| 27 |
|
\end{equation} |
| 28 |
|
A point mass interacting with other bodies moves with the |
| 29 |
|
acceleration along the direction of the force acting on it. Let |
| 30 |
< |
$F_ij$ be the force that particle $i$ exerts on particle $j$, and |
| 31 |
< |
$F_ji$ be the force that particle $j$ exerts on particle $i$. |
| 30 |
> |
$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and |
| 31 |
> |
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
| 32 |
|
Newton¡¯s third law states that |
| 33 |
|
\begin{equation} |
| 34 |
< |
F_ij = -F_ji |
| 34 |
> |
F_{ij} = -F_{ji} |
| 35 |
|
\label{introEquation:newtonThirdLaw} |
| 36 |
|
\end{equation} |
| 37 |
|
|
| 117 |
|
\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
| 118 |
|
Equations of Motion in Lagrangian Mechanics} |
| 119 |
|
|
| 120 |
< |
for a holonomic system of $f$ degrees of freedom, the equations of |
| 120 |
> |
For a holonomic system of $f$ degrees of freedom, the equations of |
| 121 |
|
motion in the Lagrangian form is |
| 122 |
|
\begin{equation} |
| 123 |
|
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - |
| 212 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
| 213 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
| 214 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
| 215 |
< |
q_i }}} \right) = 0} |
| 216 |
< |
\label{introEquation:conserveHalmitonian} |
| 215 |
> |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
| 216 |
|
\end{equation} |
| 217 |
|
|
| 218 |
< |
When studying Hamiltonian system, it is more convenient to use |
| 219 |
< |
notation |
| 218 |
> |
\section{\label{introSection:statisticalMechanics}Statistical |
| 219 |
> |
Mechanics} |
| 220 |
> |
|
| 221 |
> |
The thermodynamic behaviors and properties of Molecular Dynamics |
| 222 |
> |
simulation are governed by the principle of Statistical Mechanics. |
| 223 |
> |
The following section will give a brief introduction to some of the |
| 224 |
> |
Statistical Mechanics concepts and theorem presented in this |
| 225 |
> |
dissertation. |
| 226 |
> |
|
| 227 |
> |
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
| 228 |
> |
|
| 229 |
> |
Mathematically, phase space is the space which represents all |
| 230 |
> |
possible states. Each possible state of the system corresponds to |
| 231 |
> |
one unique point in the phase space. For mechanical systems, the |
| 232 |
> |
phase space usually consists of all possible values of position and |
| 233 |
> |
momentum variables. Consider a dynamic system in a cartesian space, |
| 234 |
> |
where each of the $6f$ coordinates and momenta is assigned to one of |
| 235 |
> |
$6f$ mutually orthogonal axes, the phase space of this system is a |
| 236 |
> |
$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , |
| 237 |
> |
\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and |
| 238 |
> |
momenta is a phase space vector. |
| 239 |
> |
|
| 240 |
> |
A microscopic state or microstate of a classical system is |
| 241 |
> |
specification of the complete phase space vector of a system at any |
| 242 |
> |
instant in time. An ensemble is defined as a collection of systems |
| 243 |
> |
sharing one or more macroscopic characteristics but each being in a |
| 244 |
> |
unique microstate. The complete ensemble is specified by giving all |
| 245 |
> |
systems or microstates consistent with the common macroscopic |
| 246 |
> |
characteristics of the ensemble. Although the state of each |
| 247 |
> |
individual system in the ensemble could be precisely described at |
| 248 |
> |
any instance in time by a suitable phase space vector, when using |
| 249 |
> |
ensembles for statistical purposes, there is no need to maintain |
| 250 |
> |
distinctions between individual systems, since the numbers of |
| 251 |
> |
systems at any time in the different states which correspond to |
| 252 |
> |
different regions of the phase space are more interesting. Moreover, |
| 253 |
> |
in the point of view of statistical mechanics, one would prefer to |
| 254 |
> |
use ensembles containing a large enough population of separate |
| 255 |
> |
members so that the numbers of systems in such different states can |
| 256 |
> |
be regarded as changing continuously as we traverse different |
| 257 |
> |
regions of the phase space. The condition of an ensemble at any time |
| 258 |
> |
can be regarded as appropriately specified by the density $\rho$ |
| 259 |
> |
with which representative points are distributed over the phase |
| 260 |
> |
space. The density of distribution for an ensemble with $f$ degrees |
| 261 |
> |
of freedom is defined as, |
| 262 |
|
\begin{equation} |
| 263 |
< |
r = r(q,p)^T |
| 263 |
> |
\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). |
| 264 |
> |
\label{introEquation:densityDistribution} |
| 265 |
|
\end{equation} |
| 266 |
< |
and to introduce a $2n \times 2n$ canonical structure matrix $J$, |
| 266 |
> |
Governed by the principles of mechanics, the phase points change |
| 267 |
> |
their value which would change the density at any time at phase |
| 268 |
> |
space. Hence, the density of distribution is also to be taken as a |
| 269 |
> |
function of the time. |
| 270 |
> |
|
| 271 |
> |
The number of systems $\delta N$ at time $t$ can be determined by, |
| 272 |
|
\begin{equation} |
| 273 |
< |
J = \left( {\begin{array}{*{20}c} |
| 274 |
< |
0 & I \\ |
| 228 |
< |
{ - I} & 0 \\ |
| 229 |
< |
\end{array}} \right) |
| 230 |
< |
\label{introEquation:canonicalMatrix} |
| 273 |
> |
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
| 274 |
> |
\label{introEquation:deltaN} |
| 275 |
|
\end{equation} |
| 276 |
< |
where $I$ is a $n \times n$ identity matrix and $J$ is a |
| 277 |
< |
skew-symmetric matrix ($ J^T = - J $). Thus, Hamiltonian system |
| 278 |
< |
can be rewritten as, |
| 276 |
> |
Assuming a large enough population of systems are exploited, we can |
| 277 |
> |
sufficiently approximate $\delta N$ without introducing |
| 278 |
> |
discontinuity when we go from one region in the phase space to |
| 279 |
> |
another. By integrating over the whole phase space, |
| 280 |
|
\begin{equation} |
| 281 |
< |
\frac{d}{{dt}}r = J\nabla _r H(r) |
| 282 |
< |
\label{introEquation:compactHamiltonian} |
| 281 |
> |
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
| 282 |
> |
\label{introEquation:totalNumberSystem} |
| 283 |
|
\end{equation} |
| 284 |
+ |
gives us an expression for the total number of the systems. Hence, |
| 285 |
+ |
the probability per unit in the phase space can be obtained by, |
| 286 |
+ |
\begin{equation} |
| 287 |
+ |
\frac{{\rho (q,p,t)}}{N} = \frac{{\rho (q,p,t)}}{{\int { \ldots \int |
| 288 |
+ |
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 289 |
+ |
\label{introEquation:unitProbability} |
| 290 |
+ |
\end{equation} |
| 291 |
+ |
With the help of Equation(\ref{introEquation:unitProbability}) and |
| 292 |
+ |
the knowledge of the system, it is possible to calculate the average |
| 293 |
+ |
value of any desired quantity which depends on the coordinates and |
| 294 |
+ |
momenta of the system. Even when the dynamics of the real system is |
| 295 |
+ |
complex, or stochastic, or even discontinuous, the average |
| 296 |
+ |
properties of the ensemble of possibilities as a whole may still |
| 297 |
+ |
remain well defined. For a classical system in thermal equilibrium |
| 298 |
+ |
with its environment, the ensemble average of a mechanical quantity, |
| 299 |
+ |
$\langle A(q , p) \rangle_t$, takes the form of an integral over the |
| 300 |
+ |
phase space of the system, |
| 301 |
+ |
\begin{equation} |
| 302 |
+ |
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
| 303 |
+ |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
| 304 |
+ |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
| 305 |
+ |
\label{introEquation:ensembelAverage} |
| 306 |
+ |
\end{equation} |
| 307 |
|
|
| 308 |
< |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 308 |
> |
There are several different types of ensembles with different |
| 309 |
> |
statistical characteristics. As a function of macroscopic |
| 310 |
> |
parameters, such as temperature \textit{etc}, partition function can |
| 311 |
> |
be used to describe the statistical properties of a system in |
| 312 |
> |
thermodynamic equilibrium. |
| 313 |
|
|
| 314 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 315 |
< |
A \emph{manifold} is an abstract mathematical space. It locally |
| 316 |
< |
looks like Euclidean space, but when viewed globally, it may have |
| 317 |
< |
more complicate structure. A good example of manifold is the surface |
| 318 |
< |
of Earth. It seems to be flat locally, but it is round if viewed as |
| 319 |
< |
a whole. A \emph{differentiable manifold} (also known as |
| 320 |
< |
\emph{smooth manifold}) is a manifold with an open cover in which |
| 321 |
< |
the covering neighborhoods are all smoothly isomorphic to one |
| 322 |
< |
another. In other words,it is possible to apply calculus on |
| 323 |
< |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
| 324 |
< |
defined as a pair $(M, \omega)$ consisting of a \emph{differentiable |
| 325 |
< |
manifold} $M$ and a close, non-degenerated, bilinear symplectic |
| 326 |
< |
form, $\omega$. One of the motivation to study \emph{symplectic |
| 327 |
< |
manifold} in Hamiltonian Mechanics is that a symplectic manifold can |
| 328 |
< |
represent all possible configurations of the system and the phase |
| 329 |
< |
space of the system can be described by it's cotangent bundle. Every |
| 330 |
< |
symplectic manifold is even dimensional. For instance, in Hamilton |
| 331 |
< |
equations, coordinate and momentum always appear in pairs. |
| 314 |
> |
As an ensemble of systems, each of which is known to be thermally |
| 315 |
> |
isolated and conserve energy, Microcanonical ensemble(NVE) has a |
| 316 |
> |
partition function like, |
| 317 |
> |
\begin{equation} |
| 318 |
> |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 319 |
> |
\end{equation} |
| 320 |
> |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
| 321 |
> |
can share its energy with a large heat reservoir. The distribution |
| 322 |
> |
of the total energy amongst the possible dynamical states is given |
| 323 |
> |
by the partition function, |
| 324 |
> |
\begin{equation} |
| 325 |
> |
\Omega (N,V,T) = e^{ - \beta A} |
| 326 |
> |
\label{introEquation:NVTPartition} |
| 327 |
> |
\end{equation} |
| 328 |
> |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 329 |
> |
TS$. Since most experiment are carried out under constant pressure |
| 330 |
> |
condition, isothermal-isobaric ensemble(NPT) play a very important |
| 331 |
> |
role in molecular simulation. The isothermal-isobaric ensemble allow |
| 332 |
> |
the system to exchange energy with a heat bath of temperature $T$ |
| 333 |
> |
and to change the volume as well. Its partition function is given as |
| 334 |
> |
\begin{equation} |
| 335 |
> |
\Delta (N,P,T) = - e^{\beta G}. |
| 336 |
> |
\label{introEquation:NPTPartition} |
| 337 |
> |
\end{equation} |
| 338 |
> |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
| 339 |
|
|
| 340 |
< |
A \emph{symplectomorphism} is also known as a \emph{canonical |
| 262 |
< |
transformation}. |
| 340 |
> |
\subsection{\label{introSection:liouville}Liouville's theorem} |
| 341 |
|
|
| 342 |
< |
Any real-valued differentiable function H on a symplectic manifold |
| 343 |
< |
can serve as an energy function or Hamiltonian. Associated to any |
| 344 |
< |
Hamiltonian is a Hamiltonian vector field; the integral curves of |
| 345 |
< |
the Hamiltonian vector field are solutions to the Hamilton-Jacobi |
| 346 |
< |
equations. The Hamiltonian vector field defines a flow on the |
| 347 |
< |
symplectic manifold, called a Hamiltonian flow or symplectomorphism. |
| 348 |
< |
By Liouville's theorem, Hamiltonian flows preserve the volume form |
| 349 |
< |
on the phase space. |
| 342 |
> |
The Liouville's theorem is the foundation on which statistical |
| 343 |
> |
mechanics rests. It describes the time evolution of phase space |
| 344 |
> |
distribution function. In order to calculate the rate of change of |
| 345 |
> |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
| 346 |
> |
consider the two faces perpendicular to the $q_1$ axis, which are |
| 347 |
> |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
| 348 |
> |
leaving the opposite face is given by the expression, |
| 349 |
> |
\begin{equation} |
| 350 |
> |
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
| 351 |
> |
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
| 352 |
> |
}}\delta q_1 } \right)\delta q_2 \ldots \delta q_f \delta p_1 |
| 353 |
> |
\ldots \delta p_f . |
| 354 |
> |
\end{equation} |
| 355 |
> |
Summing all over the phase space, we obtain |
| 356 |
> |
\begin{equation} |
| 357 |
> |
\frac{{d(\delta N)}}{{dt}} = - \sum\limits_{i = 1}^f {\left[ {\rho |
| 358 |
> |
\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} + |
| 359 |
> |
\frac{{\partial \dot p_i }}{{\partial p_i }}} \right) + \left( |
| 360 |
> |
{\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + \frac{{\partial |
| 361 |
> |
\rho }}{{\partial p_i }}\dot p_i } \right)} \right]} \delta q_1 |
| 362 |
> |
\ldots \delta q_f \delta p_1 \ldots \delta p_f . |
| 363 |
> |
\end{equation} |
| 364 |
> |
Differentiating the equations of motion in Hamiltonian formalism |
| 365 |
> |
(\ref{introEquation:motionHamiltonianCoordinate}, |
| 366 |
> |
\ref{introEquation:motionHamiltonianMomentum}), we can show, |
| 367 |
> |
\begin{equation} |
| 368 |
> |
\sum\limits_i {\left( {\frac{{\partial \dot q_i }}{{\partial q_i }} |
| 369 |
> |
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
| 370 |
> |
\end{equation} |
| 371 |
> |
which cancels the first terms of the right hand side. Furthermore, |
| 372 |
> |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
| 373 |
> |
p_f $ in both sides, we can write out Liouville's theorem in a |
| 374 |
> |
simple form, |
| 375 |
> |
\begin{equation} |
| 376 |
> |
\frac{{\partial \rho }}{{\partial t}} + \sum\limits_{i = 1}^f |
| 377 |
> |
{\left( {\frac{{\partial \rho }}{{\partial q_i }}\dot q_i + |
| 378 |
> |
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
| 379 |
> |
\label{introEquation:liouvilleTheorem} |
| 380 |
> |
\end{equation} |
| 381 |
|
|
| 382 |
< |
\subsection{\label{Construction of Symplectic Methods}} |
| 382 |
> |
Liouville's theorem states that the distribution function is |
| 383 |
> |
constant along any trajectory in phase space. In classical |
| 384 |
> |
statistical mechanics, since the number of particles in the system |
| 385 |
> |
is huge, we may be able to believe the system is stationary, |
| 386 |
> |
\begin{equation} |
| 387 |
> |
\frac{{\partial \rho }}{{\partial t}} = 0. |
| 388 |
> |
\label{introEquation:stationary} |
| 389 |
> |
\end{equation} |
| 390 |
> |
In such stationary system, the density of distribution $\rho$ can be |
| 391 |
> |
connected to the Hamiltonian $H$ through Maxwell-Boltzmann |
| 392 |
> |
distribution, |
| 393 |
> |
\begin{equation} |
| 394 |
> |
\rho \propto e^{ - \beta H} |
| 395 |
> |
\label{introEquation:densityAndHamiltonian} |
| 396 |
> |
\end{equation} |
| 397 |
|
|
| 398 |
< |
\section{\label{introSection:statisticalMechanics}Statistical |
| 399 |
< |
Mechanics} |
| 400 |
< |
|
| 401 |
< |
The thermodynamic behaviors and properties of Molecular Dynamics |
| 402 |
< |
simulation are governed by the principle of Statistical Mechanics. |
| 403 |
< |
The following section will give a brief introduction to some of the |
| 404 |
< |
Statistical Mechanics concepts presented in this dissertation. |
| 398 |
> |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
| 399 |
> |
Lets consider a region in the phase space, |
| 400 |
> |
\begin{equation} |
| 401 |
> |
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
| 402 |
> |
\end{equation} |
| 403 |
> |
If this region is small enough, the density $\rho$ can be regarded |
| 404 |
> |
as uniform over the whole phase space. Thus, the number of phase |
| 405 |
> |
points inside this region is given by, |
| 406 |
> |
\begin{equation} |
| 407 |
> |
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
| 408 |
> |
dp_1 } ..dp_f. |
| 409 |
> |
\end{equation} |
| 410 |
|
|
| 411 |
< |
\subsection{\label{introSection:ensemble}Ensemble and Phase Space} |
| 411 |
> |
\begin{equation} |
| 412 |
> |
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
| 413 |
> |
\frac{d}{{dt}}(\delta v) = 0. |
| 414 |
> |
\end{equation} |
| 415 |
> |
With the help of stationary assumption |
| 416 |
> |
(\ref{introEquation:stationary}), we obtain the principle of the |
| 417 |
> |
\emph{conservation of extension in phase space}, |
| 418 |
> |
\begin{equation} |
| 419 |
> |
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
| 420 |
> |
...dq_f dp_1 } ..dp_f = 0. |
| 421 |
> |
\label{introEquation:volumePreserving} |
| 422 |
> |
\end{equation} |
| 423 |
|
|
| 424 |
+ |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
| 425 |
+ |
|
| 426 |
+ |
Liouville's theorem can be expresses in a variety of different forms |
| 427 |
+ |
which are convenient within different contexts. For any two function |
| 428 |
+ |
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
| 429 |
+ |
bracket ${F, G}$ is defined as |
| 430 |
+ |
\begin{equation} |
| 431 |
+ |
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
| 432 |
+ |
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
| 433 |
+ |
\frac{{\partial F}}{{\partial p_i }}\frac{{\partial G}}{{\partial |
| 434 |
+ |
q_i }}} \right)}. |
| 435 |
+ |
\label{introEquation:poissonBracket} |
| 436 |
+ |
\end{equation} |
| 437 |
+ |
Substituting equations of motion in Hamiltonian formalism( |
| 438 |
+ |
\ref{introEquation:motionHamiltonianCoordinate} , |
| 439 |
+ |
\ref{introEquation:motionHamiltonianMomentum} ) into |
| 440 |
+ |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
| 441 |
+ |
theorem using Poisson bracket notion, |
| 442 |
+ |
\begin{equation} |
| 443 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
| 444 |
+ |
{\rho ,H} \right\}. |
| 445 |
+ |
\label{introEquation:liouvilleTheromInPoissin} |
| 446 |
+ |
\end{equation} |
| 447 |
+ |
Moreover, the Liouville operator is defined as |
| 448 |
+ |
\begin{equation} |
| 449 |
+ |
iL = \sum\limits_{i = 1}^f {\left( {\frac{{\partial H}}{{\partial |
| 450 |
+ |
p_i }}\frac{\partial }{{\partial q_i }} - \frac{{\partial |
| 451 |
+ |
H}}{{\partial q_i }}\frac{\partial }{{\partial p_i }}} \right)} |
| 452 |
+ |
\label{introEquation:liouvilleOperator} |
| 453 |
+ |
\end{equation} |
| 454 |
+ |
In terms of Liouville operator, Liouville's equation can also be |
| 455 |
+ |
expressed as |
| 456 |
+ |
\begin{equation} |
| 457 |
+ |
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
| 458 |
+ |
\label{introEquation:liouvilleTheoremInOperator} |
| 459 |
+ |
\end{equation} |
| 460 |
+ |
|
| 461 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
| 462 |
|
|
| 463 |
|
Various thermodynamic properties can be calculated from Molecular |
| 472 |
|
ensemble average. It states that time average and average over the |
| 473 |
|
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
| 474 |
|
\begin{equation} |
| 475 |
< |
\langle A \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 476 |
< |
\frac{1}{t}\int\limits_0^t {A(p(t),q(t))dt = \int\limits_\Gamma |
| 477 |
< |
{A(p(t),q(t))} } \rho (p(t), q(t)) dpdq |
| 475 |
> |
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 476 |
> |
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 477 |
> |
{A(q(t),p(t))} } \rho (q(t), p(t)) dqdp |
| 478 |
|
\end{equation} |
| 479 |
< |
where $\langle A \rangle_t$ is an equilibrium value of a physical |
| 480 |
< |
quantity and $\rho (p(t), q(t))$ is the equilibrium distribution |
| 481 |
< |
function. If an observation is averaged over a sufficiently long |
| 482 |
< |
time (longer than relaxation time), all accessible microstates in |
| 483 |
< |
phase space are assumed to be equally probed, giving a properly |
| 484 |
< |
weighted statistical average. This allows the researcher freedom of |
| 485 |
< |
choice when deciding how best to measure a given observable. In case |
| 486 |
< |
an ensemble averaged approach sounds most reasonable, the Monte |
| 487 |
< |
Carlo techniques\cite{metropolis:1949} can be utilized. Or if the |
| 488 |
< |
system lends itself to a time averaging approach, the Molecular |
| 489 |
< |
Dynamics techniques in Sec.~\ref{introSection:molecularDynamics} |
| 490 |
< |
will be the best choice\cite{Frenkel1996}. |
| 479 |
> |
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
| 480 |
> |
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
| 481 |
> |
distribution function. If an observation is averaged over a |
| 482 |
> |
sufficiently long time (longer than relaxation time), all accessible |
| 483 |
> |
microstates in phase space are assumed to be equally probed, giving |
| 484 |
> |
a properly weighted statistical average. This allows the researcher |
| 485 |
> |
freedom of choice when deciding how best to measure a given |
| 486 |
> |
observable. In case an ensemble averaged approach sounds most |
| 487 |
> |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
| 488 |
> |
utilized. Or if the system lends itself to a time averaging |
| 489 |
> |
approach, the Molecular Dynamics techniques in |
| 490 |
> |
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 491 |
> |
choice\cite{Frenkel1996}. |
| 492 |
> |
|
| 493 |
> |
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 494 |
> |
A variety of numerical integrators were proposed to simulate the |
| 495 |
> |
motions. They usually begin with an initial conditionals and move |
| 496 |
> |
the objects in the direction governed by the differential equations. |
| 497 |
> |
However, most of them ignore the hidden physical law contained |
| 498 |
> |
within the equations. Since 1990, geometric integrators, which |
| 499 |
> |
preserve various phase-flow invariants such as symplectic structure, |
| 500 |
> |
volume and time reversal symmetry, are developed to address this |
| 501 |
> |
issue. The velocity verlet method, which happens to be a simple |
| 502 |
> |
example of symplectic integrator, continues to gain its popularity |
| 503 |
> |
in molecular dynamics community. This fact can be partly explained |
| 504 |
> |
by its geometric nature. |
| 505 |
> |
|
| 506 |
> |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 507 |
> |
A \emph{manifold} is an abstract mathematical space. It locally |
| 508 |
> |
looks like Euclidean space, but when viewed globally, it may have |
| 509 |
> |
more complicate structure. A good example of manifold is the surface |
| 510 |
> |
of Earth. It seems to be flat locally, but it is round if viewed as |
| 511 |
> |
a whole. A \emph{differentiable manifold} (also known as |
| 512 |
> |
\emph{smooth manifold}) is a manifold with an open cover in which |
| 513 |
> |
the covering neighborhoods are all smoothly isomorphic to one |
| 514 |
> |
another. In other words,it is possible to apply calculus on |
| 515 |
> |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
| 516 |
> |
defined as a pair $(M, \omega)$ which consisting of a |
| 517 |
> |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 518 |
> |
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 519 |
> |
space $V$ is a function $\omega(x, y)$ which satisfies |
| 520 |
> |
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 521 |
> |
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 522 |
> |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
| 523 |
> |
example of symplectic form. |
| 524 |
> |
|
| 525 |
> |
One of the motivations to study \emph{symplectic manifold} in |
| 526 |
> |
Hamiltonian Mechanics is that a symplectic manifold can represent |
| 527 |
> |
all possible configurations of the system and the phase space of the |
| 528 |
> |
system can be described by it's cotangent bundle. Every symplectic |
| 529 |
> |
manifold is even dimensional. For instance, in Hamilton equations, |
| 530 |
> |
coordinate and momentum always appear in pairs. |
| 531 |
> |
|
| 532 |
> |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
| 533 |
> |
\[ |
| 534 |
> |
f : M \rightarrow N |
| 535 |
> |
\] |
| 536 |
> |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
| 537 |
> |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
| 538 |
> |
Canonical transformation is an example of symplectomorphism in |
| 539 |
> |
classical mechanics. |
| 540 |
> |
|
| 541 |
> |
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 542 |
> |
|
| 543 |
> |
For a ordinary differential system defined as |
| 544 |
> |
\begin{equation} |
| 545 |
> |
\dot x = f(x) |
| 546 |
> |
\end{equation} |
| 547 |
> |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
| 548 |
> |
\begin{equation} |
| 549 |
> |
f(r) = J\nabla _x H(r). |
| 550 |
> |
\end{equation} |
| 551 |
> |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
| 552 |
> |
matrix |
| 553 |
> |
\begin{equation} |
| 554 |
> |
J = \left( {\begin{array}{*{20}c} |
| 555 |
> |
0 & I \\ |
| 556 |
> |
{ - I} & 0 \\ |
| 557 |
> |
\end{array}} \right) |
| 558 |
> |
\label{introEquation:canonicalMatrix} |
| 559 |
> |
\end{equation} |
| 560 |
> |
where $I$ is an identity matrix. Using this notation, Hamiltonian |
| 561 |
> |
system can be rewritten as, |
| 562 |
> |
\begin{equation} |
| 563 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x) |
| 564 |
> |
\label{introEquation:compactHamiltonian} |
| 565 |
> |
\end{equation}In this case, $f$ is |
| 566 |
> |
called a \emph{Hamiltonian vector field}. |
| 567 |
> |
|
| 568 |
> |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
| 569 |
> |
\begin{equation} |
| 570 |
> |
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 571 |
> |
\end{equation} |
| 572 |
> |
The most obvious change being that matrix $J$ now depends on $x$. |
| 573 |
> |
The free rigid body is an example of Poisson system (actually a |
| 574 |
> |
Lie-Poisson system) with Hamiltonian function of angular kinetic |
| 575 |
> |
energy. |
| 576 |
> |
\begin{equation} |
| 577 |
> |
J(\pi ) = \left( {\begin{array}{*{20}c} |
| 578 |
> |
0 & {\pi _3 } & { - \pi _2 } \\ |
| 579 |
> |
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 580 |
> |
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 581 |
> |
\end{array}} \right) |
| 582 |
> |
\end{equation} |
| 583 |
> |
|
| 584 |
> |
\begin{equation} |
| 585 |
> |
H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2 |
| 586 |
> |
}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right) |
| 587 |
> |
\end{equation} |
| 588 |
> |
|
| 589 |
> |
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 590 |
> |
|
| 591 |
> |
Let $x(t)$ be the exact solution of the ODE system, |
| 592 |
> |
\begin{equation} |
| 593 |
> |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
| 594 |
> |
\end{equation} |
| 595 |
> |
The exact flow(solution) $\varphi_\tau$ is defined by |
| 596 |
> |
\[ |
| 597 |
> |
x(t+\tau) =\varphi_\tau(x(t)) |
| 598 |
> |
\] |
| 599 |
> |
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 600 |
> |
space to itself. The flow has the continuous group property, |
| 601 |
> |
\begin{equation} |
| 602 |
> |
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
| 603 |
> |
+ \tau _2 } . |
| 604 |
> |
\end{equation} |
| 605 |
> |
In particular, |
| 606 |
> |
\begin{equation} |
| 607 |
> |
\varphi _\tau \circ \varphi _{ - \tau } = I |
| 608 |
> |
\end{equation} |
| 609 |
> |
Therefore, the exact flow is self-adjoint, |
| 610 |
> |
\begin{equation} |
| 611 |
> |
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 612 |
> |
\end{equation} |
| 613 |
> |
The exact flow can also be written in terms of the of an operator, |
| 614 |
> |
\begin{equation} |
| 615 |
> |
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 616 |
> |
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 617 |
> |
\label{introEquation:exponentialOperator} |
| 618 |
> |
\end{equation} |
| 619 |
> |
|
| 620 |
> |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
| 621 |
> |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
| 622 |
> |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
| 623 |
> |
the Taylor series of $\psi_\tau$ agree to order $p$, |
| 624 |
> |
\begin{equation} |
| 625 |
> |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 626 |
> |
\end{equation} |
| 627 |
> |
|
| 628 |
> |
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 629 |
> |
|
| 630 |
> |
The hidden geometric properties of ODE and its flow play important |
| 631 |
> |
roles in numerical studies. Many of them can be found in systems |
| 632 |
> |
which occur naturally in applications. |
| 633 |
> |
|
| 634 |
> |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 635 |
> |
a \emph{symplectic} flow if it satisfies, |
| 636 |
> |
\begin{equation} |
| 637 |
> |
{\varphi '}^T J \varphi ' = J. |
| 638 |
> |
\end{equation} |
| 639 |
> |
According to Liouville's theorem, the symplectic volume is invariant |
| 640 |
> |
under a Hamiltonian flow, which is the basis for classical |
| 641 |
> |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
| 642 |
> |
field on a symplectic manifold can be shown to be a |
| 643 |
> |
symplectomorphism. As to the Poisson system, |
| 644 |
> |
\begin{equation} |
| 645 |
> |
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 646 |
> |
\end{equation} |
| 647 |
> |
is the property must be preserved by the integrator. |
| 648 |
|
|
| 649 |
+ |
It is possible to construct a \emph{volume-preserving} flow for a |
| 650 |
+ |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
| 651 |
+ |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
| 652 |
+ |
be volume-preserving. |
| 653 |
+ |
|
| 654 |
+ |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
| 655 |
+ |
will result in a new system, |
| 656 |
+ |
\[ |
| 657 |
+ |
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 658 |
+ |
\] |
| 659 |
+ |
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 660 |
+ |
In other words, the flow of this vector field is reversible if and |
| 661 |
+ |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 662 |
+ |
|
| 663 |
+ |
A \emph{first integral}, or conserved quantity of a general |
| 664 |
+ |
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 665 |
+ |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 666 |
+ |
\[ |
| 667 |
+ |
\frac{{dG(x(t))}}{{dt}} = 0. |
| 668 |
+ |
\] |
| 669 |
+ |
Using chain rule, one may obtain, |
| 670 |
+ |
\[ |
| 671 |
+ |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
| 672 |
+ |
\] |
| 673 |
+ |
which is the condition for conserving \emph{first integral}. For a |
| 674 |
+ |
canonical Hamiltonian system, the time evolution of an arbitrary |
| 675 |
+ |
smooth function $G$ is given by, |
| 676 |
+ |
\begin{equation} |
| 677 |
+ |
\begin{array}{c} |
| 678 |
+ |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 679 |
+ |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 680 |
+ |
\end{array} |
| 681 |
+ |
\label{introEquation:firstIntegral1} |
| 682 |
+ |
\end{equation} |
| 683 |
+ |
Using poisson bracket notion, Equation |
| 684 |
+ |
\ref{introEquation:firstIntegral1} can be rewritten as |
| 685 |
+ |
\[ |
| 686 |
+ |
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 687 |
+ |
\] |
| 688 |
+ |
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 689 |
+ |
integral} of a Hamiltonian system is |
| 690 |
+ |
\[ |
| 691 |
+ |
\left\{ {G,H} \right\} = 0. |
| 692 |
+ |
\] |
| 693 |
+ |
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 694 |
+ |
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 695 |
+ |
0$. |
| 696 |
+ |
|
| 697 |
+ |
|
| 698 |
+ |
When designing any numerical methods, one should always try to |
| 699 |
+ |
preserve the structural properties of the original ODE and its flow. |
| 700 |
+ |
|
| 701 |
+ |
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 702 |
+ |
A lot of well established and very effective numerical methods have |
| 703 |
+ |
been successful precisely because of their symplecticities even |
| 704 |
+ |
though this fact was not recognized when they were first |
| 705 |
+ |
constructed. The most famous example is leapfrog methods in |
| 706 |
+ |
molecular dynamics. In general, symplectic integrators can be |
| 707 |
+ |
constructed using one of four different methods. |
| 708 |
+ |
\begin{enumerate} |
| 709 |
+ |
\item Generating functions |
| 710 |
+ |
\item Variational methods |
| 711 |
+ |
\item Runge-Kutta methods |
| 712 |
+ |
\item Splitting methods |
| 713 |
+ |
\end{enumerate} |
| 714 |
+ |
|
| 715 |
+ |
Generating function tends to lead to methods which are cumbersome |
| 716 |
+ |
and difficult to use. In dissipative systems, variational methods |
| 717 |
+ |
can capture the decay of energy accurately. Since their |
| 718 |
+ |
geometrically unstable nature against non-Hamiltonian perturbations, |
| 719 |
+ |
ordinary implicit Runge-Kutta methods are not suitable for |
| 720 |
+ |
Hamiltonian system. Recently, various high-order explicit |
| 721 |
+ |
Runge--Kutta methods have been developed to overcome this |
| 722 |
+ |
instability. However, due to computational penalty involved in |
| 723 |
+ |
implementing the Runge-Kutta methods, they do not attract too much |
| 724 |
+ |
attention from Molecular Dynamics community. Instead, splitting have |
| 725 |
+ |
been widely accepted since they exploit natural decompositions of |
| 726 |
+ |
the system\cite{Tuckerman92}. |
| 727 |
+ |
|
| 728 |
+ |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
| 729 |
+ |
|
| 730 |
+ |
The main idea behind splitting methods is to decompose the discrete |
| 731 |
+ |
$\varphi_h$ as a composition of simpler flows, |
| 732 |
+ |
\begin{equation} |
| 733 |
+ |
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 734 |
+ |
\varphi _{h_n } |
| 735 |
+ |
\label{introEquation:FlowDecomposition} |
| 736 |
+ |
\end{equation} |
| 737 |
+ |
where each of the sub-flow is chosen such that each represent a |
| 738 |
+ |
simpler integration of the system. |
| 739 |
+ |
|
| 740 |
+ |
Suppose that a Hamiltonian system takes the form, |
| 741 |
+ |
\[ |
| 742 |
+ |
H = H_1 + H_2. |
| 743 |
+ |
\] |
| 744 |
+ |
Here, $H_1$ and $H_2$ may represent different physical processes of |
| 745 |
+ |
the system. For instance, they may relate to kinetic and potential |
| 746 |
+ |
energy respectively, which is a natural decomposition of the |
| 747 |
+ |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
| 748 |
+ |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
| 749 |
+ |
order is then given by the Lie-Trotter formula |
| 750 |
+ |
\begin{equation} |
| 751 |
+ |
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 752 |
+ |
\label{introEquation:firstOrderSplitting} |
| 753 |
+ |
\end{equation} |
| 754 |
+ |
where $\varphi _h$ is the result of applying the corresponding |
| 755 |
+ |
continuous $\varphi _i$ over a time $h$. By definition, as |
| 756 |
+ |
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
| 757 |
+ |
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
| 758 |
+ |
It is easy to show that any composition of symplectic flows yields a |
| 759 |
+ |
symplectic map, |
| 760 |
+ |
\begin{equation} |
| 761 |
+ |
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 762 |
+ |
'\phi ' = \phi '^T J\phi ' = J, |
| 763 |
+ |
\label{introEquation:SymplecticFlowComposition} |
| 764 |
+ |
\end{equation} |
| 765 |
+ |
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
| 766 |
+ |
splitting in this context automatically generates a symplectic map. |
| 767 |
+ |
|
| 768 |
+ |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
| 769 |
+ |
introduces local errors proportional to $h^2$, while Strang |
| 770 |
+ |
splitting gives a second-order decomposition, |
| 771 |
+ |
\begin{equation} |
| 772 |
+ |
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 773 |
+ |
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 774 |
+ |
\end{equation} |
| 775 |
+ |
which has a local error proportional to $h^3$. Sprang splitting's |
| 776 |
+ |
popularity in molecular simulation community attribute to its |
| 777 |
+ |
symmetric property, |
| 778 |
+ |
\begin{equation} |
| 779 |
+ |
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 780 |
+ |
\label{introEquation:timeReversible} |
| 781 |
+ |
\end{equation} |
| 782 |
+ |
|
| 783 |
+ |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
| 784 |
+ |
The classical equation for a system consisting of interacting |
| 785 |
+ |
particles can be written in Hamiltonian form, |
| 786 |
+ |
\[ |
| 787 |
+ |
H = T + V |
| 788 |
+ |
\] |
| 789 |
+ |
where $T$ is the kinetic energy and $V$ is the potential energy. |
| 790 |
+ |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
| 791 |
+ |
obtains the following: |
| 792 |
+ |
\begin{align} |
| 793 |
+ |
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 794 |
+ |
\frac{F[q(0)]}{m}\frac{\Delta t^2}{2}, % |
| 795 |
+ |
\label{introEquation:Lp10a} \\% |
| 796 |
+ |
% |
| 797 |
+ |
\dot{q}(\Delta t) &= \dot{q}(0) + \frac{\Delta t}{2m} |
| 798 |
+ |
\biggl [F[q(0)] + F[q(\Delta t)] \biggr]. % |
| 799 |
+ |
\label{introEquation:Lp10b} |
| 800 |
+ |
\end{align} |
| 801 |
+ |
where $F(t)$ is the force at time $t$. This integration scheme is |
| 802 |
+ |
known as \emph{velocity verlet} which is |
| 803 |
+ |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
| 804 |
+ |
time-reversible(\ref{introEquation:timeReversible}) and |
| 805 |
+ |
volume-preserving (\ref{introEquation:volumePreserving}). These |
| 806 |
+ |
geometric properties attribute to its long-time stability and its |
| 807 |
+ |
popularity in the community. However, the most commonly used |
| 808 |
+ |
velocity verlet integration scheme is written as below, |
| 809 |
+ |
\begin{align} |
| 810 |
+ |
\dot{q}\biggl (\frac{\Delta t}{2}\biggr ) &= |
| 811 |
+ |
\dot{q}(0) + \frac{\Delta t}{2m}\, F[q(0)], \label{introEquation:Lp9a}\\% |
| 812 |
+ |
% |
| 813 |
+ |
q(\Delta t) &= q(0) + \Delta t\, \dot{q}\biggl (\frac{\Delta t}{2}\biggr ),% |
| 814 |
+ |
\label{introEquation:Lp9b}\\% |
| 815 |
+ |
% |
| 816 |
+ |
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 817 |
+ |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
| 818 |
+ |
\end{align} |
| 819 |
+ |
From the preceding splitting, one can see that the integration of |
| 820 |
+ |
the equations of motion would follow: |
| 821 |
+ |
\begin{enumerate} |
| 822 |
+ |
\item calculate the velocities at the half step, $\frac{\Delta t}{2}$, from the forces calculated at the initial position. |
| 823 |
+ |
|
| 824 |
+ |
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 825 |
+ |
|
| 826 |
+ |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 827 |
+ |
|
| 828 |
+ |
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 829 |
+ |
\end{enumerate} |
| 830 |
+ |
|
| 831 |
+ |
Simply switching the order of splitting and composing, a new |
| 832 |
+ |
integrator, the \emph{position verlet} integrator, can be generated, |
| 833 |
+ |
\begin{align} |
| 834 |
+ |
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
| 835 |
+ |
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
| 836 |
+ |
\label{introEquation:positionVerlet1} \\% |
| 837 |
+ |
% |
| 838 |
+ |
q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot |
| 839 |
+ |
q(\Delta t)} \right]. % |
| 840 |
+ |
\label{introEquation:positionVerlet1} |
| 841 |
+ |
\end{align} |
| 842 |
+ |
|
| 843 |
+ |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
| 844 |
+ |
|
| 845 |
+ |
Baker-Campbell-Hausdorff formula can be used to determine the local |
| 846 |
+ |
error of splitting method in terms of commutator of the |
| 847 |
+ |
operators(\ref{introEquation:exponentialOperator}) associated with |
| 848 |
+ |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
| 849 |
+ |
$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have |
| 850 |
+ |
\begin{equation} |
| 851 |
+ |
\exp (hX + hY) = \exp (hZ) |
| 852 |
+ |
\end{equation} |
| 853 |
+ |
where |
| 854 |
+ |
\begin{equation} |
| 855 |
+ |
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
| 856 |
+ |
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
| 857 |
+ |
\end{equation} |
| 858 |
+ |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
| 859 |
+ |
\[ |
| 860 |
+ |
[X,Y] = XY - YX . |
| 861 |
+ |
\] |
| 862 |
+ |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 863 |
+ |
can obtain |
| 864 |
+ |
\begin{eqnarray*} |
| 865 |
+ |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 866 |
+ |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 867 |
+ |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 868 |
+ |
\ldots ) |
| 869 |
+ |
\end{eqnarray*} |
| 870 |
+ |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 871 |
+ |
error of Spring splitting is proportional to $h^3$. The same |
| 872 |
+ |
procedure can be applied to general splitting, of the form |
| 873 |
+ |
\begin{equation} |
| 874 |
+ |
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 875 |
+ |
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 876 |
+ |
\end{equation} |
| 877 |
+ |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
| 878 |
+ |
order method. Yoshida proposed an elegant way to compose higher |
| 879 |
+ |
order methods based on symmetric splitting. Given a symmetric second |
| 880 |
+ |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
| 881 |
+ |
method can be constructed by composing, |
| 882 |
+ |
\[ |
| 883 |
+ |
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
| 884 |
+ |
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
| 885 |
+ |
\] |
| 886 |
+ |
where $ \alpha = - \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$ and $ \beta |
| 887 |
+ |
= \frac{{2^{1/3} }}{{2 - 2^{1/3} }}$. Moreover, a symmetric |
| 888 |
+ |
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
| 889 |
+ |
\begin{equation} |
| 890 |
+ |
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
| 891 |
+ |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
| 892 |
+ |
\end{equation} |
| 893 |
+ |
, if the weights are chosen as |
| 894 |
+ |
\[ |
| 895 |
+ |
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
| 896 |
+ |
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
| 897 |
+ |
\] |
| 898 |
+ |
|
| 899 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 900 |
|
|
| 901 |
|
As a special discipline of molecular modeling, Molecular dynamics |
| 905 |
|
|
| 906 |
|
\subsection{\label{introSec:mdInit}Initialization} |
| 907 |
|
|
| 908 |
+ |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 909 |
+ |
|
| 910 |
|
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 911 |
|
|
| 912 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 913 |
|
|
| 914 |
< |
A rigid body is a body in which the distance between any two given |
| 915 |
< |
points of a rigid body remains constant regardless of external |
| 916 |
< |
forces exerted on it. A rigid body therefore conserves its shape |
| 917 |
< |
during its motion. |
| 914 |
> |
Rigid bodies are frequently involved in the modeling of different |
| 915 |
> |
areas, from engineering, physics, to chemistry. For example, |
| 916 |
> |
missiles and vehicle are usually modeled by rigid bodies. The |
| 917 |
> |
movement of the objects in 3D gaming engine or other physics |
| 918 |
> |
simulator is governed by the rigid body dynamics. In molecular |
| 919 |
> |
simulation, rigid body is used to simplify the model in |
| 920 |
> |
protein-protein docking study{\cite{Gray03}}. |
| 921 |
|
|
| 922 |
< |
Applications of dynamics of rigid bodies. |
| 922 |
> |
It is very important to develop stable and efficient methods to |
| 923 |
> |
integrate the equations of motion of orientational degrees of |
| 924 |
> |
freedom. Euler angles are the nature choice to describe the |
| 925 |
> |
rotational degrees of freedom. However, due to its singularity, the |
| 926 |
> |
numerical integration of corresponding equations of motion is very |
| 927 |
> |
inefficient and inaccurate. Although an alternative integrator using |
| 928 |
> |
different sets of Euler angles can overcome this difficulty\cite{}, |
| 929 |
> |
the computational penalty and the lost of angular momentum |
| 930 |
> |
conservation still remain. A singularity free representation |
| 931 |
> |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
| 932 |
> |
this approach suffer from the nonseparable Hamiltonian resulted from |
| 933 |
> |
quaternion representation, which prevents the symplectic algorithm |
| 934 |
> |
to be utilized. Another different approach is to apply holonomic |
| 935 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
| 936 |
> |
moves independently under the normal forces deriving from potential |
| 937 |
> |
energy and constraint forces which are used to guarantee the |
| 938 |
> |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
| 939 |
> |
algorithm converge very slowly when the number of constraint |
| 940 |
> |
increases. |
| 941 |
|
|
| 942 |
+ |
The break through in geometric literature suggests that, in order to |
| 943 |
+ |
develop a long-term integration scheme, one should preserve the |
| 944 |
+ |
symplectic structure of the flow. Introducing conjugate momentum to |
| 945 |
+ |
rotation matrix $A$ and re-formulating Hamiltonian's equation, a |
| 946 |
+ |
symplectic integrator, RSHAKE, was proposed to evolve the |
| 947 |
+ |
Hamiltonian system in a constraint manifold by iteratively |
| 948 |
+ |
satisfying the orthogonality constraint $A_t A = 1$. An alternative |
| 949 |
+ |
method using quaternion representation was developed by Omelyan. |
| 950 |
+ |
However, both of these methods are iterative and inefficient. In |
| 951 |
+ |
this section, we will present a symplectic Lie-Poisson integrator |
| 952 |
+ |
for rigid body developed by Dullweber and his |
| 953 |
+ |
coworkers\cite{Dullweber1997}. |
| 954 |
+ |
|
| 955 |
|
\subsection{\label{introSection:lieAlgebra}Lie Algebra} |
| 956 |
|
|
| 957 |
< |
\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion} |
| 957 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 958 |
|
|
| 959 |
< |
\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion} |
| 959 |
> |
\begin{equation} |
| 960 |
> |
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 961 |
> |
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 962 |
> |
\label{introEquation:RBHamiltonian} |
| 963 |
> |
\end{equation} |
| 964 |
> |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 965 |
> |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 966 |
> |
$J$, a diagonal matrix, is defined by |
| 967 |
> |
\[ |
| 968 |
> |
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 969 |
> |
\] |
| 970 |
> |
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 971 |
> |
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 972 |
> |
\begin{equation} |
| 973 |
> |
Q^T Q = 1$, \label{introEquation:orthogonalConstraint} |
| 974 |
> |
\end{equation} |
| 975 |
> |
which is used to ensure rotation matrix's orthogonality. |
| 976 |
> |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 977 |
> |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 978 |
> |
\begin{equation} |
| 979 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 980 |
> |
\label{introEquation:RBFirstOrderConstraint} |
| 981 |
> |
\end{equation} |
| 982 |
|
|
| 983 |
< |
%\subsection{\label{introSection:poissonBrackets}Poisson Brackets} |
| 983 |
> |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 984 |
> |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 985 |
> |
the equations of motion, |
| 986 |
> |
\[ |
| 987 |
> |
\begin{array}{c} |
| 988 |
> |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 989 |
> |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 990 |
> |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 991 |
> |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 992 |
> |
\end{array} |
| 993 |
> |
\] |
| 994 |
|
|
| 995 |
< |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
| 995 |
> |
In general, there are two ways to satisfy the holonomic constraints. |
| 996 |
> |
We can use constraint force provided by lagrange multiplier on the |
| 997 |
> |
normal manifold to keep the motion on constraint space. Or we can |
| 998 |
> |
simply evolve the system in constraint manifold. The two method are |
| 999 |
> |
proved to be equivalent. The holonomic constraint and equations of |
| 1000 |
> |
motions define a constraint manifold for rigid body |
| 1001 |
> |
\[ |
| 1002 |
> |
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1003 |
> |
\right\}. |
| 1004 |
> |
\] |
| 1005 |
|
|
| 1006 |
+ |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1007 |
+ |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 1008 |
+ |
transformation, the cotangent space and the phase space are |
| 1009 |
+ |
diffeomorphic. Introducing |
| 1010 |
+ |
\[ |
| 1011 |
+ |
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1012 |
+ |
\] |
| 1013 |
+ |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 1014 |
+ |
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1015 |
+ |
\[ |
| 1016 |
+ |
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1017 |
+ |
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1018 |
+ |
\] |
| 1019 |
+ |
|
| 1020 |
+ |
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1021 |
+ |
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1022 |
+ |
given as |
| 1023 |
+ |
\begin{equation} |
| 1024 |
+ |
X_i^{lab} = Q X_i + q. |
| 1025 |
+ |
\end{equation} |
| 1026 |
+ |
Therefore, potential energy $V(q,Q)$ is defined by |
| 1027 |
+ |
\[ |
| 1028 |
+ |
V(q,Q) = V(Q X_0 + q). |
| 1029 |
+ |
\] |
| 1030 |
+ |
Hence, |
| 1031 |
+ |
\[ |
| 1032 |
+ |
\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)} |
| 1033 |
+ |
\] |
| 1034 |
+ |
|
| 1035 |
+ |
\[ |
| 1036 |
+ |
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1037 |
+ |
\] |
| 1038 |
+ |
|
| 1039 |
+ |
As a common choice to describe the rotation dynamics of the rigid |
| 1040 |
+ |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1041 |
+ |
rewrite the equations of motion, |
| 1042 |
+ |
\begin{equation} |
| 1043 |
+ |
\begin{array}{l} |
| 1044 |
+ |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1045 |
+ |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1046 |
+ |
\end{array} |
| 1047 |
+ |
\label{introEqaution:RBMotionPI} |
| 1048 |
+ |
\end{equation} |
| 1049 |
+ |
, as well as holonomic constraints, |
| 1050 |
+ |
\[ |
| 1051 |
+ |
\begin{array}{l} |
| 1052 |
+ |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1053 |
+ |
Q^T Q = 1 \\ |
| 1054 |
+ |
\end{array} |
| 1055 |
+ |
\] |
| 1056 |
+ |
|
| 1057 |
+ |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1058 |
+ |
so(3)^ \star$, the hat-map isomorphism, |
| 1059 |
+ |
\begin{equation} |
| 1060 |
+ |
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1061 |
+ |
{\begin{array}{*{20}c} |
| 1062 |
+ |
0 & { - v_3 } & {v_2 } \\ |
| 1063 |
+ |
{v_3 } & 0 & { - v_1 } \\ |
| 1064 |
+ |
{ - v_2 } & {v_1 } & 0 \\ |
| 1065 |
+ |
\end{array}} \right), |
| 1066 |
+ |
\label{introEquation:hatmapIsomorphism} |
| 1067 |
+ |
\end{equation} |
| 1068 |
+ |
will let us associate the matrix products with traditional vector |
| 1069 |
+ |
operations |
| 1070 |
+ |
\[ |
| 1071 |
+ |
\hat vu = v \times u |
| 1072 |
+ |
\] |
| 1073 |
+ |
|
| 1074 |
+ |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1075 |
+ |
matrix, |
| 1076 |
+ |
\begin{equation} |
| 1077 |
+ |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1078 |
+ |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1079 |
+ |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1080 |
+ |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1081 |
+ |
\end{equation} |
| 1082 |
+ |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1083 |
+ |
\ref{introEquation:skewMatrixPI}, which implies the Lagrange |
| 1084 |
+ |
multiplier $\Lambda$ is ignored in the integration. |
| 1085 |
+ |
|
| 1086 |
+ |
Hence, applying hat-map isomorphism, we obtain the equation of |
| 1087 |
+ |
motion for angular momentum on body frame |
| 1088 |
+ |
\[ |
| 1089 |
+ |
\dot \pi = \pi \times I^{ - 1} \pi + Q^T \sum\limits_i {F_i (r,Q) |
| 1090 |
+ |
\times X_i } |
| 1091 |
+ |
\] |
| 1092 |
+ |
In the same manner, the equation of motion for rotation matrix is |
| 1093 |
+ |
given by |
| 1094 |
+ |
\[ |
| 1095 |
+ |
\dot Q = Qskew(M^{ - 1} \pi ) |
| 1096 |
+ |
\] |
| 1097 |
+ |
|
| 1098 |
+ |
The free rigid body equation is an example of a non-canonical |
| 1099 |
+ |
Hamiltonian system. |
| 1100 |
+ |
|
| 1101 |
+ |
\subsection{\label{introSection:symplecticDiscretizationRB}Symplectic Integration of Euler Equations} |
| 1102 |
+ |
|
| 1103 |
+ |
\[ |
| 1104 |
+ |
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1105 |
+ |
_{\Delta t,T} \circ \varphi _{\Delta t/2,V} |
| 1106 |
+ |
\] |
| 1107 |
+ |
|
| 1108 |
+ |
\[ |
| 1109 |
+ |
\varphi _{\Delta t,T} = \varphi _{\Delta t,R} \circ \varphi |
| 1110 |
+ |
_{\Delta t,\pi } |
| 1111 |
+ |
\] |
| 1112 |
+ |
|
| 1113 |
+ |
\[ |
| 1114 |
+ |
\varphi _{\Delta t,\pi } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1115 |
+ |
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1116 |
+ |
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1117 |
+ |
_1 } |
| 1118 |
+ |
\] |
| 1119 |
+ |
|
| 1120 |
+ |
\[ |
| 1121 |
+ |
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1122 |
+ |
_{\Delta t/2,\tau } |
| 1123 |
+ |
\] |
| 1124 |
+ |
|
| 1125 |
+ |
|
| 1126 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1127 |
|
|
| 1128 |
|
\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics} |
| 1171 |
|
\dot p &= - \frac{{\partial H}}{{\partial x}} |
| 1172 |
|
&= m\ddot x |
| 1173 |
|
&= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right)} |
| 1174 |
< |
\label{introEq:Lp5} |
| 1174 |
> |
\label{introEquation:Lp5} |
| 1175 |
|
\end{align} |
| 1176 |
|
, and |
| 1177 |
|
\begin{align} |
| 1330 |
|
|
| 1331 |
|
\subsection{\label{introSection:centersRigidBody}Centers of Rigid |
| 1332 |
|
Body} |
| 1333 |
+ |
|
| 1334 |
+ |
\section{\label{introSection:correlationFunctions}Correlation Functions} |
