31 |
|
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
32 |
|
Newton's third law states that |
33 |
|
\begin{equation} |
34 |
< |
F_{ij} = -F_{ji} |
34 |
> |
F_{ij} = -F_{ji}. |
35 |
|
\label{introEquation:newtonThirdLaw} |
36 |
|
\end{equation} |
37 |
– |
|
37 |
|
Conservation laws of Newtonian Mechanics play very important roles |
38 |
|
in solving mechanics problems. The linear momentum of a particle is |
39 |
|
conserved if it is free or it experiences no force. The second |
62 |
|
\end{equation} |
63 |
|
If there are no external torques acting on a body, the angular |
64 |
|
momentum of it is conserved. The last conservation theorem state |
65 |
< |
that if all forces are conservative, Energy |
66 |
< |
\begin{equation}E = T + V \label{introEquation:energyConservation} |
65 |
> |
that if all forces are conservative, energy is conserved, |
66 |
> |
\begin{equation}E = T + V. \label{introEquation:energyConservation} |
67 |
|
\end{equation} |
68 |
< |
is conserved. All of these conserved quantities are |
69 |
< |
important factors to determine the quality of numerical integration |
70 |
< |
schemes for rigid bodies \cite{Dullweber1997}. |
68 |
> |
All of these conserved quantities are important factors to determine |
69 |
> |
the quality of numerical integration schemes for rigid bodies |
70 |
> |
\cite{Dullweber1997}. |
71 |
|
|
72 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
73 |
|
|
74 |
< |
Newtonian Mechanics suffers from two important limitations: motions |
75 |
< |
can only be described in cartesian coordinate systems. Moreover, It |
76 |
< |
become impossible to predict analytically the properties of the |
77 |
< |
system even if we know all of the details of the interaction. In |
78 |
< |
order to overcome some of the practical difficulties which arise in |
79 |
< |
attempts to apply Newton's equation to complex system, approximate |
80 |
< |
numerical procedures may be developed. |
74 |
> |
Newtonian Mechanics suffers from a important limitation: motions can |
75 |
> |
only be described in cartesian coordinate systems which make it |
76 |
> |
impossible to predict analytically the properties of the system even |
77 |
> |
if we know all of the details of the interaction. In order to |
78 |
> |
overcome some of the practical difficulties which arise in attempts |
79 |
> |
to apply Newton's equation to complex system, approximate numerical |
80 |
> |
procedures may be developed. |
81 |
|
|
82 |
|
\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
83 |
|
Principle}} |
84 |
|
|
85 |
|
Hamilton introduced the dynamical principle upon which it is |
86 |
|
possible to base all of mechanics and most of classical physics. |
87 |
< |
Hamilton's Principle may be stated as follows, |
88 |
< |
|
89 |
< |
The actual trajectory, along which a dynamical system may move from |
90 |
< |
one point to another within a specified time, is derived by finding |
91 |
< |
the path which minimizes the time integral of the difference between |
93 |
< |
the kinetic, $K$, and potential energies, $U$. |
87 |
> |
Hamilton's Principle may be stated as follows: the actual |
88 |
> |
trajectory, along which a dynamical system may move from one point |
89 |
> |
to another within a specified time, is derived by finding the path |
90 |
> |
which minimizes the time integral of the difference between the |
91 |
> |
kinetic $K$, and potential energies $U$, |
92 |
|
\begin{equation} |
93 |
< |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
93 |
> |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}. |
94 |
|
\label{introEquation:halmitonianPrinciple1} |
95 |
|
\end{equation} |
98 |
– |
|
96 |
|
For simple mechanical systems, where the forces acting on the |
97 |
|
different parts are derivable from a potential, the Lagrangian |
98 |
|
function $L$ can be defined as the difference between the kinetic |
99 |
|
energy of the system and its potential energy, |
100 |
|
\begin{equation} |
101 |
< |
L \equiv K - U = L(q_i ,\dot q_i ) , |
101 |
> |
L \equiv K - U = L(q_i ,\dot q_i ). |
102 |
|
\label{introEquation:lagrangianDef} |
103 |
|
\end{equation} |
104 |
< |
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
104 |
> |
Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
105 |
|
\begin{equation} |
106 |
< |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
106 |
> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} . |
107 |
|
\label{introEquation:halmitonianPrinciple2} |
108 |
|
\end{equation} |
109 |
|
|
135 |
|
p_i = \frac{{\partial L}}{{\partial q_i }} |
136 |
|
\label{introEquation:generalizedMomentaDot} |
137 |
|
\end{equation} |
141 |
– |
|
138 |
|
With the help of the generalized momenta, we may now define a new |
139 |
|
quantity $H$ by the equation |
140 |
|
\begin{equation} |
142 |
|
\label{introEquation:hamiltonianDefByLagrangian} |
143 |
|
\end{equation} |
144 |
|
where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
145 |
< |
$L$ is the Lagrangian function for the system. |
146 |
< |
|
151 |
< |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
152 |
< |
one can obtain |
145 |
> |
$L$ is the Lagrangian function for the system. Differentiating |
146 |
> |
Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain |
147 |
|
\begin{equation} |
148 |
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
149 |
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
150 |
|
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
151 |
< |
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
151 |
> |
L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1} |
152 |
|
\end{equation} |
153 |
< |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
154 |
< |
second and fourth terms in the parentheses cancel. Therefore, |
153 |
> |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second |
154 |
> |
and fourth terms in the parentheses cancel. Therefore, |
155 |
|
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
156 |
|
\begin{equation} |
157 |
|
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
158 |
< |
\right)} - \frac{{\partial L}}{{\partial t}}dt |
158 |
> |
\right)} - \frac{{\partial L}}{{\partial t}}dt . |
159 |
|
\label{introEquation:diffHamiltonian2} |
160 |
|
\end{equation} |
161 |
|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
174 |
|
t}} |
175 |
|
\label{introEquation:motionHamiltonianTime} |
176 |
|
\end{equation} |
177 |
< |
|
184 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
177 |
> |
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
178 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
179 |
|
equation of motion. Due to their symmetrical formula, they are also |
180 |
|
known as the canonical equations of motions \cite{Goldstein2001}. |
188 |
|
statistical mechanics and quantum mechanics, since it treats the |
189 |
|
coordinate and its time derivative as independent variables and it |
190 |
|
only works with 1st-order differential equations\cite{Marion1990}. |
198 |
– |
|
191 |
|
In Newtonian Mechanics, a system described by conservative forces |
192 |
< |
conserves the total energy \ref{introEquation:energyConservation}. |
193 |
< |
It follows that Hamilton's equations of motion conserve the total |
194 |
< |
Hamiltonian. |
192 |
> |
conserves the total energy |
193 |
> |
(Eq.~\ref{introEquation:energyConservation}). It follows that |
194 |
> |
Hamilton's equations of motion conserve the total Hamiltonian |
195 |
|
\begin{equation} |
196 |
|
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
197 |
|
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
198 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
199 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
200 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
201 |
< |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
201 |
> |
q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian} |
202 |
|
\end{equation} |
203 |
|
|
204 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
219 |
|
momentum variables. Consider a dynamic system of $f$ particles in a |
220 |
|
cartesian space, where each of the $6f$ coordinates and momenta is |
221 |
|
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
222 |
< |
this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots |
223 |
< |
,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$ |
224 |
< |
coordinates and momenta is a phase space vector. |
225 |
< |
|
222 |
> |
this system is a $6f$ dimensional space. A point, $x = |
223 |
> |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
224 |
> |
\over q} _1 , \ldots |
225 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
226 |
> |
\over q} _f |
227 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
228 |
> |
\over p} _1 \ldots |
229 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
230 |
> |
\over p} _f )$ , with a unique set of values of $6f$ coordinates and |
231 |
> |
momenta is a phase space vector. |
232 |
|
%%%fix me |
233 |
< |
A microscopic state or microstate of a classical system is |
234 |
< |
specification of the complete phase space vector of a system at any |
237 |
< |
instant in time. An ensemble is defined as a collection of systems |
238 |
< |
sharing one or more macroscopic characteristics but each being in a |
239 |
< |
unique microstate. The complete ensemble is specified by giving all |
240 |
< |
systems or microstates consistent with the common macroscopic |
241 |
< |
characteristics of the ensemble. Although the state of each |
242 |
< |
individual system in the ensemble could be precisely described at |
243 |
< |
any instance in time by a suitable phase space vector, when using |
244 |
< |
ensembles for statistical purposes, there is no need to maintain |
245 |
< |
distinctions between individual systems, since the numbers of |
246 |
< |
systems at any time in the different states which correspond to |
247 |
< |
different regions of the phase space are more interesting. Moreover, |
248 |
< |
in the point of view of statistical mechanics, one would prefer to |
249 |
< |
use ensembles containing a large enough population of separate |
250 |
< |
members so that the numbers of systems in such different states can |
251 |
< |
be regarded as changing continuously as we traverse different |
252 |
< |
regions of the phase space. The condition of an ensemble at any time |
233 |
> |
|
234 |
> |
In statistical mechanics, the condition of an ensemble at any time |
235 |
|
can be regarded as appropriately specified by the density $\rho$ |
236 |
|
with which representative points are distributed over the phase |
237 |
|
space. The density distribution for an ensemble with $f$ degrees of |
243 |
|
Governed by the principles of mechanics, the phase points change |
244 |
|
their locations which would change the density at any time at phase |
245 |
|
space. Hence, the density distribution is also to be taken as a |
246 |
< |
function of the time. |
247 |
< |
|
266 |
< |
The number of systems $\delta N$ at time $t$ can be determined by, |
246 |
> |
function of the time. The number of systems $\delta N$ at time $t$ |
247 |
> |
can be determined by, |
248 |
|
\begin{equation} |
249 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
250 |
|
\label{introEquation:deltaN} |
277 |
|
\begin{equation} |
278 |
|
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
279 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
280 |
< |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
280 |
> |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
281 |
|
\label{introEquation:ensembelAverage} |
282 |
|
\end{equation} |
283 |
|
|
285 |
|
statistical characteristics. As a function of macroscopic |
286 |
|
parameters, such as temperature \textit{etc}, the partition function |
287 |
|
can be used to describe the statistical properties of a system in |
288 |
< |
thermodynamic equilibrium. |
289 |
< |
|
290 |
< |
As an ensemble of systems, each of which is known to be thermally |
291 |
< |
isolated and conserve energy, the Microcanonical ensemble (NVE) has |
292 |
< |
a partition function like, |
312 |
< |
\begin{equation} |
313 |
< |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
288 |
> |
thermodynamic equilibrium. As an ensemble of systems, each of which |
289 |
> |
is known to be thermally isolated and conserve energy, the |
290 |
> |
Microcanonical ensemble (NVE) has a partition function like, |
291 |
> |
\begin{equation} |
292 |
> |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
293 |
|
\end{equation} |
294 |
< |
A canonical ensemble (NVT)is an ensemble of systems, each of which |
294 |
> |
A canonical ensemble (NVT) is an ensemble of systems, each of which |
295 |
|
can share its energy with a large heat reservoir. The distribution |
296 |
|
of the total energy amongst the possible dynamical states is given |
297 |
|
by the partition function, |
298 |
|
\begin{equation} |
299 |
< |
\Omega (N,V,T) = e^{ - \beta A} |
299 |
> |
\Omega (N,V,T) = e^{ - \beta A}. |
300 |
|
\label{introEquation:NVTPartition} |
301 |
|
\end{equation} |
302 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
353 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
354 |
|
\label{introEquation:liouvilleTheorem} |
355 |
|
\end{equation} |
377 |
– |
|
356 |
|
Liouville's theorem states that the distribution function is |
357 |
|
constant along any trajectory in phase space. In classical |
358 |
|
statistical mechanics, since the number of members in an ensemble is |
409 |
|
q_i }}} \right)}. |
410 |
|
\label{introEquation:poissonBracket} |
411 |
|
\end{equation} |
412 |
< |
Substituting equations of motion in Hamiltonian formalism( |
413 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
414 |
< |
Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into |
412 |
> |
Substituting equations of motion in Hamiltonian formalism |
413 |
> |
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
414 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into |
415 |
|
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
416 |
|
Liouville's theorem using Poisson bracket notion, |
417 |
|
\begin{equation} |
445 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
446 |
|
Hypothesis makes a connection between time average and the ensemble |
447 |
|
average. It states that the time average and average over the |
448 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
448 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
449 |
|
\begin{equation} |
450 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
451 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
459 |
|
a properly weighted statistical average. This allows the researcher |
460 |
|
freedom of choice when deciding how best to measure a given |
461 |
|
observable. In case an ensemble averaged approach sounds most |
462 |
< |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
462 |
> |
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
463 |
|
utilized. Or if the system lends itself to a time averaging |
464 |
|
approach, the Molecular Dynamics techniques in |
465 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
474 |
|
geometric integrators, which preserve various phase-flow invariants |
475 |
|
such as symplectic structure, volume and time reversal symmetry, are |
476 |
|
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
477 |
< |
Leimkuhler1999}. The velocity verlet method, which happens to be a |
477 |
> |
Leimkuhler1999}. The velocity Verlet method, which happens to be a |
478 |
|
simple example of symplectic integrator, continues to gain |
479 |
|
popularity in the molecular dynamics community. This fact can be |
480 |
|
partly explained by its geometric nature. |
494 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
495 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
496 |
|
$\omega(x, x) = 0$. The cross product operation in vector field is |
497 |
< |
an example of symplectic form. |
498 |
< |
|
499 |
< |
One of the motivations to study \emph{symplectic manifolds} in |
500 |
< |
Hamiltonian Mechanics is that a symplectic manifold can represent |
501 |
< |
all possible configurations of the system and the phase space of the |
502 |
< |
system can be described by it's cotangent bundle. Every symplectic |
503 |
< |
manifold is even dimensional. For instance, in Hamilton equations, |
526 |
< |
coordinate and momentum always appear in pairs. |
497 |
> |
an example of symplectic form. One of the motivations to study |
498 |
> |
\emph{symplectic manifolds} in Hamiltonian Mechanics is that a |
499 |
> |
symplectic manifold can represent all possible configurations of the |
500 |
> |
system and the phase space of the system can be described by it's |
501 |
> |
cotangent bundle. Every symplectic manifold is even dimensional. For |
502 |
> |
instance, in Hamilton equations, coordinate and momentum always |
503 |
> |
appear in pairs. |
504 |
|
|
505 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
506 |
|
|
509 |
|
\dot x = f(x) |
510 |
|
\end{equation} |
511 |
|
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
512 |
+ |
$f(r) = J\nabla _x H(r)$. Here, $H = H (q, p)$ is Hamiltonian |
513 |
+ |
function and $J$ is the skew-symmetric matrix |
514 |
|
\begin{equation} |
536 |
– |
f(r) = J\nabla _x H(r). |
537 |
– |
\end{equation} |
538 |
– |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
539 |
– |
matrix |
540 |
– |
\begin{equation} |
515 |
|
J = \left( {\begin{array}{*{20}c} |
516 |
|
0 & I \\ |
517 |
|
{ - I} & 0 \\ |
521 |
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
522 |
|
system can be rewritten as, |
523 |
|
\begin{equation} |
524 |
< |
\frac{d}{{dt}}x = J\nabla _x H(x) |
524 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x). |
525 |
|
\label{introEquation:compactHamiltonian} |
526 |
|
\end{equation}In this case, $f$ is |
527 |
< |
called a \emph{Hamiltonian vector field}. |
528 |
< |
|
555 |
< |
Another generalization of Hamiltonian dynamics is Poisson |
556 |
< |
Dynamics\cite{Olver1986}, |
527 |
> |
called a \emph{Hamiltonian vector field}. Another generalization of |
528 |
> |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
529 |
|
\begin{equation} |
530 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
531 |
|
\end{equation} |
533 |
|
|
534 |
|
\subsection{\label{introSection:exactFlow}Exact Flow} |
535 |
|
|
536 |
< |
Let $x(t)$ be the exact solution of the ODE system, |
537 |
< |
\begin{equation} |
538 |
< |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
539 |
< |
\end{equation} |
540 |
< |
The exact flow(solution) $\varphi_\tau$ is defined by |
569 |
< |
\[ |
570 |
< |
x(t+\tau) =\varphi_\tau(x(t)) |
536 |
> |
Let $x(t)$ be the exact solution of the ODE |
537 |
> |
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
538 |
> |
define its exact flow(solution) $\varphi_\tau$ |
539 |
> |
\[ x(t+\tau) |
540 |
> |
=\varphi_\tau(x(t)) |
541 |
|
\] |
542 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
543 |
|
space to itself. The flow has the continuous group property, |
559 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
560 |
|
\label{introEquation:exponentialOperator} |
561 |
|
\end{equation} |
592 |
– |
|
562 |
|
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
563 |
< |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
563 |
> |
Instead, we use an approximate map, $\psi_\tau$, which is usually |
564 |
|
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
565 |
|
the Taylor series of $\psi_\tau$ agree to order $p$, |
566 |
|
\begin{equation} |
567 |
< |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
567 |
> |
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
568 |
|
\end{equation} |
569 |
|
|
570 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
571 |
|
|
572 |
< |
The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE |
573 |
< |
and its flow play important roles in numerical studies. Many of them |
574 |
< |
can be found in systems which occur naturally in applications. |
606 |
< |
|
572 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
573 |
> |
ODE and its flow play important roles in numerical studies. Many of |
574 |
> |
them can be found in systems which occur naturally in applications. |
575 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
576 |
|
a \emph{symplectic} flow if it satisfies, |
577 |
|
\begin{equation} |
585 |
|
\begin{equation} |
586 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
587 |
|
\end{equation} |
588 |
< |
is the property must be preserved by the integrator. |
589 |
< |
|
590 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
591 |
< |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
592 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
593 |
< |
be volume-preserving. |
626 |
< |
|
627 |
< |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
628 |
< |
will result in a new system, |
588 |
> |
is the property that must be preserved by the integrator. It is |
589 |
> |
possible to construct a \emph{volume-preserving} flow for a source |
590 |
> |
free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det |
591 |
> |
d\varphi = 1$. One can show easily that a symplectic flow will be |
592 |
> |
volume-preserving. Changing the variables $y = h(x)$ in an ODE |
593 |
> |
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
594 |
|
\[ |
595 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
596 |
|
\] |
597 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
598 |
|
In other words, the flow of this vector field is reversible if and |
599 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
600 |
< |
|
636 |
< |
A \emph{first integral}, or conserved quantity of a general |
599 |
> |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
600 |
> |
\emph{first integral}, or conserved quantity of a general |
601 |
|
differential function is a function $ G:R^{2d} \to R^d $ which is |
602 |
|
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
603 |
|
\[ |
605 |
|
\] |
606 |
|
Using chain rule, one may obtain, |
607 |
|
\[ |
608 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
608 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
609 |
|
\] |
610 |
|
which is the condition for conserving \emph{first integral}. For a |
611 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
612 |
|
smooth function $G$ is given by, |
649 |
– |
|
613 |
|
\begin{eqnarray} |
614 |
< |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
615 |
< |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
614 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
615 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
616 |
|
\label{introEquation:firstIntegral1} |
617 |
|
\end{eqnarray} |
618 |
< |
|
619 |
< |
|
657 |
< |
Using poisson bracket notion, Equation |
658 |
< |
\ref{introEquation:firstIntegral1} can be rewritten as |
618 |
> |
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1} |
619 |
> |
can be rewritten as |
620 |
|
\[ |
621 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
622 |
|
\] |
623 |
|
Therefore, the sufficient condition for $G$ to be the \emph{first |
624 |
< |
integral} of a Hamiltonian system is |
664 |
< |
\[ |
665 |
< |
\left\{ {G,H} \right\} = 0. |
666 |
< |
\] |
624 |
> |
integral} of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ |
625 |
|
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
626 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
627 |
< |
0$. |
670 |
< |
|
671 |
< |
When designing any numerical methods, one should always try to |
627 |
> |
0$. When designing any numerical methods, one should always try to |
628 |
|
preserve the structural properties of the original ODE and its flow. |
629 |
|
|
630 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
631 |
|
A lot of well established and very effective numerical methods have |
632 |
|
been successful precisely because of their symplecticities even |
633 |
|
though this fact was not recognized when they were first |
634 |
< |
constructed. The most famous example is the Verlet-leapfrog methods |
634 |
> |
constructed. The most famous example is the Verlet-leapfrog method |
635 |
|
in molecular dynamics. In general, symplectic integrators can be |
636 |
|
constructed using one of four different methods. |
637 |
|
\begin{enumerate} |
640 |
|
\item Runge-Kutta methods |
641 |
|
\item Splitting methods |
642 |
|
\end{enumerate} |
687 |
– |
|
643 |
|
Generating function\cite{Channell1990} tends to lead to methods |
644 |
|
which are cumbersome and difficult to use. In dissipative systems, |
645 |
|
variational methods can capture the decay of energy |
646 |
|
accurately\cite{Kane2000}. Since their geometrically unstable nature |
647 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
648 |
|
methods are not suitable for Hamiltonian system. Recently, various |
649 |
< |
high-order explicit Runge-Kutta methods |
650 |
< |
\cite{Owren1992,Chen2003}have been developed to overcome this |
651 |
< |
instability. However, due to computational penalty involved in |
652 |
< |
implementing the Runge-Kutta methods, they have not attracted much |
653 |
< |
attention from the Molecular Dynamics community. Instead, splitting |
654 |
< |
methods have been widely accepted since they exploit natural |
655 |
< |
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
649 |
> |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
650 |
> |
have been developed to overcome this instability. However, due to |
651 |
> |
computational penalty involved in implementing the Runge-Kutta |
652 |
> |
methods, they have not attracted much attention from the Molecular |
653 |
> |
Dynamics community. Instead, splitting methods have been widely |
654 |
> |
accepted since they exploit natural decompositions of the |
655 |
> |
system\cite{Tuckerman1992, McLachlan1998}. |
656 |
|
|
657 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
658 |
|
|
664 |
|
\label{introEquation:FlowDecomposition} |
665 |
|
\end{equation} |
666 |
|
where each of the sub-flow is chosen such that each represent a |
667 |
< |
simpler integration of the system. |
668 |
< |
|
714 |
< |
Suppose that a Hamiltonian system takes the form, |
667 |
> |
simpler integration of the system. Suppose that a Hamiltonian system |
668 |
> |
takes the form, |
669 |
|
\[ |
670 |
|
H = H_1 + H_2. |
671 |
|
\] |
693 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
694 |
|
splitting in this context automatically generates a symplectic map. |
695 |
|
|
696 |
< |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
697 |
< |
introduces local errors proportional to $h^2$, while Strang |
698 |
< |
splitting gives a second-order decomposition, |
696 |
> |
The Lie-Trotter |
697 |
> |
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
698 |
> |
local errors proportional to $h^2$, while Strang splitting gives a |
699 |
> |
second-order decomposition, |
700 |
|
\begin{equation} |
701 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
702 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
707 |
|
\begin{equation} |
708 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
709 |
|
\label{introEquation:timeReversible} |
710 |
< |
\end{equation},appendixFig:architecture |
710 |
> |
\end{equation} |
711 |
|
|
712 |
< |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Example of Splitting Method}} |
712 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
713 |
|
The classical equation for a system consisting of interacting |
714 |
|
particles can be written in Hamiltonian form, |
715 |
|
\[ |
716 |
|
H = T + V |
717 |
|
\] |
718 |
|
where $T$ is the kinetic energy and $V$ is the potential energy. |
719 |
< |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
719 |
> |
Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one |
720 |
|
obtains the following: |
721 |
|
\begin{align} |
722 |
|
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
743 |
|
\label{introEquation:Lp9b}\\% |
744 |
|
% |
745 |
|
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
746 |
< |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
746 |
> |
\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c} |
747 |
|
\end{align} |
748 |
|
From the preceding splitting, one can see that the integration of |
749 |
|
the equations of motion would follow: |
752 |
|
|
753 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
754 |
|
|
755 |
< |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
755 |
> |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
756 |
|
|
757 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
758 |
|
\end{enumerate} |
759 |
< |
|
760 |
< |
Simply switching the order of splitting and composing, a new |
761 |
< |
integrator, the \emph{position verlet} integrator, can be generated, |
759 |
> |
By simply switching the order of the propagators in the splitting |
760 |
> |
and composing a new integrator, the \emph{position verlet} |
761 |
> |
integrator, can be generated, |
762 |
|
\begin{align} |
763 |
|
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
764 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
771 |
|
|
772 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
773 |
|
|
774 |
< |
Baker-Campbell-Hausdorff formula can be used to determine the local |
775 |
< |
error of splitting method in terms of commutator of the |
774 |
> |
The Baker-Campbell-Hausdorff formula can be used to determine the |
775 |
> |
local error of splitting method in terms of the commutator of the |
776 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
777 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
777 |
> |
the sub-flow. For operators $hX$ and $hY$ which are associated with |
778 |
|
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
779 |
|
\begin{equation} |
780 |
|
\exp (hX + hY) = \exp (hZ) |
788 |
|
\[ |
789 |
|
[X,Y] = XY - YX . |
790 |
|
\] |
791 |
< |
Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to |
792 |
< |
Sprang splitting, we can obtain |
791 |
> |
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
792 |
> |
to the Sprang splitting, we can obtain |
793 |
|
\begin{eqnarray*} |
794 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
795 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
796 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
796 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots |
797 |
> |
). |
798 |
|
\end{eqnarray*} |
799 |
< |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
799 |
> |
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
800 |
|
error of Spring splitting is proportional to $h^3$. The same |
801 |
< |
procedure can be applied to general splitting, of the form |
801 |
> |
procedure can be applied to a general splitting of the form |
802 |
|
\begin{equation} |
803 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
804 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
805 |
|
\end{equation} |
806 |
< |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
807 |
< |
order method. Yoshida proposed an elegant way to compose higher |
806 |
> |
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
807 |
> |
order methods. Yoshida proposed an elegant way to compose higher |
808 |
|
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
809 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
810 |
|
fourth-order symmetric method can be constructed by composing, |
817 |
|
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
818 |
|
\begin{equation} |
819 |
|
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
820 |
< |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
820 |
> |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)}, |
821 |
|
\end{equation} |
822 |
< |
, if the weights are chosen as |
822 |
> |
if the weights are chosen as |
823 |
|
\[ |
824 |
|
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
825 |
|
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
857 |
|
These three individual steps will be covered in the following |
858 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
859 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
860 |
< |
will discusses issues in production run. |
860 |
> |
will discusse issues in production run. |
861 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
862 |
|
trajectory analysis. |
863 |
|
|
870 |
|
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
871 |
|
thousands of crystal structures of molecules are discovered every |
872 |
|
year, many more remain unknown due to the difficulties of |
873 |
< |
purification and crystallization. Even for the molecule with known |
874 |
< |
structure, some important information is missing. For example, the |
873 |
> |
purification and crystallization. Even for molecules with known |
874 |
> |
structure, some important information is missing. For example, a |
875 |
|
missing hydrogen atom which acts as donor in hydrogen bonding must |
876 |
|
be added. Moreover, in order to include electrostatic interaction, |
877 |
|
one may need to specify the partial charges for individual atoms. |
878 |
|
Under some circumstances, we may even need to prepare the system in |
879 |
< |
a special setup. For instance, when studying transport phenomenon in |
880 |
< |
membrane system, we may prepare the lipids in bilayer structure |
881 |
< |
instead of placing lipids randomly in solvent, since we are not |
882 |
< |
interested in self-aggregation and it takes a long time to happen. |
879 |
> |
a special configuration. For instance, when studying transport |
880 |
> |
phenomenon in membrane systems, we may prepare the lipids in a |
881 |
> |
bilayer structure instead of placing lipids randomly in solvent, |
882 |
> |
since we are not interested in the slow self-aggregation process. |
883 |
|
|
884 |
|
\subsubsection{\textbf{Minimization}} |
885 |
|
|
886 |
|
It is quite possible that some of molecules in the system from |
887 |
< |
preliminary preparation may be overlapped with each other. This |
888 |
< |
close proximity leads to high potential energy which consequently |
889 |
< |
jeopardizes any molecular dynamics simulations. To remove these |
890 |
< |
steric overlaps, one typically performs energy minimization to find |
891 |
< |
a more reasonable conformation. Several energy minimization methods |
892 |
< |
have been developed to exploit the energy surface and to locate the |
893 |
< |
local minimum. While converging slowly near the minimum, steepest |
894 |
< |
descent method is extremely robust when systems are far from |
895 |
< |
harmonic. Thus, it is often used to refine structure from |
896 |
< |
crystallographic data. Relied on the gradient or hessian, advanced |
897 |
< |
methods like conjugate gradient and Newton-Raphson converge rapidly |
898 |
< |
to a local minimum, while become unstable if the energy surface is |
899 |
< |
far from quadratic. Another factor must be taken into account, when |
887 |
> |
preliminary preparation may be overlapping with each other. This |
888 |
> |
close proximity leads to high initial potential energy which |
889 |
> |
consequently jeopardizes any molecular dynamics simulations. To |
890 |
> |
remove these steric overlaps, one typically performs energy |
891 |
> |
minimization to find a more reasonable conformation. Several energy |
892 |
> |
minimization methods have been developed to exploit the energy |
893 |
> |
surface and to locate the local minimum. While converging slowly |
894 |
> |
near the minimum, steepest descent method is extremely robust when |
895 |
> |
systems are strongly anharmonic. Thus, it is often used to refine |
896 |
> |
structure from crystallographic data. Relied on the gradient or |
897 |
> |
hessian, advanced methods like Newton-Raphson converge rapidly to a |
898 |
> |
local minimum, but become unstable if the energy surface is far from |
899 |
> |
quadratic. Another factor that must be taken into account, when |
900 |
|
choosing energy minimization method, is the size of the system. |
901 |
|
Steepest descent and conjugate gradient can deal with models of any |
902 |
< |
size. Because of the limit of computation power to calculate hessian |
903 |
< |
matrix and insufficient storage capacity to store them, most |
904 |
< |
Newton-Raphson methods can not be used with very large models. |
902 |
> |
size. Because of the limits on computer memory to store the hessian |
903 |
> |
matrix and the computing power needed to diagonalized these |
904 |
> |
matrices, most Newton-Raphson methods can not be used with very |
905 |
> |
large systems. |
906 |
|
|
907 |
|
\subsubsection{\textbf{Heating}} |
908 |
|
|
909 |
|
Typically, Heating is performed by assigning random velocities |
910 |
< |
according to a Gaussian distribution for a temperature. Beginning at |
911 |
< |
a lower temperature and gradually increasing the temperature by |
912 |
< |
assigning greater random velocities, we end up with setting the |
913 |
< |
temperature of the system to a final temperature at which the |
914 |
< |
simulation will be conducted. In heating phase, we should also keep |
915 |
< |
the system from drifting or rotating as a whole. Equivalently, the |
916 |
< |
net linear momentum and angular momentum of the system should be |
917 |
< |
shifted to zero. |
910 |
> |
according to a Maxwell-Boltzman distribution for a desired |
911 |
> |
temperature. Beginning at a lower temperature and gradually |
912 |
> |
increasing the temperature by assigning larger random velocities, we |
913 |
> |
end up with setting the temperature of the system to a final |
914 |
> |
temperature at which the simulation will be conducted. In heating |
915 |
> |
phase, we should also keep the system from drifting or rotating as a |
916 |
> |
whole. To do this, the net linear momentum and angular momentum of |
917 |
> |
the system is shifted to zero after each resampling from the Maxwell |
918 |
> |
-Boltzman distribution. |
919 |
|
|
920 |
|
\subsubsection{\textbf{Equilibration}} |
921 |
|
|
931 |
|
|
932 |
|
\subsection{\label{introSection:production}Production} |
933 |
|
|
934 |
< |
Production run is the most important step of the simulation, in |
934 |
> |
The production run is the most important step of the simulation, in |
935 |
|
which the equilibrated structure is used as a starting point and the |
936 |
|
motions of the molecules are collected for later analysis. In order |
937 |
|
to capture the macroscopic properties of the system, the molecular |
938 |
< |
dynamics simulation must be performed in correct and efficient way. |
938 |
> |
dynamics simulation must be performed by sampling correctly and |
939 |
> |
efficiently from the relevant thermodynamic ensemble. |
940 |
|
|
941 |
|
The most expensive part of a molecular dynamics simulation is the |
942 |
|
calculation of non-bonded forces, such as van der Waals force and |
943 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
944 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
945 |
|
which making large simulations prohibitive in the absence of any |
946 |
< |
computation saving techniques. |
947 |
< |
|
948 |
< |
A natural approach to avoid system size issue is to represent the |
949 |
< |
bulk behavior by a finite number of the particles. However, this |
950 |
< |
approach will suffer from the surface effect. To offset this, |
992 |
< |
\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc}) |
946 |
> |
algorithmic tricks. A natural approach to avoid system size issues |
947 |
> |
is to represent the bulk behavior by a finite number of the |
948 |
> |
particles. However, this approach will suffer from the surface |
949 |
> |
effect at the edges of the simulation. To offset this, |
950 |
> |
\textit{Periodic boundary conditions} (see Fig.~\ref{introFig:pbc}) |
951 |
|
is developed to simulate bulk properties with a relatively small |
952 |
|
number of particles. In this method, the simulation box is |
953 |
|
replicated throughout space to form an infinite lattice. During the |
966 |
|
|
967 |
|
%cutoff and minimum image convention |
968 |
|
Another important technique to improve the efficiency of force |
969 |
< |
evaluation is to apply cutoff where particles farther than a |
970 |
< |
predetermined distance, are not included in the calculation |
969 |
> |
evaluation is to apply spherical cutoff where particles farther than |
970 |
> |
a predetermined distance are not included in the calculation |
971 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
972 |
|
discontinuity in the potential energy curve. Fortunately, one can |
973 |
< |
shift the potential to ensure the potential curve go smoothly to |
974 |
< |
zero at the cutoff radius. Cutoff strategy works pretty well for |
975 |
< |
Lennard-Jones interaction because of its short range nature. |
976 |
< |
However, simply truncating the electrostatic interaction with the |
977 |
< |
use of cutoff has been shown to lead to severe artifacts in |
978 |
< |
simulations. Ewald summation, in which the slowly conditionally |
979 |
< |
convergent Coulomb potential is transformed into direct and |
980 |
< |
reciprocal sums with rapid and absolute convergence, has proved to |
981 |
< |
minimize the periodicity artifacts in liquid simulations. Taking the |
982 |
< |
advantages of the fast Fourier transform (FFT) for calculating |
983 |
< |
discrete Fourier transforms, the particle mesh-based |
973 |
> |
shift simple radial potential to ensure the potential curve go |
974 |
> |
smoothly to zero at the cutoff radius. The cutoff strategy works |
975 |
> |
well for Lennard-Jones interaction because of its short range |
976 |
> |
nature. However, simply truncating the electrostatic interaction |
977 |
> |
with the use of cutoffs has been shown to lead to severe artifacts |
978 |
> |
in simulations. The Ewald summation, in which the slowly decaying |
979 |
> |
Coulomb potential is transformed into direct and reciprocal sums |
980 |
> |
with rapid and absolute convergence, has proved to minimize the |
981 |
> |
periodicity artifacts in liquid simulations. Taking the advantages |
982 |
> |
of the fast Fourier transform (FFT) for calculating discrete Fourier |
983 |
> |
transforms, the particle mesh-based |
984 |
|
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
985 |
< |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast |
986 |
< |
multipole method}\cite{Greengard1987, Greengard1994}, which treats |
987 |
< |
Coulombic interaction exactly at short range, and approximate the |
988 |
< |
potential at long range through multipolar expansion. In spite of |
989 |
< |
their wide acceptances at the molecular simulation community, these |
990 |
< |
two methods are hard to be implemented correctly and efficiently. |
991 |
< |
Instead, we use a damped and charge-neutralized Coulomb potential |
992 |
< |
method developed by Wolf and his coworkers\cite{Wolf1999}. The |
993 |
< |
shifted Coulomb potential for particle $i$ and particle $j$ at |
994 |
< |
distance $r_{rj}$ is given by: |
985 |
> |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
986 |
> |
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
987 |
> |
which treats Coulombic interactions exactly at short range, and |
988 |
> |
approximate the potential at long range through multipolar |
989 |
> |
expansion. In spite of their wide acceptance at the molecular |
990 |
> |
simulation community, these two methods are difficult to implement |
991 |
> |
correctly and efficiently. Instead, we use a damped and |
992 |
> |
charge-neutralized Coulomb potential method developed by Wolf and |
993 |
> |
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
994 |
> |
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
995 |
|
\begin{equation} |
996 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
997 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1013 |
|
|
1014 |
|
\subsection{\label{introSection:Analysis} Analysis} |
1015 |
|
|
1016 |
< |
Recently, advanced visualization technique are widely applied to |
1016 |
> |
Recently, advanced visualization technique have become applied to |
1017 |
|
monitor the motions of molecules. Although the dynamics of the |
1018 |
|
system can be described qualitatively from animation, quantitative |
1019 |
< |
trajectory analysis are more appreciable. According to the |
1020 |
< |
principles of Statistical Mechanics, |
1019 |
> |
trajectory analysis are more useful. According to the principles of |
1020 |
> |
Statistical Mechanics in |
1021 |
|
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1022 |
< |
thermodynamics properties, analyze fluctuations of structural |
1022 |
> |
thermodynamic properties, analyze fluctuations of structural |
1023 |
|
parameters, and investigate time-dependent processes of the molecule |
1024 |
|
from the trajectories. |
1025 |
|
|
1026 |
< |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamics Properties}} |
1026 |
> |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
1027 |
|
|
1028 |
< |
Thermodynamics properties, which can be expressed in terms of some |
1028 |
> |
Thermodynamic properties, which can be expressed in terms of some |
1029 |
|
function of the coordinates and momenta of all particles in the |
1030 |
|
system, can be directly computed from molecular dynamics. The usual |
1031 |
|
way to measure the pressure is based on virial theorem of Clausius |
1048 |
|
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
1049 |
|
|
1050 |
|
Structural Properties of a simple fluid can be described by a set of |
1051 |
< |
distribution functions. Among these functions,\emph{pair |
1051 |
> |
distribution functions. Among these functions,the \emph{pair |
1052 |
|
distribution function}, also known as \emph{radial distribution |
1053 |
< |
function}, is of most fundamental importance to liquid-state theory. |
1054 |
< |
Pair distribution function can be gathered by Fourier transforming |
1055 |
< |
raw data from a series of neutron diffraction experiments and |
1056 |
< |
integrating over the surface factor \cite{Powles1973}. The |
1057 |
< |
experiment result can serve as a criterion to justify the |
1058 |
< |
correctness of the theory. Moreover, various equilibrium |
1059 |
< |
thermodynamic and structural properties can also be expressed in |
1060 |
< |
terms of radial distribution function \cite{Allen1987}. |
1061 |
< |
|
1104 |
< |
A pair distribution functions $g(r)$ gives the probability that a |
1053 |
> |
function}, is of most fundamental importance to liquid theory. |
1054 |
> |
Experimentally, pair distribution function can be gathered by |
1055 |
> |
Fourier transforming raw data from a series of neutron diffraction |
1056 |
> |
experiments and integrating over the surface factor |
1057 |
> |
\cite{Powles1973}. The experimental results can serve as a criterion |
1058 |
> |
to justify the correctness of a liquid model. Moreover, various |
1059 |
> |
equilibrium thermodynamic and structural properties can also be |
1060 |
> |
expressed in terms of radial distribution function \cite{Allen1987}. |
1061 |
> |
The pair distribution functions $g(r)$ gives the probability that a |
1062 |
|
particle $i$ will be located at a distance $r$ from a another |
1063 |
|
particle $j$ in the system |
1064 |
< |
\[ |
1064 |
> |
\begin{equation} |
1065 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1066 |
< |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1067 |
< |
\] |
1066 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
1067 |
> |
(r)}{\rho}. |
1068 |
> |
\end{equation} |
1069 |
|
Note that the delta function can be replaced by a histogram in |
1070 |
< |
computer simulation. Figure |
1071 |
< |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1072 |
< |
distribution function for the liquid argon system. The occurrence of |
1115 |
< |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1116 |
< |
to find particles at certain radial values than at others. This is a |
1117 |
< |
result of the attractive interaction at such distances. Because of |
1118 |
< |
the strong repulsive forces at short distance, the probability of |
1119 |
< |
locating particles at distances less than about 2.5{\AA} from each |
1120 |
< |
other is essentially zero. |
1070 |
> |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
1071 |
> |
the height of these peaks gradually decreases to 1 as the liquid of |
1072 |
> |
large distance approaches the bulk density. |
1073 |
|
|
1122 |
– |
%\begin{figure} |
1123 |
– |
%\centering |
1124 |
– |
%\includegraphics[width=\linewidth]{pdf.eps} |
1125 |
– |
%\caption[Pair distribution function for the liquid argon |
1126 |
– |
%]{Pair distribution function for the liquid argon} |
1127 |
– |
%\label{introFigure:pairDistributionFunction} |
1128 |
– |
%\end{figure} |
1074 |
|
|
1075 |
|
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
1076 |
|
Properties}} |
1077 |
|
|
1078 |
|
Time-dependent properties are usually calculated using \emph{time |
1079 |
< |
correlation function}, which correlates random variables $A$ and $B$ |
1080 |
< |
at two different time |
1079 |
> |
correlation functions}, which correlate random variables $A$ and $B$ |
1080 |
> |
at two different times, |
1081 |
|
\begin{equation} |
1082 |
|
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1083 |
|
\label{introEquation:timeCorrelationFunction} |
1084 |
|
\end{equation} |
1085 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
1086 |
< |
function is called \emph{auto correlation function}. One example of |
1087 |
< |
auto correlation function is velocity auto-correlation function |
1088 |
< |
which is directly related to transport properties of molecular |
1089 |
< |
liquids: |
1086 |
> |
function is called an \emph{autocorrelation function}. One example |
1087 |
> |
of an auto correlation function is the velocity auto-correlation |
1088 |
> |
function which is directly related to transport properties of |
1089 |
> |
molecular liquids: |
1090 |
|
\[ |
1091 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1092 |
|
\right\rangle } dt |
1093 |
|
\] |
1094 |
< |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1095 |
< |
function which is averaging over time origins and over all the |
1096 |
< |
atoms, dipole autocorrelation are calculated for the entire system. |
1097 |
< |
The dipole autocorrelation function is given by: |
1094 |
> |
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
1095 |
> |
function, which is averaging over time origins and over all the |
1096 |
> |
atoms, the dipole autocorrelation functions are calculated for the |
1097 |
> |
entire system. The dipole autocorrelation function is given by: |
1098 |
|
\[ |
1099 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1100 |
|
\right\rangle |
1102 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
1103 |
|
by |
1104 |
|
\[ |
1105 |
< |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1105 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
1106 |
|
\] |
1107 |
|
In principle, many time correlation functions can be related with |
1108 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
1111 |
|
each frequency using the following relationship: |
1112 |
|
\[ |
1113 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1114 |
< |
i2\pi vt} dt} |
1114 |
> |
i2\pi vt} dt}. |
1115 |
|
\] |
1116 |
|
|
1117 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1120 |
|
areas, from engineering, physics, to chemistry. For example, |
1121 |
|
missiles and vehicle are usually modeled by rigid bodies. The |
1122 |
|
movement of the objects in 3D gaming engine or other physics |
1123 |
< |
simulator is governed by the rigid body dynamics. In molecular |
1124 |
< |
simulation, rigid body is used to simplify the model in |
1125 |
< |
protein-protein docking study\cite{Gray2003}. |
1123 |
> |
simulator is governed by rigid body dynamics. In molecular |
1124 |
> |
simulations, rigid bodies are used to simplify protein-protein |
1125 |
> |
docking studies\cite{Gray2003}. |
1126 |
|
|
1127 |
|
It is very important to develop stable and efficient methods to |
1128 |
< |
integrate the equations of motion of orientational degrees of |
1129 |
< |
freedom. Euler angles are the nature choice to describe the |
1130 |
< |
rotational degrees of freedom. However, due to its singularity, the |
1131 |
< |
numerical integration of corresponding equations of motion is very |
1132 |
< |
inefficient and inaccurate. Although an alternative integrator using |
1133 |
< |
different sets of Euler angles can overcome this |
1134 |
< |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
1135 |
< |
of angular momentum conservation still remain. A singularity free |
1136 |
< |
representation utilizing quaternions was developed by Evans in |
1137 |
< |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
1138 |
< |
nonseparable Hamiltonian resulted from quaternion representation, |
1139 |
< |
which prevents the symplectic algorithm to be utilized. Another |
1140 |
< |
different approach is to apply holonomic constraints to the atoms |
1141 |
< |
belonging to the rigid body. Each atom moves independently under the |
1142 |
< |
normal forces deriving from potential energy and constraint forces |
1143 |
< |
which are used to guarantee the rigidness. However, due to their |
1144 |
< |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
1145 |
< |
when the number of constraint increases\cite{Ryckaert1977, |
1146 |
< |
Andersen1983}. |
1128 |
> |
integrate the equations of motion for orientational degrees of |
1129 |
> |
freedom. Euler angles are the natural choice to describe the |
1130 |
> |
rotational degrees of freedom. However, due to $\frac {1}{sin |
1131 |
> |
\theta}$ singularities, the numerical integration of corresponding |
1132 |
> |
equations of motion is very inefficient and inaccurate. Although an |
1133 |
> |
alternative integrator using multiple sets of Euler angles can |
1134 |
> |
overcome this difficulty\cite{Barojas1973}, the computational |
1135 |
> |
penalty and the loss of angular momentum conservation still remain. |
1136 |
> |
A singularity-free representation utilizing quaternions was |
1137 |
> |
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
1138 |
> |
approach uses a nonseparable Hamiltonian resulting from the |
1139 |
> |
quaternion representation, which prevents the symplectic algorithm |
1140 |
> |
to be utilized. Another different approach is to apply holonomic |
1141 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
1142 |
> |
moves independently under the normal forces deriving from potential |
1143 |
> |
energy and constraint forces which are used to guarantee the |
1144 |
> |
rigidness. However, due to their iterative nature, the SHAKE and |
1145 |
> |
Rattle algorithms also converge very slowly when the number of |
1146 |
> |
constraints increases\cite{Ryckaert1977, Andersen1983}. |
1147 |
|
|
1148 |
< |
The break through in geometric literature suggests that, in order to |
1148 |
> |
A break-through in geometric literature suggests that, in order to |
1149 |
|
develop a long-term integration scheme, one should preserve the |
1150 |
< |
symplectic structure of the flow. Introducing conjugate momentum to |
1151 |
< |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
1152 |
< |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
1153 |
< |
the Hamiltonian system in a constraint manifold by iteratively |
1154 |
< |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
1155 |
< |
method using quaternion representation was developed by |
1156 |
< |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
1157 |
< |
iterative and inefficient. In this section, we will present a |
1150 |
> |
symplectic structure of the flow. By introducing a conjugate |
1151 |
> |
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
1152 |
> |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
1153 |
> |
proposed to evolve the Hamiltonian system in a constraint manifold |
1154 |
> |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
1155 |
> |
An alternative method using the quaternion representation was |
1156 |
> |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
1157 |
> |
methods are iterative and inefficient. In this section, we descibe a |
1158 |
|
symplectic Lie-Poisson integrator for rigid body developed by |
1159 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1160 |
|
|
1161 |
< |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
1162 |
< |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
1161 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
1162 |
> |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
1163 |
|
function |
1164 |
|
\begin{equation} |
1165 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
1173 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
1174 |
|
\] |
1175 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
1176 |
< |
constrained Hamiltonian equation subjects to a holonomic constraint, |
1176 |
> |
constrained Hamiltonian equation is subjected to a holonomic |
1177 |
> |
constraint, |
1178 |
|
\begin{equation} |
1179 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1180 |
|
\end{equation} |
1181 |
< |
which is used to ensure rotation matrix's orthogonality. |
1182 |
< |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
1183 |
< |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
1181 |
> |
which is used to ensure rotation matrix's unitarity. Differentiating |
1182 |
> |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
1183 |
> |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
1184 |
|
\begin{equation} |
1185 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1186 |
|
\label{introEquation:RBFirstOrderConstraint} |
1187 |
|
\end{equation} |
1242 |
– |
|
1188 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1189 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1190 |
|
the equations of motion, |
1246 |
– |
|
1191 |
|
\begin{eqnarray} |
1192 |
< |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1193 |
< |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1194 |
< |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1192 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
1193 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\ |
1194 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
1195 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1196 |
|
\end{eqnarray} |
1253 |
– |
|
1197 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1198 |
< |
We can use constraint force provided by lagrange multiplier on the |
1199 |
< |
normal manifold to keep the motion on constraint space. Or we can |
1200 |
< |
simply evolve the system in constraint manifold. These two methods |
1201 |
< |
are proved to be equivalent. The holonomic constraint and equations |
1202 |
< |
of motions define a constraint manifold for rigid body |
1198 |
> |
We can use a constraint force provided by a Lagrange multiplier on |
1199 |
> |
the normal manifold to keep the motion on constraint space. Or we |
1200 |
> |
can simply evolve the system on the constraint manifold. These two |
1201 |
> |
methods have been proved to be equivalent. The holonomic constraint |
1202 |
> |
and equations of motions define a constraint manifold for rigid |
1203 |
> |
bodies |
1204 |
|
\[ |
1205 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1206 |
|
\right\}. |
1207 |
|
\] |
1264 |
– |
|
1208 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
1209 |
< |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
1209 |
> |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
1210 |
> |
rotation group $SO(3)$. However, it turns out that under symplectic |
1211 |
|
transformation, the cotangent space and the phase space are |
1212 |
< |
diffeomorphic. Introducing |
1212 |
> |
diffeomorphic. By introducing |
1213 |
|
\[ |
1214 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
1215 |
|
\] |
1219 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
1220 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
1221 |
|
\] |
1278 |
– |
|
1222 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
1223 |
|
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
1224 |
|
given as |
1237 |
|
\[ |
1238 |
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
1239 |
|
\] |
1240 |
< |
respectively. |
1241 |
< |
|
1242 |
< |
As a common choice to describe the rotation dynamics of the rigid |
1300 |
< |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
1301 |
< |
rewrite the equations of motion, |
1240 |
> |
respectively. As a common choice to describe the rotation dynamics |
1241 |
> |
of the rigid body, the angular momentum on the body fixed frame $\Pi |
1242 |
> |
= Q^t P$ is introduced to rewrite the equations of motion, |
1243 |
|
\begin{equation} |
1244 |
|
\begin{array}{l} |
1245 |
< |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1246 |
< |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
1245 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda, \\ |
1246 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1}, \\ |
1247 |
|
\end{array} |
1248 |
|
\label{introEqaution:RBMotionPI} |
1249 |
|
\end{equation} |
1250 |
< |
, as well as holonomic constraints, |
1251 |
< |
\[ |
1252 |
< |
\begin{array}{l} |
1312 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
1313 |
< |
Q^T Q = 1 \\ |
1314 |
< |
\end{array} |
1315 |
< |
\] |
1316 |
< |
|
1317 |
< |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1318 |
< |
so(3)^ \star$, the hat-map isomorphism, |
1250 |
> |
as well as holonomic constraints $\Pi J^{ - 1} + J^{ - 1} \Pi ^t = |
1251 |
> |
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a |
1252 |
> |
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism, |
1253 |
|
\begin{equation} |
1254 |
|
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1255 |
|
{\begin{array}{*{20}c} |
1262 |
|
will let us associate the matrix products with traditional vector |
1263 |
|
operations |
1264 |
|
\[ |
1265 |
< |
\hat vu = v \times u |
1265 |
> |
\hat vu = v \times u. |
1266 |
|
\] |
1267 |
< |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1267 |
> |
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
1268 |
|
matrix, |
1269 |
+ |
\begin{eqnarray} |
1270 |
+ |
(\dot \Pi - \dot \Pi ^T )&= &(\Pi - \Pi ^T )(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
1271 |
+ |
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1272 |
+ |
(\Lambda - \Lambda ^T ). \label{introEquation:skewMatrixPI} |
1273 |
+ |
\end{eqnarray} |
1274 |
+ |
Since $\Lambda$ is symmetric, the last term of |
1275 |
+ |
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
1276 |
+ |
Lagrange multiplier $\Lambda$ is absent from the equations of |
1277 |
+ |
motion. This unique property eliminates the requirement of |
1278 |
+ |
iterations which can not be avoided in other methods\cite{Kol1997, |
1279 |
+ |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
1280 |
+ |
equation of motion for angular momentum on body frame |
1281 |
|
\begin{equation} |
1336 |
– |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
1337 |
– |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
1338 |
– |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1339 |
– |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
1340 |
– |
\end{equation} |
1341 |
– |
Since $\Lambda$ is symmetric, the last term of Equation |
1342 |
– |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1343 |
– |
multiplier $\Lambda$ is absent from the equations of motion. This |
1344 |
– |
unique property eliminate the requirement of iterations which can |
1345 |
– |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
1346 |
– |
|
1347 |
– |
Applying hat-map isomorphism, we obtain the equation of motion for |
1348 |
– |
angular momentum on body frame |
1349 |
– |
\begin{equation} |
1282 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
1283 |
|
F_i (r,Q)} \right) \times X_i }. |
1284 |
|
\label{introEquation:bodyAngularMotion} |
1286 |
|
In the same manner, the equation of motion for rotation matrix is |
1287 |
|
given by |
1288 |
|
\[ |
1289 |
< |
\dot Q = Qskew(I^{ - 1} \pi ) |
1289 |
> |
\dot Q = Qskew(I^{ - 1} \pi ). |
1290 |
|
\] |
1291 |
|
|
1292 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
1293 |
|
Lie-Poisson Integrator for Free Rigid Body} |
1294 |
|
|
1295 |
< |
If there is not external forces exerted on the rigid body, the only |
1296 |
< |
contribution to the rotational is from the kinetic potential (the |
1297 |
< |
first term of \ref{introEquation:bodyAngularMotion}). The free rigid |
1298 |
< |
body is an example of Lie-Poisson system with Hamiltonian function |
1295 |
> |
If there are no external forces exerted on the rigid body, the only |
1296 |
> |
contribution to the rotational motion is from the kinetic energy |
1297 |
> |
(the first term of \ref{introEquation:bodyAngularMotion}). The free |
1298 |
> |
rigid body is an example of a Lie-Poisson system with Hamiltonian |
1299 |
> |
function |
1300 |
|
\begin{equation} |
1301 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
1302 |
|
\label{introEquation:rotationalKineticRB} |
1308 |
|
0 & {\pi _3 } & { - \pi _2 } \\ |
1309 |
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
1310 |
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
1311 |
< |
\end{array}} \right) |
1311 |
> |
\end{array}} \right). |
1312 |
|
\end{equation} |
1313 |
|
Thus, the dynamics of free rigid body is governed by |
1314 |
|
\begin{equation} |
1315 |
< |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
1315 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
1316 |
|
\end{equation} |
1317 |
< |
|
1318 |
< |
One may notice that each $T_i^r$ in Equation |
1319 |
< |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1387 |
< |
instance, the equations of motion due to $T_1^r$ are given by |
1317 |
> |
One may notice that each $T_i^r$ in |
1318 |
> |
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly. |
1319 |
> |
For instance, the equations of motion due to $T_1^r$ are given by |
1320 |
|
\begin{equation} |
1321 |
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
1322 |
|
\label{introEqaution:RBMotionSingleTerm} |
1323 |
|
\end{equation} |
1324 |
< |
where |
1324 |
> |
with |
1325 |
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
1326 |
|
0 & 0 & 0 \\ |
1327 |
|
0 & 0 & {\pi _1 } \\ |
1328 |
|
0 & { - \pi _1 } & 0 \\ |
1329 |
|
\end{array}} \right). |
1330 |
|
\] |
1331 |
< |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1331 |
> |
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is |
1332 |
|
\[ |
1333 |
|
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
1334 |
|
Q(0)e^{\Delta tR_1 } |
1342 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
1343 |
|
\] |
1344 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
1345 |
< |
tR_1 }$, we can use Cayley transformation, |
1345 |
> |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
1346 |
> |
propagator, |
1347 |
|
\[ |
1348 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1349 |
< |
) |
1349 |
> |
). |
1350 |
|
\] |
1351 |
|
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1352 |
< |
manner. |
1353 |
< |
|
1421 |
< |
In order to construct a second-order symplectic method, we split the |
1422 |
< |
angular kinetic Hamiltonian function can into five terms |
1352 |
> |
manner. In order to construct a second-order symplectic method, we |
1353 |
> |
split the angular kinetic Hamiltonian function into five terms |
1354 |
|
\[ |
1355 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1356 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1357 |
< |
(\pi _1 ) |
1358 |
< |
\]. |
1359 |
< |
Concatenating flows corresponding to these five terms, we can obtain |
1360 |
< |
an symplectic integrator, |
1357 |
> |
(\pi _1 ). |
1358 |
> |
\] |
1359 |
> |
By concatenating the propagators corresponding to these five terms, |
1360 |
> |
we can obtain an symplectic integrator, |
1361 |
|
\[ |
1362 |
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
1363 |
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
1364 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1365 |
|
_1 }. |
1366 |
|
\] |
1436 |
– |
|
1367 |
|
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
1368 |
|
$F(\pi )$ and $G(\pi )$ is defined by |
1369 |
|
\[ |
1370 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
1371 |
< |
) |
1371 |
> |
). |
1372 |
|
\] |
1373 |
|
If the Poisson bracket of a function $F$ with an arbitrary smooth |
1374 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
1379 |
|
then by the chain rule |
1380 |
|
\[ |
1381 |
|
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
1382 |
< |
}}{2})\pi |
1382 |
> |
}}{2})\pi. |
1383 |
|
\] |
1384 |
< |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
1384 |
> |
Thus, $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel |
1385 |
> |
\pi |
1386 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
1387 |
< |
Lie-Poisson integrator is found to be extremely efficient and stable |
1388 |
< |
which can be explained by the fact the small angle approximation is |
1389 |
< |
used and the norm of the angular momentum is conserved. |
1387 |
> |
Lie-Poisson integrator is found to be both extremely efficient and |
1388 |
> |
stable. These properties can be explained by the fact the small |
1389 |
> |
angle approximation is used and the norm of the angular momentum is |
1390 |
> |
conserved. |
1391 |
|
|
1392 |
|
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
1393 |
|
Splitting for Rigid Body} |
1394 |
|
|
1395 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
1396 |
< |
energy and potential energy, |
1397 |
< |
\[ |
1398 |
< |
H = T(p,\pi ) + V(q,Q) |
1467 |
< |
\] |
1468 |
< |
The equations of motion corresponding to potential energy and |
1469 |
< |
kinetic energy are listed in the below table, |
1396 |
> |
energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations |
1397 |
> |
of motion corresponding to potential energy and kinetic energy are |
1398 |
> |
listed in the below table, |
1399 |
|
\begin{table} |
1400 |
< |
\caption{Equations of motion due to Potential and Kinetic Energies} |
1400 |
> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
1401 |
|
\begin{center} |
1402 |
|
\begin{tabular}{|l|l|} |
1403 |
|
\hline |
1411 |
|
\end{tabular} |
1412 |
|
\end{center} |
1413 |
|
\end{table} |
1414 |
< |
A second-order symplectic method is now obtained by the |
1415 |
< |
composition of the flow maps, |
1414 |
> |
A second-order symplectic method is now obtained by the composition |
1415 |
> |
of the position and velocity propagators, |
1416 |
|
\[ |
1417 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
1418 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
1419 |
|
\] |
1420 |
|
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
1421 |
< |
sub-flows which corresponding to force and torque respectively, |
1421 |
> |
sub-propagators which corresponding to force and torque |
1422 |
> |
respectively, |
1423 |
|
\[ |
1424 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
1425 |
|
_{\Delta t/2,\tau }. |
1426 |
|
\] |
1427 |
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
1428 |
< |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
1429 |
< |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
1430 |
< |
|
1431 |
< |
Furthermore, kinetic potential can be separated to translational |
1502 |
< |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
1428 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order |
1429 |
> |
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the |
1430 |
> |
kinetic energy can be separated to translational kinetic term, $T^t |
1431 |
> |
(p)$, and rotational kinetic term, $T^r (\pi )$, |
1432 |
|
\begin{equation} |
1433 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
1434 |
|
\end{equation} |
1435 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
1436 |
< |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
1437 |
< |
corresponding flow maps are given by |
1436 |
> |
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore, |
1437 |
> |
the corresponding propagators are given by |
1438 |
|
\[ |
1439 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
1440 |
|
_{\Delta t,T^r }. |
1441 |
|
\] |
1442 |
< |
Finally, we obtain the overall symplectic flow maps for free moving |
1443 |
< |
rigid body |
1444 |
< |
\begin{equation} |
1445 |
< |
\begin{array}{c} |
1446 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
1447 |
< |
\circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \\ |
1519 |
< |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1520 |
< |
\end{array} |
1442 |
> |
Finally, we obtain the overall symplectic propagators for freely |
1443 |
> |
moving rigid bodies |
1444 |
> |
\begin{eqnarray} |
1445 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
1446 |
> |
& & \circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \notag\\ |
1447 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1448 |
|
\label{introEquation:overallRBFlowMaps} |
1449 |
< |
\end{equation} |
1449 |
> |
\end{eqnarray} |
1450 |
|
|
1451 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1452 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
1453 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
1454 |
|
has been applied in a variety of studies. This section will review |
1455 |
< |
the theory of Langevin dynamics simulation. A brief derivation of |
1456 |
< |
generalized Langevin equation will be given first. Follow that, we |
1457 |
< |
will discuss the physical meaning of the terms appearing in the |
1458 |
< |
equation as well as the calculation of friction tensor from |
1459 |
< |
hydrodynamics theory. |
1455 |
> |
the theory of Langevin dynamics. A brief derivation of generalized |
1456 |
> |
Langevin equation will be given first. Following that, we will |
1457 |
> |
discuss the physical meaning of the terms appearing in the equation |
1458 |
> |
as well as the calculation of friction tensor from hydrodynamics |
1459 |
> |
theory. |
1460 |
|
|
1461 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
1462 |
|
|
1463 |
< |
Harmonic bath model, in which an effective set of harmonic |
1463 |
> |
A harmonic bath model, in which an effective set of harmonic |
1464 |
|
oscillators are used to mimic the effect of a linearly responding |
1465 |
|
environment, has been widely used in quantum chemistry and |
1466 |
|
statistical mechanics. One of the successful applications of |
1467 |
< |
Harmonic bath model is the derivation of Deriving Generalized |
1468 |
< |
Langevin Dynamics. Lets consider a system, in which the degree of |
1467 |
> |
Harmonic bath model is the derivation of the Generalized Langevin |
1468 |
> |
Dynamics (GLE). Lets consider a system, in which the degree of |
1469 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
1470 |
|
Hamiltonian of the form |
1471 |
|
\begin{equation} |
1472 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
1473 |
|
\label{introEquation:bathGLE}. |
1474 |
|
\end{equation} |
1475 |
< |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
1476 |
< |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
1475 |
> |
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated |
1476 |
> |
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
1477 |
|
\[ |
1478 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
1479 |
|
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
1481 |
|
\] |
1482 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
1483 |
|
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
1484 |
< |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
1484 |
> |
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath |
1485 |
|
coupling, |
1486 |
|
\[ |
1487 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
1488 |
|
\] |
1489 |
< |
where $g_\alpha$ are the coupling constants between the bath and the |
1490 |
< |
coordinate $x$. Introducing |
1489 |
> |
where $g_\alpha$ are the coupling constants between the bath |
1490 |
> |
coordinates ($x_ \alpha$) and the system coordinate ($x$). |
1491 |
> |
Introducing |
1492 |
|
\[ |
1493 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1494 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1495 |
< |
\] and combining the last two terms in Equation |
1496 |
< |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
1569 |
< |
Hamiltonian as |
1495 |
> |
\] |
1496 |
> |
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as |
1497 |
|
\[ |
1498 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1499 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1500 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1501 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1501 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}}. |
1502 |
|
\] |
1503 |
|
Since the first two terms of the new Hamiltonian depend only on the |
1504 |
|
system coordinates, we can get the equations of motion for |
1505 |
< |
Generalized Langevin Dynamics by Hamilton's equations |
1579 |
< |
\ref{introEquation:motionHamiltonianCoordinate, |
1580 |
< |
introEquation:motionHamiltonianMomentum}, |
1505 |
> |
Generalized Langevin Dynamics by Hamilton's equations, |
1506 |
|
\begin{equation} |
1507 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
1508 |
|
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
1515 |
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
1516 |
|
\label{introEquation:bathMotionGLE} |
1517 |
|
\end{equation} |
1593 |
– |
|
1518 |
|
In order to derive an equation for $x$, the dynamics of the bath |
1519 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
1520 |
|
transform which is particularly useful in solving linear ordinary |
1521 |
< |
differential equations, Laplace transform is the appropriate tool to |
1522 |
< |
solve this problem. The basic idea is to transform the difficult |
1521 |
> |
differential equations,the Laplace transform is the appropriate tool |
1522 |
> |
to solve this problem. The basic idea is to transform the difficult |
1523 |
|
differential equations into simple algebra problems which can be |
1524 |
< |
solved easily. Then applying inverse Laplace transform, also known |
1525 |
< |
as the Bromwich integral, we can retrieve the solutions of the |
1526 |
< |
original problems. |
1527 |
< |
|
1604 |
< |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
1605 |
< |
transform of f(t) is a new function defined as |
1524 |
> |
solved easily. Then, by applying the inverse Laplace transform, also |
1525 |
> |
known as the Bromwich integral, we can retrieve the solutions of the |
1526 |
> |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
1527 |
> |
$, the Laplace transform of $f(t)$ is a new function defined as |
1528 |
|
\[ |
1529 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1530 |
|
\] |
1531 |
|
where $p$ is real and $L$ is called the Laplace Transform |
1532 |
|
Operator. Below are some important properties of Laplace transform |
1611 |
– |
|
1533 |
|
\begin{eqnarray*} |
1534 |
|
L(x + y) & = & L(x) + L(y) \\ |
1535 |
|
L(ax) & = & aL(x) \\ |
1537 |
|
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
1538 |
|
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
1539 |
|
\end{eqnarray*} |
1540 |
< |
|
1620 |
< |
|
1621 |
< |
Applying Laplace transform to the bath coordinates, we obtain |
1540 |
> |
Applying the Laplace transform to the bath coordinates, we obtain |
1541 |
|
\begin{eqnarray*} |
1542 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1543 |
< |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1542 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x), \\ |
1543 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
1544 |
|
\end{eqnarray*} |
1626 |
– |
|
1545 |
|
By the same way, the system coordinates become |
1546 |
|
\begin{eqnarray*} |
1547 |
< |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1548 |
< |
& & \mbox{} - \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1547 |
> |
mL(\ddot x) & = & |
1548 |
> |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1549 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}. |
1550 |
|
\end{eqnarray*} |
1632 |
– |
|
1551 |
|
With the help of some relatively important inverse Laplace |
1552 |
|
transformations: |
1553 |
|
\[ |
1557 |
|
L(1) = \frac{1}{p} \\ |
1558 |
|
\end{array} |
1559 |
|
\] |
1560 |
< |
, we obtain |
1560 |
> |
we obtain |
1561 |
|
\begin{eqnarray*} |
1562 |
|
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
1563 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1605 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
1606 |
|
implies it is completely deterministic within the context of a |
1607 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
1608 |
< |
uncorrelated to $x$ and $\dot x$, |
1609 |
< |
\[ |
1610 |
< |
\begin{array}{l} |
1611 |
< |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
1612 |
< |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
1695 |
< |
\end{array} |
1696 |
< |
\] |
1697 |
< |
This property is what we expect from a truly random process. As long |
1698 |
< |
as the model, which is gaussian distribution in general, chosen for |
1699 |
< |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
1700 |
< |
still remains. |
1701 |
< |
|
1608 |
> |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
1609 |
> |
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
1610 |
> |
0.$ This property is what we expect from a truly random process. As |
1611 |
> |
long as the model chosen for $R(t)$ was a gaussian distribution in |
1612 |
> |
general, the stochastic nature of the GLE still remains. |
1613 |
|
%dynamic friction kernel |
1614 |
|
The convolution integral |
1615 |
|
\[ |
1624 |
|
\[ |
1625 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
1626 |
|
\] |
1627 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1627 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1628 |
|
\[ |
1629 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
1630 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
1631 |
|
\] |
1632 |
< |
which can be used to describe dynamic caging effect. The other |
1633 |
< |
extreme is the bath that responds infinitely quickly to motions in |
1634 |
< |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
1635 |
< |
time: |
1632 |
> |
which can be used to describe the effect of dynamic caging in |
1633 |
> |
viscous solvents. The other extreme is the bath that responds |
1634 |
> |
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
1635 |
> |
taken as a $delta$ function in time: |
1636 |
|
\[ |
1637 |
|
\xi (t) = 2\xi _0 \delta (t) |
1638 |
|
\] |
1641 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
1642 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
1643 |
|
\] |
1644 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1644 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1645 |
|
\begin{equation} |
1646 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
1647 |
|
x(t) + R(t) \label{introEquation:LangevinEquation} |
1654 |
|
|
1655 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
1656 |
|
|
1657 |
< |
Defining a new set of coordinates, |
1657 |
> |
Defining a new set of coordinates |
1658 |
|
\[ |
1659 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1660 |
< |
^2 }}x(0) |
1661 |
< |
\], |
1660 |
> |
^2 }}x(0), |
1661 |
> |
\] |
1662 |
|
we can rewrite $R(T)$ as |
1663 |
|
\[ |
1664 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1665 |
|
\] |
1666 |
|
And since the $q$ coordinates are harmonic oscillators, |
1756 |
– |
|
1667 |
|
\begin{eqnarray*} |
1668 |
|
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1669 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1672 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1673 |
|
& = &kT\xi (t) \\ |
1674 |
|
\end{eqnarray*} |
1765 |
– |
|
1675 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1676 |
|
\begin{equation} |
1677 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1678 |
< |
\label{introEquation:secondFluctuationDissipation}. |
1678 |
> |
\label{introEquation:secondFluctuationDissipation}, |
1679 |
|
\end{equation} |
1680 |
< |
In effect, it acts as a constraint on the possible ways in which one |
1681 |
< |
can model the random force and friction kernel. |
1680 |
> |
which acts as a constraint on the possible ways in which one can |
1681 |
> |
model the random force and friction kernel. |