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# Line 6 | Line 6 | behind classical mechanics. Firstly, One can determine
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 | Newton¡¯s first law defines a class of inertial frames
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 | N \equiv r \times F \label{introEquation:torqueDefinit
46   \end{equation}
47   The torque $\tau$ with respect to the same origin is defined to be
48   \begin{equation}
49 < N \equiv r \times F \label{introEquation:torqueDefinition}
49 > \tau \equiv r \times F \label{introEquation:torqueDefinition}
50   \end{equation}
51   Differentiating Eq.~\ref{introEquation:angularMomentumDefinition},
52   \[
# Line 59 | Line 59 | thus,
59   \]
60   thus,
61   \begin{equation}
62 < \dot L = r \times \dot p = N
62 > \dot L = r \times \dot p = \tau
63   \end{equation}
64   If there are no external torques acting on a body, the angular
65   momentum of it is conserved. The last conservation theorem state
# Line 68 | Line 68 | scheme for rigid body \cite{Dullweber1997}.
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
116 > For a system of $f$ degrees of freedom, the equations of motion in
117 > the Lagrangian form is
118   \begin{equation}
119   \frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} -
120   \frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f
# Line 132 | Line 128 | independent of generalized velocities, the generalized
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 | function of the generalized velocities $\dot q_i$ and
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 | momentum variables. Consider a dynamic system in a car
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 + %%%fix me
235   A microscopic state or microstate of a classical system is
236   specification of the complete phase space vector of a system at any
237   instant in time. An ensemble is defined as a collection of systems
# Line 257 | Line 252 | space. The density of distribution for an ensemble wit
252   regions of the phase space. The condition of an ensemble at any time
253   can be regarded as appropriately specified by the density $\rho$
254   with which representative points are distributed over the phase
255 < space. The density of distribution for an ensemble with $f$ degrees
256 < of freedom is defined as,
255 > space. The density distribution for an ensemble with $f$ degrees of
256 > freedom is defined as,
257   \begin{equation}
258   \rho  = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t).
259   \label{introEquation:densityDistribution}
260   \end{equation}
261   Governed by the principles of mechanics, the phase points change
262 < their value which would change the density at any time at phase
263 < space. Hence, the density of distribution is also to be taken as a
262 > their locations which would change the density at any time at phase
263 > space. Hence, the density distribution is also to be taken as a
264   function of the time.
265  
266   The number of systems $\delta N$ at time $t$ can be determined by,
# Line 273 | Line 268 | Assuming a large enough population of systems are expl
268   \delta N = \rho (q,p,t)dq_1  \ldots dq_f dp_1  \ldots dp_f.
269   \label{introEquation:deltaN}
270   \end{equation}
271 < Assuming a large enough population of systems are exploited, we can
272 < sufficiently approximate $\delta N$ without introducing
273 < discontinuity when we go from one region in the phase space to
274 < another. By integrating over the whole phase space,
271 > Assuming a large enough population of systems, we can sufficiently
272 > approximate $\delta N$ without introducing discontinuity when we go
273 > from one region in the phase space to another. By integrating over
274 > the whole phase space,
275   \begin{equation}
276   N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f
277   \label{introEquation:totalNumberSystem}
# Line 288 | Line 283 | With the help of Equation(\ref{introEquation:unitProba
283   {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}.
284   \label{introEquation:unitProbability}
285   \end{equation}
286 < With the help of Equation(\ref{introEquation:unitProbability}) and
287 < the knowledge of the system, it is possible to calculate the average
286 > With the help of Eq.~\ref{introEquation:unitProbability} and the
287 > knowledge of the system, it is possible to calculate the average
288   value of any desired quantity which depends on the coordinates and
289   momenta of the system. Even when the dynamics of the real system is
290   complex, or stochastic, or even discontinuous, the average
291 < properties of the ensemble of possibilities as a whole may still
292 < remain well defined. For a classical system in thermal equilibrium
293 < with its environment, the ensemble average of a mechanical quantity,
294 < $\langle A(q , p) \rangle_t$, takes the form of an integral over the
295 < phase space of the system,
291 > properties of the ensemble of possibilities as a whole remaining
292 > well defined. For a classical system in thermal equilibrium with its
293 > environment, the ensemble average of a mechanical quantity, $\langle
294 > A(q , p) \rangle_t$, takes the form of an integral over the phase
295 > space of the system,
296   \begin{equation}
297   \langle  A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho
298   (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho
# Line 307 | Line 302 | parameters, such as temperature \textit{etc}, partitio
302  
303   There are several different types of ensembles with different
304   statistical characteristics. As a function of macroscopic
305 < parameters, such as temperature \textit{etc}, partition function can
306 < be used to describe the statistical properties of a system in
305 > parameters, such as temperature \textit{etc}, the partition function
306 > can be used to describe the statistical properties of a system in
307   thermodynamic equilibrium.
308  
309   As an ensemble of systems, each of which is known to be thermally
310 < isolated and conserve energy, Microcanonical ensemble(NVE) has a
311 < partition function like,
310 > isolated and conserve energy, the Microcanonical ensemble (NVE) has
311 > a partition function like,
312   \begin{equation}
313   \Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
314   \end{equation}
315 < A canonical ensemble(NVT)is an ensemble of systems, each of which
315 > A canonical ensemble (NVT)is an ensemble of systems, each of which
316   can share its energy with a large heat reservoir. The distribution
317   of the total energy amongst the possible dynamical states is given
318   by the partition function,
# Line 326 | Line 321 | TS$. Since most experiment are carried out under const
321   \label{introEquation:NVTPartition}
322   \end{equation}
323   Here, $A$ is the Helmholtz free energy which is defined as $ A = U -
324 < TS$. Since most experiment are carried out under constant pressure
325 < condition, isothermal-isobaric ensemble(NPT) play a very important
326 < role in molecular simulation. The isothermal-isobaric ensemble allow
327 < the system to exchange energy with a heat bath of temperature $T$
328 < and to change the volume as well. Its partition function is given as
324 > TS$. Since most experiments are carried out under constant pressure
325 > condition, the isothermal-isobaric ensemble (NPT) plays a very
326 > important role in molecular simulations. The isothermal-isobaric
327 > ensemble allow the system to exchange energy with a heat bath of
328 > temperature $T$ and to change the volume as well. Its partition
329 > function is given as
330   \begin{equation}
331   \Delta (N,P,T) =  - e^{\beta G}.
332   \label{introEquation:NPTPartition}
# Line 339 | Line 335 | The Liouville's theorem is the foundation on which sta
335  
336   \subsection{\label{introSection:liouville}Liouville's theorem}
337  
338 < The Liouville's theorem is the foundation on which statistical
339 < mechanics rests. It describes the time evolution of phase space
338 > Liouville's theorem is the foundation on which statistical mechanics
339 > rests. It describes the time evolution of the phase space
340   distribution function. In order to calculate the rate of change of
341 < $\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we
342 < consider the two faces perpendicular to the $q_1$ axis, which are
343 < located at $q_1$ and $q_1 + \delta q_1$, the number of phase points
344 < leaving the opposite face is given by the expression,
341 > $\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider
342 > the two faces perpendicular to the $q_1$ axis, which are located at
343 > $q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the
344 > opposite face is given by the expression,
345   \begin{equation}
346   \left( {\rho  + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 }
347   \right)\left( {\dot q_1  + \frac{{\partial \dot q_1 }}{{\partial q_1
# Line 369 | Line 365 | divining $ \delta q_1  \ldots \delta q_f \delta p_1  \
365   + \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)}  = 0 ,
366   \end{equation}
367   which cancels the first terms of the right hand side. Furthermore,
368 < divining $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
368 > dividing $ \delta q_1  \ldots \delta q_f \delta p_1  \ldots \delta
369   p_f $ in both sides, we can write out Liouville's theorem in a
370   simple form,
371   \begin{equation}
# Line 381 | Line 377 | statistical mechanics, since the number of particles i
377  
378   Liouville's theorem states that the distribution function is
379   constant along any trajectory in phase space. In classical
380 < statistical mechanics, since the number of particles in the system
381 < is huge, we may be able to believe the system is stationary,
380 > statistical mechanics, since the number of members in an ensemble is
381 > huge and constant, we can assume the local density has no reason
382 > (other than classical mechanics) to change,
383   \begin{equation}
384   \frac{{\partial \rho }}{{\partial t}} = 0.
385   \label{introEquation:stationary}
# Line 395 | Line 392 | distribution,
392   \label{introEquation:densityAndHamiltonian}
393   \end{equation}
394  
395 < \subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space}
395 > \subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}}
396   Lets consider a region in the phase space,
397   \begin{equation}
398   \delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f .
399   \end{equation}
400   If this region is small enough, the density $\rho$ can be regarded
401 < as uniform over the whole phase space. Thus, the number of phase
402 < points inside this region is given by,
401 > as uniform over the whole integral. Thus, the number of phase points
402 > inside this region is given by,
403   \begin{equation}
404   \delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f
405   dp_1 } ..dp_f.
# Line 414 | Line 411 | With the help of stationary assumption
411   \end{equation}
412   With the help of stationary assumption
413   (\ref{introEquation:stationary}), we obtain the principle of the
414 < \emph{conservation of extension in phase space},
414 > \emph{conservation of volume in phase space},
415   \begin{equation}
416   \frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 }
417   ...dq_f dp_1 } ..dp_f  = 0.
418   \label{introEquation:volumePreserving}
419   \end{equation}
420  
421 < \subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms}
421 > \subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}}
422  
423   Liouville's theorem can be expresses in a variety of different forms
424   which are convenient within different contexts. For any two function
# Line 435 | Line 432 | Substituting equations of motion in Hamiltonian formal
432   \label{introEquation:poissonBracket}
433   \end{equation}
434   Substituting equations of motion in Hamiltonian formalism(
435 < \ref{introEquation:motionHamiltonianCoordinate} ,
436 < \ref{introEquation:motionHamiltonianMomentum} ) into
437 < (\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's
438 < theorem using Poisson bracket notion,
435 > Eq.~\ref{introEquation:motionHamiltonianCoordinate} ,
436 > Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into
437 > (Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite
438 > Liouville's theorem using Poisson bracket notion,
439   \begin{equation}
440   \left( {\frac{{\partial \rho }}{{\partial t}}} \right) =  - \left\{
441   {\rho ,H} \right\}.
# Line 463 | Line 460 | simulation and the quality of the underlying model. Ho
460   Various thermodynamic properties can be calculated from Molecular
461   Dynamics simulation. By comparing experimental values with the
462   calculated properties, one can determine the accuracy of the
463 < simulation and the quality of the underlying model. However, both of
464 < experiment and computer simulation are usually performed during a
463 > simulation and the quality of the underlying model. However, both
464 > experiments and computer simulations are usually performed during a
465   certain time interval and the measurements are averaged over a
466   period of them which is different from the average behavior of
467 < many-body system in Statistical Mechanics. Fortunately, Ergodic
468 < Hypothesis is proposed to make a connection between time average and
469 < ensemble average. It states that time average and average over the
467 > many-body system in Statistical Mechanics. Fortunately, the Ergodic
468 > Hypothesis makes a connection between time average and the ensemble
469 > average. It states that the time average and average over the
470   statistical ensemble are identical \cite{Frenkel1996, Leach2001}.
471   \begin{equation}
472   \langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
# Line 491 | Line 488 | A variety of numerical integrators were proposed to si
488   choice\cite{Frenkel1996}.
489  
490   \section{\label{introSection:geometricIntegratos}Geometric Integrators}
491 < A variety of numerical integrators were proposed to simulate the
492 < motions. They usually begin with an initial conditionals and move
493 < the objects in the direction governed by the differential equations.
494 < However, most of them ignore the hidden physical law contained
495 < within the equations. Since 1990, geometric integrators, which
496 < preserve various phase-flow invariants such as symplectic structure,
497 < volume and time reversal symmetry, are developed to address this
498 < issue\cite{Dullweber1997, McLachlan1998, Leimkuhler1999}. The
499 < velocity verlet method, which happens to be a simple example of
500 < symplectic integrator, continues to gain its popularity in molecular
501 < dynamics community. This fact can be partly explained by its
502 < geometric nature.
491 > A variety of numerical integrators have been proposed to simulate
492 > the motions of atoms in MD simulation. They usually begin with
493 > initial conditionals and move the objects in the direction governed
494 > by the differential equations. However, most of them ignore the
495 > hidden physical laws contained within the equations. Since 1990,
496 > geometric integrators, which preserve various phase-flow invariants
497 > such as symplectic structure, volume and time reversal symmetry, are
498 > developed to address this issue\cite{Dullweber1997, McLachlan1998,
499 > Leimkuhler1999}. The velocity verlet method, which happens to be a
500 > simple example of symplectic integrator, continues to gain
501 > popularity in the molecular dynamics community. This fact can be
502 > partly explained by its geometric nature.
503  
504 < \subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
505 < A \emph{manifold} is an abstract mathematical space. It locally
506 < looks like Euclidean space, but when viewed globally, it may have
507 < more complicate structure. A good example of manifold is the surface
508 < of Earth. It seems to be flat locally, but it is round if viewed as
509 < a whole. A \emph{differentiable manifold} (also known as
510 < \emph{smooth manifold}) is a manifold with an open cover in which
511 < the covering neighborhoods are all smoothly isomorphic to one
512 < 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
504 > \subsection{\label{introSection:symplecticManifold}Symplectic Manifolds}
505 > A \emph{manifold} is an abstract mathematical space. It looks
506 > locally like Euclidean space, but when viewed globally, it may have
507 > more complicated structure. A good example of manifold is the
508 > surface of Earth. It seems to be flat locally, but it is round if
509 > viewed as a whole. A \emph{differentiable manifold} (also known as
510 > \emph{smooth manifold}) is a manifold on which it is possible to
511 > apply calculus on \emph{differentiable manifold}. A \emph{symplectic
512 > manifold} is defined as a pair $(M, \omega)$ which consists of a
513   \emph{differentiable manifold} $M$ and a close, non-degenerated,
514   bilinear symplectic form, $\omega$. A symplectic form on a vector
515   space $V$ is a function $\omega(x, y)$ which satisfies
516   $\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+
517   \lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and
518 < $\omega(x, x) = 0$. Cross product operation in vector field is an
519 < example of symplectic form.
518 > $\omega(x, x) = 0$. The cross product operation in vector field is
519 > an example of symplectic form.
520  
521 < One of the motivations to study \emph{symplectic manifold} in
521 > One of the motivations to study \emph{symplectic manifolds} in
522   Hamiltonian Mechanics is that a symplectic manifold can represent
523   all possible configurations of the system and the phase space of the
524   system can be described by it's cotangent bundle. Every symplectic
525   manifold is even dimensional. For instance, in Hamilton equations,
526   coordinate and momentum always appear in pairs.
527  
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 –
528   \subsection{\label{introSection:ODE}Ordinary Differential Equations}
529  
530 < For a ordinary differential system defined as
530 > For an ordinary differential system defined as
531   \begin{equation}
532   \dot x = f(x)
533   \end{equation}
534 < where $x = x(q,p)^T$, this system is canonical Hamiltonian, if
534 > where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if
535   \begin{equation}
536   f(r) = J\nabla _x H(r).
537   \end{equation}
# Line 689 | Line 675 | constructed. The most famous example is leapfrog metho
675   A lot of well established and very effective numerical methods have
676   been successful precisely because of their symplecticities even
677   though this fact was not recognized when they were first
678 < constructed. The most famous example is leapfrog methods in
679 < molecular dynamics. In general, symplectic integrators can be
678 > constructed. The most famous example is the Verlet-leapfrog methods
679 > in molecular dynamics. In general, symplectic integrators can be
680   constructed using one of four different methods.
681   \begin{enumerate}
682   \item Generating functions
# Line 708 | Line 694 | implementing the Runge-Kutta methods, they do not attr
694   high-order explicit Runge-Kutta methods
695   \cite{Owren1992,Chen2003}have been developed to overcome this
696   instability. However, due to computational penalty involved in
697 < implementing the Runge-Kutta methods, they do not attract too much
698 < attention from Molecular Dynamics community. Instead, splitting have
699 < been widely accepted since they exploit natural decompositions of
700 < the system\cite{Tuckerman1992, McLachlan1998}.
697 > implementing the Runge-Kutta methods, they have not attracted much
698 > attention from the Molecular Dynamics community. Instead, splitting
699 > methods have been widely accepted since they exploit natural
700 > decompositions of the system\cite{Tuckerman1992, McLachlan1998}.
701  
702 < \subsubsection{\label{introSection:splittingMethod}Splitting Method}
702 > \subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}}
703  
704   The main idea behind splitting methods is to decompose the discrete
705   $\varphi_h$ as a composition of simpler flows,
# Line 734 | Line 720 | order is then given by the Lie-Trotter formula
720   energy respectively, which is a natural decomposition of the
721   problem. If $H_1$ and $H_2$ can be integrated using exact flows
722   $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first
723 < order is then given by the Lie-Trotter formula
723 > order expression is then given by the Lie-Trotter formula
724   \begin{equation}
725   \varphi _h  = \varphi _{1,h}  \circ \varphi _{2,h},
726   \label{introEquation:firstOrderSplitting}
# Line 760 | Line 746 | which has a local error proportional to $h^3$. Sprang
746   \varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
747   _{1,h/2} , \label{introEquation:secondOrderSplitting}
748   \end{equation}
749 < which has a local error proportional to $h^3$. Sprang splitting's
750 < popularity in molecular simulation community attribute to its
751 < symmetric property,
749 > which has a local error proportional to $h^3$. The Sprang
750 > splitting's popularity in molecular simulation community attribute
751 > to its symmetric property,
752   \begin{equation}
753   \varphi _h^{ - 1} = \varphi _{ - h}.
754   \label{introEquation:timeReversible}
755 < \end{equation}
755 > \end{equation},appendixFig:architecture
756  
757 < \subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method}
757 > \subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Example of Splitting Method}}
758   The classical equation for a system consisting of interacting
759   particles can be written in Hamiltonian form,
760   \[
# Line 828 | Line 814 | q(\Delta t)} \right]. %
814   \label{introEquation:positionVerlet2}
815   \end{align}
816  
817 < \subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
817 > \subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}}
818  
819   Baker-Campbell-Hausdorff formula can be used to determine the local
820   error of splitting method in terms of commutator of the
# Line 914 | Line 900 | initialization of a simulation. Sec.~\ref{introSec:pro
900   \end{enumerate}
901   These three individual steps will be covered in the following
902   sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
903 < initialization of a simulation. Sec.~\ref{introSec:production} will
904 < discusses issues in production run. Sec.~\ref{introSection:Analysis}
905 < provides the theoretical tools for trajectory analysis.
903 > initialization of a simulation. Sec.~\ref{introSection:production}
904 > will discusses issues in production run.
905 > Sec.~\ref{introSection:Analysis} provides the theoretical tools for
906 > trajectory analysis.
907  
908   \subsection{\label{introSec:initialSystemSettings}Initialization}
909  
910 < \subsubsection{Preliminary preparation}
910 > \subsubsection{\textbf{Preliminary preparation}}
911  
912   When selecting the starting structure of a molecule for molecular
913   simulation, one may retrieve its Cartesian coordinates from public
# Line 938 | Line 925 | interested in self-aggregation and it takes a long tim
925   instead of placing lipids randomly in solvent, since we are not
926   interested in self-aggregation and it takes a long time to happen.
927  
928 < \subsubsection{Minimization}
928 > \subsubsection{\textbf{Minimization}}
929  
930   It is quite possible that some of molecules in the system from
931   preliminary preparation may be overlapped with each other. This
# Line 960 | Line 947 | Newton-Raphson methods can not be used with very large
947   matrix and insufficient storage capacity to store them, most
948   Newton-Raphson methods can not be used with very large models.
949  
950 < \subsubsection{Heating}
950 > \subsubsection{\textbf{Heating}}
951  
952   Typically, Heating is performed by assigning random velocities
953   according to a Gaussian distribution for a temperature. Beginning at
# Line 972 | Line 959 | shifted to zero.
959   net linear momentum and angular momentum of the system should be
960   shifted to zero.
961  
962 < \subsubsection{Equilibration}
962 > \subsubsection{\textbf{Equilibration}}
963  
964   The purpose of equilibration is to allow the system to evolve
965   spontaneously for a period of time and reach equilibrium. The
# Line 1078 | Line 1065 | from the trajectories.
1065   parameters, and investigate time-dependent processes of the molecule
1066   from the trajectories.
1067  
1068 < \subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties}
1068 > \subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamics Properties}}
1069  
1070   Thermodynamics properties, which can be expressed in terms of some
1071   function of the coordinates and momenta of all particles in the
# Line 1100 | Line 1087 | P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\
1087   < j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
1088   \end{equation}
1089  
1090 < \subsubsection{\label{introSection:structuralProperties}Structural Properties}
1090 > \subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}}
1091  
1092   Structural Properties of a simple fluid can be described by a set of
1093   distribution functions. Among these functions,\emph{pair
# Line 1140 | Line 1127 | other is essentially zero.
1127   %\label{introFigure:pairDistributionFunction}
1128   %\end{figure}
1129  
1130 < \subsubsection{\label{introSection:timeDependentProperties}Time-dependent
1131 < Properties}
1130 > \subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent
1131 > Properties}}
1132  
1133   Time-dependent properties are usually calculated using \emph{time
1134   correlation function}, which correlates random variables $A$ and $B$
# Line 1343 | Line 1330 | operations
1330   \[
1331   \hat vu = v \times u
1332   \]
1346 –
1333   Using \ref{introEqaution:RBMotionPI}, one can construct a skew
1334   matrix,
1335   \begin{equation}
1336 < (\mathop \Pi \limits^ \bullet   - \mathop \Pi \limits^ \bullet  ^T
1336 > (\mathop \Pi \limits^ \bullet   - \mathop \Pi \limits^ {\bullet  ^T}
1337   ){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi  + \Pi J^{
1338   - 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]} -
1339   (\Lambda  - \Lambda ^T ) . \label{introEquation:skewMatrixPI}
# Line 1376 | Line 1362 | first term of \ref{ introEquation:bodyAngularMotion}).
1362  
1363   If there is not external forces exerted on the rigid body, the only
1364   contribution to the rotational is from the kinetic potential (the
1365 < first term of \ref{ introEquation:bodyAngularMotion}). The free
1366 < rigid body is an example of Lie-Poisson system with Hamiltonian
1381 < function
1365 > first term of \ref{introEquation:bodyAngularMotion}). The free rigid
1366 > body is an example of Lie-Poisson system with Hamiltonian function
1367   \begin{equation}
1368   T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
1369   \label{introEquation:rotationalKineticRB}
# Line 1697 | Line 1682 | which is known as the \emph{generalized Langevin equat
1682   \end{equation}
1683   which is known as the \emph{generalized Langevin equation}.
1684  
1685 < \subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel}
1685 > \subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}}
1686  
1687   One may notice that $R(t)$ depends only on initial conditions, which
1688   implies it is completely deterministic within the context of a
# Line 1752 | Line 1737 | or be determined by Stokes' law for regular shaped par
1737   \end{equation}
1738   which is known as the Langevin equation. The static friction
1739   coefficient $\xi _0$ can either be calculated from spectral density
1740 < or be determined by Stokes' law for regular shaped particles.A
1740 > or be determined by Stokes' law for regular shaped particles. A
1741   briefly review on calculating friction tensor for arbitrary shaped
1742   particles is given in Sec.~\ref{introSection:frictionTensor}.
1743  
1744 < \subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
1744 > \subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}}
1745  
1746   Defining a new set of coordinates,
1747   \[
# Line 1785 | Line 1770 | can model the random force and friction kernel.
1770   \end{equation}
1771   In effect, it acts as a constraint on the possible ways in which one
1772   can model the random force and friction kernel.
1788 –
1789 – \subsection{\label{introSection:frictionTensor} Friction Tensor}
1790 – Theoretically, the friction kernel can be determined using velocity
1791 – autocorrelation function. However, this approach become impractical
1792 – when the system become more and more complicate. Instead, various
1793 – approaches based on hydrodynamics have been developed to calculate
1794 – the friction coefficients. The friction effect is isotropic in
1795 – Equation, $\zeta$ can be taken as a scalar. In general, friction
1796 – tensor $\Xi$ is a $6\times 6$ matrix given by
1797 – \[
1798 – \Xi  = \left( {\begin{array}{*{20}c}
1799 –   {\Xi _{}^{tt} } & {\Xi _{}^{rt} }  \\
1800 –   {\Xi _{}^{tr} } & {\Xi _{}^{rr} }  \\
1801 – \end{array}} \right).
1802 – \]
1803 – Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction
1804 – tensor and rotational resistance (friction) tensor respectively,
1805 – while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $
1806 – {\Xi^{rt} }$ is rotation-translation coupling tensor. When a
1807 – particle moves in a fluid, it may experience friction force or
1808 – torque along the opposite direction of the velocity or angular
1809 – velocity,
1810 – \[
1811 – \left( \begin{array}{l}
1812 – F_R  \\
1813 – \tau _R  \\
1814 – \end{array} \right) =  - \left( {\begin{array}{*{20}c}
1815 –   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
1816 –   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
1817 – \end{array}} \right)\left( \begin{array}{l}
1818 – v \\
1819 – w \\
1820 – \end{array} \right)
1821 – \]
1822 – where $F_r$ is the friction force and $\tau _R$ is the friction
1823 – toque.
1824 –
1825 – \subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape}
1826 –
1827 – For a spherical particle, the translational and rotational friction
1828 – constant can be calculated from Stoke's law,
1829 – \[
1830 – \Xi ^{tt}  = \left( {\begin{array}{*{20}c}
1831 –   {6\pi \eta R} & 0 & 0  \\
1832 –   0 & {6\pi \eta R} & 0  \\
1833 –   0 & 0 & {6\pi \eta R}  \\
1834 – \end{array}} \right)
1835 – \]
1836 – and
1837 – \[
1838 – \Xi ^{rr}  = \left( {\begin{array}{*{20}c}
1839 –   {8\pi \eta R^3 } & 0 & 0  \\
1840 –   0 & {8\pi \eta R^3 } & 0  \\
1841 –   0 & 0 & {8\pi \eta R^3 }  \\
1842 – \end{array}} \right)
1843 – \]
1844 – where $\eta$ is the viscosity of the solvent and $R$ is the
1845 – hydrodynamics radius.
1846 –
1847 – Other non-spherical shape, such as cylinder and ellipsoid
1848 – \textit{etc}, are widely used as reference for developing new
1849 – hydrodynamics theory, because their properties can be calculated
1850 – exactly. In 1936, Perrin extended Stokes's law to general ellipsoid,
1851 – also called a triaxial ellipsoid, which is given in Cartesian
1852 – coordinates by\cite{Perrin1934, Perrin1936}
1853 – \[
1854 – \frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2
1855 – }} = 1
1856 – \]
1857 – where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately,
1858 – due to the complexity of the elliptic integral, only the ellipsoid
1859 – with the restriction of two axes having to be equal, \textit{i.e.}
1860 – prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved
1861 – exactly. Introducing an elliptic integral parameter $S$ for prolate,
1862 – \[
1863 – S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
1864 – } }}{b},
1865 – \]
1866 – and oblate,
1867 – \[
1868 – S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
1869 – }}{a}
1870 – \],
1871 – one can write down the translational and rotational resistance
1872 – tensors
1873 – \[
1874 – \begin{array}{l}
1875 – \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
1876 – \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
1877 – \end{array},
1878 – \]
1879 – and
1880 – \[
1881 – \begin{array}{l}
1882 – \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
1883 – \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
1884 – \end{array}.
1885 – \]
1886 –
1887 – \subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape}
1888 –
1889 – Unlike spherical and other regular shaped molecules, there is not
1890 – analytical solution for friction tensor of any arbitrary shaped
1891 – rigid molecules. The ellipsoid of revolution model and general
1892 – triaxial ellipsoid model have been used to approximate the
1893 – hydrodynamic properties of rigid bodies. However, since the mapping
1894 – from all possible ellipsoidal space, $r$-space, to all possible
1895 – combination of rotational diffusion coefficients, $D$-space is not
1896 – unique\cite{Wegener1979} as well as the intrinsic coupling between
1897 – translational and rotational motion of rigid body, general ellipsoid
1898 – is not always suitable for modeling arbitrarily shaped rigid
1899 – molecule. A number of studies have been devoted to determine the
1900 – friction tensor for irregularly shaped rigid bodies using more
1901 – advanced method where the molecule of interest was modeled by
1902 – combinations of spheres(beads)\cite{Carrasco1999} and the
1903 – hydrodynamics properties of the molecule can be calculated using the
1904 – hydrodynamic interaction tensor. Let us consider a rigid assembly of
1905 – $N$ beads immersed in a continuous medium. Due to hydrodynamics
1906 – interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different
1907 – than its unperturbed velocity $v_i$,
1908 – \[
1909 – v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
1910 – \]
1911 – where $F_i$ is the frictional force, and $T_{ij}$ is the
1912 – hydrodynamic interaction tensor. The friction force of $i$th bead is
1913 – proportional to its ``net'' velocity
1914 – \begin{equation}
1915 – F_i  = \zeta _i v_i  - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }.
1916 – \label{introEquation:tensorExpression}
1917 – \end{equation}
1918 – This equation is the basis for deriving the hydrodynamic tensor. In
1919 – 1930, Oseen and Burgers gave a simple solution to Equation
1920 – \ref{introEquation:tensorExpression}
1921 – \begin{equation}
1922 – T_{ij}  = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij}
1923 – R_{ij}^T }}{{R_{ij}^2 }}} \right).
1924 – \label{introEquation:oseenTensor}
1925 – \end{equation}
1926 – Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
1927 – A second order expression for element of different size was
1928 – introduced by Rotne and Prager\cite{Rotne1969} and improved by
1929 – Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977},
1930 – \begin{equation}
1931 – T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
1932 – \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
1933 – _i^2  + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} -
1934 – \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
1935 – \label{introEquation:RPTensorNonOverlapped}
1936 – \end{equation}
1937 – Both of the Equation \ref{introEquation:oseenTensor} and Equation
1938 – \ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij}
1939 – \ge \sigma _i  + \sigma _j$. An alternative expression for
1940 – overlapping beads with the same radius, $\sigma$, is given by
1941 – \begin{equation}
1942 – T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
1943 – \frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I +
1944 – \frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
1945 – \label{introEquation:RPTensorOverlapped}
1946 – \end{equation}
1947 –
1948 – To calculate the resistance tensor at an arbitrary origin $O$, we
1949 – construct a $3N \times 3N$ matrix consisting of $N \times N$
1950 – $B_{ij}$ blocks
1951 – \begin{equation}
1952 – B = \left( {\begin{array}{*{20}c}
1953 –   {B_{11} } &  \ldots  & {B_{1N} }  \\
1954 –    \vdots  &  \ddots  &  \vdots   \\
1955 –   {B_{N1} } &  \cdots  & {B_{NN} }  \\
1956 – \end{array}} \right),
1957 – \end{equation}
1958 – where $B_{ij}$ is given by
1959 – \[
1960 – B_{ij}  = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij}
1961 – )T_{ij}
1962 – \]
1963 – where $\delta _{ij}$ is Kronecker delta function. Inverting matrix
1964 – $B$, we obtain
1965 –
1966 – \[
1967 – C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
1968 –   {C_{11} } &  \ldots  & {C_{1N} }  \\
1969 –    \vdots  &  \ddots  &  \vdots   \\
1970 –   {C_{N1} } &  \cdots  & {C_{NN} }  \\
1971 – \end{array}} \right)
1972 – \]
1973 – , which can be partitioned into $N \times N$ $3 \times 3$ block
1974 – $C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$
1975 – \[
1976 – U_i  = \left( {\begin{array}{*{20}c}
1977 –   0 & { - z_i } & {y_i }  \\
1978 –   {z_i } & 0 & { - x_i }  \\
1979 –   { - y_i } & {x_i } & 0  \\
1980 – \end{array}} \right)
1981 – \]
1982 – where $x_i$, $y_i$, $z_i$ are the components of the vector joining
1983 – bead $i$ and origin $O$. Hence, the elements of resistance tensor at
1984 – arbitrary origin $O$ can be written as
1985 – \begin{equation}
1986 – \begin{array}{l}
1987 – \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\
1988 – \Xi _{}^{tr}  = \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
1989 – \Xi _{}^{rr}  =  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j  \\
1990 – \end{array}
1991 – \label{introEquation:ResistanceTensorArbitraryOrigin}
1992 – \end{equation}
1993 –
1994 – The resistance tensor depends on the origin to which they refer. The
1995 – proper location for applying friction force is the center of
1996 – resistance (reaction), at which the trace of rotational resistance
1997 – tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of
1998 – resistance is defined as an unique point of the rigid body at which
1999 – the translation-rotation coupling tensor are symmetric,
2000 – \begin{equation}
2001 – \Xi^{tr}  = \left( {\Xi^{tr} } \right)^T
2002 – \label{introEquation:definitionCR}
2003 – \end{equation}
2004 – Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin},
2005 – we can easily find out that the translational resistance tensor is
2006 – origin independent, while the rotational resistance tensor and
2007 – translation-rotation coupling resistance tensor depend on the
2008 – origin. Given resistance tensor at an arbitrary origin $O$, and a
2009 – vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can
2010 – obtain the resistance tensor at $P$ by
2011 – \begin{equation}
2012 – \begin{array}{l}
2013 – \Xi _P^{tt}  = \Xi _O^{tt}  \\
2014 – \Xi _P^{tr}  = \Xi _P^{rt}  = \Xi _O^{tr}  - U_{OP} \Xi _O^{tt}  \\
2015 – \Xi _P^{rr}  = \Xi _O^{rr}  - U_{OP} \Xi _O^{tt} U_{OP}  + \Xi _O^{tr} U_{OP}  - U_{OP} \Xi _O^{tr} ^{^T }  \\
2016 – \end{array}
2017 – \label{introEquation:resistanceTensorTransformation}
2018 – \end{equation}
2019 – where
2020 – \[
2021 – U_{OP}  = \left( {\begin{array}{*{20}c}
2022 –   0 & { - z_{OP} } & {y_{OP} }  \\
2023 –   {z_i } & 0 & { - x_{OP} }  \\
2024 –   { - y_{OP} } & {x_{OP} } & 0  \\
2025 – \end{array}} \right)
2026 – \]
2027 – Using Equations \ref{introEquation:definitionCR} and
2028 – \ref{introEquation:resistanceTensorTransformation}, one can locate
2029 – the position of center of resistance,
2030 – \begin{eqnarray*}
2031 – \left( \begin{array}{l}
2032 – x_{OR}  \\
2033 – y_{OR}  \\
2034 – z_{OR}  \\
2035 – \end{array} \right) & = &\left( {\begin{array}{*{20}c}
2036 –   {(\Xi _O^{rr} )_{yy}  + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} }  \\
2037 –   { - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz}  + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} }  \\
2038 –   { - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx}  + (\Xi _O^{rr} )_{yy} }  \\
2039 – \end{array}} \right)^{ - 1}  \\
2040 –  & & \left( \begin{array}{l}
2041 – (\Xi _O^{tr} )_{yz}  - (\Xi _O^{tr} )_{zy}  \\
2042 – (\Xi _O^{tr} )_{zx}  - (\Xi _O^{tr} )_{xz}  \\
2043 – (\Xi _O^{tr} )_{xy}  - (\Xi _O^{tr} )_{yx}  \\
2044 – \end{array} \right) \\
2045 – \end{eqnarray*}
2046 –
2047 –
2048 –
2049 – where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
2050 – joining center of resistance $R$ and origin $O$.

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