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 |
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} |
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 |
|
\[ |
59 |
|
\] |
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 |
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} |
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 |
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} |
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 |
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}. |
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 |
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, |
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} |
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 |
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, |
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} |
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 |
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} |
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} |
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. |
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 |
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\}. |
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 } |
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} |
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 |
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, |
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} |
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 |
|
\[ |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 $B$ |
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 |
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 |
|
\[ |
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. |
1787 |
– |
|
1788 |
– |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1789 |
– |
Theoretically, the friction kernel can be determined using velocity |
1790 |
– |
autocorrelation function. However, this approach become impractical |
1791 |
– |
when the system become more and more complicate. Instead, various |
1792 |
– |
approaches based on hydrodynamics have been developed to calculate |
1793 |
– |
the friction coefficients. The friction effect is isotropic in |
1794 |
– |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
1795 |
– |
tensor $\Xi$ is a $6\times 6$ matrix given by |
1796 |
– |
\[ |
1797 |
– |
\Xi = \left( {\begin{array}{*{20}c} |
1798 |
– |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
1799 |
– |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
1800 |
– |
\end{array}} \right). |
1801 |
– |
\] |
1802 |
– |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
1803 |
– |
tensor and rotational resistance (friction) tensor respectively, |
1804 |
– |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
1805 |
– |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
1806 |
– |
particle moves in a fluid, it may experience friction force or |
1807 |
– |
torque along the opposite direction of the velocity or angular |
1808 |
– |
velocity, |
1809 |
– |
\[ |
1810 |
– |
\left( \begin{array}{l} |
1811 |
– |
F_R \\ |
1812 |
– |
\tau _R \\ |
1813 |
– |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
1814 |
– |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
1815 |
– |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
1816 |
– |
\end{array}} \right)\left( \begin{array}{l} |
1817 |
– |
v \\ |
1818 |
– |
w \\ |
1819 |
– |
\end{array} \right) |
1820 |
– |
\] |
1821 |
– |
where $F_r$ is the friction force and $\tau _R$ is the friction |
1822 |
– |
toque. |
1823 |
– |
|
1824 |
– |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
1825 |
– |
|
1826 |
– |
For a spherical particle, the translational and rotational friction |
1827 |
– |
constant can be calculated from Stoke's law, |
1828 |
– |
\[ |
1829 |
– |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
1830 |
– |
{6\pi \eta R} & 0 & 0 \\ |
1831 |
– |
0 & {6\pi \eta R} & 0 \\ |
1832 |
– |
0 & 0 & {6\pi \eta R} \\ |
1833 |
– |
\end{array}} \right) |
1834 |
– |
\] |
1835 |
– |
and |
1836 |
– |
\[ |
1837 |
– |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
1838 |
– |
{8\pi \eta R^3 } & 0 & 0 \\ |
1839 |
– |
0 & {8\pi \eta R^3 } & 0 \\ |
1840 |
– |
0 & 0 & {8\pi \eta R^3 } \\ |
1841 |
– |
\end{array}} \right) |
1842 |
– |
\] |
1843 |
– |
where $\eta$ is the viscosity of the solvent and $R$ is the |
1844 |
– |
hydrodynamics radius. |
1845 |
– |
|
1846 |
– |
Other non-spherical shape, such as cylinder and ellipsoid |
1847 |
– |
\textit{etc}, are widely used as reference for developing new |
1848 |
– |
hydrodynamics theory, because their properties can be calculated |
1849 |
– |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
1850 |
– |
also called a triaxial ellipsoid, which is given in Cartesian |
1851 |
– |
coordinates by\cite{Perrin1934, Perrin1936} |
1852 |
– |
\[ |
1853 |
– |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
1854 |
– |
}} = 1 |
1855 |
– |
\] |
1856 |
– |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
1857 |
– |
due to the complexity of the elliptic integral, only the ellipsoid |
1858 |
– |
with the restriction of two axes having to be equal, \textit{i.e.} |
1859 |
– |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
1860 |
– |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
1861 |
– |
\[ |
1862 |
– |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
1863 |
– |
} }}{b}, |
1864 |
– |
\] |
1865 |
– |
and oblate, |
1866 |
– |
\[ |
1867 |
– |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
1868 |
– |
}}{a} |
1869 |
– |
\], |
1870 |
– |
one can write down the translational and rotational resistance |
1871 |
– |
tensors |
1872 |
– |
\[ |
1873 |
– |
\begin{array}{l} |
1874 |
– |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
1875 |
– |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
1876 |
– |
\end{array}, |
1877 |
– |
\] |
1878 |
– |
and |
1879 |
– |
\[ |
1880 |
– |
\begin{array}{l} |
1881 |
– |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
1882 |
– |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
1883 |
– |
\end{array}. |
1884 |
– |
\] |
1885 |
– |
|
1886 |
– |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
1887 |
– |
|
1888 |
– |
Unlike spherical and other regular shaped molecules, there is not |
1889 |
– |
analytical solution for friction tensor of any arbitrary shaped |
1890 |
– |
rigid molecules. The ellipsoid of revolution model and general |
1891 |
– |
triaxial ellipsoid model have been used to approximate the |
1892 |
– |
hydrodynamic properties of rigid bodies. However, since the mapping |
1893 |
– |
from all possible ellipsoidal space, $r$-space, to all possible |
1894 |
– |
combination of rotational diffusion coefficients, $D$-space is not |
1895 |
– |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
1896 |
– |
translational and rotational motion of rigid body, general ellipsoid |
1897 |
– |
is not always suitable for modeling arbitrarily shaped rigid |
1898 |
– |
molecule. A number of studies have been devoted to determine the |
1899 |
– |
friction tensor for irregularly shaped rigid bodies using more |
1900 |
– |
advanced method where the molecule of interest was modeled by |
1901 |
– |
combinations of spheres(beads)\cite{Carrasco1999} and the |
1902 |
– |
hydrodynamics properties of the molecule can be calculated using the |
1903 |
– |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
1904 |
– |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
1905 |
– |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
1906 |
– |
than its unperturbed velocity $v_i$, |
1907 |
– |
\[ |
1908 |
– |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
1909 |
– |
\] |
1910 |
– |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
1911 |
– |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
1912 |
– |
proportional to its ``net'' velocity |
1913 |
– |
\begin{equation} |
1914 |
– |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
1915 |
– |
\label{introEquation:tensorExpression} |
1916 |
– |
\end{equation} |
1917 |
– |
This equation is the basis for deriving the hydrodynamic tensor. In |
1918 |
– |
1930, Oseen and Burgers gave a simple solution to Equation |
1919 |
– |
\ref{introEquation:tensorExpression} |
1920 |
– |
\begin{equation} |
1921 |
– |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
1922 |
– |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
1923 |
– |
\label{introEquation:oseenTensor} |
1924 |
– |
\end{equation} |
1925 |
– |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
1926 |
– |
A second order expression for element of different size was |
1927 |
– |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
1928 |
– |
Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
1929 |
– |
\begin{equation} |
1930 |
– |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
1931 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
1932 |
– |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
1933 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
1934 |
– |
\label{introEquation:RPTensorNonOverlapped} |
1935 |
– |
\end{equation} |
1936 |
– |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
1937 |
– |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
1938 |
– |
\ge \sigma _i + \sigma _j$. An alternative expression for |
1939 |
– |
overlapping beads with the same radius, $\sigma$, is given by |
1940 |
– |
\begin{equation} |
1941 |
– |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
1942 |
– |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
1943 |
– |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
1944 |
– |
\label{introEquation:RPTensorOverlapped} |
1945 |
– |
\end{equation} |
1946 |
– |
|
1947 |
– |
To calculate the resistance tensor at an arbitrary origin $O$, we |
1948 |
– |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
1949 |
– |
$B_{ij}$ blocks |
1950 |
– |
\begin{equation} |
1951 |
– |
B = \left( {\begin{array}{*{20}c} |
1952 |
– |
{B_{11} } & \ldots & {B_{1N} } \\ |
1953 |
– |
\vdots & \ddots & \vdots \\ |
1954 |
– |
{B_{N1} } & \cdots & {B_{NN} } \\ |
1955 |
– |
\end{array}} \right), |
1956 |
– |
\end{equation} |
1957 |
– |
where $B_{ij}$ is given by |
1958 |
– |
\[ |
1959 |
– |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
1960 |
– |
)T_{ij} |
1961 |
– |
\] |
1962 |
– |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
1963 |
– |
$B$, we obtain |
1964 |
– |
|
1965 |
– |
\[ |
1966 |
– |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
1967 |
– |
{C_{11} } & \ldots & {C_{1N} } \\ |
1968 |
– |
\vdots & \ddots & \vdots \\ |
1969 |
– |
{C_{N1} } & \cdots & {C_{NN} } \\ |
1970 |
– |
\end{array}} \right) |
1971 |
– |
\] |
1972 |
– |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
1973 |
– |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
1974 |
– |
\[ |
1975 |
– |
U_i = \left( {\begin{array}{*{20}c} |
1976 |
– |
0 & { - z_i } & {y_i } \\ |
1977 |
– |
{z_i } & 0 & { - x_i } \\ |
1978 |
– |
{ - y_i } & {x_i } & 0 \\ |
1979 |
– |
\end{array}} \right) |
1980 |
– |
\] |
1981 |
– |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
1982 |
– |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
1983 |
– |
arbitrary origin $O$ can be written as |
1984 |
– |
\begin{equation} |
1985 |
– |
\begin{array}{l} |
1986 |
– |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
1987 |
– |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
1988 |
– |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
1989 |
– |
\end{array} |
1990 |
– |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
1991 |
– |
\end{equation} |
1992 |
– |
|
1993 |
– |
The resistance tensor depends on the origin to which they refer. The |
1994 |
– |
proper location for applying friction force is the center of |
1995 |
– |
resistance (reaction), at which the trace of rotational resistance |
1996 |
– |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
1997 |
– |
resistance is defined as an unique point of the rigid body at which |
1998 |
– |
the translation-rotation coupling tensor are symmetric, |
1999 |
– |
\begin{equation} |
2000 |
– |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
2001 |
– |
\label{introEquation:definitionCR} |
2002 |
– |
\end{equation} |
2003 |
– |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
2004 |
– |
we can easily find out that the translational resistance tensor is |
2005 |
– |
origin independent, while the rotational resistance tensor and |
2006 |
– |
translation-rotation coupling resistance tensor depend on the |
2007 |
– |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
2008 |
– |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
2009 |
– |
obtain the resistance tensor at $P$ by |
2010 |
– |
\begin{equation} |
2011 |
– |
\begin{array}{l} |
2012 |
– |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
2013 |
– |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
2014 |
– |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
2015 |
– |
\end{array} |
2016 |
– |
\label{introEquation:resistanceTensorTransformation} |
2017 |
– |
\end{equation} |
2018 |
– |
where |
2019 |
– |
\[ |
2020 |
– |
U_{OP} = \left( {\begin{array}{*{20}c} |
2021 |
– |
0 & { - z_{OP} } & {y_{OP} } \\ |
2022 |
– |
{z_i } & 0 & { - x_{OP} } \\ |
2023 |
– |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
2024 |
– |
\end{array}} \right) |
2025 |
– |
\] |
2026 |
– |
Using Equations \ref{introEquation:definitionCR} and |
2027 |
– |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
2028 |
– |
the position of center of resistance, |
2029 |
– |
\begin{eqnarray*} |
2030 |
– |
\left( \begin{array}{l} |
2031 |
– |
x_{OR} \\ |
2032 |
– |
y_{OR} \\ |
2033 |
– |
z_{OR} \\ |
2034 |
– |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
2035 |
– |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
2036 |
– |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
2037 |
– |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
2038 |
– |
\end{array}} \right)^{ - 1} \\ |
2039 |
– |
& & \left( \begin{array}{l} |
2040 |
– |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
2041 |
– |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
2042 |
– |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
2043 |
– |
\end{array} \right) \\ |
2044 |
– |
\end{eqnarray*} |
2045 |
– |
|
2046 |
– |
|
2047 |
– |
|
2048 |
– |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
2049 |
– |
joining center of resistance $R$ and origin $O$. |