
1064 < \[
1064 > \begin{equation}
1065   g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
1066   \ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho
1067   (r)}{\rho}.
1068 < \]
1068 > \end{equation}
1069   Note that the delta function can be replaced by a histogram in
1070   computer simulation. Peaks in $g(r)$ represent solvent shells, and
1071   the height of these peaks gradually decreases to 1 as the liquid of
# Line 1133 | Line 1102 | u_{tot} (t) = \sum\limits_i {u_i (t)}
1102   Here $u_{tot}$ is the net dipole of the entire system and is given
1103   by
1104   \[
1105 < u_{tot} (t) = \sum\limits_i {u_i (t)}
1105 > u_{tot} (t) = \sum\limits_i {u_i (t)}.
1106   \]
1107   In principle, many time correlation functions can be related with
1108   Fourier transforms of the infrared, Raman, and inelastic neutron
# Line 1142 | Line 1111 | i2\pi vt} dt}
1111   each frequency using the following relationship:
1112   \[
1113   \hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
1114 < i2\pi vt} dt}
1114 > i2\pi vt} dt}.
1115   \]
1116  
1117   \section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
# Line 1210 | Line 1179 | which is used to ensure rotation matrix's unitarity. D
1179   Q^T Q = 1, \label{introEquation:orthogonalConstraint}
1180   \end{equation}
1181   which is used to ensure rotation matrix's unitarity. Differentiating
1182 < \ref{introEquation:orthogonalConstraint} and using Equation
1183 < \ref{introEquation:RBMotionMomentum}, one may obtain,
1182 > Eq.~\ref{introEquation:orthogonalConstraint} and using
1183 > Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain,
1184   \begin{equation}
1185   Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
1186   \label{introEquation:RBFirstOrderConstraint}
1187   \end{equation}
1219 –
1188   Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
1189   \ref{introEquation:motionHamiltonianMomentum}), one can write down
1190   the equations of motion,
1223 –
1191   \begin{eqnarray}
1192 < \frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\
1193 < \frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
1194 < \frac{{dQ}}{{dt}} & = & PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
1192 > \frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\
1193 > \frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\
1194 > \frac{{dQ}}{{dt}} & = & PJ^{ - 1},  \label{introEquation:RBMotionRotation}\\
1195   \frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}
1196   \end{eqnarray}
1230 –
1197   In general, there are two ways to satisfy the holonomic constraints.
1198   We can use a constraint force provided by a Lagrange multiplier on
1199   the normal manifold to keep the motion on constraint space. Or we
# Line 1239 | Line 1205 | M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1}
1205   M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
1206   \right\}.
1207   \]
1242 –
1208   Unfortunately, this constraint manifold is not the cotangent bundle
1209   $T^* SO(3)$ which can be consider as a symplectic manifold on Lie
1210   rotation group $SO(3)$. However, it turns out that under symplectic
# Line 1254 | Line 1219 | T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \t
1219   T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
1220   1,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
1221   \]
1257 –
1222   For a body fixed vector $X_i$ with respect to the center of mass of
1223   the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
1224   given as
# Line 1273 | Line 1237 | respectively.
1237   \[
1238   \nabla _Q V(q,Q) = F(q,Q)X_i^t
1239   \]
1240 < respectively.
1241 <
1242 < As a common choice to describe the rotation dynamics of the rigid
1279 < body, the angular momentum on the body fixed frame $\Pi  = Q^t P$ is
1280 < introduced to rewrite the equations of motion,
1240 > respectively. As a common choice to describe the rotation dynamics
1241 > of the rigid body, the angular momentum on the body fixed frame $\Pi
1242 > = Q^t P$ is introduced to rewrite the equations of motion,
1243   \begin{equation}
1244   \begin{array}{l}
1245 < \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda  \\
1246 < \dot Q  = Q\Pi {\rm{ }}J^{ - 1}  \\
1245 > \dot \Pi  = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda,  \\
1246 > \dot Q  = Q\Pi {\rm{ }}J^{ - 1},  \\
1247   \end{array}
1248   \label{introEqaution:RBMotionPI}
1249   \end{equation}
1250 < , as well as holonomic constraints,
1251 < \[
1252 < \begin{array}{l}
1291 < \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
1292 < Q^T Q = 1 \\
1293 < \end{array}
1294 < \]
1295 <
1296 < For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
1297 < so(3)^ \star$, the hat-map isomorphism,
1250 > as well as holonomic constraints $\Pi J^{ - 1}  + J^{ - 1} \Pi ^t  =
1251 > 0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a
1252 > matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism,
1253   \begin{equation}
1254   v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
1255   {\begin{array}{*{20}c}
# Line 1307 | Line 1262 | operations
1262   will let us associate the matrix products with traditional vector
1263   operations
1264   \[
1265 < \hat vu = v \times u
1265 > \hat vu = v \times u.
1266   \]
1267 < Using \ref{introEqaution:RBMotionPI}, one can construct a skew
1267 > Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew
1268   matrix,
1269 <
1270 < \begin{eqnarry*}
1271 < (\dot \Pi  - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{
1272 < }}(J^{ - 1} \Pi  + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i
1273 < (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  - (\Lambda  - \Lambda ^T ).
1274 < \label{introEquation:skewMatrixPI}
1275 < \end{eqnarray*}
1276 <
1277 < Since $\Lambda$ is symmetric, the last term of Equation
1278 < \ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
1279 < multiplier $\Lambda$ is absent from the equations of motion. This
1280 < unique property eliminates the requirement of iterations which can
1326 < not be avoided in other methods\cite{Kol1997, Omelyan1998}.
1327 <
1328 < Applying the hat-map isomorphism, we obtain the equation of motion
1329 < for angular momentum on body frame
1269 > \begin{eqnarray}
1270 > (\dot \Pi  - \dot \Pi ^T )&= &(\Pi  - \Pi ^T )(J^{ - 1} \Pi  + \Pi J^{ - 1} ) \notag \\
1271 > & & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]}  -
1272 > (\Lambda  - \Lambda ^T ). \label{introEquation:skewMatrixPI}
1273 > \end{eqnarray}
1274 > Since $\Lambda$ is symmetric, the last term of
1275 > Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the
1276 > Lagrange multiplier $\Lambda$ is absent from the equations of
1277 > motion. This unique property eliminates the requirement of
1278 > iterations which can not be avoided in other methods\cite{Kol1997,
1279 > Omelyan1998}. Applying the hat-map isomorphism, we obtain the
1280 > equation of motion for angular momentum on body frame
1281   \begin{equation}
1282   \dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
1283   F_i (r,Q)} \right) \times X_i }.
# Line 1335 | Line 1286 | given by
1286   In the same manner, the equation of motion for rotation matrix is
1287   given by
1288   \[
1289 < \dot Q = Qskew(I^{ - 1} \pi )
1289 > \dot Q = Qskew(I^{ - 1} \pi ).
1290   \]
1291  
1292   \subsection{\label{introSection:SymplecticFreeRB}Symplectic
# Line 1357 | Line 1308 | J(\pi ) = \left( {\begin{array}{*{20}c}
1308     0 & {\pi _3 } & { - \pi _2 }  \\
1309     { - \pi _3 } & 0 & {\pi _1 }  \\
1310     {\pi _2 } & { - \pi _1 } & 0  \\
1311 < \end{array}} \right)
1311 > \end{array}} \right).
1312   \end{equation}
1313   Thus, the dynamics of free rigid body is governed by
1314   \begin{equation}
1315 < \frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
1315 > \frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi ).
1316   \end{equation}
1317 <
1318 < One may notice that each $T_i^r$ in Equation
1319 < \ref{introEquation:rotationalKineticRB} can be solved exactly. For
1369 < instance, the equations of motion due to $T_1^r$ are given by
1317 > One may notice that each $T_i^r$ in
1318 > Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly.
1319 > For instance, the equations of motion due to $T_1^r$ are given by
1320   \begin{equation}
1321   \frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
1322   \label{introEqaution:RBMotionSingleTerm}
1323   \end{equation}
1324 < where
1324 > with
1325   \[ R_1  = \left( {\begin{array}{*{20}c}
1326     0 & 0 & 0  \\
1327     0 & 0 & {\pi _1 }  \\
1328     0 & { - \pi _1 } & 0  \\
1329   \end{array}} \right).
1330   \]
1331 < The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is
1331 > The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is
1332   \[
1333   \pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
1334   Q(0)e^{\Delta tR_1 }
# Line 1396 | Line 1346 | e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1
1346   propagator,
1347   \[
1348   e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
1349 < )
1349 > ).
1350   \]
1351   The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
1352   manner. In order to construct a second-order symplectic method, we
1353 < split the angular kinetic Hamiltonian function can into five terms
1353 > split the angular kinetic Hamiltonian function into five terms
1354   \[
1355   T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
1356   ) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
# Line 1414 | Line 1364 | _1 }.
1364   \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
1365   _1 }.
1366   \]
1417 –
1367   The non-canonical Lie-Poisson bracket ${F, G}$ of two function
1368   $F(\pi )$ and $G(\pi )$ is defined by
1369   \[
1370   \{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
1371 < )
1371 > ).
1372   \]
1373   If the Poisson bracket of a function $F$ with an arbitrary smooth
1374   function $G$ is zero, $F$ is a \emph{Casimir}, which is the
# Line 1430 | Line 1379 | then by the chain rule
1379   then by the chain rule
1380   \[
1381   \nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
1382 < }}{2})\pi
1382 > }}{2})\pi.
1383   \]
1384 < Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
1384 > Thus, $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel
1385 > \pi
1386   \parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
1387   Lie-Poisson integrator is found to be both extremely efficient and
1388   stable. These properties can be explained by the fact the small
# Line 1443 | Line 1393 | energy and potential energy,
1393   Splitting for Rigid Body}
1394  
1395   The Hamiltonian of rigid body can be separated in terms of kinetic
1396 < energy and potential energy,
1397 < \[
1398 < H = T(p,\pi ) + V(q,Q)
1449 < \]
1450 < The equations of motion corresponding to potential energy and
1451 < kinetic energy are listed in the below table,
1396 > energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations
1397 > of motion corresponding to potential energy and kinetic energy are
1398 > listed in the below table,
1399   \begin{table}
1400 < \caption{Equations of motion due to Potential and Kinetic Energies}
1400 > \caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES}
1401   \begin{center}
1402   \begin{tabular}{|l|l|}
1403    \hline
# Line 1486 | Line 1433 | defined by \ref{introEquation:rotationalKineticRB}. Th
1433   T(p,\pi ) =T^t (p) + T^r (\pi ).
1434   \end{equation}
1435   where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
1436 < defined by \ref{introEquation:rotationalKineticRB}. Therefore, the
1437 < corresponding propagators are given by
1436 > defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore,
1437 > the corresponding propagators are given by
1438   \[
1439   \varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
1440   _{\Delta t,T^r }.
1441   \]
1442   Finally, we obtain the overall symplectic propagators for freely
1443   moving rigid bodies
1444 < \begin{equation}
1445 < \begin{array}{c}
1446 < \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
1447 <  \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 }  \\
1501 <  \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
1502 < \end{array}
1444 > \begin{eqnarray}
1445 > \varphi _{\Delta t}  &=& \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \notag\\
1446 >  & & \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\\
1447 >  & & \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
1448   \label{introEquation:overallRBFlowMaps}
1449 < \end{equation}
1449 > \end{eqnarray}
1450  
1451   \section{\label{introSection:langevinDynamics}Langevin Dynamics}
1452   As an alternative to newtonian dynamics, Langevin dynamics, which
# Line 1547 | Line 1492 | W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\a
1492   \[
1493   W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
1494   }}{{2m_\alpha  w_\alpha ^2 }}} x^2
1495 < \] and combining the last two terms in Equation
1496 < \ref{introEquation:bathGLE}, we may rewrite the Harmonic bath
1552 < Hamiltonian as
1495 > \]
1496 > and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as
1497   \[
1498   H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
1499   {\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
1500   w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
1501 < w_\alpha ^2 }}x} \right)^2 } \right\}}
1501 > w_\alpha ^2 }}x} \right)^2 } \right\}}.
1502   \]
1503   Since the first two terms of the new Hamiltonian depend only on the
1504   system coordinates, we can get the equations of motion for
# Line 1571 | Line 1515 | m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x
1515   \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
1516   \label{introEquation:bathMotionGLE}
1517   \end{equation}
1574 –
1518   In order to derive an equation for $x$, the dynamics of the bath
1519   variables $x_\alpha$ must be solved exactly first. As an integral
1520   transform which is particularly useful in solving linear ordinary
# Line 1580 | Line 1523 | original problems.
1523   differential equations into simple algebra problems which can be
1524   solved easily. Then, by applying the inverse Laplace transform, also
1525   known as the Bromwich integral, we can retrieve the solutions of the
1526 < original problems.
1527 <
1585 < Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace
1586 < transform of f(t) is a new function defined as
1526 > original problems. Let $f(t)$ be a function defined on $ [0,\infty )
1527 > $, the Laplace transform of $f(t)$ is a new function defined as
1528   \[
1529   L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
1530   \]
1531   where  $p$ is real and  $L$ is called the Laplace Transform
1532   Operator. Below are some important properties of Laplace transform
1592 –
1533   \begin{eqnarray*}
1534   L(x + y)  & = & L(x) + L(y) \\
1535   L(ax)     & = & aL(x) \\
# Line 1597 | Line 1537 | Operator. Below are some important properties of Lapla
1537   L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
1538   L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
1539   \end{eqnarray*}
1600 –
1601 –
1540   Applying the Laplace transform to the bath coordinates, we obtain
1541   \begin{eqnarray*}
1542 < 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) \\
1543 < L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }} \\
1542 > 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), \\
1543 > L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}. \\
1544   \end{eqnarray*}
1607 –
1545   By the same way, the system coordinates become
1546   \begin{eqnarray*}
1547 < mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
1548 <  & & \mbox{} - \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\}}  \\
1547 > mL(\ddot x) & = &
1548 >  - \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\}}  \\
1549 >  & & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}.
1550   \end{eqnarray*}
1613 –
1551   With the help of some relatively important inverse Laplace
1552   transformations:
1553   \[
# Line 1620 | Line 1557 | transformations:
1557   L(1) = \frac{1}{p} \\
1558   \end{array}
1559   \]
1560 < , we obtain
1560 > we obtain
1561   \begin{eqnarray*}
1562   m\ddot x & =  & - \frac{{\partial W(x)}}{{\partial x}} -
1563   \sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
# Line 1668 | Line 1605 | uncorrelated to $x$ and $\dot x$,
1605   One may notice that $R(t)$ depends only on initial conditions, which
1606   implies it is completely deterministic within the context of a
1607   harmonic bath. However, it is easy to verify that $R(t)$ is totally
1608 < uncorrelated to $x$ and $\dot x$,
1609 < \[
1610 < \begin{array}{l}
1611 < \left\langle {x(t)R(t)} \right\rangle  = 0, \\
1675 < \left\langle {\dot x(t)R(t)} \right\rangle  = 0. \\
1676 < \end{array}
1677 < \]
1678 < This property is what we expect from a truly random process. As long
1679 < as the model chosen for $R(t)$ was a gaussian distribution in
1608 > uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)}
1609 > \right\rangle  = 0, \left\langle {\dot x(t)R(t)} \right\rangle  =
1610 > 0.$ This property is what we expect from a truly random process. As
1611 > long as the model chosen for $R(t)$ was a gaussian distribution in
1612   general, the stochastic nature of the GLE still remains.
1681 –
1613   %dynamic friction kernel
1614   The convolution integral
1615   \[
# Line 1693 | Line 1624 | and Equation \ref{introEuqation:GeneralizedLangevinDyn
1624   \[
1625   \int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
1626   \]
1627 < and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
1627 > and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1628   \[
1629   m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
1630   \frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
# Line 1710 | Line 1641 | and Equation \ref{introEuqation:GeneralizedLangevinDyn
1641   \int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
1642   {\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
1643   \]
1644 < and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
1644 > and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes
1645   \begin{equation}
1646   m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
1647   x(t) + R(t) \label{introEquation:LangevinEquation}
# Line 1723 | Line 1654 | Defining a new set of coordinates,
1654  
1655   \subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
1656  
1657 < Defining a new set of coordinates,
1657 > Defining a new set of coordinates
1658   \[
1659   q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
1660 < ^2 }}x(0)
1661 < \],
1660 > ^2 }}x(0),
1661 > \]
1662   we can rewrite $R(T)$ as
1663   \[
1664   R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
1665   \]
1666   And since the $q$ coordinates are harmonic oscillators,
1736 –
1667   \begin{eqnarray*}
1668   \left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
1669   \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
# Line 1742 | Line 1672 | And since the $q$ coordinates are harmonic oscillators
1672    & = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
1673    & = &kT\xi (t) \\
1674   \end{eqnarray*}
1745 –
1675   Thus, we recover the \emph{second fluctuation dissipation theorem}
1676   \begin{equation}
1677   \xi (t) = \left\langle {R(t)R(0)} \right\rangle
1678 < \label{introEquation:secondFluctuationDissipation}.
1678 > \label{introEquation:secondFluctuationDissipation},
1679   \end{equation}
1680 < In effect, it acts as a constraint on the possible ways in which one
1681 < can model the random force and friction kernel.
1680 > which acts as a constraint on the possible ways in which one can
1681 > model the random force and friction kernel.

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