3 |
|
\section{\label{introSection:classicalMechanics}Classical |
4 |
|
Mechanics} |
5 |
|
|
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 |
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 |
13 |
< |
specification of this state; Finally, the specification of the state |
14 |
< |
when further combine with the laws of mechanics will also be |
15 |
< |
sufficient to predict the future behavior of the system. |
6 |
> |
Using equations of motion derived from Classical Mechanics, |
7 |
> |
Molecular Dynamics simulations are carried out by integrating the |
8 |
> |
equations of motion for a given system of particles. There are three |
9 |
> |
fundamental ideas behind classical mechanics. Firstly, one can |
10 |
> |
determine the state of a mechanical system at any time of interest; |
11 |
> |
Secondly, all the mechanical properties of the system at that time |
12 |
> |
can be determined by combining the knowledge of the properties of |
13 |
> |
the system with the specification of this state; Finally, the |
14 |
> |
specification of the state when further combined with the laws of |
15 |
> |
mechanics will also be sufficient to predict the future behavior of |
16 |
> |
the system. |
17 |
|
|
18 |
|
\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
19 |
|
The discovery of Newton's three laws of mechanics which govern the |
32 |
|
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
33 |
|
Newton's third law states that |
34 |
|
\begin{equation} |
35 |
< |
F_{ij} = -F_{ji} |
35 |
> |
F_{ij} = -F_{ji}. |
36 |
|
\label{introEquation:newtonThirdLaw} |
37 |
|
\end{equation} |
37 |
– |
|
38 |
|
Conservation laws of Newtonian Mechanics play very important roles |
39 |
|
in solving mechanics problems. The linear momentum of a particle is |
40 |
|
conserved if it is free or it experiences no force. The second |
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 |
66 |
< |
that if all forces are conservative, Energy |
67 |
< |
\begin{equation}E = T + V \label{introEquation:energyConservation} |
66 |
> |
that if all forces are conservative, energy is conserved, |
67 |
> |
\begin{equation}E = T + V. \label{introEquation:energyConservation} |
68 |
|
\end{equation} |
69 |
< |
is conserved. All of these conserved quantities are |
70 |
< |
important factors to determine the quality of numerical integration |
71 |
< |
schemes for rigid bodies \cite{Dullweber1997}. |
69 |
> |
All of these conserved quantities are important factors to determine |
70 |
> |
the quality of numerical integration schemes for rigid bodies |
71 |
> |
\cite{Dullweber1997}. |
72 |
|
|
73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
74 |
|
|
75 |
< |
Newtonian Mechanics suffers from two important limitations: motions |
76 |
< |
can only be described in cartesian coordinate systems. Moreover, it |
77 |
< |
becomes 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. |
75 |
> |
Newtonian Mechanics suffers from an important limitation: motion can |
76 |
> |
only be described in cartesian coordinate systems which make it |
77 |
> |
impossible to predict analytically the properties of the system even |
78 |
> |
if we know all of the details of the interaction. In order to |
79 |
> |
overcome some of the practical difficulties which arise in attempts |
80 |
> |
to apply Newton's equation to complex systems, approximate numerical |
81 |
> |
procedures may be developed. |
82 |
|
|
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 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, $U$. |
88 |
> |
Hamilton's Principle may be stated as follows: the trajectory, along |
89 |
> |
which a dynamical system may move from one point to another within a |
90 |
> |
specified time, is derived by finding the path which minimizes the |
91 |
> |
time integral of the difference between the kinetic $K$, and |
92 |
> |
potential energies $U$, |
93 |
|
\begin{equation} |
94 |
< |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
94 |
> |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}. |
95 |
|
\label{introEquation:halmitonianPrinciple1} |
96 |
|
\end{equation} |
98 |
– |
|
97 |
|
For simple mechanical systems, where the forces acting on the |
98 |
|
different parts are derivable from a potential, the Lagrangian |
99 |
|
function $L$ can be defined as the difference between the kinetic |
100 |
|
energy of the system and its potential energy, |
101 |
|
\begin{equation} |
102 |
< |
L \equiv K - U = L(q_i ,\dot q_i ) , |
102 |
> |
L \equiv K - U = L(q_i ,\dot q_i ). |
103 |
|
\label{introEquation:lagrangianDef} |
104 |
|
\end{equation} |
105 |
< |
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
105 |
> |
Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
106 |
|
\begin{equation} |
107 |
< |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
107 |
> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} . |
108 |
|
\label{introEquation:halmitonianPrinciple2} |
109 |
|
\end{equation} |
110 |
|
|
136 |
|
p_i = \frac{{\partial L}}{{\partial q_i }} |
137 |
|
\label{introEquation:generalizedMomentaDot} |
138 |
|
\end{equation} |
141 |
– |
|
139 |
|
With the help of the generalized momenta, we may now define a new |
140 |
|
quantity $H$ by the equation |
141 |
|
\begin{equation} |
143 |
|
\label{introEquation:hamiltonianDefByLagrangian} |
144 |
|
\end{equation} |
145 |
|
where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
146 |
< |
$L$ is the Lagrangian function for the system. |
147 |
< |
|
151 |
< |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
152 |
< |
one can obtain |
146 |
> |
$L$ is the Lagrangian function for the system. Differentiating |
147 |
> |
Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain |
148 |
|
\begin{equation} |
149 |
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
150 |
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
151 |
|
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
152 |
< |
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
152 |
> |
L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1} |
153 |
|
\end{equation} |
154 |
< |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
155 |
< |
second and fourth terms in the parentheses cancel. Therefore, |
154 |
> |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second |
155 |
> |
and fourth terms in the parentheses cancel. Therefore, |
156 |
|
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
157 |
|
\begin{equation} |
158 |
|
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
159 |
< |
\right)} - \frac{{\partial L}}{{\partial t}}dt |
159 |
> |
\right)} - \frac{{\partial L}}{{\partial t}}dt . |
160 |
|
\label{introEquation:diffHamiltonian2} |
161 |
|
\end{equation} |
162 |
|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
175 |
|
t}} |
176 |
|
\label{introEquation:motionHamiltonianTime} |
177 |
|
\end{equation} |
178 |
< |
|
184 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
178 |
> |
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
179 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
180 |
|
equation of motion. Due to their symmetrical formula, they are also |
181 |
|
known as the canonical equations of motions \cite{Goldstein2001}. |
189 |
|
statistical mechanics and quantum mechanics, since it treats the |
190 |
|
coordinate and its time derivative as independent variables and it |
191 |
|
only works with 1st-order differential equations\cite{Marion1990}. |
198 |
– |
|
192 |
|
In Newtonian Mechanics, a system described by conservative forces |
193 |
< |
conserves the total energy \ref{introEquation:energyConservation}. |
194 |
< |
It follows that Hamilton's equations of motion conserve the total |
195 |
< |
Hamiltonian. |
193 |
> |
conserves the total energy |
194 |
> |
(Eq.~\ref{introEquation:energyConservation}). It follows that |
195 |
> |
Hamilton's equations of motion conserve the total Hamiltonian |
196 |
|
\begin{equation} |
197 |
|
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
198 |
|
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
199 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
200 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
201 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
202 |
< |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
202 |
> |
q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian} |
203 |
|
\end{equation} |
204 |
|
|
205 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
214 |
|
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
215 |
|
|
216 |
|
Mathematically, phase space is the space which represents all |
217 |
< |
possible states. Each possible state of the system corresponds to |
218 |
< |
one unique point in the phase space. For mechanical systems, the |
219 |
< |
phase space usually consists of all possible values of position and |
220 |
< |
momentum variables. Consider a dynamic system of $f$ particles in a |
221 |
< |
cartesian space, where each of the $6f$ coordinates and momenta is |
222 |
< |
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
223 |
< |
this system is a $6f$ dimensional space. A point, $x = (\rightarrow |
224 |
< |
q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow |
225 |
< |
p_f )$, with a unique set of values of $6f$ coordinates and momenta |
226 |
< |
is a phase space vector. |
217 |
> |
possible states of a system. Each possible state of the system |
218 |
> |
corresponds to one unique point in the phase space. For mechanical |
219 |
> |
systems, the phase space usually consists of all possible values of |
220 |
> |
position and momentum variables. Consider a dynamic system of $f$ |
221 |
> |
particles in a cartesian space, where each of the $6f$ coordinates |
222 |
> |
and momenta is assigned to one of $6f$ mutually orthogonal axes, the |
223 |
> |
phase space of this system is a $6f$ dimensional space. A point, $x |
224 |
> |
= |
225 |
> |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
226 |
> |
\over q} _1 , \ldots |
227 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
228 |
> |
\over q} _f |
229 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
230 |
> |
\over p} _1 \ldots |
231 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
232 |
> |
\over p} _f )$ , with a unique set of values of $6f$ coordinates and |
233 |
> |
momenta is a phase space vector. |
234 |
|
%%%fix me |
235 |
|
|
236 |
|
In statistical mechanics, the condition of an ensemble at any time |
243 |
|
\label{introEquation:densityDistribution} |
244 |
|
\end{equation} |
245 |
|
Governed by the principles of mechanics, the phase points change |
246 |
< |
their locations which would change the density at any time at phase |
246 |
> |
their locations which changes the density at any time at phase |
247 |
|
space. Hence, the density distribution is also to be taken as a |
248 |
< |
function of the time. |
249 |
< |
|
250 |
< |
The number of systems $\delta N$ at time $t$ can be determined by, |
248 |
> |
function of the time. The number of systems $\delta N$ at time $t$ |
249 |
> |
can be determined by, |
250 |
|
\begin{equation} |
251 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
252 |
|
\label{introEquation:deltaN} |
253 |
|
\end{equation} |
254 |
< |
Assuming a large enough population of systems, we can sufficiently |
254 |
> |
Assuming enough copies of the systems, we can sufficiently |
255 |
|
approximate $\delta N$ without introducing discontinuity when we go |
256 |
|
from one region in the phase space to another. By integrating over |
257 |
|
the whole phase space, |
259 |
|
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
260 |
|
\label{introEquation:totalNumberSystem} |
261 |
|
\end{equation} |
262 |
< |
gives us an expression for the total number of the systems. Hence, |
263 |
< |
the probability per unit in the phase space can be obtained by, |
262 |
> |
gives us an expression for the total number of copies. Hence, the |
263 |
> |
probability per unit volume in the phase space can be obtained by, |
264 |
|
\begin{equation} |
265 |
|
\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int |
266 |
|
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
269 |
|
With the help of Eq.~\ref{introEquation:unitProbability} and the |
270 |
|
knowledge of the system, it is possible to calculate the average |
271 |
|
value of any desired quantity which depends on the coordinates and |
272 |
< |
momenta of the system. Even when the dynamics of the real system is |
272 |
> |
momenta of the system. Even when the dynamics of the real system are |
273 |
|
complex, or stochastic, or even discontinuous, the average |
274 |
< |
properties of the ensemble of possibilities as a whole remaining |
275 |
< |
well defined. For a classical system in thermal equilibrium with its |
274 |
> |
properties of the ensemble of possibilities as a whole remain well |
275 |
> |
defined. For a classical system in thermal equilibrium with its |
276 |
|
environment, the ensemble average of a mechanical quantity, $\langle |
277 |
|
A(q , p) \rangle_t$, takes the form of an integral over the phase |
278 |
|
space of the system, |
279 |
|
\begin{equation} |
280 |
|
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
281 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
282 |
< |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
282 |
> |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
283 |
|
\label{introEquation:ensembelAverage} |
284 |
|
\end{equation} |
285 |
|
|
287 |
|
statistical characteristics. As a function of macroscopic |
288 |
|
parameters, such as temperature \textit{etc}, the partition function |
289 |
|
can be used to describe the statistical properties of a system in |
290 |
< |
thermodynamic equilibrium. |
291 |
< |
|
292 |
< |
As an ensemble of systems, each of which is known to be thermally |
294 |
< |
isolated and conserve energy, the Microcanonical ensemble (NVE) has |
295 |
< |
a partition function like, |
290 |
> |
thermodynamic equilibrium. As an ensemble of systems, each of which |
291 |
> |
is known to be thermally isolated and conserve energy, the |
292 |
> |
Microcanonical ensemble (NVE) has a partition function like, |
293 |
|
\begin{equation} |
294 |
< |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
294 |
> |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
295 |
|
\end{equation} |
296 |
< |
A canonical ensemble (NVT)is an ensemble of systems, each of which |
296 |
> |
A canonical ensemble (NVT) is an ensemble of systems, each of which |
297 |
|
can share its energy with a large heat reservoir. The distribution |
298 |
|
of the total energy amongst the possible dynamical states is given |
299 |
|
by the partition function, |
300 |
|
\begin{equation} |
301 |
< |
\Omega (N,V,T) = e^{ - \beta A} |
301 |
> |
\Omega (N,V,T) = e^{ - \beta A}. |
302 |
|
\label{introEquation:NVTPartition} |
303 |
|
\end{equation} |
304 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
355 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
356 |
|
\label{introEquation:liouvilleTheorem} |
357 |
|
\end{equation} |
361 |
– |
|
358 |
|
Liouville's theorem states that the distribution function is |
359 |
|
constant along any trajectory in phase space. In classical |
360 |
< |
statistical mechanics, since the number of members in an ensemble is |
361 |
< |
huge and constant, we can assume the local density has no reason |
362 |
< |
(other than classical mechanics) to change, |
360 |
> |
statistical mechanics, since the number of system copies in an |
361 |
> |
ensemble is huge and constant, we can assume the local density has |
362 |
> |
no reason (other than classical mechanics) to change, |
363 |
|
\begin{equation} |
364 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
365 |
|
\label{introEquation:stationary} |
389 |
|
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
390 |
|
\frac{d}{{dt}}(\delta v) = 0. |
391 |
|
\end{equation} |
392 |
< |
With the help of stationary assumption |
393 |
< |
(\ref{introEquation:stationary}), we obtain the principle of the |
392 |
> |
With the help of the stationary assumption |
393 |
> |
(Eq.~\ref{introEquation:stationary}), we obtain the principle of |
394 |
|
\emph{conservation of volume in phase space}, |
395 |
|
\begin{equation} |
396 |
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
400 |
|
|
401 |
|
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
402 |
|
|
403 |
< |
Liouville's theorem can be expresses in a variety of different forms |
403 |
> |
Liouville's theorem can be expressed in a variety of different forms |
404 |
|
which are convenient within different contexts. For any two function |
405 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
406 |
|
bracket ${F, G}$ is defined as |
411 |
|
q_i }}} \right)}. |
412 |
|
\label{introEquation:poissonBracket} |
413 |
|
\end{equation} |
414 |
< |
Substituting equations of motion in Hamiltonian formalism( |
415 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
416 |
< |
Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into |
414 |
> |
Substituting equations of motion in Hamiltonian formalism |
415 |
> |
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
416 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into |
417 |
|
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
418 |
|
Liouville's theorem using Poisson bracket notion, |
419 |
|
\begin{equation} |
434 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
435 |
|
\label{introEquation:liouvilleTheoremInOperator} |
436 |
|
\end{equation} |
437 |
< |
|
437 |
> |
which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$. |
438 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
439 |
|
|
440 |
|
Various thermodynamic properties can be calculated from Molecular |
443 |
|
simulation and the quality of the underlying model. However, both |
444 |
|
experiments and computer simulations are usually performed during a |
445 |
|
certain time interval and the measurements are averaged over a |
446 |
< |
period of them which is different from the average behavior of |
446 |
> |
period of time which is different from the average behavior of |
447 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
448 |
|
Hypothesis makes a connection between time average and the ensemble |
449 |
|
average. It states that the time average and average over the |
450 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
450 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
451 |
|
\begin{equation} |
452 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
453 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
456 |
|
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
457 |
|
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
458 |
|
distribution function. If an observation is averaged over a |
459 |
< |
sufficiently long time (longer than relaxation time), all accessible |
460 |
< |
microstates in phase space are assumed to be equally probed, giving |
461 |
< |
a properly weighted statistical average. This allows the researcher |
462 |
< |
freedom of choice when deciding how best to measure a given |
463 |
< |
observable. In case an ensemble averaged approach sounds most |
464 |
< |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
459 |
> |
sufficiently long time (longer than the relaxation time), all |
460 |
> |
accessible microstates in phase space are assumed to be equally |
461 |
> |
probed, giving a properly weighted statistical average. This allows |
462 |
> |
the researcher freedom of choice when deciding how best to measure a |
463 |
> |
given observable. In case an ensemble averaged approach sounds most |
464 |
> |
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
465 |
|
utilized. Or if the system lends itself to a time averaging |
466 |
|
approach, the Molecular Dynamics techniques in |
467 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
474 |
|
by the differential equations. However, most of them ignore the |
475 |
|
hidden physical laws contained within the equations. Since 1990, |
476 |
|
geometric integrators, which preserve various phase-flow invariants |
477 |
< |
such as symplectic structure, volume and time reversal symmetry, are |
478 |
< |
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
479 |
< |
Leimkuhler1999}. The velocity Verlet method, which happens to be a |
480 |
< |
simple example of symplectic integrator, continues to gain |
481 |
< |
popularity in the molecular dynamics community. This fact can be |
482 |
< |
partly explained by its geometric nature. |
477 |
> |
such as symplectic structure, volume and time reversal symmetry, |
478 |
> |
were developed to address this issue\cite{Dullweber1997, |
479 |
> |
McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which |
480 |
> |
happens to be a simple example of symplectic integrator, continues |
481 |
> |
to gain popularity in the molecular dynamics community. This fact |
482 |
> |
can be partly explained by its geometric nature. |
483 |
|
|
484 |
|
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
485 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
488 |
|
surface of Earth. It seems to be flat locally, but it is round if |
489 |
|
viewed as a whole. A \emph{differentiable manifold} (also known as |
490 |
|
\emph{smooth manifold}) is a manifold on which it is possible to |
491 |
< |
apply calculus on \emph{differentiable manifold}. A \emph{symplectic |
492 |
< |
manifold} is defined as a pair $(M, \omega)$ which consists of a |
491 |
> |
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
492 |
> |
defined as a pair $(M, \omega)$ which consists of a |
493 |
|
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
494 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
495 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
496 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
497 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
498 |
< |
$\omega(x, x) = 0$. The cross product operation in vector field is |
499 |
< |
an example of symplectic form. |
498 |
> |
$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in |
499 |
> |
vector field is an example of symplectic form. One of the |
500 |
> |
motivations to study \emph{symplectic manifolds} in Hamiltonian |
501 |
> |
Mechanics is that a symplectic manifold can represent all possible |
502 |
> |
configurations of the system and the phase space of the system can |
503 |
> |
be described by it's cotangent bundle\cite{Jost2002}. Every |
504 |
> |
symplectic manifold is even dimensional. For instance, in Hamilton |
505 |
> |
equations, coordinate and momentum always appear in pairs. |
506 |
|
|
505 |
– |
One of the motivations to study \emph{symplectic manifolds} in |
506 |
– |
Hamiltonian Mechanics is that a symplectic manifold can represent |
507 |
– |
all possible configurations of the system and the phase space of the |
508 |
– |
system can be described by it's cotangent bundle. Every symplectic |
509 |
– |
manifold is even dimensional. For instance, in Hamilton equations, |
510 |
– |
coordinate and momentum always appear in pairs. |
511 |
– |
|
507 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
508 |
|
|
509 |
|
For an ordinary differential system defined as |
511 |
|
\dot x = f(x) |
512 |
|
\end{equation} |
513 |
|
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
514 |
+ |
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian |
515 |
+ |
function and $J$ is the skew-symmetric matrix |
516 |
|
\begin{equation} |
520 |
– |
f(r) = J\nabla _x H(r). |
521 |
– |
\end{equation} |
522 |
– |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
523 |
– |
matrix |
524 |
– |
\begin{equation} |
517 |
|
J = \left( {\begin{array}{*{20}c} |
518 |
|
0 & I \\ |
519 |
|
{ - I} & 0 \\ |
523 |
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
524 |
|
system can be rewritten as, |
525 |
|
\begin{equation} |
526 |
< |
\frac{d}{{dt}}x = J\nabla _x H(x) |
526 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x). |
527 |
|
\label{introEquation:compactHamiltonian} |
528 |
|
\end{equation}In this case, $f$ is |
529 |
< |
called a \emph{Hamiltonian vector field}. |
530 |
< |
|
539 |
< |
Another generalization of Hamiltonian dynamics is Poisson |
540 |
< |
Dynamics\cite{Olver1986}, |
529 |
> |
called a \emph{Hamiltonian vector field}. Another generalization of |
530 |
> |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
531 |
|
\begin{equation} |
532 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
533 |
|
\end{equation} |
534 |
|
The most obvious change being that matrix $J$ now depends on $x$. |
535 |
|
|
536 |
< |
\subsection{\label{introSection:exactFlow}Exact Flow} |
536 |
> |
\subsection{\label{introSection:exactFlow}Exact Propagator} |
537 |
|
|
538 |
< |
Let $x(t)$ be the exact solution of the ODE system, |
539 |
< |
\begin{equation} |
540 |
< |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
541 |
< |
\end{equation} |
542 |
< |
The exact flow(solution) $\varphi_\tau$ is defined by |
553 |
< |
\[ |
554 |
< |
x(t+\tau) =\varphi_\tau(x(t)) |
538 |
> |
Let $x(t)$ be the exact solution of the ODE |
539 |
> |
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
540 |
> |
define its exact propagator(solution) $\varphi_\tau$ |
541 |
> |
\[ x(t+\tau) |
542 |
> |
=\varphi_\tau(x(t)) |
543 |
|
\] |
544 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
545 |
< |
space to itself. The flow has the continuous group property, |
545 |
> |
space to itself. The propagator has the continuous group property, |
546 |
|
\begin{equation} |
547 |
|
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
548 |
|
+ \tau _2 } . |
551 |
|
\begin{equation} |
552 |
|
\varphi _\tau \circ \varphi _{ - \tau } = I |
553 |
|
\end{equation} |
554 |
< |
Therefore, the exact flow is self-adjoint, |
554 |
> |
Therefore, the exact propagator is self-adjoint, |
555 |
|
\begin{equation} |
556 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
557 |
|
\end{equation} |
558 |
< |
The exact flow can also be written in terms of the of an operator, |
558 |
> |
The exact propagator can also be written in terms of operator, |
559 |
|
\begin{equation} |
560 |
|
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
561 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
562 |
|
\label{introEquation:exponentialOperator} |
563 |
|
\end{equation} |
564 |
< |
|
565 |
< |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
566 |
< |
Instead, we use an approximate map, $\psi_\tau$, which is usually |
567 |
< |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
568 |
< |
the Taylor series of $\psi_\tau$ agree to order $p$, |
564 |
> |
In most cases, it is not easy to find the exact propagator |
565 |
> |
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
566 |
> |
which is usually called an integrator. The order of an integrator |
567 |
> |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
568 |
> |
order $p$, |
569 |
|
\begin{equation} |
570 |
|
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
571 |
|
\end{equation} |
573 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
574 |
|
|
575 |
|
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
576 |
< |
ODE and its flow play important roles in numerical studies. Many of |
577 |
< |
them can be found in systems which occur naturally in applications. |
578 |
< |
|
579 |
< |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
592 |
< |
a \emph{symplectic} flow if it satisfies, |
576 |
> |
ODE and its propagator play important roles in numerical studies. |
577 |
> |
Many of them can be found in systems which occur naturally in |
578 |
> |
applications. Let $\varphi$ be the propagator of Hamiltonian vector |
579 |
> |
field, $\varphi$ is a \emph{symplectic} propagator if it satisfies, |
580 |
|
\begin{equation} |
581 |
|
{\varphi '}^T J \varphi ' = J. |
582 |
|
\end{equation} |
583 |
|
According to Liouville's theorem, the symplectic volume is invariant |
584 |
< |
under a Hamiltonian flow, which is the basis for classical |
585 |
< |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
586 |
< |
field on a symplectic manifold can be shown to be a |
584 |
> |
under a Hamiltonian propagator, which is the basis for classical |
585 |
> |
statistical mechanics. Furthermore, the propagator of a Hamiltonian |
586 |
> |
vector field on a symplectic manifold can be shown to be a |
587 |
|
symplectomorphism. As to the Poisson system, |
588 |
|
\begin{equation} |
589 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
590 |
|
\end{equation} |
591 |
< |
is the property that must be preserved by the integrator. |
592 |
< |
|
593 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
594 |
< |
source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ |
595 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
596 |
< |
be volume-preserving. |
597 |
< |
|
611 |
< |
Changing the variables $y = h(x)$ in an ODE |
612 |
< |
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
591 |
> |
is the property that must be preserved by the integrator. It is |
592 |
> |
possible to construct a \emph{volume-preserving} propagator for a |
593 |
> |
source free ODE ($ \nabla \cdot f = 0 $), if the propagator |
594 |
> |
satisfies $ \det d\varphi = 1$. One can show easily that a |
595 |
> |
symplectic propagator will be volume-preserving. Changing the |
596 |
> |
variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will |
597 |
> |
result in a new system, |
598 |
|
\[ |
599 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
600 |
|
\] |
601 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
602 |
< |
In other words, the flow of this vector field is reversible if and |
603 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
604 |
< |
|
605 |
< |
A \emph{first integral}, or conserved quantity of a general |
606 |
< |
differential function is a function $ G:R^{2d} \to R^d $ which is |
622 |
< |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
602 |
> |
In other words, the propagator of this vector field is reversible if |
603 |
> |
and only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
604 |
> |
conserved quantity of a general differential function is a function |
605 |
> |
$ G:R^{2d} \to R^d $ which is constant for all solutions of the ODE |
606 |
> |
$\frac{{dx}}{{dt}} = f(x)$ , |
607 |
|
\[ |
608 |
|
\frac{{dG(x(t))}}{{dt}} = 0. |
609 |
|
\] |
610 |
< |
Using chain rule, one may obtain, |
610 |
> |
Using the chain rule, one may obtain, |
611 |
|
\[ |
612 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
612 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
613 |
|
\] |
614 |
< |
which is the condition for conserving \emph{first integral}. For a |
615 |
< |
canonical Hamiltonian system, the time evolution of an arbitrary |
616 |
< |
smooth function $G$ is given by, |
633 |
< |
|
614 |
> |
which is the condition for conserved quantities. For a canonical |
615 |
> |
Hamiltonian system, the time evolution of an arbitrary smooth |
616 |
> |
function $G$ is given by, |
617 |
|
\begin{eqnarray} |
618 |
< |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
619 |
< |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
618 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
619 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
620 |
|
\label{introEquation:firstIntegral1} |
621 |
|
\end{eqnarray} |
622 |
< |
|
623 |
< |
|
641 |
< |
Using poisson bracket notion, Equation |
642 |
< |
\ref{introEquation:firstIntegral1} can be rewritten as |
622 |
> |
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1} |
623 |
> |
can be rewritten as |
624 |
|
\[ |
625 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
626 |
|
\] |
627 |
< |
Therefore, the sufficient condition for $G$ to be the \emph{first |
628 |
< |
integral} of a Hamiltonian system is |
629 |
< |
\[ |
630 |
< |
\left\{ {G,H} \right\} = 0. |
650 |
< |
\] |
651 |
< |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
652 |
< |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
653 |
< |
0$. |
654 |
< |
|
627 |
> |
Therefore, the sufficient condition for $G$ to be a conserved |
628 |
> |
quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As |
629 |
> |
is well known, the Hamiltonian (or energy) H of a Hamiltonian system |
630 |
> |
is a conserved quantity, which is due to the fact $\{ H,H\} = 0$. |
631 |
|
When designing any numerical methods, one should always try to |
632 |
< |
preserve the structural properties of the original ODE and its flow. |
632 |
> |
preserve the structural properties of the original ODE and its |
633 |
> |
propagator. |
634 |
|
|
635 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
636 |
|
A lot of well established and very effective numerical methods have |
637 |
< |
been successful precisely because of their symplecticities even |
637 |
> |
been successful precisely because of their symplectic nature even |
638 |
|
though this fact was not recognized when they were first |
639 |
|
constructed. The most famous example is the Verlet-leapfrog method |
640 |
|
in molecular dynamics. In general, symplectic integrators can be |
645 |
|
\item Runge-Kutta methods |
646 |
|
\item Splitting methods |
647 |
|
\end{enumerate} |
648 |
< |
|
672 |
< |
Generating function\cite{Channell1990} tends to lead to methods |
648 |
> |
Generating functions\cite{Channell1990} tend to lead to methods |
649 |
|
which are cumbersome and difficult to use. In dissipative systems, |
650 |
|
variational methods can capture the decay of energy |
651 |
< |
accurately\cite{Kane2000}. Since their geometrically unstable nature |
651 |
> |
accurately\cite{Kane2000}. Since they are geometrically unstable |
652 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
653 |
|
methods are not suitable for Hamiltonian system. Recently, various |
654 |
< |
high-order explicit Runge-Kutta methods |
655 |
< |
\cite{Owren1992,Chen2003}have been developed to overcome this |
656 |
< |
instability. However, due to computational penalty involved in |
657 |
< |
implementing the Runge-Kutta methods, they have not attracted much |
658 |
< |
attention from the Molecular Dynamics community. Instead, splitting |
659 |
< |
methods have been widely accepted since they exploit natural |
660 |
< |
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
654 |
> |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
655 |
> |
have been developed to overcome this instability. However, due to |
656 |
> |
computational penalty involved in implementing the Runge-Kutta |
657 |
> |
methods, they have not attracted much attention from the Molecular |
658 |
> |
Dynamics community. Instead, splitting methods have been widely |
659 |
> |
accepted since they exploit natural decompositions of the |
660 |
> |
system\cite{Tuckerman1992, McLachlan1998}. |
661 |
|
|
662 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
663 |
|
|
664 |
|
The main idea behind splitting methods is to decompose the discrete |
665 |
< |
$\varphi_h$ as a composition of simpler flows, |
665 |
> |
$\varphi_h$ as a composition of simpler propagators, |
666 |
|
\begin{equation} |
667 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
668 |
|
\varphi _{h_n } |
669 |
|
\label{introEquation:FlowDecomposition} |
670 |
|
\end{equation} |
671 |
< |
where each of the sub-flow is chosen such that each represent a |
672 |
< |
simpler integration of the system. |
673 |
< |
|
698 |
< |
Suppose that a Hamiltonian system takes the form, |
671 |
> |
where each of the sub-propagator is chosen such that each represent |
672 |
> |
a simpler integration of the system. Suppose that a Hamiltonian |
673 |
> |
system takes the form, |
674 |
|
\[ |
675 |
|
H = H_1 + H_2. |
676 |
|
\] |
677 |
|
Here, $H_1$ and $H_2$ may represent different physical processes of |
678 |
|
the system. For instance, they may relate to kinetic and potential |
679 |
|
energy respectively, which is a natural decomposition of the |
680 |
< |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
681 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
682 |
< |
order expression is then given by the Lie-Trotter formula |
680 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact |
681 |
> |
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
682 |
> |
simple first order expression is then given by the Lie-Trotter |
683 |
> |
formula |
684 |
|
\begin{equation} |
685 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
686 |
|
\label{introEquation:firstOrderSplitting} |
689 |
|
continuous $\varphi _i$ over a time $h$. By definition, as |
690 |
|
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
691 |
|
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
692 |
< |
It is easy to show that any composition of symplectic flows yields a |
693 |
< |
symplectic map, |
692 |
> |
It is easy to show that any composition of symplectic propagators |
693 |
> |
yields a symplectic map, |
694 |
|
\begin{equation} |
695 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
696 |
|
'\phi ' = \phi '^T J\phi ' = J, |
698 |
|
\end{equation} |
699 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
700 |
|
splitting in this context automatically generates a symplectic map. |
701 |
< |
|
702 |
< |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
703 |
< |
introduces local errors proportional to $h^2$, while Strang |
704 |
< |
splitting gives a second-order decomposition, |
701 |
> |
The Lie-Trotter |
702 |
> |
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
703 |
> |
local errors proportional to $h^2$, while the Strang splitting gives |
704 |
> |
a second-order decomposition, |
705 |
|
\begin{equation} |
706 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
707 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
708 |
|
\end{equation} |
709 |
< |
which has a local error proportional to $h^3$. The Sprang |
709 |
> |
which has a local error proportional to $h^3$. The Strang |
710 |
|
splitting's popularity in molecular simulation community attribute |
711 |
|
to its symmetric property, |
712 |
|
\begin{equation} |
761 |
|
|
762 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
763 |
|
\end{enumerate} |
788 |
– |
|
764 |
|
By simply switching the order of the propagators in the splitting |
765 |
|
and composing a new integrator, the \emph{position verlet} |
766 |
|
integrator, can be generated, |
777 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
778 |
|
|
779 |
|
The Baker-Campbell-Hausdorff formula can be used to determine the |
780 |
< |
local error of splitting method in terms of the commutator of the |
780 |
> |
local error of a splitting method in terms of the commutator of the |
781 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
782 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associated with |
783 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
782 |
> |
the sub-propagator. For operators $hX$ and $hY$ which are associated |
783 |
> |
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
784 |
|
\begin{equation} |
785 |
|
\exp (hX + hY) = \exp (hZ) |
786 |
|
\end{equation} |
789 |
|
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
790 |
|
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
791 |
|
\end{equation} |
792 |
< |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
792 |
> |
Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by |
793 |
|
\[ |
794 |
|
[X,Y] = XY - YX . |
795 |
|
\] |
796 |
|
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
797 |
< |
to the Sprang splitting, we can obtain |
797 |
> |
to the Strang splitting, we can obtain |
798 |
|
\begin{eqnarray*} |
799 |
|
\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 \\ |
800 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
801 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
801 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots |
802 |
> |
). |
803 |
|
\end{eqnarray*} |
804 |
< |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0,\] the dominant local |
805 |
< |
error of Spring splitting is proportional to $h^3$. The same |
806 |
< |
procedure can be applied to a general splitting, of the form |
804 |
> |
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
805 |
> |
error of Strang splitting is proportional to $h^3$. The same |
806 |
> |
procedure can be applied to a general splitting of the form |
807 |
|
\begin{equation} |
808 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
809 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
838 |
|
dynamical information. The basic idea of molecular dynamics is that |
839 |
|
macroscopic properties are related to microscopic behavior and |
840 |
|
microscopic behavior can be calculated from the trajectories in |
841 |
< |
simulations. For instance, instantaneous temperature of an |
842 |
< |
Hamiltonian system of $N$ particle can be measured by |
841 |
> |
simulations. For instance, instantaneous temperature of a |
842 |
> |
Hamiltonian system of $N$ particles can be measured by |
843 |
|
\[ |
844 |
|
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
845 |
|
\] |
846 |
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
847 |
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
848 |
< |
the boltzman constant. |
848 |
> |
the Boltzman constant. |
849 |
|
|
850 |
|
A typical molecular dynamics run consists of three essential steps: |
851 |
|
\begin{enumerate} |
862 |
|
These three individual steps will be covered in the following |
863 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
864 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
865 |
< |
will discusse issues in production run. |
865 |
> |
will discuss issues of production runs. |
866 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
867 |
< |
trajectory analysis. |
867 |
> |
analysis of trajectories. |
868 |
|
|
869 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
870 |
|
|
876 |
|
thousands of crystal structures of molecules are discovered every |
877 |
|
year, many more remain unknown due to the difficulties of |
878 |
|
purification and crystallization. Even for molecules with known |
879 |
< |
structure, some important information is missing. For example, a |
879 |
> |
structures, some important information is missing. For example, a |
880 |
|
missing hydrogen atom which acts as donor in hydrogen bonding must |
881 |
< |
be added. Moreover, in order to include electrostatic interaction, |
881 |
> |
be added. Moreover, in order to include electrostatic interactions, |
882 |
|
one may need to specify the partial charges for individual atoms. |
883 |
|
Under some circumstances, we may even need to prepare the system in |
884 |
|
a special configuration. For instance, when studying transport |
898 |
|
surface and to locate the local minimum. While converging slowly |
899 |
|
near the minimum, steepest descent method is extremely robust when |
900 |
|
systems are strongly anharmonic. Thus, it is often used to refine |
901 |
< |
structure from crystallographic data. Relied on the gradient or |
902 |
< |
hessian, advanced methods like Newton-Raphson converge rapidly to a |
903 |
< |
local minimum, but become unstable if the energy surface is far from |
901 |
> |
structures from crystallographic data. Relying on the Hessian, |
902 |
> |
advanced methods like Newton-Raphson converge rapidly to a local |
903 |
> |
minimum, but become unstable if the energy surface is far from |
904 |
|
quadratic. Another factor that must be taken into account, when |
905 |
|
choosing energy minimization method, is the size of the system. |
906 |
|
Steepest descent and conjugate gradient can deal with models of any |
907 |
|
size. Because of the limits on computer memory to store the hessian |
908 |
< |
matrix and the computing power needed to diagonalized these |
909 |
< |
matrices, most Newton-Raphson methods can not be used with very |
934 |
< |
large systems. |
908 |
> |
matrix and the computing power needed to diagonalize these matrices, |
909 |
> |
most Newton-Raphson methods can not be used with very large systems. |
910 |
|
|
911 |
|
\subsubsection{\textbf{Heating}} |
912 |
|
|
913 |
< |
Typically, Heating is performed by assigning random velocities |
913 |
> |
Typically, heating is performed by assigning random velocities |
914 |
|
according to a Maxwell-Boltzman distribution for a desired |
915 |
|
temperature. Beginning at a lower temperature and gradually |
916 |
|
increasing the temperature by assigning larger random velocities, we |
917 |
< |
end up with setting the temperature of the system to a final |
918 |
< |
temperature at which the simulation will be conducted. In heating |
919 |
< |
phase, we should also keep the system from drifting or rotating as a |
920 |
< |
whole. To do this, the net linear momentum and angular momentum of |
921 |
< |
the system is shifted to zero after each resampling from the Maxwell |
922 |
< |
-Boltzman distribution. |
917 |
> |
end up setting the temperature of the system to a final temperature |
918 |
> |
at which the simulation will be conducted. In heating phase, we |
919 |
> |
should also keep the system from drifting or rotating as a whole. To |
920 |
> |
do this, the net linear momentum and angular momentum of the system |
921 |
> |
is shifted to zero after each resampling from the Maxwell -Boltzman |
922 |
> |
distribution. |
923 |
|
|
924 |
|
\subsubsection{\textbf{Equilibration}} |
925 |
|
|
946 |
|
calculation of non-bonded forces, such as van der Waals force and |
947 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
948 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
949 |
< |
which making large simulations prohibitive in the absence of any |
950 |
< |
algorithmic tricks. |
951 |
< |
|
952 |
< |
A natural approach to avoid system size issues is to represent the |
953 |
< |
bulk behavior by a finite number of the particles. However, this |
954 |
< |
approach will suffer from the surface effect at the edges of the |
955 |
< |
simulation. To offset this, \textit{Periodic boundary conditions} |
956 |
< |
(see Fig.~\ref{introFig:pbc}) is developed to simulate bulk |
957 |
< |
properties with a relatively small number of particles. In this |
958 |
< |
method, the simulation box is replicated throughout space to form an |
959 |
< |
infinite lattice. During the simulation, when a particle moves in |
960 |
< |
the primary cell, its image in other cells move in exactly the same |
961 |
< |
direction with exactly the same orientation. Thus, as a particle |
987 |
< |
leaves the primary cell, one of its images will enter through the |
988 |
< |
opposite face. |
949 |
> |
which makes large simulations prohibitive in the absence of any |
950 |
> |
algorithmic tricks. A natural approach to avoid system size issues |
951 |
> |
is to represent the bulk behavior by a finite number of the |
952 |
> |
particles. However, this approach will suffer from surface effects |
953 |
> |
at the edges of the simulation. To offset this, \textit{Periodic |
954 |
> |
boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to |
955 |
> |
simulate bulk properties with a relatively small number of |
956 |
> |
particles. In this method, the simulation box is replicated |
957 |
> |
throughout space to form an infinite lattice. During the simulation, |
958 |
> |
when a particle moves in the primary cell, its image in other cells |
959 |
> |
move in exactly the same direction with exactly the same |
960 |
> |
orientation. Thus, as a particle leaves the primary cell, one of its |
961 |
> |
images will enter through the opposite face. |
962 |
|
\begin{figure} |
963 |
|
\centering |
964 |
|
\includegraphics[width=\linewidth]{pbc.eps} |
970 |
|
|
971 |
|
%cutoff and minimum image convention |
972 |
|
Another important technique to improve the efficiency of force |
973 |
< |
evaluation is to apply spherical cutoff where particles farther than |
974 |
< |
a predetermined distance are not included in the calculation |
973 |
> |
evaluation is to apply spherical cutoffs where particles farther |
974 |
> |
than a predetermined distance are not included in the calculation |
975 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
976 |
|
discontinuity in the potential energy curve. Fortunately, one can |
977 |
< |
shift simple radial potential to ensure the potential curve go |
977 |
> |
shift a simple radial potential to ensure the potential curve go |
978 |
|
smoothly to zero at the cutoff radius. The cutoff strategy works |
979 |
|
well for Lennard-Jones interaction because of its short range |
980 |
|
nature. However, simply truncating the electrostatic interaction |
1020 |
|
Recently, advanced visualization technique have become applied to |
1021 |
|
monitor the motions of molecules. Although the dynamics of the |
1022 |
|
system can be described qualitatively from animation, quantitative |
1023 |
< |
trajectory analysis are more useful. According to the principles of |
1024 |
< |
Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics}, |
1025 |
< |
one can compute thermodynamic properties, analyze fluctuations of |
1026 |
< |
structural parameters, and investigate time-dependent processes of |
1027 |
< |
the molecule from the trajectories. |
1023 |
> |
trajectory analysis is more useful. According to the principles of |
1024 |
> |
Statistical Mechanics in |
1025 |
> |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1026 |
> |
thermodynamic properties, analyze fluctuations of structural |
1027 |
> |
parameters, and investigate time-dependent processes of the molecule |
1028 |
> |
from the trajectories. |
1029 |
|
|
1030 |
|
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
1031 |
|
|
1055 |
|
distribution functions. Among these functions,the \emph{pair |
1056 |
|
distribution function}, also known as \emph{radial distribution |
1057 |
|
function}, is of most fundamental importance to liquid theory. |
1058 |
< |
Experimentally, pair distribution function can be gathered by |
1058 |
> |
Experimentally, pair distribution functions can be gathered by |
1059 |
|
Fourier transforming raw data from a series of neutron diffraction |
1060 |
|
experiments and integrating over the surface factor |
1061 |
|
\cite{Powles1973}. The experimental results can serve as a criterion |
1062 |
|
to justify the correctness of a liquid model. Moreover, various |
1063 |
|
equilibrium thermodynamic and structural properties can also be |
1064 |
< |
expressed in terms of radial distribution function \cite{Allen1987}. |
1065 |
< |
|
1066 |
< |
The pair distribution functions $g(r)$ gives the probability that a |
1067 |
< |
particle $i$ will be located at a distance $r$ from a another |
1068 |
< |
particle $j$ in the system |
1095 |
< |
\[ |
1064 |
> |
expressed in terms of the radial distribution function |
1065 |
> |
\cite{Allen1987}. The pair distribution functions $g(r)$ gives the |
1066 |
> |
probability that a particle $i$ will be located at a distance $r$ |
1067 |
> |
from a another particle $j$ in the system |
1068 |
> |
\begin{equation} |
1069 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1070 |
|
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
1071 |
|
(r)}{\rho}. |
1072 |
< |
\] |
1072 |
> |
\end{equation} |
1073 |
|
Note that the delta function can be replaced by a histogram in |
1074 |
|
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
1075 |
|
the height of these peaks gradually decreases to 1 as the liquid of |
1096 |
|
\right\rangle } dt |
1097 |
|
\] |
1098 |
|
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
1099 |
< |
function, which is averaging over time origins and over all the |
1100 |
< |
atoms, the dipole autocorrelation functions are calculated for the |
1099 |
> |
function, which is averaged over time origins and over all the |
1100 |
> |
atoms, the dipole autocorrelation functions is calculated for the |
1101 |
|
entire system. The dipole autocorrelation function is given by: |
1102 |
|
\[ |
1103 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1106 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
1107 |
|
by |
1108 |
|
\[ |
1109 |
< |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1109 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
1110 |
|
\] |
1111 |
< |
In principle, many time correlation functions can be related with |
1111 |
> |
In principle, many time correlation functions can be related to |
1112 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
1113 |
|
scattering spectra of molecular liquids. In practice, one can |
1114 |
< |
extract the IR spectrum from the intensity of dipole fluctuation at |
1115 |
< |
each frequency using the following relationship: |
1114 |
> |
extract the IR spectrum from the intensity of the molecular dipole |
1115 |
> |
fluctuation at each frequency using the following relationship: |
1116 |
|
\[ |
1117 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1118 |
< |
i2\pi vt} dt} |
1118 |
> |
i2\pi vt} dt}. |
1119 |
|
\] |
1120 |
|
|
1121 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1122 |
|
|
1123 |
|
Rigid bodies are frequently involved in the modeling of different |
1124 |
|
areas, from engineering, physics, to chemistry. For example, |
1125 |
< |
missiles and vehicle are usually modeled by rigid bodies. The |
1126 |
< |
movement of the objects in 3D gaming engine or other physics |
1127 |
< |
simulator is governed by rigid body dynamics. In molecular |
1125 |
> |
missiles and vehicles are usually modeled by rigid bodies. The |
1126 |
> |
movement of the objects in 3D gaming engines or other physics |
1127 |
> |
simulators is governed by rigid body dynamics. In molecular |
1128 |
|
simulations, rigid bodies are used to simplify protein-protein |
1129 |
|
docking studies\cite{Gray2003}. |
1130 |
|
|
1133 |
|
freedom. Euler angles are the natural choice to describe the |
1134 |
|
rotational degrees of freedom. However, due to $\frac {1}{sin |
1135 |
|
\theta}$ singularities, the numerical integration of corresponding |
1136 |
< |
equations of motion is very inefficient and inaccurate. Although an |
1137 |
< |
alternative integrator using multiple sets of Euler angles can |
1138 |
< |
overcome this difficulty\cite{Barojas1973}, the computational |
1139 |
< |
penalty and the loss of angular momentum conservation still remain. |
1140 |
< |
A singularity-free representation utilizing quaternions was |
1141 |
< |
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
1142 |
< |
approach uses a nonseparable Hamiltonian resulting from the |
1143 |
< |
quaternion representation, which prevents the symplectic algorithm |
1144 |
< |
to be utilized. Another different approach is to apply holonomic |
1145 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
1146 |
< |
moves independently under the normal forces deriving from potential |
1147 |
< |
energy and constraint forces which are used to guarantee the |
1148 |
< |
rigidness. However, due to their iterative nature, the SHAKE and |
1149 |
< |
Rattle algorithms also converge very slowly when the number of |
1150 |
< |
constraints increases\cite{Ryckaert1977, Andersen1983}. |
1136 |
> |
equations of these motion is very inefficient and inaccurate. |
1137 |
> |
Although an alternative integrator using multiple sets of Euler |
1138 |
> |
angles can overcome this difficulty\cite{Barojas1973}, the |
1139 |
> |
computational penalty and the loss of angular momentum conservation |
1140 |
> |
still remain. A singularity-free representation utilizing |
1141 |
> |
quaternions was developed by Evans in 1977\cite{Evans1977}. |
1142 |
> |
Unfortunately, this approach uses a nonseparable Hamiltonian |
1143 |
> |
resulting from the quaternion representation, which prevents the |
1144 |
> |
symplectic algorithm from being utilized. Another different approach |
1145 |
> |
is to apply holonomic constraints to the atoms belonging to the |
1146 |
> |
rigid body. Each atom moves independently under the normal forces |
1147 |
> |
deriving from potential energy and constraint forces which are used |
1148 |
> |
to guarantee the rigidness. However, due to their iterative nature, |
1149 |
> |
the SHAKE and Rattle algorithms also converge very slowly when the |
1150 |
> |
number of constraints increases\cite{Ryckaert1977, Andersen1983}. |
1151 |
|
|
1152 |
|
A break-through in geometric literature suggests that, in order to |
1153 |
|
develop a long-term integration scheme, one should preserve the |
1154 |
< |
symplectic structure of the flow. By introducing a conjugate |
1154 |
> |
symplectic structure of the propagator. By introducing a conjugate |
1155 |
|
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
1156 |
|
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
1157 |
|
proposed to evolve the Hamiltonian system in a constraint manifold |
1159 |
|
An alternative method using the quaternion representation was |
1160 |
|
developed by Omelyan\cite{Omelyan1998}. However, both of these |
1161 |
|
methods are iterative and inefficient. In this section, we descibe a |
1162 |
< |
symplectic Lie-Poisson integrator for rigid body developed by |
1162 |
> |
symplectic Lie-Poisson integrator for rigid bodies developed by |
1163 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
1164 |
|
|
1165 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
1183 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1184 |
|
\end{equation} |
1185 |
|
which is used to ensure rotation matrix's unitarity. Differentiating |
1186 |
< |
\ref{introEquation:orthogonalConstraint} and using Equation |
1187 |
< |
\ref{introEquation:RBMotionMomentum}, one may obtain, |
1186 |
> |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
1187 |
> |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
1188 |
|
\begin{equation} |
1189 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1190 |
|
\label{introEquation:RBFirstOrderConstraint} |
1191 |
|
\end{equation} |
1219 |
– |
|
1192 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
1193 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1194 |
|
the equations of motion, |
1223 |
– |
|
1195 |
|
\begin{eqnarray} |
1196 |
< |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1197 |
< |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1198 |
< |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1196 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
1197 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\ |
1198 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
1199 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1200 |
|
\end{eqnarray} |
1230 |
– |
|
1201 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1202 |
|
We can use a constraint force provided by a Lagrange multiplier on |
1203 |
|
the normal manifold to keep the motion on constraint space. Or we |
1209 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
1210 |
|
\right\}. |
1211 |
|
\] |
1242 |
– |
|
1212 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
1213 |
|
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
1214 |
|
rotation group $SO(3)$. However, it turns out that under symplectic |
1223 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
1224 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
1225 |
|
\] |
1257 |
– |
|
1226 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
1227 |
|
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
1228 |
|
given as |
1241 |
|
\[ |
1242 |
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
1243 |
|
\] |
1244 |
< |
respectively. |
1245 |
< |
|
1246 |
< |
As a common choice to describe the rotation dynamics of the rigid |
1279 |
< |
body, the angular momentum on the body fixed frame $\Pi = Q^t P$ is |
1280 |
< |
introduced to rewrite the equations of motion, |
1244 |
> |
respectively. As a common choice to describe the rotation dynamics |
1245 |
> |
of the rigid body, the angular momentum on the body fixed frame $\Pi |
1246 |
> |
= Q^t P$ is introduced to rewrite the equations of motion, |
1247 |
|
\begin{equation} |
1248 |
|
\begin{array}{l} |
1249 |
< |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
1250 |
< |
\dot Q = Q\Pi {\rm{ }}J^{ - 1} \\ |
1249 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda, \\ |
1250 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1}, \\ |
1251 |
|
\end{array} |
1252 |
|
\label{introEqaution:RBMotionPI} |
1253 |
|
\end{equation} |
1254 |
< |
, as well as holonomic constraints, |
1255 |
< |
\[ |
1256 |
< |
\begin{array}{l} |
1291 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
1292 |
< |
Q^T Q = 1 \\ |
1293 |
< |
\end{array} |
1294 |
< |
\] |
1295 |
< |
|
1296 |
< |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1297 |
< |
so(3)^ \star$, the hat-map isomorphism, |
1254 |
> |
as well as holonomic constraints $\Pi J^{ - 1} + J^{ - 1} \Pi ^t = |
1255 |
> |
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a |
1256 |
> |
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism, |
1257 |
|
\begin{equation} |
1258 |
|
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1259 |
|
{\begin{array}{*{20}c} |
1266 |
|
will let us associate the matrix products with traditional vector |
1267 |
|
operations |
1268 |
|
\[ |
1269 |
< |
\hat vu = v \times u |
1269 |
> |
\hat vu = v \times u. |
1270 |
|
\] |
1271 |
< |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
1271 |
> |
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
1272 |
|
matrix, |
1273 |
< |
|
1274 |
< |
\begin{eqnarray*} |
1275 |
< |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
1276 |
< |
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
1277 |
< |
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
1278 |
< |
\label{introEquation:skewMatrixPI} |
1279 |
< |
\end{eqnarray*} |
1280 |
< |
|
1281 |
< |
Since $\Lambda$ is symmetric, the last term of Equation |
1282 |
< |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
1283 |
< |
multiplier $\Lambda$ is absent from the equations of motion. This |
1284 |
< |
unique property eliminates the requirement of iterations which can |
1326 |
< |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
1327 |
< |
|
1328 |
< |
Applying the hat-map isomorphism, we obtain the equation of motion |
1329 |
< |
for angular momentum on body frame |
1273 |
> |
\begin{eqnarray} |
1274 |
> |
(\dot \Pi - \dot \Pi ^T )&= &(\Pi - \Pi ^T )(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
1275 |
> |
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1276 |
> |
(\Lambda - \Lambda ^T ). \label{introEquation:skewMatrixPI} |
1277 |
> |
\end{eqnarray} |
1278 |
> |
Since $\Lambda$ is symmetric, the last term of |
1279 |
> |
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
1280 |
> |
Lagrange multiplier $\Lambda$ is absent from the equations of |
1281 |
> |
motion. This unique property eliminates the requirement of |
1282 |
> |
iterations which can not be avoided in other methods\cite{Kol1997, |
1283 |
> |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
1284 |
> |
equation of motion for angular momentum on body frame |
1285 |
|
\begin{equation} |
1286 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
1287 |
|
F_i (r,Q)} \right) \times X_i }. |
1290 |
|
In the same manner, the equation of motion for rotation matrix is |
1291 |
|
given by |
1292 |
|
\[ |
1293 |
< |
\dot Q = Qskew(I^{ - 1} \pi ) |
1293 |
> |
\dot Q = Qskew(I^{ - 1} \pi ). |
1294 |
|
\] |
1295 |
|
|
1296 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
1312 |
|
0 & {\pi _3 } & { - \pi _2 } \\ |
1313 |
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
1314 |
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
1315 |
< |
\end{array}} \right) |
1315 |
> |
\end{array}} \right). |
1316 |
|
\end{equation} |
1317 |
|
Thus, the dynamics of free rigid body is governed by |
1318 |
|
\begin{equation} |
1319 |
< |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
1319 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
1320 |
|
\end{equation} |
1321 |
< |
|
1322 |
< |
One may notice that each $T_i^r$ in Equation |
1323 |
< |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1369 |
< |
instance, the equations of motion due to $T_1^r$ are given by |
1321 |
> |
One may notice that each $T_i^r$ in |
1322 |
> |
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly. |
1323 |
> |
For instance, the equations of motion due to $T_1^r$ are given by |
1324 |
|
\begin{equation} |
1325 |
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
1326 |
|
\label{introEqaution:RBMotionSingleTerm} |
1327 |
|
\end{equation} |
1328 |
< |
where |
1328 |
> |
with |
1329 |
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
1330 |
|
0 & 0 & 0 \\ |
1331 |
|
0 & 0 & {\pi _1 } \\ |
1332 |
|
0 & { - \pi _1 } & 0 \\ |
1333 |
|
\end{array}} \right). |
1334 |
|
\] |
1335 |
< |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1335 |
> |
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is |
1336 |
|
\[ |
1337 |
|
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
1338 |
|
Q(0)e^{\Delta tR_1 } |
1350 |
|
propagator, |
1351 |
|
\[ |
1352 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
1353 |
< |
) |
1353 |
> |
). |
1354 |
|
\] |
1355 |
< |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1355 |
> |
The propagator maps for $T_2^r$ and $T_3^r$ can be found in the same |
1356 |
|
manner. In order to construct a second-order symplectic method, we |
1357 |
< |
split the angular kinetic Hamiltonian function can into five terms |
1357 |
> |
split the angular kinetic Hamiltonian function into five terms |
1358 |
|
\[ |
1359 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1360 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1368 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
1369 |
|
_1 }. |
1370 |
|
\] |
1417 |
– |
|
1371 |
|
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
1372 |
|
$F(\pi )$ and $G(\pi )$ is defined by |
1373 |
|
\[ |
1374 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
1375 |
< |
) |
1375 |
> |
). |
1376 |
|
\] |
1377 |
|
If the Poisson bracket of a function $F$ with an arbitrary smooth |
1378 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
1383 |
|
then by the chain rule |
1384 |
|
\[ |
1385 |
|
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
1386 |
< |
}}{2})\pi |
1386 |
> |
}}{2})\pi. |
1387 |
|
\] |
1388 |
< |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
1388 |
> |
Thus, $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel |
1389 |
> |
\pi |
1390 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
1391 |
|
Lie-Poisson integrator is found to be both extremely efficient and |
1392 |
|
stable. These properties can be explained by the fact the small |
1397 |
|
Splitting for Rigid Body} |
1398 |
|
|
1399 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
1400 |
< |
energy and potential energy, |
1401 |
< |
\[ |
1402 |
< |
H = T(p,\pi ) + V(q,Q) |
1449 |
< |
\] |
1450 |
< |
The equations of motion corresponding to potential energy and |
1451 |
< |
kinetic energy are listed in the below table, |
1400 |
> |
energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations |
1401 |
> |
of motion corresponding to potential energy and kinetic energy are |
1402 |
> |
listed in the below table, |
1403 |
|
\begin{table} |
1404 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
1405 |
|
\begin{center} |
1437 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
1438 |
|
\end{equation} |
1439 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
1440 |
< |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
1441 |
< |
corresponding propagators are given by |
1440 |
> |
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore, |
1441 |
> |
the corresponding propagators are given by |
1442 |
|
\[ |
1443 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
1444 |
|
_{\Delta t,T^r }. |
1445 |
|
\] |
1446 |
|
Finally, we obtain the overall symplectic propagators for freely |
1447 |
|
moving rigid bodies |
1448 |
< |
\begin{equation} |
1449 |
< |
\begin{array}{c} |
1450 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
1451 |
< |
\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 } \\ |
1501 |
< |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1502 |
< |
\end{array} |
1448 |
> |
\begin{eqnarray} |
1449 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
1450 |
> |
& & \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\\ |
1451 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1452 |
|
\label{introEquation:overallRBFlowMaps} |
1453 |
< |
\end{equation} |
1453 |
> |
\end{eqnarray} |
1454 |
|
|
1455 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1456 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
1496 |
|
\[ |
1497 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
1498 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
1499 |
< |
\] and combining the last two terms in Equation |
1500 |
< |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
1552 |
< |
Hamiltonian as |
1499 |
> |
\] |
1500 |
> |
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as |
1501 |
|
\[ |
1502 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
1503 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
1504 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
1505 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
1505 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}}. |
1506 |
|
\] |
1507 |
|
Since the first two terms of the new Hamiltonian depend only on the |
1508 |
|
system coordinates, we can get the equations of motion for |
1519 |
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
1520 |
|
\label{introEquation:bathMotionGLE} |
1521 |
|
\end{equation} |
1574 |
– |
|
1522 |
|
In order to derive an equation for $x$, the dynamics of the bath |
1523 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
1524 |
|
transform which is particularly useful in solving linear ordinary |
1527 |
|
differential equations into simple algebra problems which can be |
1528 |
|
solved easily. Then, by applying the inverse Laplace transform, also |
1529 |
|
known as the Bromwich integral, we can retrieve the solutions of the |
1530 |
< |
original problems. |
1531 |
< |
|
1585 |
< |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
1586 |
< |
transform of f(t) is a new function defined as |
1530 |
> |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
1531 |
> |
$, the Laplace transform of $f(t)$ is a new function defined as |
1532 |
|
\[ |
1533 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1534 |
|
\] |
1535 |
|
where $p$ is real and $L$ is called the Laplace Transform |
1536 |
|
Operator. Below are some important properties of Laplace transform |
1592 |
– |
|
1537 |
|
\begin{eqnarray*} |
1538 |
|
L(x + y) & = & L(x) + L(y) \\ |
1539 |
|
L(ax) & = & aL(x) \\ |
1541 |
|
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
1542 |
|
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
1543 |
|
\end{eqnarray*} |
1600 |
– |
|
1601 |
– |
|
1544 |
|
Applying the Laplace transform to the bath coordinates, we obtain |
1545 |
|
\begin{eqnarray*} |
1546 |
< |
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) \\ |
1547 |
< |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1546 |
> |
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), \\ |
1547 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
1548 |
|
\end{eqnarray*} |
1607 |
– |
|
1549 |
|
By the same way, the system coordinates become |
1550 |
|
\begin{eqnarray*} |
1551 |
< |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
1552 |
< |
& & \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\}} \\ |
1551 |
> |
mL(\ddot x) & = & |
1552 |
> |
- \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\}} \\ |
1553 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}. |
1554 |
|
\end{eqnarray*} |
1613 |
– |
|
1555 |
|
With the help of some relatively important inverse Laplace |
1556 |
|
transformations: |
1557 |
|
\[ |
1561 |
|
L(1) = \frac{1}{p} \\ |
1562 |
|
\end{array} |
1563 |
|
\] |
1564 |
< |
, we obtain |
1564 |
> |
we obtain |
1565 |
|
\begin{eqnarray*} |
1566 |
|
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
1567 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
1609 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
1610 |
|
implies it is completely deterministic within the context of a |
1611 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
1612 |
< |
uncorrelated to $x$ and $\dot x$, |
1613 |
< |
\[ |
1614 |
< |
\begin{array}{l} |
1615 |
< |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
1675 |
< |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
1676 |
< |
\end{array} |
1677 |
< |
\] |
1678 |
< |
This property is what we expect from a truly random process. As long |
1679 |
< |
as the model chosen for $R(t)$ was a gaussian distribution in |
1612 |
> |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
1613 |
> |
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
1614 |
> |
0.$ This property is what we expect from a truly random process. As |
1615 |
> |
long as the model chosen for $R(t)$ was a gaussian distribution in |
1616 |
|
general, the stochastic nature of the GLE still remains. |
1681 |
– |
|
1617 |
|
%dynamic friction kernel |
1618 |
|
The convolution integral |
1619 |
|
\[ |
1628 |
|
\[ |
1629 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
1630 |
|
\] |
1631 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1631 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1632 |
|
\[ |
1633 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
1634 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
1645 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
1646 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
1647 |
|
\] |
1648 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1648 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
1649 |
|
\begin{equation} |
1650 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
1651 |
|
x(t) + R(t) \label{introEquation:LangevinEquation} |
1658 |
|
|
1659 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
1660 |
|
|
1661 |
< |
Defining a new set of coordinates, |
1661 |
> |
Defining a new set of coordinates |
1662 |
|
\[ |
1663 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1664 |
< |
^2 }}x(0) |
1665 |
< |
\], |
1664 |
> |
^2 }}x(0), |
1665 |
> |
\] |
1666 |
|
we can rewrite $R(T)$ as |
1667 |
|
\[ |
1668 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1669 |
|
\] |
1670 |
|
And since the $q$ coordinates are harmonic oscillators, |
1736 |
– |
|
1671 |
|
\begin{eqnarray*} |
1672 |
|
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
1673 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
1676 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
1677 |
|
& = &kT\xi (t) \\ |
1678 |
|
\end{eqnarray*} |
1745 |
– |
|
1679 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1680 |
|
\begin{equation} |
1681 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1682 |
< |
\label{introEquation:secondFluctuationDissipation}. |
1682 |
> |
\label{introEquation:secondFluctuationDissipation}, |
1683 |
|
\end{equation} |
1684 |
< |
In effect, it acts as a constraint on the possible ways in which one |
1685 |
< |
can model the random force and friction kernel. |
1684 |
> |
which acts as a constraint on the possible ways in which one can |
1685 |
> |
model the random force and friction kernel. |