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 |
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 |
|
A microscopic state or microstate of a classical system is |
235 |
|
specification of the complete phase space vector of a system at any |
251 |
|
regions of the phase space. The condition of an ensemble at any time |
252 |
|
can be regarded as appropriately specified by the density $\rho$ |
253 |
|
with which representative points are distributed over the phase |
254 |
< |
space. The density of distribution for an ensemble with $f$ degrees |
255 |
< |
of freedom is defined as, |
254 |
> |
space. The density distribution for an ensemble with $f$ degrees of |
255 |
> |
freedom is defined as, |
256 |
|
\begin{equation} |
257 |
|
\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). |
258 |
|
\label{introEquation:densityDistribution} |
259 |
|
\end{equation} |
260 |
|
Governed by the principles of mechanics, the phase points change |
261 |
< |
their value which would change the density at any time at phase |
262 |
< |
space. Hence, the density of distribution is also to be taken as a |
261 |
> |
their locations which would change the density at any time at phase |
262 |
> |
space. Hence, the density distribution is also to be taken as a |
263 |
|
function of the time. |
264 |
|
|
265 |
|
The number of systems $\delta N$ at time $t$ can be determined by, |
267 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
268 |
|
\label{introEquation:deltaN} |
269 |
|
\end{equation} |
270 |
< |
Assuming a large enough population of systems are exploited, we can |
271 |
< |
sufficiently approximate $\delta N$ without introducing |
272 |
< |
discontinuity when we go from one region in the phase space to |
273 |
< |
another. By integrating over the whole phase space, |
270 |
> |
Assuming a large enough population of systems, we can sufficiently |
271 |
> |
approximate $\delta N$ without introducing discontinuity when we go |
272 |
> |
from one region in the phase space to another. By integrating over |
273 |
> |
the whole phase space, |
274 |
|
\begin{equation} |
275 |
|
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
276 |
|
\label{introEquation:totalNumberSystem} |
287 |
|
value of any desired quantity which depends on the coordinates and |
288 |
|
momenta of the system. Even when the dynamics of the real system is |
289 |
|
complex, or stochastic, or even discontinuous, the average |
290 |
< |
properties of the ensemble of possibilities as a whole may still |
291 |
< |
remain well defined. For a classical system in thermal equilibrium |
292 |
< |
with its environment, the ensemble average of a mechanical quantity, |
293 |
< |
$\langle A(q , p) \rangle_t$, takes the form of an integral over the |
294 |
< |
phase space of the system, |
290 |
> |
properties of the ensemble of possibilities as a whole remaining |
291 |
> |
well defined. For a classical system in thermal equilibrium with its |
292 |
> |
environment, the ensemble average of a mechanical quantity, $\langle |
293 |
> |
A(q , p) \rangle_t$, takes the form of an integral over the phase |
294 |
> |
space of the system, |
295 |
|
\begin{equation} |
296 |
|
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
297 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
301 |
|
|
302 |
|
There are several different types of ensembles with different |
303 |
|
statistical characteristics. As a function of macroscopic |
304 |
< |
parameters, such as temperature \textit{etc}, partition function can |
305 |
< |
be used to describe the statistical properties of a system in |
304 |
> |
parameters, such as temperature \textit{etc}, the partition function |
305 |
> |
can be used to describe the statistical properties of a system in |
306 |
|
thermodynamic equilibrium. |
307 |
|
|
308 |
|
As an ensemble of systems, each of which is known to be thermally |
309 |
< |
isolated and conserve energy, Microcanonical ensemble(NVE) has a |
309 |
> |
isolated and conserve energy, the Microcanonical ensemble(NVE) has a |
310 |
|
partition function like, |
311 |
|
\begin{equation} |
312 |
|
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
320 |
|
\label{introEquation:NVTPartition} |
321 |
|
\end{equation} |
322 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
323 |
< |
TS$. Since most experiment are carried out under constant pressure |
324 |
< |
condition, isothermal-isobaric ensemble(NPT) play a very important |
325 |
< |
role in molecular simulation. The isothermal-isobaric ensemble allow |
326 |
< |
the system to exchange energy with a heat bath of temperature $T$ |
327 |
< |
and to change the volume as well. Its partition function is given as |
323 |
> |
TS$. Since most experiments are carried out under constant pressure |
324 |
> |
condition, the isothermal-isobaric ensemble(NPT) plays a very |
325 |
> |
important role in molecular simulations. The isothermal-isobaric |
326 |
> |
ensemble allow the system to exchange energy with a heat bath of |
327 |
> |
temperature $T$ and to change the volume as well. Its partition |
328 |
> |
function is given as |
329 |
|
\begin{equation} |
330 |
|
\Delta (N,P,T) = - e^{\beta G}. |
331 |
|
\label{introEquation:NPTPartition} |
334 |
|
|
335 |
|
\subsection{\label{introSection:liouville}Liouville's theorem} |
336 |
|
|
337 |
< |
The Liouville's theorem is the foundation on which statistical |
338 |
< |
mechanics rests. It describes the time evolution of phase space |
337 |
> |
Liouville's theorem is the foundation on which statistical mechanics |
338 |
> |
rests. It describes the time evolution of the phase space |
339 |
|
distribution function. In order to calculate the rate of change of |
340 |
|
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
341 |
|
consider the two faces perpendicular to the $q_1$ axis, which are |
364 |
|
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
365 |
|
\end{equation} |
366 |
|
which cancels the first terms of the right hand side. Furthermore, |
367 |
< |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
367 |
> |
dividing $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
368 |
|
p_f $ in both sides, we can write out Liouville's theorem in a |
369 |
|
simple form, |
370 |
|
\begin{equation} |
390 |
|
\label{introEquation:densityAndHamiltonian} |
391 |
|
\end{equation} |
392 |
|
|
393 |
< |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
393 |
> |
\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}} |
394 |
|
Lets consider a region in the phase space, |
395 |
|
\begin{equation} |
396 |
|
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
397 |
|
\end{equation} |
398 |
|
If this region is small enough, the density $\rho$ can be regarded |
399 |
< |
as uniform over the whole phase space. Thus, the number of phase |
400 |
< |
points inside this region is given by, |
399 |
> |
as uniform over the whole integral. Thus, the number of phase points |
400 |
> |
inside this region is given by, |
401 |
|
\begin{equation} |
402 |
|
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
403 |
|
dp_1 } ..dp_f. |
409 |
|
\end{equation} |
410 |
|
With the help of stationary assumption |
411 |
|
(\ref{introEquation:stationary}), we obtain the principle of the |
412 |
< |
\emph{conservation of extension in phase space}, |
412 |
> |
\emph{conservation of volume in phase space}, |
413 |
|
\begin{equation} |
414 |
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
415 |
|
...dq_f dp_1 } ..dp_f = 0. |
416 |
|
\label{introEquation:volumePreserving} |
417 |
|
\end{equation} |
418 |
|
|
419 |
< |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
419 |
> |
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
420 |
|
|
421 |
|
Liouville's theorem can be expresses in a variety of different forms |
422 |
|
which are convenient within different contexts. For any two function |
458 |
|
Various thermodynamic properties can be calculated from Molecular |
459 |
|
Dynamics simulation. By comparing experimental values with the |
460 |
|
calculated properties, one can determine the accuracy of the |
461 |
< |
simulation and the quality of the underlying model. However, both of |
462 |
< |
experiment and computer simulation are usually performed during a |
461 |
> |
simulation and the quality of the underlying model. However, both |
462 |
> |
experiments and computer simulations are usually performed during a |
463 |
|
certain time interval and the measurements are averaged over a |
464 |
|
period of them which is different from the average behavior of |
465 |
< |
many-body system in Statistical Mechanics. Fortunately, Ergodic |
466 |
< |
Hypothesis is proposed to make a connection between time average and |
467 |
< |
ensemble average. It states that time average and average over the |
468 |
< |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
465 |
> |
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
466 |
> |
Hypothesis makes a connection between time average and the ensemble |
467 |
> |
average. It states that the time average and average over the |
468 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
469 |
|
\begin{equation} |
470 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
471 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
479 |
|
a properly weighted statistical average. This allows the researcher |
480 |
|
freedom of choice when deciding how best to measure a given |
481 |
|
observable. In case an ensemble averaged approach sounds most |
482 |
< |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
482 |
> |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
483 |
|
utilized. Or if the system lends itself to a time averaging |
484 |
|
approach, the Molecular Dynamics techniques in |
485 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
486 |
|
choice\cite{Frenkel1996}. |
487 |
|
|
488 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
489 |
< |
A variety of numerical integrators were proposed to simulate the |
490 |
< |
motions. They usually begin with an initial conditionals and move |
491 |
< |
the objects in the direction governed by the differential equations. |
492 |
< |
However, most of them ignore the hidden physical law contained |
493 |
< |
within the equations. Since 1990, geometric integrators, which |
494 |
< |
preserve various phase-flow invariants such as symplectic structure, |
495 |
< |
volume and time reversal symmetry, are developed to address this |
496 |
< |
issue. The velocity verlet method, which happens to be a simple |
497 |
< |
example of symplectic integrator, continues to gain its popularity |
498 |
< |
in molecular dynamics community. This fact can be partly explained |
499 |
< |
by its geometric nature. |
489 |
> |
A variety of numerical integrators have been proposed to simulate |
490 |
> |
the motions of atoms in MD simulation. They usually begin with |
491 |
> |
initial conditionals and move the objects in the direction governed |
492 |
> |
by the differential equations. However, most of them ignore the |
493 |
> |
hidden physical laws contained within the equations. Since 1990, |
494 |
> |
geometric integrators, which preserve various phase-flow invariants |
495 |
> |
such as symplectic structure, volume and time reversal symmetry, are |
496 |
> |
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
497 |
> |
Leimkuhler1999}. The velocity verlet method, which happens to be a |
498 |
> |
simple example of symplectic integrator, continues to gain |
499 |
> |
popularity in the molecular dynamics community. This fact can be |
500 |
> |
partly explained by its geometric nature. |
501 |
|
|
502 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
503 |
< |
A \emph{manifold} is an abstract mathematical space. It locally |
504 |
< |
looks like Euclidean space, but when viewed globally, it may have |
505 |
< |
more complicate structure. A good example of manifold is the surface |
506 |
< |
of Earth. It seems to be flat locally, but it is round if viewed as |
507 |
< |
a whole. A \emph{differentiable manifold} (also known as |
508 |
< |
\emph{smooth manifold}) is a manifold with an open cover in which |
509 |
< |
the covering neighborhoods are all smoothly isomorphic to one |
510 |
< |
another. In other words,it is possible to apply calculus on |
515 |
< |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
516 |
< |
defined as a pair $(M, \omega)$ which consisting of a |
502 |
> |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
503 |
> |
A \emph{manifold} is an abstract mathematical space. It looks |
504 |
> |
locally like Euclidean space, but when viewed globally, it may have |
505 |
> |
more complicated structure. A good example of manifold is the |
506 |
> |
surface of Earth. It seems to be flat locally, but it is round if |
507 |
> |
viewed as a whole. A \emph{differentiable manifold} (also known as |
508 |
> |
\emph{smooth manifold}) is a manifold on which it is possible to |
509 |
> |
apply calculus on \emph{differentiable manifold}. A \emph{symplectic |
510 |
> |
manifold} is defined as a pair $(M, \omega)$ which consists of a |
511 |
|
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
512 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
513 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
514 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
515 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
516 |
< |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
517 |
< |
example of symplectic form. |
516 |
> |
$\omega(x, x) = 0$. The cross product operation in vector field is |
517 |
> |
an example of symplectic form. |
518 |
|
|
519 |
< |
One of the motivations to study \emph{symplectic manifold} in |
519 |
> |
One of the motivations to study \emph{symplectic manifolds} in |
520 |
|
Hamiltonian Mechanics is that a symplectic manifold can represent |
521 |
|
all possible configurations of the system and the phase space of the |
522 |
|
system can be described by it's cotangent bundle. Every symplectic |
523 |
|
manifold is even dimensional. For instance, in Hamilton equations, |
524 |
|
coordinate and momentum always appear in pairs. |
525 |
|
|
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 |
– |
|
526 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
527 |
|
|
528 |
< |
For a ordinary differential system defined as |
528 |
> |
For an ordinary differential system defined as |
529 |
|
\begin{equation} |
530 |
|
\dot x = f(x) |
531 |
|
\end{equation} |
532 |
< |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
532 |
> |
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
533 |
|
\begin{equation} |
534 |
|
f(r) = J\nabla _x H(r). |
535 |
|
\end{equation} |
550 |
|
\end{equation}In this case, $f$ is |
551 |
|
called a \emph{Hamiltonian vector field}. |
552 |
|
|
553 |
< |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
553 |
> |
Another generalization of Hamiltonian dynamics is Poisson |
554 |
> |
Dynamics\cite{Olver1986}, |
555 |
|
\begin{equation} |
556 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
557 |
|
\end{equation} |
598 |
|
|
599 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
600 |
|
|
601 |
< |
The hidden geometric properties of ODE and its flow play important |
602 |
< |
roles in numerical studies. Many of them can be found in systems |
603 |
< |
which occur naturally in applications. |
601 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE |
602 |
> |
and its flow play important roles in numerical studies. Many of them |
603 |
> |
can be found in systems which occur naturally in applications. |
604 |
|
|
605 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
606 |
|
a \emph{symplectic} flow if it satisfies, |
644 |
|
which is the condition for conserving \emph{first integral}. For a |
645 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
646 |
|
smooth function $G$ is given by, |
647 |
< |
\begin{equation} |
648 |
< |
\begin{array}{c} |
649 |
< |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
650 |
< |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
665 |
< |
\end{array} |
647 |
> |
|
648 |
> |
\begin{eqnarray} |
649 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
650 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
651 |
|
\label{introEquation:firstIntegral1} |
652 |
< |
\end{equation} |
652 |
> |
\end{eqnarray} |
653 |
> |
|
654 |
> |
|
655 |
|
Using poisson bracket notion, Equation |
656 |
|
\ref{introEquation:firstIntegral1} can be rewritten as |
657 |
|
\[ |
666 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
667 |
|
0$. |
668 |
|
|
669 |
< |
|
683 |
< |
When designing any numerical methods, one should always try to |
669 |
> |
When designing any numerical methods, one should always try to |
670 |
|
preserve the structural properties of the original ODE and its flow. |
671 |
|
|
672 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
673 |
|
A lot of well established and very effective numerical methods have |
674 |
|
been successful precisely because of their symplecticities even |
675 |
|
though this fact was not recognized when they were first |
676 |
< |
constructed. The most famous example is leapfrog methods in |
677 |
< |
molecular dynamics. In general, symplectic integrators can be |
676 |
> |
constructed. The most famous example is the Verlet-leapfrog methods |
677 |
> |
in molecular dynamics. In general, symplectic integrators can be |
678 |
|
constructed using one of four different methods. |
679 |
|
\begin{enumerate} |
680 |
|
\item Generating functions |
683 |
|
\item Splitting methods |
684 |
|
\end{enumerate} |
685 |
|
|
686 |
< |
Generating function tends to lead to methods which are cumbersome |
687 |
< |
and difficult to use. In dissipative systems, variational methods |
688 |
< |
can capture the decay of energy accurately. Since their |
689 |
< |
geometrically unstable nature against non-Hamiltonian perturbations, |
690 |
< |
ordinary implicit Runge-Kutta methods are not suitable for |
691 |
< |
Hamiltonian system. Recently, various high-order explicit |
692 |
< |
Runge--Kutta methods have been developed to overcome this |
686 |
> |
Generating function\cite{Channell1990} tends to lead to methods |
687 |
> |
which are cumbersome and difficult to use. In dissipative systems, |
688 |
> |
variational methods can capture the decay of energy |
689 |
> |
accurately\cite{Kane2000}. Since their geometrically unstable nature |
690 |
> |
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
691 |
> |
methods are not suitable for Hamiltonian system. Recently, various |
692 |
> |
high-order explicit Runge-Kutta methods |
693 |
> |
\cite{Owren1992,Chen2003}have been developed to overcome this |
694 |
|
instability. However, due to computational penalty involved in |
695 |
< |
implementing the Runge-Kutta methods, they do not attract too much |
696 |
< |
attention from Molecular Dynamics community. Instead, splitting have |
697 |
< |
been widely accepted since they exploit natural decompositions of |
698 |
< |
the system\cite{Tuckerman92}. |
695 |
> |
implementing the Runge-Kutta methods, they have not attracted much |
696 |
> |
attention from the Molecular Dynamics community. Instead, splitting |
697 |
> |
methods have been widely accepted since they exploit natural |
698 |
> |
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
699 |
|
|
700 |
< |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
700 |
> |
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
701 |
|
|
702 |
|
The main idea behind splitting methods is to decompose the discrete |
703 |
|
$\varphi_h$ as a composition of simpler flows, |
718 |
|
energy respectively, which is a natural decomposition of the |
719 |
|
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
720 |
|
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
721 |
< |
order is then given by the Lie-Trotter formula |
721 |
> |
order expression is then given by the Lie-Trotter formula |
722 |
|
\begin{equation} |
723 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
724 |
|
\label{introEquation:firstOrderSplitting} |
744 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
745 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
746 |
|
\end{equation} |
747 |
< |
which has a local error proportional to $h^3$. Sprang splitting's |
748 |
< |
popularity in molecular simulation community attribute to its |
749 |
< |
symmetric property, |
747 |
> |
which has a local error proportional to $h^3$. The Sprang |
748 |
> |
splitting's popularity in molecular simulation community attribute |
749 |
> |
to its symmetric property, |
750 |
|
\begin{equation} |
751 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
752 |
|
\label{introEquation:timeReversible} |
753 |
< |
\end{equation} |
753 |
> |
\end{equation},appendixFig:architecture |
754 |
|
|
755 |
< |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
755 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Example of Splitting Method}} |
756 |
|
The classical equation for a system consisting of interacting |
757 |
|
particles can be written in Hamiltonian form, |
758 |
|
\[ |
809 |
|
% |
810 |
|
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
811 |
|
q(\Delta t)} \right]. % |
812 |
< |
\label{introEquation:positionVerlet1} |
812 |
> |
\label{introEquation:positionVerlet2} |
813 |
|
\end{align} |
814 |
|
|
815 |
< |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
815 |
> |
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
816 |
|
|
817 |
|
Baker-Campbell-Hausdorff formula can be used to determine the local |
818 |
|
error of splitting method in terms of commutator of the |
819 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
820 |
|
the sub-flow. For operators $hX$ and $hY$ which are associate to |
821 |
< |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
821 |
> |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
822 |
|
\begin{equation} |
823 |
|
\exp (hX + hY) = \exp (hZ) |
824 |
|
\end{equation} |
831 |
|
\[ |
832 |
|
[X,Y] = XY - YX . |
833 |
|
\] |
834 |
< |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
835 |
< |
can obtain |
834 |
> |
Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to |
835 |
> |
Sprang splitting, we can obtain |
836 |
|
\begin{eqnarray*} |
837 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
838 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
839 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
853 |
< |
\ldots ) |
837 |
> |
\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 \\ |
838 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
839 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
840 |
|
\end{eqnarray*} |
841 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
842 |
|
error of Spring splitting is proportional to $h^3$. The same |
845 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
846 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
847 |
|
\end{equation} |
848 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
848 |
> |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
849 |
|
order method. Yoshida proposed an elegant way to compose higher |
850 |
< |
order methods based on symmetric splitting. Given a symmetric second |
851 |
< |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
852 |
< |
method can be constructed by composing, |
850 |
> |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
851 |
> |
a symmetric second order base method $ \varphi _h^{(2)} $, a |
852 |
> |
fourth-order symmetric method can be constructed by composing, |
853 |
|
\[ |
854 |
|
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
855 |
|
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
869 |
|
|
870 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
871 |
|
|
872 |
< |
As a special discipline of molecular modeling, Molecular dynamics |
873 |
< |
has proven to be a powerful tool for studying the functions of |
874 |
< |
biological systems, providing structural, thermodynamic and |
875 |
< |
dynamical information. |
872 |
> |
As one of the principal tools of molecular modeling, Molecular |
873 |
> |
dynamics has proven to be a powerful tool for studying the functions |
874 |
> |
of biological systems, providing structural, thermodynamic and |
875 |
> |
dynamical information. The basic idea of molecular dynamics is that |
876 |
> |
macroscopic properties are related to microscopic behavior and |
877 |
> |
microscopic behavior can be calculated from the trajectories in |
878 |
> |
simulations. For instance, instantaneous temperature of an |
879 |
> |
Hamiltonian system of $N$ particle can be measured by |
880 |
> |
\[ |
881 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
882 |
> |
\] |
883 |
> |
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
884 |
> |
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
885 |
> |
the boltzman constant. |
886 |
|
|
887 |
< |
\subsection{\label{introSec:mdInit}Initialization} |
887 |
> |
A typical molecular dynamics run consists of three essential steps: |
888 |
> |
\begin{enumerate} |
889 |
> |
\item Initialization |
890 |
> |
\begin{enumerate} |
891 |
> |
\item Preliminary preparation |
892 |
> |
\item Minimization |
893 |
> |
\item Heating |
894 |
> |
\item Equilibration |
895 |
> |
\end{enumerate} |
896 |
> |
\item Production |
897 |
> |
\item Analysis |
898 |
> |
\end{enumerate} |
899 |
> |
These three individual steps will be covered in the following |
900 |
> |
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
901 |
> |
initialization of a simulation. Sec.~\ref{introSection:production} |
902 |
> |
will discusses issues in production run. |
903 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
904 |
> |
trajectory analysis. |
905 |
|
|
906 |
< |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
906 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
907 |
|
|
908 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
908 |
> |
\subsubsection{\textbf{Preliminary preparation}} |
909 |
|
|
910 |
+ |
When selecting the starting structure of a molecule for molecular |
911 |
+ |
simulation, one may retrieve its Cartesian coordinates from public |
912 |
+ |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
913 |
+ |
thousands of crystal structures of molecules are discovered every |
914 |
+ |
year, many more remain unknown due to the difficulties of |
915 |
+ |
purification and crystallization. Even for the molecule with known |
916 |
+ |
structure, some important information is missing. For example, the |
917 |
+ |
missing hydrogen atom which acts as donor in hydrogen bonding must |
918 |
+ |
be added. Moreover, in order to include electrostatic interaction, |
919 |
+ |
one may need to specify the partial charges for individual atoms. |
920 |
+ |
Under some circumstances, we may even need to prepare the system in |
921 |
+ |
a special setup. For instance, when studying transport phenomenon in |
922 |
+ |
membrane system, we may prepare the lipids in bilayer structure |
923 |
+ |
instead of placing lipids randomly in solvent, since we are not |
924 |
+ |
interested in self-aggregation and it takes a long time to happen. |
925 |
+ |
|
926 |
+ |
\subsubsection{\textbf{Minimization}} |
927 |
+ |
|
928 |
+ |
It is quite possible that some of molecules in the system from |
929 |
+ |
preliminary preparation may be overlapped with each other. This |
930 |
+ |
close proximity leads to high potential energy which consequently |
931 |
+ |
jeopardizes any molecular dynamics simulations. To remove these |
932 |
+ |
steric overlaps, one typically performs energy minimization to find |
933 |
+ |
a more reasonable conformation. Several energy minimization methods |
934 |
+ |
have been developed to exploit the energy surface and to locate the |
935 |
+ |
local minimum. While converging slowly near the minimum, steepest |
936 |
+ |
descent method is extremely robust when systems are far from |
937 |
+ |
harmonic. Thus, it is often used to refine structure from |
938 |
+ |
crystallographic data. Relied on the gradient or hessian, advanced |
939 |
+ |
methods like conjugate gradient and Newton-Raphson converge rapidly |
940 |
+ |
to a local minimum, while become unstable if the energy surface is |
941 |
+ |
far from quadratic. Another factor must be taken into account, when |
942 |
+ |
choosing energy minimization method, is the size of the system. |
943 |
+ |
Steepest descent and conjugate gradient can deal with models of any |
944 |
+ |
size. Because of the limit of computation power to calculate hessian |
945 |
+ |
matrix and insufficient storage capacity to store them, most |
946 |
+ |
Newton-Raphson methods can not be used with very large models. |
947 |
+ |
|
948 |
+ |
\subsubsection{\textbf{Heating}} |
949 |
+ |
|
950 |
+ |
Typically, Heating is performed by assigning random velocities |
951 |
+ |
according to a Gaussian distribution for a temperature. Beginning at |
952 |
+ |
a lower temperature and gradually increasing the temperature by |
953 |
+ |
assigning greater random velocities, we end up with setting the |
954 |
+ |
temperature of the system to a final temperature at which the |
955 |
+ |
simulation will be conducted. In heating phase, we should also keep |
956 |
+ |
the system from drifting or rotating as a whole. Equivalently, the |
957 |
+ |
net linear momentum and angular momentum of the system should be |
958 |
+ |
shifted to zero. |
959 |
+ |
|
960 |
+ |
\subsubsection{\textbf{Equilibration}} |
961 |
+ |
|
962 |
+ |
The purpose of equilibration is to allow the system to evolve |
963 |
+ |
spontaneously for a period of time and reach equilibrium. The |
964 |
+ |
procedure is continued until various statistical properties, such as |
965 |
+ |
temperature, pressure, energy, volume and other structural |
966 |
+ |
properties \textit{etc}, become independent of time. Strictly |
967 |
+ |
speaking, minimization and heating are not necessary, provided the |
968 |
+ |
equilibration process is long enough. However, these steps can serve |
969 |
+ |
as a means to arrive at an equilibrated structure in an effective |
970 |
+ |
way. |
971 |
+ |
|
972 |
+ |
\subsection{\label{introSection:production}Production} |
973 |
+ |
|
974 |
+ |
Production run is the most important step of the simulation, in |
975 |
+ |
which the equilibrated structure is used as a starting point and the |
976 |
+ |
motions of the molecules are collected for later analysis. In order |
977 |
+ |
to capture the macroscopic properties of the system, the molecular |
978 |
+ |
dynamics simulation must be performed in correct and efficient way. |
979 |
+ |
|
980 |
+ |
The most expensive part of a molecular dynamics simulation is the |
981 |
+ |
calculation of non-bonded forces, such as van der Waals force and |
982 |
+ |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
983 |
+ |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
984 |
+ |
which making large simulations prohibitive in the absence of any |
985 |
+ |
computation saving techniques. |
986 |
+ |
|
987 |
+ |
A natural approach to avoid system size issue is to represent the |
988 |
+ |
bulk behavior by a finite number of the particles. However, this |
989 |
+ |
approach will suffer from the surface effect. To offset this, |
990 |
+ |
\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc}) |
991 |
+ |
is developed to simulate bulk properties with a relatively small |
992 |
+ |
number of particles. In this method, the simulation box is |
993 |
+ |
replicated throughout space to form an infinite lattice. During the |
994 |
+ |
simulation, when a particle moves in the primary cell, its image in |
995 |
+ |
other cells move in exactly the same direction with exactly the same |
996 |
+ |
orientation. Thus, as a particle leaves the primary cell, one of its |
997 |
+ |
images will enter through the opposite face. |
998 |
+ |
\begin{figure} |
999 |
+ |
\centering |
1000 |
+ |
\includegraphics[width=\linewidth]{pbc.eps} |
1001 |
+ |
\caption[An illustration of periodic boundary conditions]{A 2-D |
1002 |
+ |
illustration of periodic boundary conditions. As one particle leaves |
1003 |
+ |
the left of the simulation box, an image of it enters the right.} |
1004 |
+ |
\label{introFig:pbc} |
1005 |
+ |
\end{figure} |
1006 |
+ |
|
1007 |
+ |
%cutoff and minimum image convention |
1008 |
+ |
Another important technique to improve the efficiency of force |
1009 |
+ |
evaluation is to apply cutoff where particles farther than a |
1010 |
+ |
predetermined distance, are not included in the calculation |
1011 |
+ |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1012 |
+ |
discontinuity in the potential energy curve. Fortunately, one can |
1013 |
+ |
shift the potential to ensure the potential curve go smoothly to |
1014 |
+ |
zero at the cutoff radius. Cutoff strategy works pretty well for |
1015 |
+ |
Lennard-Jones interaction because of its short range nature. |
1016 |
+ |
However, simply truncating the electrostatic interaction with the |
1017 |
+ |
use of cutoff has been shown to lead to severe artifacts in |
1018 |
+ |
simulations. Ewald summation, in which the slowly conditionally |
1019 |
+ |
convergent Coulomb potential is transformed into direct and |
1020 |
+ |
reciprocal sums with rapid and absolute convergence, has proved to |
1021 |
+ |
minimize the periodicity artifacts in liquid simulations. Taking the |
1022 |
+ |
advantages of the fast Fourier transform (FFT) for calculating |
1023 |
+ |
discrete Fourier transforms, the particle mesh-based |
1024 |
+ |
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
1025 |
+ |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast |
1026 |
+ |
multipole method}\cite{Greengard1987, Greengard1994}, which treats |
1027 |
+ |
Coulombic interaction exactly at short range, and approximate the |
1028 |
+ |
potential at long range through multipolar expansion. In spite of |
1029 |
+ |
their wide acceptances at the molecular simulation community, these |
1030 |
+ |
two methods are hard to be implemented correctly and efficiently. |
1031 |
+ |
Instead, we use a damped and charge-neutralized Coulomb potential |
1032 |
+ |
method developed by Wolf and his coworkers\cite{Wolf1999}. The |
1033 |
+ |
shifted Coulomb potential for particle $i$ and particle $j$ at |
1034 |
+ |
distance $r_{rj}$ is given by: |
1035 |
+ |
\begin{equation} |
1036 |
+ |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1037 |
+ |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1038 |
+ |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
1039 |
+ |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
1040 |
+ |
\end{equation} |
1041 |
+ |
where $\alpha$ is the convergence parameter. Due to the lack of |
1042 |
+ |
inherent periodicity and rapid convergence,this method is extremely |
1043 |
+ |
efficient and easy to implement. |
1044 |
+ |
\begin{figure} |
1045 |
+ |
\centering |
1046 |
+ |
\includegraphics[width=\linewidth]{shifted_coulomb.eps} |
1047 |
+ |
\caption[An illustration of shifted Coulomb potential]{An |
1048 |
+ |
illustration of shifted Coulomb potential.} |
1049 |
+ |
\label{introFigure:shiftedCoulomb} |
1050 |
+ |
\end{figure} |
1051 |
+ |
|
1052 |
+ |
%multiple time step |
1053 |
+ |
|
1054 |
+ |
\subsection{\label{introSection:Analysis} Analysis} |
1055 |
+ |
|
1056 |
+ |
Recently, advanced visualization technique are widely applied to |
1057 |
+ |
monitor the motions of molecules. Although the dynamics of the |
1058 |
+ |
system can be described qualitatively from animation, quantitative |
1059 |
+ |
trajectory analysis are more appreciable. According to the |
1060 |
+ |
principles of Statistical Mechanics, |
1061 |
+ |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1062 |
+ |
thermodynamics properties, analyze fluctuations of structural |
1063 |
+ |
parameters, and investigate time-dependent processes of the molecule |
1064 |
+ |
from the trajectories. |
1065 |
+ |
|
1066 |
+ |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamics Properties}} |
1067 |
+ |
|
1068 |
+ |
Thermodynamics properties, which can be expressed in terms of some |
1069 |
+ |
function of the coordinates and momenta of all particles in the |
1070 |
+ |
system, can be directly computed from molecular dynamics. The usual |
1071 |
+ |
way to measure the pressure is based on virial theorem of Clausius |
1072 |
+ |
which states that the virial is equal to $-3Nk_BT$. For a system |
1073 |
+ |
with forces between particles, the total virial, $W$, contains the |
1074 |
+ |
contribution from external pressure and interaction between the |
1075 |
+ |
particles: |
1076 |
+ |
\[ |
1077 |
+ |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
1078 |
+ |
f_{ij} } } \right\rangle |
1079 |
+ |
\] |
1080 |
+ |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
1081 |
+ |
distance $r_{ij}$. Thus, the expression for the pressure is given |
1082 |
+ |
by: |
1083 |
+ |
\begin{equation} |
1084 |
+ |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
1085 |
+ |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1086 |
+ |
\end{equation} |
1087 |
+ |
|
1088 |
+ |
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
1089 |
+ |
|
1090 |
+ |
Structural Properties of a simple fluid can be described by a set of |
1091 |
+ |
distribution functions. Among these functions,\emph{pair |
1092 |
+ |
distribution function}, also known as \emph{radial distribution |
1093 |
+ |
function}, is of most fundamental importance to liquid-state theory. |
1094 |
+ |
Pair distribution function can be gathered by Fourier transforming |
1095 |
+ |
raw data from a series of neutron diffraction experiments and |
1096 |
+ |
integrating over the surface factor \cite{Powles1973}. The |
1097 |
+ |
experiment result can serve as a criterion to justify the |
1098 |
+ |
correctness of the theory. Moreover, various equilibrium |
1099 |
+ |
thermodynamic and structural properties can also be expressed in |
1100 |
+ |
terms of radial distribution function \cite{Allen1987}. |
1101 |
+ |
|
1102 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
1103 |
+ |
particle $i$ will be located at a distance $r$ from a another |
1104 |
+ |
particle $j$ in the system |
1105 |
+ |
\[ |
1106 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1107 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1108 |
+ |
\] |
1109 |
+ |
Note that the delta function can be replaced by a histogram in |
1110 |
+ |
computer simulation. Figure |
1111 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1112 |
+ |
distribution function for the liquid argon system. The occurrence of |
1113 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1114 |
+ |
to find particles at certain radial values than at others. This is a |
1115 |
+ |
result of the attractive interaction at such distances. Because of |
1116 |
+ |
the strong repulsive forces at short distance, the probability of |
1117 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
1118 |
+ |
other is essentially zero. |
1119 |
+ |
|
1120 |
+ |
%\begin{figure} |
1121 |
+ |
%\centering |
1122 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
1123 |
+ |
%\caption[Pair distribution function for the liquid argon |
1124 |
+ |
%]{Pair distribution function for the liquid argon} |
1125 |
+ |
%\label{introFigure:pairDistributionFunction} |
1126 |
+ |
%\end{figure} |
1127 |
+ |
|
1128 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
1129 |
+ |
Properties}} |
1130 |
+ |
|
1131 |
+ |
Time-dependent properties are usually calculated using \emph{time |
1132 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
1133 |
+ |
at two different time |
1134 |
+ |
\begin{equation} |
1135 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1136 |
+ |
\label{introEquation:timeCorrelationFunction} |
1137 |
+ |
\end{equation} |
1138 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
1139 |
+ |
function is called \emph{auto correlation function}. One example of |
1140 |
+ |
auto correlation function is velocity auto-correlation function |
1141 |
+ |
which is directly related to transport properties of molecular |
1142 |
+ |
liquids: |
1143 |
+ |
\[ |
1144 |
+ |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1145 |
+ |
\right\rangle } dt |
1146 |
+ |
\] |
1147 |
+ |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1148 |
+ |
function which is averaging over time origins and over all the |
1149 |
+ |
atoms, dipole autocorrelation are calculated for the entire system. |
1150 |
+ |
The dipole autocorrelation function is given by: |
1151 |
+ |
\[ |
1152 |
+ |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1153 |
+ |
\right\rangle |
1154 |
+ |
\] |
1155 |
+ |
Here $u_{tot}$ is the net dipole of the entire system and is given |
1156 |
+ |
by |
1157 |
+ |
\[ |
1158 |
+ |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1159 |
+ |
\] |
1160 |
+ |
In principle, many time correlation functions can be related with |
1161 |
+ |
Fourier transforms of the infrared, Raman, and inelastic neutron |
1162 |
+ |
scattering spectra of molecular liquids. In practice, one can |
1163 |
+ |
extract the IR spectrum from the intensity of dipole fluctuation at |
1164 |
+ |
each frequency using the following relationship: |
1165 |
+ |
\[ |
1166 |
+ |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1167 |
+ |
i2\pi vt} dt} |
1168 |
+ |
\] |
1169 |
+ |
|
1170 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1171 |
|
|
1172 |
|
Rigid bodies are frequently involved in the modeling of different |
1175 |
|
movement of the objects in 3D gaming engine or other physics |
1176 |
|
simulator is governed by the rigid body dynamics. In molecular |
1177 |
|
simulation, rigid body is used to simplify the model in |
1178 |
< |
protein-protein docking study{\cite{Gray03}}. |
1178 |
> |
protein-protein docking study\cite{Gray2003}. |
1179 |
|
|
1180 |
|
It is very important to develop stable and efficient methods to |
1181 |
|
integrate the equations of motion of orientational degrees of |
1183 |
|
rotational degrees of freedom. However, due to its singularity, the |
1184 |
|
numerical integration of corresponding equations of motion is very |
1185 |
|
inefficient and inaccurate. Although an alternative integrator using |
1186 |
< |
different sets of Euler angles can overcome this difficulty\cite{}, |
1187 |
< |
the computational penalty and the lost of angular momentum |
1188 |
< |
conservation still remain. A singularity free representation |
1189 |
< |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
1190 |
< |
this approach suffer from the nonseparable Hamiltonian resulted from |
1191 |
< |
quaternion representation, which prevents the symplectic algorithm |
1192 |
< |
to be utilized. Another different approach is to apply holonomic |
1193 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
1194 |
< |
moves independently under the normal forces deriving from potential |
1195 |
< |
energy and constraint forces which are used to guarantee the |
1196 |
< |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
1197 |
< |
algorithm converge very slowly when the number of constraint |
1198 |
< |
increases. |
1186 |
> |
different sets of Euler angles can overcome this |
1187 |
> |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
1188 |
> |
of angular momentum conservation still remain. A singularity free |
1189 |
> |
representation utilizing quaternions was developed by Evans in |
1190 |
> |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
1191 |
> |
nonseparable Hamiltonian resulted from quaternion representation, |
1192 |
> |
which prevents the symplectic algorithm to be utilized. Another |
1193 |
> |
different approach is to apply holonomic constraints to the atoms |
1194 |
> |
belonging to the rigid body. Each atom moves independently under the |
1195 |
> |
normal forces deriving from potential energy and constraint forces |
1196 |
> |
which are used to guarantee the rigidness. However, due to their |
1197 |
> |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
1198 |
> |
when the number of constraint increases\cite{Ryckaert1977, |
1199 |
> |
Andersen1983}. |
1200 |
|
|
1201 |
|
The break through in geometric literature suggests that, in order to |
1202 |
|
develop a long-term integration scheme, one should preserve the |
1203 |
|
symplectic structure of the flow. Introducing conjugate momentum to |
1204 |
< |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
1205 |
< |
symplectic integrator, RSHAKE, was proposed to evolve the |
1206 |
< |
Hamiltonian system in a constraint manifold by iteratively |
1207 |
< |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
1208 |
< |
method using quaternion representation was developed by Omelyan. |
1209 |
< |
However, both of these methods are iterative and inefficient. In |
1210 |
< |
this section, we will present a symplectic Lie-Poisson integrator |
1211 |
< |
for rigid body developed by Dullweber and his |
1212 |
< |
coworkers\cite{Dullweber1997} in depth. |
1204 |
> |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
1205 |
> |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
1206 |
> |
the Hamiltonian system in a constraint manifold by iteratively |
1207 |
> |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
1208 |
> |
method using quaternion representation was developed by |
1209 |
> |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
1210 |
> |
iterative and inefficient. In this section, we will present a |
1211 |
> |
symplectic Lie-Poisson integrator for rigid body developed by |
1212 |
> |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1213 |
|
|
1214 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
1215 |
|
The motion of the rigid body is Hamiltonian with the Hamiltonian |
1228 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1229 |
|
constrained Hamiltonian equation subjects to a holonomic constraint, |
1230 |
|
\begin{equation} |
1231 |
< |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
1231 |
> |
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1232 |
|
\end{equation} |
1233 |
|
which is used to ensure rotation matrix's orthogonality. |
1234 |
|
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1241 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1242 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1243 |
|
the equations of motion, |
1244 |
< |
\[ |
1245 |
< |
\begin{array}{c} |
1246 |
< |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1247 |
< |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1248 |
< |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1249 |
< |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
1250 |
< |
\end{array} |
977 |
< |
\] |
1244 |
> |
|
1245 |
> |
\begin{eqnarray} |
1246 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1247 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1248 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1249 |
> |
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1250 |
> |
\end{eqnarray} |
1251 |
|
|
1252 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1253 |
|
We can use constraint force provided by lagrange multiplier on the |
1254 |
|
normal manifold to keep the motion on constraint space. Or we can |
1255 |
< |
simply evolve the system in constraint manifold. The two method are |
1256 |
< |
proved to be equivalent. The holonomic constraint and equations of |
1257 |
< |
motions define a constraint manifold for rigid body |
1255 |
> |
simply evolve the system in constraint manifold. These two methods |
1256 |
> |
are proved to be equivalent. The holonomic constraint and equations |
1257 |
> |
of motions define a constraint manifold for rigid body |
1258 |
|
\[ |
1259 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1260 |
|
\right\}. |
1328 |
|
\[ |
1329 |
|
\hat vu = v \times u |
1330 |
|
\] |
1058 |
– |
|
1331 |
|
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1332 |
|
matrix, |
1333 |
|
\begin{equation} |
1334 |
< |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
1334 |
> |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
1335 |
|
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
1336 |
|
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1337 |
|
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
1340 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1341 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
1342 |
|
unique property eliminate the requirement of iterations which can |
1343 |
< |
not be avoided in other methods\cite{}. |
1343 |
> |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
1344 |
|
|
1345 |
|
Applying hat-map isomorphism, we obtain the equation of motion for |
1346 |
|
angular momentum on body frame |
1360 |
|
|
1361 |
|
If there is not external forces exerted on the rigid body, the only |
1362 |
|
contribution to the rotational is from the kinetic potential (the |
1363 |
< |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
1364 |
< |
rigid body is an example of Lie-Poisson system with Hamiltonian |
1093 |
< |
function |
1363 |
> |
first term of \ref{introEquation:bodyAngularMotion}). The free rigid |
1364 |
> |
body is an example of Lie-Poisson system with Hamiltonian function |
1365 |
|
\begin{equation} |
1366 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1367 |
|
\label{introEquation:rotationalKineticRB} |
1407 |
|
0 & { - \sin \theta _1 } & {\cos \theta _1 } \\ |
1408 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1409 |
|
\] |
1410 |
< |
To reduce the cost of computing expensive functions in e^{\Delta |
1411 |
< |
tR_1 }, we can use Cayley transformation, |
1410 |
> |
To reduce the cost of computing expensive functions in $e^{\Delta |
1411 |
> |
tR_1 }$, we can use Cayley transformation, |
1412 |
|
\[ |
1413 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1414 |
|
) |
1415 |
|
\] |
1416 |
< |
|
1146 |
< |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
1416 |
> |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1417 |
|
manner. |
1418 |
|
|
1419 |
|
In order to construct a second-order symplectic method, we split the |
1465 |
|
\] |
1466 |
|
The equations of motion corresponding to potential energy and |
1467 |
|
kinetic energy are listed in the below table, |
1468 |
+ |
\begin{table} |
1469 |
+ |
\caption{Equations of motion due to Potential and Kinetic Energies} |
1470 |
|
\begin{center} |
1471 |
|
\begin{tabular}{|l|l|} |
1472 |
|
\hline |
1479 |
|
\hline |
1480 |
|
\end{tabular} |
1481 |
|
\end{center} |
1482 |
< |
A second-order symplectic method is now obtained by the composition |
1483 |
< |
of the flow maps, |
1482 |
> |
\end{table} |
1483 |
> |
A second-order symplectic method is now obtained by the |
1484 |
> |
composition of the flow maps, |
1485 |
|
\[ |
1486 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1487 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1488 |
|
\] |
1489 |
< |
Moreover, \varphi _{\Delta t/2,V} can be divided into two sub-flows |
1490 |
< |
which corresponding to force and torque respectively, |
1489 |
> |
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
1490 |
> |
sub-flows which corresponding to force and torque respectively, |
1491 |
|
\[ |
1492 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1493 |
|
_{\Delta t/2,\tau }. |
1494 |
|
\] |
1495 |
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
1496 |
|
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
1497 |
< |
order inside \varphi _{\Delta t/2,V} does not matter. |
1497 |
> |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
1498 |
|
|
1499 |
|
Furthermore, kinetic potential can be separated to translational |
1500 |
|
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
1524 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
1525 |
|
has been applied in a variety of studies. This section will review |
1526 |
|
the theory of Langevin dynamics simulation. A brief derivation of |
1527 |
< |
generalized Langevin Dynamics will be given first. Follow that, we |
1527 |
> |
generalized Langevin equation will be given first. Follow that, we |
1528 |
|
will discuss the physical meaning of the terms appearing in the |
1529 |
|
equation as well as the calculation of friction tensor from |
1530 |
|
hydrodynamics theory. |
1531 |
|
|
1532 |
< |
\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics} |
1532 |
> |
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
1533 |
|
|
1534 |
+ |
Harmonic bath model, in which an effective set of harmonic |
1535 |
+ |
oscillators are used to mimic the effect of a linearly responding |
1536 |
+ |
environment, has been widely used in quantum chemistry and |
1537 |
+ |
statistical mechanics. One of the successful applications of |
1538 |
+ |
Harmonic bath model is the derivation of Deriving Generalized |
1539 |
+ |
Langevin Dynamics. Lets consider a system, in which the degree of |
1540 |
+ |
freedom $x$ is assumed to couple to the bath linearly, giving a |
1541 |
+ |
Hamiltonian of the form |
1542 |
|
\begin{equation} |
1543 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1544 |
< |
\label{introEquation:bathGLE} |
1544 |
> |
\label{introEquation:bathGLE}. |
1545 |
|
\end{equation} |
1546 |
< |
where $H_B$ is harmonic bath Hamiltonian, |
1546 |
> |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
1547 |
> |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
1548 |
|
\[ |
1549 |
< |
H_B =\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1550 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha w_\alpha ^2 } \right\}} |
1549 |
> |
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1550 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
1551 |
> |
\right\}} |
1552 |
|
\] |
1553 |
< |
and $\Delta U$ is bilinear system-bath coupling, |
1553 |
> |
where the index $\alpha$ runs over all the bath degrees of freedom, |
1554 |
> |
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
1555 |
> |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
1556 |
> |
coupling, |
1557 |
|
\[ |
1558 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1559 |
|
\] |
1560 |
< |
Completing the square, |
1560 |
> |
where $g_\alpha$ are the coupling constants between the bath and the |
1561 |
> |
coordinate $x$. Introducing |
1562 |
|
\[ |
1563 |
< |
H_B + \Delta U = \sum\limits_{\alpha = 1}^N {\left\{ |
1564 |
< |
{\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1565 |
< |
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1566 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} - \sum\limits_{\alpha = |
1567 |
< |
1}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1281 |
< |
\] |
1282 |
< |
and putting it back into Eq.~\ref{introEquation:bathGLE}, |
1563 |
> |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1564 |
> |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1565 |
> |
\] and combining the last two terms in Equation |
1566 |
> |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
1567 |
> |
Hamiltonian as |
1568 |
|
\[ |
1569 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1570 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1571 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1572 |
|
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1573 |
|
\] |
1289 |
– |
where |
1290 |
– |
\[ |
1291 |
– |
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1292 |
– |
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1293 |
– |
\] |
1574 |
|
Since the first two terms of the new Hamiltonian depend only on the |
1575 |
|
system coordinates, we can get the equations of motion for |
1576 |
|
Generalized Langevin Dynamics by Hamilton's equations |
1577 |
|
\ref{introEquation:motionHamiltonianCoordinate, |
1578 |
|
introEquation:motionHamiltonianMomentum}, |
1579 |
< |
\begin{align} |
1580 |
< |
\dot p &= - \frac{{\partial H}}{{\partial x}} |
1581 |
< |
&= m\ddot x |
1582 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
1583 |
< |
\label{introEquation:Lp5} |
1584 |
< |
\end{align} |
1585 |
< |
, and |
1586 |
< |
\begin{align} |
1587 |
< |
\dot p_\alpha &= - \frac{{\partial H}}{{\partial x_\alpha }} |
1588 |
< |
&= m\ddot x_\alpha |
1589 |
< |
&= \- m_\alpha w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha}}{{m_\alpha w_\alpha ^2 }}x} \right) |
1590 |
< |
\end{align} |
1579 |
> |
\begin{equation} |
1580 |
> |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
1581 |
> |
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
1582 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)}, |
1583 |
> |
\label{introEquation:coorMotionGLE} |
1584 |
> |
\end{equation} |
1585 |
> |
and |
1586 |
> |
\begin{equation} |
1587 |
> |
m\ddot x_\alpha = - m_\alpha w_\alpha ^2 \left( {x_\alpha - |
1588 |
> |
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
1589 |
> |
\label{introEquation:bathMotionGLE} |
1590 |
> |
\end{equation} |
1591 |
|
|
1592 |
< |
\subsection{\label{introSection:laplaceTransform}The Laplace Transform} |
1592 |
> |
In order to derive an equation for $x$, the dynamics of the bath |
1593 |
> |
variables $x_\alpha$ must be solved exactly first. As an integral |
1594 |
> |
transform which is particularly useful in solving linear ordinary |
1595 |
> |
differential equations, Laplace transform is the appropriate tool to |
1596 |
> |
solve this problem. The basic idea is to transform the difficult |
1597 |
> |
differential equations into simple algebra problems which can be |
1598 |
> |
solved easily. Then applying inverse Laplace transform, also known |
1599 |
> |
as the Bromwich integral, we can retrieve the solutions of the |
1600 |
> |
original problems. |
1601 |
|
|
1602 |
+ |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
1603 |
+ |
transform of f(t) is a new function defined as |
1604 |
|
\[ |
1605 |
< |
L(x) = \int_0^\infty {x(t)e^{ - pt} dt} |
1605 |
> |
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1606 |
|
\] |
1607 |
+ |
where $p$ is real and $L$ is called the Laplace Transform |
1608 |
+ |
Operator. Below are some important properties of Laplace transform |
1609 |
|
|
1610 |
< |
\[ |
1611 |
< |
L(x + y) = L(x) + L(y) |
1612 |
< |
\] |
1610 |
> |
\begin{eqnarray*} |
1611 |
> |
L(x + y) & = & L(x) + L(y) \\ |
1612 |
> |
L(ax) & = & aL(x) \\ |
1613 |
> |
L(\dot x) & = & pL(x) - px(0) \\ |
1614 |
> |
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
1615 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
1616 |
> |
\end{eqnarray*} |
1617 |
|
|
1322 |
– |
\[ |
1323 |
– |
L(ax) = aL(x) |
1324 |
– |
\] |
1618 |
|
|
1619 |
< |
\[ |
1620 |
< |
L(\dot x) = pL(x) - px(0) |
1621 |
< |
\] |
1619 |
> |
Applying Laplace transform to the bath coordinates, we obtain |
1620 |
> |
\begin{eqnarray*} |
1621 |
> |
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) \\ |
1622 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1623 |
> |
\end{eqnarray*} |
1624 |
|
|
1625 |
< |
\[ |
1626 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) |
1627 |
< |
\] |
1625 |
> |
By the same way, the system coordinates become |
1626 |
> |
\begin{eqnarray*} |
1627 |
> |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1628 |
> |
& & \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\}} \\ |
1629 |
> |
\end{eqnarray*} |
1630 |
|
|
1631 |
+ |
With the help of some relatively important inverse Laplace |
1632 |
+ |
transformations: |
1633 |
|
\[ |
1634 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) |
1635 |
< |
\] |
1636 |
< |
|
1637 |
< |
Some relatively important transformation, |
1638 |
< |
\[ |
1340 |
< |
L(\cos at) = \frac{p}{{p^2 + a^2 }} |
1634 |
> |
\begin{array}{c} |
1635 |
> |
L(\cos at) = \frac{p}{{p^2 + a^2 }} \\ |
1636 |
> |
L(\sin at) = \frac{a}{{p^2 + a^2 }} \\ |
1637 |
> |
L(1) = \frac{1}{p} \\ |
1638 |
> |
\end{array} |
1639 |
|
\] |
1640 |
< |
|
1641 |
< |
\[ |
1642 |
< |
L(\sin at) = \frac{a}{{p^2 + a^2 }} |
1345 |
< |
\] |
1346 |
< |
|
1347 |
< |
\[ |
1348 |
< |
L(1) = \frac{1}{p} |
1349 |
< |
\] |
1350 |
< |
|
1351 |
< |
First, the bath coordinates, |
1352 |
< |
\[ |
1353 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega |
1354 |
< |
_\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha |
1355 |
< |
}}L(x) |
1356 |
< |
\] |
1357 |
< |
\[ |
1358 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + |
1359 |
< |
px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} |
1360 |
< |
\] |
1361 |
< |
Then, the system coordinates, |
1362 |
< |
\begin{align} |
1363 |
< |
mL(\ddot x) &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1364 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ {\frac{{\frac{{g_\alpha |
1365 |
< |
}}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha |
1366 |
< |
(0)}}{{p^2 + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha |
1367 |
< |
}}\omega _\alpha ^2 L(x)} \right\}} |
1368 |
< |
% |
1369 |
< |
&= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} - |
1370 |
< |
\sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) |
1371 |
< |
- \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) |
1372 |
< |
- \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} |
1373 |
< |
\end{align} |
1374 |
< |
Then, the inverse transform, |
1375 |
< |
|
1376 |
< |
\begin{align} |
1377 |
< |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
1640 |
> |
, we obtain |
1641 |
> |
\begin{eqnarray*} |
1642 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
1643 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1644 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
1645 |
< |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
1646 |
< |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
1647 |
< |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
1648 |
< |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
1649 |
< |
% |
1650 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1645 |
> |
_\alpha t)\dot x(t - \tau )d\tau } } \right\}} \\ |
1646 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1647 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1648 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1649 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1650 |
> |
\end{eqnarray*} |
1651 |
> |
\begin{eqnarray*} |
1652 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
1653 |
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1654 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
1655 |
< |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
1656 |
< |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
1657 |
< |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
1658 |
< |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
1659 |
< |
(\omega _\alpha t)} \right\}} |
1660 |
< |
\end{align} |
1661 |
< |
|
1655 |
> |
t)\dot x(t - \tau )d} \tau } \\ |
1656 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
1657 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
1658 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
1659 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
1660 |
> |
\end{eqnarray*} |
1661 |
> |
Introducing a \emph{dynamic friction kernel} |
1662 |
|
\begin{equation} |
1663 |
+ |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1664 |
+ |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
1665 |
+ |
\label{introEquation:dynamicFrictionKernelDefinition} |
1666 |
+ |
\end{equation} |
1667 |
+ |
and \emph{a random force} |
1668 |
+ |
\begin{equation} |
1669 |
+ |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
1670 |
+ |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
1671 |
+ |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
1672 |
+ |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t), |
1673 |
+ |
\label{introEquation:randomForceDefinition} |
1674 |
+ |
\end{equation} |
1675 |
+ |
the equation of motion can be rewritten as |
1676 |
+ |
\begin{equation} |
1677 |
|
m\ddot x = - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi |
1678 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
1679 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
1680 |
|
\end{equation} |
1681 |
< |
%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and |
1401 |
< |
%$W$ is the potential of mean force. $W(x) = - kT\ln p(x)$ |
1402 |
< |
\[ |
1403 |
< |
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
1404 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha t)} |
1405 |
< |
\] |
1406 |
< |
For an infinite harmonic bath, we can use the spectral density and |
1407 |
< |
an integral over frequencies. |
1681 |
> |
which is known as the \emph{generalized Langevin equation}. |
1682 |
|
|
1683 |
+ |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
1684 |
+ |
|
1685 |
+ |
One may notice that $R(t)$ depends only on initial conditions, which |
1686 |
+ |
implies it is completely deterministic within the context of a |
1687 |
+ |
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
1688 |
+ |
uncorrelated to $x$ and $\dot x$, |
1689 |
|
\[ |
1690 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {\left( {g_\alpha x_\alpha (0) |
1691 |
< |
- \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}x(0)} |
1692 |
< |
\right)\cos (\omega _\alpha t)} + \frac{{\dot x_\alpha |
1693 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t) |
1690 |
> |
\begin{array}{l} |
1691 |
> |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
1692 |
> |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
1693 |
> |
\end{array} |
1694 |
|
\] |
1695 |
< |
The random forces depend only on initial conditions. |
1695 |
> |
This property is what we expect from a truly random process. As long |
1696 |
> |
as the model, which is gaussian distribution in general, chosen for |
1697 |
> |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
1698 |
> |
still remains. |
1699 |
|
|
1700 |
< |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
1701 |
< |
So we can define a new set of coordinates, |
1700 |
> |
%dynamic friction kernel |
1701 |
> |
The convolution integral |
1702 |
|
\[ |
1703 |
< |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1421 |
< |
^2 }}x(0) |
1703 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } |
1704 |
|
\] |
1705 |
< |
This makes |
1705 |
> |
depends on the entire history of the evolution of $x$, which implies |
1706 |
> |
that the bath retains memory of previous motions. In other words, |
1707 |
> |
the bath requires a finite time to respond to change in the motion |
1708 |
> |
of the system. For a sluggish bath which responds slowly to changes |
1709 |
> |
in the system coordinate, we may regard $\xi(t)$ as a constant |
1710 |
> |
$\xi(t) = \Xi_0$. Hence, the convolution integral becomes |
1711 |
|
\[ |
1712 |
< |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)} |
1712 |
> |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
1713 |
|
\] |
1714 |
< |
And since the $q$ coordinates are harmonic oscillators, |
1714 |
> |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1715 |
|
\[ |
1716 |
< |
\begin{array}{l} |
1717 |
< |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1431 |
< |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1432 |
< |
\end{array} |
1716 |
> |
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
1717 |
> |
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
1718 |
|
\] |
1719 |
+ |
which can be used to describe dynamic caging effect. The other |
1720 |
+ |
extreme is the bath that responds infinitely quickly to motions in |
1721 |
+ |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
1722 |
+ |
time: |
1723 |
+ |
\[ |
1724 |
+ |
\xi (t) = 2\xi _0 \delta (t) |
1725 |
+ |
\] |
1726 |
+ |
Hence, the convolution integral becomes |
1727 |
+ |
\[ |
1728 |
+ |
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
1729 |
+ |
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
1730 |
+ |
\] |
1731 |
+ |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1732 |
+ |
\begin{equation} |
1733 |
+ |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
1734 |
+ |
x(t) + R(t) \label{introEquation:LangevinEquation} |
1735 |
+ |
\end{equation} |
1736 |
+ |
which is known as the Langevin equation. The static friction |
1737 |
+ |
coefficient $\xi _0$ can either be calculated from spectral density |
1738 |
+ |
or be determined by Stokes' law for regular shaped particles.A |
1739 |
+ |
briefly review on calculating friction tensor for arbitrary shaped |
1740 |
+ |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
1741 |
|
|
1742 |
< |
\begin{align} |
1436 |
< |
\left\langle {R(t)R(0)} \right\rangle &= \sum\limits_\alpha |
1437 |
< |
{\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha |
1438 |
< |
(t)q_\beta (0)} \right\rangle } } |
1439 |
< |
% |
1440 |
< |
&= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} |
1441 |
< |
\right\rangle \cos (\omega _\alpha t)} |
1442 |
< |
% |
1443 |
< |
&= kT\xi (t) |
1444 |
< |
\end{align} |
1742 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
1743 |
|
|
1744 |
+ |
Defining a new set of coordinates, |
1745 |
+ |
\[ |
1746 |
+ |
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1747 |
+ |
^2 }}x(0) |
1748 |
+ |
\], |
1749 |
+ |
we can rewrite $R(T)$ as |
1750 |
+ |
\[ |
1751 |
+ |
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1752 |
+ |
\] |
1753 |
+ |
And since the $q$ coordinates are harmonic oscillators, |
1754 |
+ |
|
1755 |
+ |
\begin{eqnarray*} |
1756 |
+ |
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1757 |
+ |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1758 |
+ |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
1759 |
+ |
\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 } } \\ |
1760 |
+ |
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1761 |
+ |
& = &kT\xi (t) \\ |
1762 |
+ |
\end{eqnarray*} |
1763 |
+ |
|
1764 |
+ |
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1765 |
|
\begin{equation} |
1766 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1767 |
< |
\label{introEquation:secondFluctuationDissipation} |
1767 |
> |
\label{introEquation:secondFluctuationDissipation}. |
1768 |
|
\end{equation} |
1769 |
+ |
In effect, it acts as a constraint on the possible ways in which one |
1770 |
+ |
can model the random force and friction kernel. |
1771 |
|
|
1772 |
|
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
1773 |
|
Theoretically, the friction kernel can be determined using velocity |
1775 |
|
when the system become more and more complicate. Instead, various |
1776 |
|
approaches based on hydrodynamics have been developed to calculate |
1777 |
|
the friction coefficients. The friction effect is isotropic in |
1778 |
< |
Equation, \zeta can be taken as a scalar. In general, friction |
1779 |
< |
tensor \Xi is a $6\times 6$ matrix given by |
1778 |
> |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
1779 |
> |
tensor $\Xi$ is a $6\times 6$ matrix given by |
1780 |
|
\[ |
1781 |
|
\Xi = \left( {\begin{array}{*{20}c} |
1782 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
1805 |
|
where $F_r$ is the friction force and $\tau _R$ is the friction |
1806 |
|
toque. |
1807 |
|
|
1808 |
< |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
1808 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
1809 |
|
|
1810 |
|
For a spherical particle, the translational and rotational friction |
1811 |
|
constant can be calculated from Stoke's law, |
1832 |
|
hydrodynamics theory, because their properties can be calculated |
1833 |
|
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
1834 |
|
also called a triaxial ellipsoid, which is given in Cartesian |
1835 |
< |
coordinates by |
1835 |
> |
coordinates by\cite{Perrin1934, Perrin1936} |
1836 |
|
\[ |
1837 |
|
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
1838 |
|
}} = 1 |
1867 |
|
\end{array}. |
1868 |
|
\] |
1869 |
|
|
1870 |
< |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
1870 |
> |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
1871 |
|
|
1872 |
|
Unlike spherical and other regular shaped molecules, there is not |
1873 |
|
analytical solution for friction tensor of any arbitrary shaped |
1876 |
|
hydrodynamic properties of rigid bodies. However, since the mapping |
1877 |
|
from all possible ellipsoidal space, $r$-space, to all possible |
1878 |
|
combination of rotational diffusion coefficients, $D$-space is not |
1879 |
< |
unique\cite{Wegener79} as well as the intrinsic coupling between |
1880 |
< |
translational and rotational motion of rigid body\cite{}, general |
1881 |
< |
ellipsoid is not always suitable for modeling arbitrarily shaped |
1882 |
< |
rigid molecule. A number of studies have been devoted to determine |
1883 |
< |
the friction tensor for irregularly shaped rigid bodies using more |
1884 |
< |
advanced method\cite{} where the molecule of interest was modeled by |
1885 |
< |
combinations of spheres(beads)\cite{} and the hydrodynamics |
1886 |
< |
properties of the molecule can be calculated using the hydrodynamic |
1887 |
< |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
1888 |
< |
immersed in a continuous medium. Due to hydrodynamics interaction, |
1889 |
< |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
1890 |
< |
unperturbed velocity $v_i$, |
1879 |
> |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
1880 |
> |
translational and rotational motion of rigid body, general ellipsoid |
1881 |
> |
is not always suitable for modeling arbitrarily shaped rigid |
1882 |
> |
molecule. A number of studies have been devoted to determine the |
1883 |
> |
friction tensor for irregularly shaped rigid bodies using more |
1884 |
> |
advanced method where the molecule of interest was modeled by |
1885 |
> |
combinations of spheres(beads)\cite{Carrasco1999} and the |
1886 |
> |
hydrodynamics properties of the molecule can be calculated using the |
1887 |
> |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
1888 |
> |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
1889 |
> |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
1890 |
> |
than its unperturbed velocity $v_i$, |
1891 |
|
\[ |
1892 |
|
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
1893 |
|
\] |
1908 |
|
\end{equation} |
1909 |
|
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
1910 |
|
A second order expression for element of different size was |
1911 |
< |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
1912 |
< |
la Torre and Bloomfield, |
1911 |
> |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
1912 |
> |
Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
1913 |
|
\begin{equation} |
1914 |
|
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
1915 |
|
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
1943 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
1944 |
|
)T_{ij} |
1945 |
|
\] |
1946 |
< |
where \delta _{ij} is Kronecker delta function. Inverting matrix |
1946 |
> |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
1947 |
|
$B$, we obtain |
1948 |
|
|
1949 |
|
\[ |
1987 |
|
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
1988 |
|
we can easily find out that the translational resistance tensor is |
1989 |
|
origin independent, while the rotational resistance tensor and |
1990 |
< |
translation-rotation coupling resistance tensor do depend on the |
1990 |
> |
translation-rotation coupling resistance tensor depend on the |
1991 |
|
origin. Given resistance tensor at an arbitrary origin $O$, and a |
1992 |
|
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
1993 |
|
obtain the resistance tensor at $P$ by |
1995 |
|
\begin{array}{l} |
1996 |
|
\Xi _P^{tt} = \Xi _O^{tt} \\ |
1997 |
|
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
1998 |
< |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
1998 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
1999 |
|
\end{array} |
2000 |
|
\label{introEquation:resistanceTensorTransformation} |
2001 |
|
\end{equation} |
2010 |
|
Using Equations \ref{introEquation:definitionCR} and |
2011 |
|
\ref{introEquation:resistanceTensorTransformation}, one can locate |
2012 |
|
the position of center of resistance, |
2013 |
< |
\[ |
2014 |
< |
\left( \begin{array}{l} |
2013 |
> |
\begin{eqnarray*} |
2014 |
> |
\left( \begin{array}{l} |
2015 |
|
x_{OR} \\ |
2016 |
|
y_{OR} \\ |
2017 |
|
z_{OR} \\ |
2018 |
< |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
2018 |
> |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
2019 |
|
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
2020 |
|
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
2021 |
|
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
2022 |
< |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
2022 |
> |
\end{array}} \right)^{ - 1} \\ |
2023 |
> |
& & \left( \begin{array}{l} |
2024 |
|
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
2025 |
|
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
2026 |
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
2027 |
< |
\end{array} \right). |
2028 |
< |
\] |
2027 |
> |
\end{array} \right) \\ |
2028 |
> |
\end{eqnarray*} |
2029 |
> |
|
2030 |
> |
|
2031 |
> |
|
2032 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
2033 |
|
joining center of resistance $R$ and origin $O$. |
1709 |
– |
|
1710 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |