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{tolman79}. |
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 |
184 |
|
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
185 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
186 |
|
equation of motion. Due to their symmetrical formula, they are also |
187 |
< |
known as the canonical equations of motions \cite{Goldstein01}. |
187 |
> |
known as the canonical equations of motions \cite{Goldstein2001}. |
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{Marion90}. |
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 |
470 |
< |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
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 } |
473 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
481 |
|
a properly weighted statistical average. This allows the researcher |
482 |
|
freedom of choice when deciding how best to measure a given |
483 |
|
observable. In case an ensemble averaged approach sounds most |
484 |
< |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
484 |
> |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
485 |
|
utilized. Or if the system lends itself to a time averaging |
486 |
|
approach, the Molecular Dynamics techniques in |
487 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
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. The velocity verlet method, which happens to be a simple |
499 |
< |
example of symplectic integrator, continues to gain its popularity |
500 |
< |
in molecular dynamics community. This fact can be partly explained |
501 |
< |
by its 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 |
515 |
< |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
516 |
< |
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 |
|
|
532 |
– |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
533 |
– |
\[ |
534 |
– |
f : M \rightarrow N |
535 |
– |
\] |
536 |
– |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
537 |
– |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
538 |
– |
Canonical transformation is an example of symplectomorphism in |
539 |
– |
classical mechanics. |
540 |
– |
|
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} |
552 |
|
\end{equation}In this case, $f$ is |
553 |
|
called a \emph{Hamiltonian vector field}. |
554 |
|
|
555 |
< |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
555 |
> |
Another generalization of Hamiltonian dynamics is Poisson |
556 |
> |
Dynamics\cite{Olver1986}, |
557 |
|
\begin{equation} |
558 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
559 |
|
\end{equation} |
600 |
|
|
601 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
602 |
|
|
603 |
< |
The hidden geometric properties of ODE and its flow play important |
604 |
< |
roles in numerical studies. Many of them can be found in systems |
605 |
< |
which occur naturally in applications. |
603 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE |
604 |
> |
and its flow play important roles in numerical studies. Many of them |
605 |
> |
can be found in systems which occur naturally in applications. |
606 |
|
|
607 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
608 |
|
a \emph{symplectic} flow if it satisfies, |
646 |
|
which is the condition for conserving \emph{first integral}. For a |
647 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
648 |
|
smooth function $G$ is given by, |
649 |
< |
\begin{equation} |
650 |
< |
\begin{array}{c} |
651 |
< |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
652 |
< |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
665 |
< |
\end{array} |
649 |
> |
|
650 |
> |
\begin{eqnarray} |
651 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
652 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
653 |
|
\label{introEquation:firstIntegral1} |
654 |
< |
\end{equation} |
654 |
> |
\end{eqnarray} |
655 |
> |
|
656 |
> |
|
657 |
|
Using poisson bracket notion, Equation |
658 |
|
\ref{introEquation:firstIntegral1} can be rewritten as |
659 |
|
\[ |
668 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
669 |
|
0$. |
670 |
|
|
671 |
< |
|
683 |
< |
When designing any numerical methods, one should always try to |
671 |
> |
When designing any numerical methods, one should always try to |
672 |
|
preserve the structural properties of the original ODE and its flow. |
673 |
|
|
674 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
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 |
685 |
|
\item Splitting methods |
686 |
|
\end{enumerate} |
687 |
|
|
688 |
< |
Generating function tends to lead to methods which are cumbersome |
689 |
< |
and difficult to use. In dissipative systems, variational methods |
690 |
< |
can capture the decay of energy accurately. Since their |
691 |
< |
geometrically unstable nature against non-Hamiltonian perturbations, |
692 |
< |
ordinary implicit Runge-Kutta methods are not suitable for |
693 |
< |
Hamiltonian system. Recently, various high-order explicit |
694 |
< |
Runge--Kutta methods have been developed to overcome this |
688 |
> |
Generating function\cite{Channell1990} tends to lead to methods |
689 |
> |
which are cumbersome and difficult to use. In dissipative systems, |
690 |
> |
variational methods can capture the decay of energy |
691 |
> |
accurately\cite{Kane2000}. Since their geometrically unstable nature |
692 |
> |
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
693 |
> |
methods are not suitable for Hamiltonian system. Recently, various |
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{Tuckerman92}. |
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 |
821 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
822 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
823 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
823 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
824 |
|
\begin{equation} |
825 |
|
\exp (hX + hY) = \exp (hZ) |
826 |
|
\end{equation} |
833 |
|
\[ |
834 |
|
[X,Y] = XY - YX . |
835 |
|
\] |
836 |
< |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
837 |
< |
can obtain |
836 |
> |
Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to |
837 |
> |
Sprang splitting, we can obtain |
838 |
|
\begin{eqnarray*} |
839 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
840 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
841 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
853 |
< |
\ldots ) |
839 |
> |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
840 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
841 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
842 |
|
\end{eqnarray*} |
843 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
844 |
|
error of Spring splitting is proportional to $h^3$. The same |
847 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
848 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
849 |
|
\end{equation} |
850 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
850 |
> |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
851 |
|
order method. Yoshida proposed an elegant way to compose higher |
852 |
< |
order methods based on symmetric splitting. Given a symmetric second |
853 |
< |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
854 |
< |
method can be constructed by composing, |
852 |
> |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
853 |
> |
a symmetric second order base method $ \varphi _h^{(2)} $, a |
854 |
> |
fourth-order symmetric method can be constructed by composing, |
855 |
|
\[ |
856 |
|
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
857 |
|
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
871 |
|
|
872 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
873 |
|
|
874 |
< |
As a special discipline of molecular modeling, Molecular dynamics |
875 |
< |
has proven to be a powerful tool for studying the functions of |
876 |
< |
biological systems, providing structural, thermodynamic and |
877 |
< |
dynamical information. |
874 |
> |
As one of the principal tools of molecular modeling, Molecular |
875 |
> |
dynamics has proven to be a powerful tool for studying the functions |
876 |
> |
of biological systems, providing structural, thermodynamic and |
877 |
> |
dynamical information. The basic idea of molecular dynamics is that |
878 |
> |
macroscopic properties are related to microscopic behavior and |
879 |
> |
microscopic behavior can be calculated from the trajectories in |
880 |
> |
simulations. For instance, instantaneous temperature of an |
881 |
> |
Hamiltonian system of $N$ particle can be measured by |
882 |
> |
\[ |
883 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
884 |
> |
\] |
885 |
> |
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
886 |
> |
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
887 |
> |
the boltzman constant. |
888 |
|
|
889 |
< |
One of the principal tools for modeling proteins, nucleic acids and |
890 |
< |
their complexes. Stability of proteins Folding of proteins. |
891 |
< |
Molecular recognition by:proteins, DNA, RNA, lipids, hormones STP, |
892 |
< |
etc. Enzyme reactions Rational design of biologically active |
893 |
< |
molecules (drug design) Small and large-scale conformational |
894 |
< |
changes. determination and construction of 3D structures (homology, |
895 |
< |
Xray diffraction, NMR) Dynamic processes such as ion transport in |
896 |
< |
biological systems. |
889 |
> |
A typical molecular dynamics run consists of three essential steps: |
890 |
> |
\begin{enumerate} |
891 |
> |
\item Initialization |
892 |
> |
\begin{enumerate} |
893 |
> |
\item Preliminary preparation |
894 |
> |
\item Minimization |
895 |
> |
\item Heating |
896 |
> |
\item Equilibration |
897 |
> |
\end{enumerate} |
898 |
> |
\item Production |
899 |
> |
\item Analysis |
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{introSection:production} |
904 |
> |
will discusses issues in production run. |
905 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
906 |
> |
trajectory analysis. |
907 |
|
|
908 |
< |
Macroscopic properties are related to microscopic behavior. |
908 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
909 |
|
|
910 |
< |
Time dependent (and independent) microscopic behavior of a molecule |
903 |
< |
can be calculated by molecular dynamics simulations. |
910 |
> |
\subsubsection{\textbf{Preliminary preparation}} |
911 |
|
|
912 |
< |
\subsection{\label{introSec:mdInit}Initialization} |
912 |
> |
When selecting the starting structure of a molecule for molecular |
913 |
> |
simulation, one may retrieve its Cartesian coordinates from public |
914 |
> |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
915 |
> |
thousands of crystal structures of molecules are discovered every |
916 |
> |
year, many more remain unknown due to the difficulties of |
917 |
> |
purification and crystallization. Even for the molecule with known |
918 |
> |
structure, some important information is missing. For example, the |
919 |
> |
missing hydrogen atom which acts as donor in hydrogen bonding must |
920 |
> |
be added. Moreover, in order to include electrostatic interaction, |
921 |
> |
one may need to specify the partial charges for individual atoms. |
922 |
> |
Under some circumstances, we may even need to prepare the system in |
923 |
> |
a special setup. For instance, when studying transport phenomenon in |
924 |
> |
membrane system, we may prepare the lipids in bilayer structure |
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 |
< |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
928 |
> |
\subsubsection{\textbf{Minimization}} |
929 |
|
|
930 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
930 |
> |
It is quite possible that some of molecules in the system from |
931 |
> |
preliminary preparation may be overlapped with each other. This |
932 |
> |
close proximity leads to high potential energy which consequently |
933 |
> |
jeopardizes any molecular dynamics simulations. To remove these |
934 |
> |
steric overlaps, one typically performs energy minimization to find |
935 |
> |
a more reasonable conformation. Several energy minimization methods |
936 |
> |
have been developed to exploit the energy surface and to locate the |
937 |
> |
local minimum. While converging slowly near the minimum, steepest |
938 |
> |
descent method is extremely robust when systems are far from |
939 |
> |
harmonic. Thus, it is often used to refine structure from |
940 |
> |
crystallographic data. Relied on the gradient or hessian, advanced |
941 |
> |
methods like conjugate gradient and Newton-Raphson converge rapidly |
942 |
> |
to a local minimum, while become unstable if the energy surface is |
943 |
> |
far from quadratic. Another factor must be taken into account, when |
944 |
> |
choosing energy minimization method, is the size of the system. |
945 |
> |
Steepest descent and conjugate gradient can deal with models of any |
946 |
> |
size. Because of the limit of computation power to calculate hessian |
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{\textbf{Heating}} |
951 |
> |
|
952 |
> |
Typically, Heating is performed by assigning random velocities |
953 |
> |
according to a Gaussian distribution for a temperature. Beginning at |
954 |
> |
a lower temperature and gradually increasing the temperature by |
955 |
> |
assigning greater random velocities, we end up with setting the |
956 |
> |
temperature of the system to a final temperature at which the |
957 |
> |
simulation will be conducted. In heating phase, we should also keep |
958 |
> |
the system from drifting or rotating as a whole. Equivalently, the |
959 |
> |
net linear momentum and angular momentum of the system should be |
960 |
> |
shifted to zero. |
961 |
> |
|
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 |
966 |
> |
procedure is continued until various statistical properties, such as |
967 |
> |
temperature, pressure, energy, volume and other structural |
968 |
> |
properties \textit{etc}, become independent of time. Strictly |
969 |
> |
speaking, minimization and heating are not necessary, provided the |
970 |
> |
equilibration process is long enough. However, these steps can serve |
971 |
> |
as a means to arrive at an equilibrated structure in an effective |
972 |
> |
way. |
973 |
> |
|
974 |
> |
\subsection{\label{introSection:production}Production} |
975 |
> |
|
976 |
> |
Production run is the most important step of the simulation, in |
977 |
> |
which the equilibrated structure is used as a starting point and the |
978 |
> |
motions of the molecules are collected for later analysis. In order |
979 |
> |
to capture the macroscopic properties of the system, the molecular |
980 |
> |
dynamics simulation must be performed in correct and efficient way. |
981 |
> |
|
982 |
> |
The most expensive part of a molecular dynamics simulation is the |
983 |
> |
calculation of non-bonded forces, such as van der Waals force and |
984 |
> |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
985 |
> |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
986 |
> |
which making large simulations prohibitive in the absence of any |
987 |
> |
computation saving techniques. |
988 |
> |
|
989 |
> |
A natural approach to avoid system size issue is to represent the |
990 |
> |
bulk behavior by a finite number of the particles. However, this |
991 |
> |
approach will suffer from the surface effect. To offset this, |
992 |
> |
\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc}) |
993 |
> |
is developed to simulate bulk properties with a relatively small |
994 |
> |
number of particles. In this method, the simulation box is |
995 |
> |
replicated throughout space to form an infinite lattice. During the |
996 |
> |
simulation, when a particle moves in the primary cell, its image in |
997 |
> |
other cells move in exactly the same direction with exactly the same |
998 |
> |
orientation. Thus, as a particle leaves the primary cell, one of its |
999 |
> |
images will enter through the opposite face. |
1000 |
> |
\begin{figure} |
1001 |
> |
\centering |
1002 |
> |
\includegraphics[width=\linewidth]{pbc.eps} |
1003 |
> |
\caption[An illustration of periodic boundary conditions]{A 2-D |
1004 |
> |
illustration of periodic boundary conditions. As one particle leaves |
1005 |
> |
the left of the simulation box, an image of it enters the right.} |
1006 |
> |
\label{introFig:pbc} |
1007 |
> |
\end{figure} |
1008 |
> |
|
1009 |
> |
%cutoff and minimum image convention |
1010 |
> |
Another important technique to improve the efficiency of force |
1011 |
> |
evaluation is to apply cutoff where particles farther than a |
1012 |
> |
predetermined distance, are not included in the calculation |
1013 |
> |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1014 |
> |
discontinuity in the potential energy curve. Fortunately, one can |
1015 |
> |
shift the potential to ensure the potential curve go smoothly to |
1016 |
> |
zero at the cutoff radius. Cutoff strategy works pretty well for |
1017 |
> |
Lennard-Jones interaction because of its short range nature. |
1018 |
> |
However, simply truncating the electrostatic interaction with the |
1019 |
> |
use of cutoff has been shown to lead to severe artifacts in |
1020 |
> |
simulations. Ewald summation, in which the slowly conditionally |
1021 |
> |
convergent Coulomb potential is transformed into direct and |
1022 |
> |
reciprocal sums with rapid and absolute convergence, has proved to |
1023 |
> |
minimize the periodicity artifacts in liquid simulations. Taking the |
1024 |
> |
advantages of the fast Fourier transform (FFT) for calculating |
1025 |
> |
discrete Fourier transforms, the particle mesh-based |
1026 |
> |
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
1027 |
> |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast |
1028 |
> |
multipole method}\cite{Greengard1987, Greengard1994}, which treats |
1029 |
> |
Coulombic interaction exactly at short range, and approximate the |
1030 |
> |
potential at long range through multipolar expansion. In spite of |
1031 |
> |
their wide acceptances at the molecular simulation community, these |
1032 |
> |
two methods are hard to be implemented correctly and efficiently. |
1033 |
> |
Instead, we use a damped and charge-neutralized Coulomb potential |
1034 |
> |
method developed by Wolf and his coworkers\cite{Wolf1999}. The |
1035 |
> |
shifted Coulomb potential for particle $i$ and particle $j$ at |
1036 |
> |
distance $r_{rj}$ is given by: |
1037 |
> |
\begin{equation} |
1038 |
> |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1039 |
> |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1040 |
> |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
1041 |
> |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
1042 |
> |
\end{equation} |
1043 |
> |
where $\alpha$ is the convergence parameter. Due to the lack of |
1044 |
> |
inherent periodicity and rapid convergence,this method is extremely |
1045 |
> |
efficient and easy to implement. |
1046 |
> |
\begin{figure} |
1047 |
> |
\centering |
1048 |
> |
\includegraphics[width=\linewidth]{shifted_coulomb.eps} |
1049 |
> |
\caption[An illustration of shifted Coulomb potential]{An |
1050 |
> |
illustration of shifted Coulomb potential.} |
1051 |
> |
\label{introFigure:shiftedCoulomb} |
1052 |
> |
\end{figure} |
1053 |
> |
|
1054 |
> |
%multiple time step |
1055 |
> |
|
1056 |
> |
\subsection{\label{introSection:Analysis} Analysis} |
1057 |
> |
|
1058 |
> |
Recently, advanced visualization technique are widely applied to |
1059 |
> |
monitor the motions of molecules. Although the dynamics of the |
1060 |
> |
system can be described qualitatively from animation, quantitative |
1061 |
> |
trajectory analysis are more appreciable. According to the |
1062 |
> |
principles of Statistical Mechanics, |
1063 |
> |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1064 |
> |
thermodynamics properties, analyze fluctuations of structural |
1065 |
> |
parameters, and investigate time-dependent processes of the molecule |
1066 |
> |
from the trajectories. |
1067 |
> |
|
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 |
1072 |
> |
system, can be directly computed from molecular dynamics. The usual |
1073 |
> |
way to measure the pressure is based on virial theorem of Clausius |
1074 |
> |
which states that the virial is equal to $-3Nk_BT$. For a system |
1075 |
> |
with forces between particles, the total virial, $W$, contains the |
1076 |
> |
contribution from external pressure and interaction between the |
1077 |
> |
particles: |
1078 |
> |
\[ |
1079 |
> |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
1080 |
> |
f_{ij} } } \right\rangle |
1081 |
> |
\] |
1082 |
> |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
1083 |
> |
distance $r_{ij}$. Thus, the expression for the pressure is given |
1084 |
> |
by: |
1085 |
> |
\begin{equation} |
1086 |
> |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
1087 |
> |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1088 |
> |
\end{equation} |
1089 |
|
|
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 |
1094 |
+ |
distribution function}, also known as \emph{radial distribution |
1095 |
+ |
function}, is of most fundamental importance to liquid-state theory. |
1096 |
+ |
Pair distribution function can be gathered by Fourier transforming |
1097 |
+ |
raw data from a series of neutron diffraction experiments and |
1098 |
+ |
integrating over the surface factor \cite{Powles1973}. The |
1099 |
+ |
experiment result can serve as a criterion to justify the |
1100 |
+ |
correctness of the theory. Moreover, various equilibrium |
1101 |
+ |
thermodynamic and structural properties can also be expressed in |
1102 |
+ |
terms of radial distribution function \cite{Allen1987}. |
1103 |
+ |
|
1104 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
1105 |
+ |
particle $i$ will be located at a distance $r$ from a another |
1106 |
+ |
particle $j$ in the system |
1107 |
+ |
\[ |
1108 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1109 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1110 |
+ |
\] |
1111 |
+ |
Note that the delta function can be replaced by a histogram in |
1112 |
+ |
computer simulation. Figure |
1113 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1114 |
+ |
distribution function for the liquid argon system. The occurrence of |
1115 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1116 |
+ |
to find particles at certain radial values than at others. This is a |
1117 |
+ |
result of the attractive interaction at such distances. Because of |
1118 |
+ |
the strong repulsive forces at short distance, the probability of |
1119 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
1120 |
+ |
other is essentially zero. |
1121 |
+ |
|
1122 |
+ |
%\begin{figure} |
1123 |
+ |
%\centering |
1124 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
1125 |
+ |
%\caption[Pair distribution function for the liquid argon |
1126 |
+ |
%]{Pair distribution function for the liquid argon} |
1127 |
+ |
%\label{introFigure:pairDistributionFunction} |
1128 |
+ |
%\end{figure} |
1129 |
+ |
|
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 $B$ |
1135 |
+ |
at two different time |
1136 |
+ |
\begin{equation} |
1137 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1138 |
+ |
\label{introEquation:timeCorrelationFunction} |
1139 |
+ |
\end{equation} |
1140 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
1141 |
+ |
function is called \emph{auto correlation function}. One example of |
1142 |
+ |
auto correlation function is velocity auto-correlation function |
1143 |
+ |
which is directly related to transport properties of molecular |
1144 |
+ |
liquids: |
1145 |
+ |
\[ |
1146 |
+ |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1147 |
+ |
\right\rangle } dt |
1148 |
+ |
\] |
1149 |
+ |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1150 |
+ |
function which is averaging over time origins and over all the |
1151 |
+ |
atoms, dipole autocorrelation are calculated for the entire system. |
1152 |
+ |
The dipole autocorrelation function is given by: |
1153 |
+ |
\[ |
1154 |
+ |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1155 |
+ |
\right\rangle |
1156 |
+ |
\] |
1157 |
+ |
Here $u_{tot}$ is the net dipole of the entire system and is given |
1158 |
+ |
by |
1159 |
+ |
\[ |
1160 |
+ |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1161 |
+ |
\] |
1162 |
+ |
In principle, many time correlation functions can be related with |
1163 |
+ |
Fourier transforms of the infrared, Raman, and inelastic neutron |
1164 |
+ |
scattering spectra of molecular liquids. In practice, one can |
1165 |
+ |
extract the IR spectrum from the intensity of dipole fluctuation at |
1166 |
+ |
each frequency using the following relationship: |
1167 |
+ |
\[ |
1168 |
+ |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1169 |
+ |
i2\pi vt} dt} |
1170 |
+ |
\] |
1171 |
+ |
|
1172 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1173 |
|
|
1174 |
|
Rigid bodies are frequently involved in the modeling of different |
1177 |
|
movement of the objects in 3D gaming engine or other physics |
1178 |
|
simulator is governed by the rigid body dynamics. In molecular |
1179 |
|
simulation, rigid body is used to simplify the model in |
1180 |
< |
protein-protein docking study{\cite{Gray03}}. |
1180 |
> |
protein-protein docking study\cite{Gray2003}. |
1181 |
|
|
1182 |
|
It is very important to develop stable and efficient methods to |
1183 |
|
integrate the equations of motion of orientational degrees of |
1185 |
|
rotational degrees of freedom. However, due to its singularity, the |
1186 |
|
numerical integration of corresponding equations of motion is very |
1187 |
|
inefficient and inaccurate. Although an alternative integrator using |
1188 |
< |
different sets of Euler angles can overcome this difficulty\cite{}, |
1189 |
< |
the computational penalty and the lost of angular momentum |
1190 |
< |
conservation still remain. A singularity free representation |
1191 |
< |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
1192 |
< |
this approach suffer from the nonseparable Hamiltonian resulted from |
1193 |
< |
quaternion representation, which prevents the symplectic algorithm |
1194 |
< |
to be utilized. Another different approach is to apply holonomic |
1195 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
1196 |
< |
moves independently under the normal forces deriving from potential |
1197 |
< |
energy and constraint forces which are used to guarantee the |
1198 |
< |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
1199 |
< |
algorithm converge very slowly when the number of constraint |
1200 |
< |
increases. |
1188 |
> |
different sets of Euler angles can overcome this |
1189 |
> |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
1190 |
> |
of angular momentum conservation still remain. A singularity free |
1191 |
> |
representation utilizing quaternions was developed by Evans in |
1192 |
> |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
1193 |
> |
nonseparable Hamiltonian resulted from quaternion representation, |
1194 |
> |
which prevents the symplectic algorithm to be utilized. Another |
1195 |
> |
different approach is to apply holonomic constraints to the atoms |
1196 |
> |
belonging to the rigid body. Each atom moves independently under the |
1197 |
> |
normal forces deriving from potential energy and constraint forces |
1198 |
> |
which are used to guarantee the rigidness. However, due to their |
1199 |
> |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
1200 |
> |
when the number of constraint increases\cite{Ryckaert1977, |
1201 |
> |
Andersen1983}. |
1202 |
|
|
1203 |
|
The break through in geometric literature suggests that, in order to |
1204 |
|
develop a long-term integration scheme, one should preserve the |
1205 |
|
symplectic structure of the flow. Introducing conjugate momentum to |
1206 |
|
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
1207 |
< |
symplectic integrator, RSHAKE, was proposed to evolve the |
1208 |
< |
Hamiltonian system in a constraint manifold by iteratively |
1207 |
> |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
1208 |
> |
the Hamiltonian system in a constraint manifold by iteratively |
1209 |
|
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
1210 |
< |
method using quaternion representation was developed by Omelyan. |
1211 |
< |
However, both of these methods are iterative and inefficient. In |
1212 |
< |
this section, we will present a symplectic Lie-Poisson integrator |
1213 |
< |
for rigid body developed by Dullweber and his |
1214 |
< |
coworkers\cite{Dullweber1997} in depth. |
1210 |
> |
method using quaternion representation was developed by |
1211 |
> |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
1212 |
> |
iterative and inefficient. In this section, we will present a |
1213 |
> |
symplectic Lie-Poisson integrator for rigid body developed by |
1214 |
> |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1215 |
|
|
1216 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
1217 |
|
The motion of the rigid body is Hamiltonian with the Hamiltonian |
1230 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1231 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
1232 |
|
\begin{equation} |
1233 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
1233 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1234 |
|
\end{equation} |
1235 |
|
which is used to ensure rotation matrix's orthogonality. |
1236 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1243 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1244 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1245 |
|
the equations of motion, |
984 |
– |
\[ |
985 |
– |
\begin{array}{c} |
986 |
– |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
987 |
– |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
988 |
– |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
989 |
– |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
990 |
– |
\end{array} |
991 |
– |
\] |
1246 |
|
|
1247 |
+ |
\begin{eqnarray} |
1248 |
+ |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1249 |
+ |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1250 |
+ |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1251 |
+ |
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1252 |
+ |
\end{eqnarray} |
1253 |
+ |
|
1254 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1255 |
|
We can use constraint force provided by lagrange multiplier on the |
1256 |
|
normal manifold to keep the motion on constraint space. Or we can |
1257 |
< |
simply evolve the system in constraint manifold. The two method are |
1258 |
< |
proved to be equivalent. The holonomic constraint and equations of |
1259 |
< |
motions define a constraint manifold for rigid body |
1257 |
> |
simply evolve the system in constraint manifold. These two methods |
1258 |
> |
are proved to be equivalent. The holonomic constraint and equations |
1259 |
> |
of motions define a constraint manifold for rigid body |
1260 |
|
\[ |
1261 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1262 |
|
\right\}. |
1330 |
|
\[ |
1331 |
|
\hat vu = v \times u |
1332 |
|
\] |
1072 |
– |
|
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} |
1342 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1343 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
1344 |
|
unique property eliminate the requirement of iterations which can |
1345 |
< |
not be avoided in other methods\cite{}. |
1345 |
> |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
1346 |
|
|
1347 |
|
Applying hat-map isomorphism, we obtain the equation of motion for |
1348 |
|
angular momentum on body frame |
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 |
1107 |
< |
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} |
1415 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1416 |
|
) |
1417 |
|
\] |
1418 |
< |
|
1160 |
< |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
1418 |
> |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1419 |
|
manner. |
1420 |
|
|
1421 |
|
In order to construct a second-order symplectic method, we split the |
1467 |
|
\] |
1468 |
|
The equations of motion corresponding to potential energy and |
1469 |
|
kinetic energy are listed in the below table, |
1470 |
+ |
\begin{table} |
1471 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
1472 |
|
\begin{center} |
1473 |
|
\begin{tabular}{|l|l|} |
1474 |
|
\hline |
1481 |
|
\hline |
1482 |
|
\end{tabular} |
1483 |
|
\end{center} |
1484 |
< |
A second-order symplectic method is now obtained by the composition |
1485 |
< |
of the flow maps, |
1484 |
> |
\end{table} |
1485 |
> |
A second-order symplectic method is now obtained by the |
1486 |
> |
composition of the flow maps, |
1487 |
|
\[ |
1488 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1489 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1608 |
|
\] |
1609 |
|
where $p$ is real and $L$ is called the Laplace Transform |
1610 |
|
Operator. Below are some important properties of Laplace transform |
1350 |
– |
\begin{equation} |
1351 |
– |
\begin{array}{c} |
1352 |
– |
L(x + y) = L(x) + L(y) \\ |
1353 |
– |
L(ax) = aL(x) \\ |
1354 |
– |
L(\dot x) = pL(x) - px(0) \\ |
1355 |
– |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
1356 |
– |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
1357 |
– |
\end{array} |
1358 |
– |
\end{equation} |
1611 |
|
|
1612 |
+ |
\begin{eqnarray*} |
1613 |
+ |
L(x + y) & = & L(x) + L(y) \\ |
1614 |
+ |
L(ax) & = & aL(x) \\ |
1615 |
+ |
L(\dot x) & = & pL(x) - px(0) \\ |
1616 |
+ |
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
1617 |
+ |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
1618 |
+ |
\end{eqnarray*} |
1619 |
+ |
|
1620 |
+ |
|
1621 |
|
Applying Laplace transform to the bath coordinates, we obtain |
1622 |
< |
\[ |
1623 |
< |
\begin{array}{c} |
1624 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1625 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1626 |
< |
\end{array} |
1366 |
< |
\] |
1622 |
> |
\begin{eqnarray*} |
1623 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1624 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1625 |
> |
\end{eqnarray*} |
1626 |
> |
|
1627 |
|
By the same way, the system coordinates become |
1628 |
< |
\[ |
1629 |
< |
\begin{array}{c} |
1630 |
< |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1631 |
< |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1372 |
< |
\end{array} |
1373 |
< |
\] |
1628 |
> |
\begin{eqnarray*} |
1629 |
> |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1630 |
> |
& & \mbox{} - \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1631 |
> |
\end{eqnarray*} |
1632 |
|
|
1633 |
|
With the help of some relatively important inverse Laplace |
1634 |
|
transformations: |
1640 |
|
\end{array} |
1641 |
|
\] |
1642 |
|
, we obtain |
1643 |
< |
\begin{align} |
1644 |
< |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
1643 |
> |
\begin{eqnarray*} |
1644 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
1645 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1646 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
1647 |
< |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
1648 |
< |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
1649 |
< |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
1650 |
< |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
1651 |
< |
% |
1652 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1647 |
> |
_\alpha t)\dot x(t - \tau )d\tau } } \right\}} \\ |
1648 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1649 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1650 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1651 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1652 |
> |
\end{eqnarray*} |
1653 |
> |
\begin{eqnarray*} |
1654 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1655 |
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1656 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1657 |
< |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
1658 |
< |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
1659 |
< |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
1660 |
< |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
1661 |
< |
(\omega _\alpha t)} \right\}} |
1662 |
< |
\end{align} |
1403 |
< |
|
1657 |
> |
t)\dot x(t - \tau )d} \tau } \\ |
1658 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1659 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1660 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1661 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1662 |
> |
\end{eqnarray*} |
1663 |
|
Introducing a \emph{dynamic friction kernel} |
1664 |
|
\begin{equation} |
1665 |
|
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
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 section \ref{introSection:frictionTensor}. |
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 |
|
\[ |
1753 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1754 |
|
\] |
1755 |
|
And since the $q$ coordinates are harmonic oscillators, |
1756 |
< |
\[ |
1757 |
< |
\begin{array}{c} |
1758 |
< |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1759 |
< |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1760 |
< |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1761 |
< |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1762 |
< |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1763 |
< |
= kT\xi (t) \\ |
1764 |
< |
\end{array} |
1765 |
< |
\] |
1756 |
> |
|
1757 |
> |
\begin{eqnarray*} |
1758 |
> |
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1759 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1760 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1761 |
> |
\left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
1762 |
> |
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1763 |
> |
& = &kT\xi (t) \\ |
1764 |
> |
\end{eqnarray*} |
1765 |
> |
|
1766 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1767 |
|
\begin{equation} |
1768 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
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. |
1514 |
– |
|
1515 |
– |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1516 |
– |
Theoretically, the friction kernel can be determined using velocity |
1517 |
– |
autocorrelation function. However, this approach become impractical |
1518 |
– |
when the system become more and more complicate. Instead, various |
1519 |
– |
approaches based on hydrodynamics have been developed to calculate |
1520 |
– |
the friction coefficients. The friction effect is isotropic in |
1521 |
– |
Equation, \zeta can be taken as a scalar. In general, friction |
1522 |
– |
tensor \Xi is a $6\times 6$ matrix given by |
1523 |
– |
\[ |
1524 |
– |
\Xi = \left( {\begin{array}{*{20}c} |
1525 |
– |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
1526 |
– |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
1527 |
– |
\end{array}} \right). |
1528 |
– |
\] |
1529 |
– |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
1530 |
– |
tensor and rotational resistance (friction) tensor respectively, |
1531 |
– |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
1532 |
– |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
1533 |
– |
particle moves in a fluid, it may experience friction force or |
1534 |
– |
torque along the opposite direction of the velocity or angular |
1535 |
– |
velocity, |
1536 |
– |
\[ |
1537 |
– |
\left( \begin{array}{l} |
1538 |
– |
F_R \\ |
1539 |
– |
\tau _R \\ |
1540 |
– |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
1541 |
– |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
1542 |
– |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
1543 |
– |
\end{array}} \right)\left( \begin{array}{l} |
1544 |
– |
v \\ |
1545 |
– |
w \\ |
1546 |
– |
\end{array} \right) |
1547 |
– |
\] |
1548 |
– |
where $F_r$ is the friction force and $\tau _R$ is the friction |
1549 |
– |
toque. |
1550 |
– |
|
1551 |
– |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
1552 |
– |
|
1553 |
– |
For a spherical particle, the translational and rotational friction |
1554 |
– |
constant can be calculated from Stoke's law, |
1555 |
– |
\[ |
1556 |
– |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
1557 |
– |
{6\pi \eta R} & 0 & 0 \\ |
1558 |
– |
0 & {6\pi \eta R} & 0 \\ |
1559 |
– |
0 & 0 & {6\pi \eta R} \\ |
1560 |
– |
\end{array}} \right) |
1561 |
– |
\] |
1562 |
– |
and |
1563 |
– |
\[ |
1564 |
– |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
1565 |
– |
{8\pi \eta R^3 } & 0 & 0 \\ |
1566 |
– |
0 & {8\pi \eta R^3 } & 0 \\ |
1567 |
– |
0 & 0 & {8\pi \eta R^3 } \\ |
1568 |
– |
\end{array}} \right) |
1569 |
– |
\] |
1570 |
– |
where $\eta$ is the viscosity of the solvent and $R$ is the |
1571 |
– |
hydrodynamics radius. |
1572 |
– |
|
1573 |
– |
Other non-spherical shape, such as cylinder and ellipsoid |
1574 |
– |
\textit{etc}, are widely used as reference for developing new |
1575 |
– |
hydrodynamics theory, because their properties can be calculated |
1576 |
– |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
1577 |
– |
also called a triaxial ellipsoid, which is given in Cartesian |
1578 |
– |
coordinates by |
1579 |
– |
\[ |
1580 |
– |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
1581 |
– |
}} = 1 |
1582 |
– |
\] |
1583 |
– |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
1584 |
– |
due to the complexity of the elliptic integral, only the ellipsoid |
1585 |
– |
with the restriction of two axes having to be equal, \textit{i.e.} |
1586 |
– |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
1587 |
– |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
1588 |
– |
\[ |
1589 |
– |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
1590 |
– |
} }}{b}, |
1591 |
– |
\] |
1592 |
– |
and oblate, |
1593 |
– |
\[ |
1594 |
– |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
1595 |
– |
}}{a} |
1596 |
– |
\], |
1597 |
– |
one can write down the translational and rotational resistance |
1598 |
– |
tensors |
1599 |
– |
\[ |
1600 |
– |
\begin{array}{l} |
1601 |
– |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
1602 |
– |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
1603 |
– |
\end{array}, |
1604 |
– |
\] |
1605 |
– |
and |
1606 |
– |
\[ |
1607 |
– |
\begin{array}{l} |
1608 |
– |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
1609 |
– |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
1610 |
– |
\end{array}. |
1611 |
– |
\] |
1612 |
– |
|
1613 |
– |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
1614 |
– |
|
1615 |
– |
Unlike spherical and other regular shaped molecules, there is not |
1616 |
– |
analytical solution for friction tensor of any arbitrary shaped |
1617 |
– |
rigid molecules. The ellipsoid of revolution model and general |
1618 |
– |
triaxial ellipsoid model have been used to approximate the |
1619 |
– |
hydrodynamic properties of rigid bodies. However, since the mapping |
1620 |
– |
from all possible ellipsoidal space, $r$-space, to all possible |
1621 |
– |
combination of rotational diffusion coefficients, $D$-space is not |
1622 |
– |
unique\cite{Wegener79} as well as the intrinsic coupling between |
1623 |
– |
translational and rotational motion of rigid body\cite{}, general |
1624 |
– |
ellipsoid is not always suitable for modeling arbitrarily shaped |
1625 |
– |
rigid molecule. A number of studies have been devoted to determine |
1626 |
– |
the friction tensor for irregularly shaped rigid bodies using more |
1627 |
– |
advanced method\cite{} where the molecule of interest was modeled by |
1628 |
– |
combinations of spheres(beads)\cite{} and the hydrodynamics |
1629 |
– |
properties of the molecule can be calculated using the hydrodynamic |
1630 |
– |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
1631 |
– |
immersed in a continuous medium. Due to hydrodynamics interaction, |
1632 |
– |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
1633 |
– |
unperturbed velocity $v_i$, |
1634 |
– |
\[ |
1635 |
– |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
1636 |
– |
\] |
1637 |
– |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
1638 |
– |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
1639 |
– |
proportional to its ``net'' velocity |
1640 |
– |
\begin{equation} |
1641 |
– |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
1642 |
– |
\label{introEquation:tensorExpression} |
1643 |
– |
\end{equation} |
1644 |
– |
This equation is the basis for deriving the hydrodynamic tensor. In |
1645 |
– |
1930, Oseen and Burgers gave a simple solution to Equation |
1646 |
– |
\ref{introEquation:tensorExpression} |
1647 |
– |
\begin{equation} |
1648 |
– |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
1649 |
– |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
1650 |
– |
\label{introEquation:oseenTensor} |
1651 |
– |
\end{equation} |
1652 |
– |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
1653 |
– |
A second order expression for element of different size was |
1654 |
– |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
1655 |
– |
la Torre and Bloomfield, |
1656 |
– |
\begin{equation} |
1657 |
– |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
1658 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
1659 |
– |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
1660 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
1661 |
– |
\label{introEquation:RPTensorNonOverlapped} |
1662 |
– |
\end{equation} |
1663 |
– |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
1664 |
– |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
1665 |
– |
\ge \sigma _i + \sigma _j$. An alternative expression for |
1666 |
– |
overlapping beads with the same radius, $\sigma$, is given by |
1667 |
– |
\begin{equation} |
1668 |
– |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
1669 |
– |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
1670 |
– |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
1671 |
– |
\label{introEquation:RPTensorOverlapped} |
1672 |
– |
\end{equation} |
1673 |
– |
|
1674 |
– |
To calculate the resistance tensor at an arbitrary origin $O$, we |
1675 |
– |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
1676 |
– |
$B_{ij}$ blocks |
1677 |
– |
\begin{equation} |
1678 |
– |
B = \left( {\begin{array}{*{20}c} |
1679 |
– |
{B_{11} } & \ldots & {B_{1N} } \\ |
1680 |
– |
\vdots & \ddots & \vdots \\ |
1681 |
– |
{B_{N1} } & \cdots & {B_{NN} } \\ |
1682 |
– |
\end{array}} \right), |
1683 |
– |
\end{equation} |
1684 |
– |
where $B_{ij}$ is given by |
1685 |
– |
\[ |
1686 |
– |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
1687 |
– |
)T_{ij} |
1688 |
– |
\] |
1689 |
– |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
1690 |
– |
$B$, we obtain |
1691 |
– |
|
1692 |
– |
\[ |
1693 |
– |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
1694 |
– |
{C_{11} } & \ldots & {C_{1N} } \\ |
1695 |
– |
\vdots & \ddots & \vdots \\ |
1696 |
– |
{C_{N1} } & \cdots & {C_{NN} } \\ |
1697 |
– |
\end{array}} \right) |
1698 |
– |
\] |
1699 |
– |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
1700 |
– |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
1701 |
– |
\[ |
1702 |
– |
U_i = \left( {\begin{array}{*{20}c} |
1703 |
– |
0 & { - z_i } & {y_i } \\ |
1704 |
– |
{z_i } & 0 & { - x_i } \\ |
1705 |
– |
{ - y_i } & {x_i } & 0 \\ |
1706 |
– |
\end{array}} \right) |
1707 |
– |
\] |
1708 |
– |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
1709 |
– |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
1710 |
– |
arbitrary origin $O$ can be written as |
1711 |
– |
\begin{equation} |
1712 |
– |
\begin{array}{l} |
1713 |
– |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
1714 |
– |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
1715 |
– |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
1716 |
– |
\end{array} |
1717 |
– |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
1718 |
– |
\end{equation} |
1719 |
– |
|
1720 |
– |
The resistance tensor depends on the origin to which they refer. The |
1721 |
– |
proper location for applying friction force is the center of |
1722 |
– |
resistance (reaction), at which the trace of rotational resistance |
1723 |
– |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
1724 |
– |
resistance is defined as an unique point of the rigid body at which |
1725 |
– |
the translation-rotation coupling tensor are symmetric, |
1726 |
– |
\begin{equation} |
1727 |
– |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
1728 |
– |
\label{introEquation:definitionCR} |
1729 |
– |
\end{equation} |
1730 |
– |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
1731 |
– |
we can easily find out that the translational resistance tensor is |
1732 |
– |
origin independent, while the rotational resistance tensor and |
1733 |
– |
translation-rotation coupling resistance tensor depend on the |
1734 |
– |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
1735 |
– |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
1736 |
– |
obtain the resistance tensor at $P$ by |
1737 |
– |
\begin{equation} |
1738 |
– |
\begin{array}{l} |
1739 |
– |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
1740 |
– |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
1741 |
– |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
1742 |
– |
\end{array} |
1743 |
– |
\label{introEquation:resistanceTensorTransformation} |
1744 |
– |
\end{equation} |
1745 |
– |
where |
1746 |
– |
\[ |
1747 |
– |
U_{OP} = \left( {\begin{array}{*{20}c} |
1748 |
– |
0 & { - z_{OP} } & {y_{OP} } \\ |
1749 |
– |
{z_i } & 0 & { - x_{OP} } \\ |
1750 |
– |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
1751 |
– |
\end{array}} \right) |
1752 |
– |
\] |
1753 |
– |
Using Equations \ref{introEquation:definitionCR} and |
1754 |
– |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
1755 |
– |
the position of center of resistance, |
1756 |
– |
\[ |
1757 |
– |
\left( \begin{array}{l} |
1758 |
– |
x_{OR} \\ |
1759 |
– |
y_{OR} \\ |
1760 |
– |
z_{OR} \\ |
1761 |
– |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
1762 |
– |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
1763 |
– |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
1764 |
– |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
1765 |
– |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
1766 |
– |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
1767 |
– |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
1768 |
– |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
1769 |
– |
\end{array} \right). |
1770 |
– |
\] |
1771 |
– |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
1772 |
– |
joining center of resistance $R$ and origin $O$. |
1773 |
– |
|
1774 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |