| 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 |
– |
There are several different types of ensembles with different |
| 288 |
– |
statistical characteristics. As a function of macroscopic |
| 289 |
– |
parameters, such as temperature \textit{etc}, the partition function |
| 290 |
– |
can be used to describe the statistical properties of a system in |
| 291 |
– |
thermodynamic equilibrium. |
| 292 |
– |
|
| 293 |
– |
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, |
| 296 |
– |
\begin{equation} |
| 297 |
– |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 298 |
– |
\end{equation} |
| 299 |
– |
A canonical ensemble (NVT)is an ensemble of systems, each of which |
| 300 |
– |
can share its energy with a large heat reservoir. The distribution |
| 301 |
– |
of the total energy amongst the possible dynamical states is given |
| 302 |
– |
by the partition function, |
| 303 |
– |
\begin{equation} |
| 304 |
– |
\Omega (N,V,T) = e^{ - \beta A} |
| 305 |
– |
\label{introEquation:NVTPartition} |
| 306 |
– |
\end{equation} |
| 307 |
– |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 308 |
– |
TS$. Since most experiments are carried out under constant pressure |
| 309 |
– |
condition, the isothermal-isobaric ensemble (NPT) plays a very |
| 310 |
– |
important role in molecular simulations. The isothermal-isobaric |
| 311 |
– |
ensemble allow the system to exchange energy with a heat bath of |
| 312 |
– |
temperature $T$ and to change the volume as well. Its partition |
| 313 |
– |
function is given as |
| 314 |
– |
\begin{equation} |
| 315 |
– |
\Delta (N,P,T) = - e^{\beta G}. |
| 316 |
– |
\label{introEquation:NPTPartition} |
| 317 |
– |
\end{equation} |
| 318 |
– |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
| 319 |
– |
|
| 286 |
|
\subsection{\label{introSection:liouville}Liouville's theorem} |
| 287 |
|
|
| 288 |
|
Liouville's theorem is the foundation on which statistical mechanics |
| 324 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
| 325 |
|
\label{introEquation:liouvilleTheorem} |
| 326 |
|
\end{equation} |
| 361 |
– |
|
| 327 |
|
Liouville's theorem states that the distribution function is |
| 328 |
|
constant along any trajectory in phase space. In classical |
| 329 |
< |
statistical mechanics, since the number of members in an ensemble is |
| 330 |
< |
huge and constant, we can assume the local density has no reason |
| 331 |
< |
(other than classical mechanics) to change, |
| 329 |
> |
statistical mechanics, since the number of system copies in an |
| 330 |
> |
ensemble is huge and constant, we can assume the local density has |
| 331 |
> |
no reason (other than classical mechanics) to change, |
| 332 |
|
\begin{equation} |
| 333 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
| 334 |
|
\label{introEquation:stationary} |
| 358 |
|
\frac{{d(\delta N)}}{{dt}} = \frac{{d\rho }}{{dt}}\delta v + \rho |
| 359 |
|
\frac{d}{{dt}}(\delta v) = 0. |
| 360 |
|
\end{equation} |
| 361 |
< |
With the help of stationary assumption |
| 362 |
< |
(\ref{introEquation:stationary}), we obtain the principle of the |
| 361 |
> |
With the help of the stationary assumption |
| 362 |
> |
(Eq.~\ref{introEquation:stationary}), we obtain the principle of |
| 363 |
|
\emph{conservation of volume in phase space}, |
| 364 |
|
\begin{equation} |
| 365 |
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
| 369 |
|
|
| 370 |
|
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
| 371 |
|
|
| 372 |
< |
Liouville's theorem can be expresses in a variety of different forms |
| 372 |
> |
Liouville's theorem can be expressed in a variety of different forms |
| 373 |
|
which are convenient within different contexts. For any two function |
| 374 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
| 375 |
|
bracket ${F, G}$ is defined as |
| 380 |
|
q_i }}} \right)}. |
| 381 |
|
\label{introEquation:poissonBracket} |
| 382 |
|
\end{equation} |
| 383 |
< |
Substituting equations of motion in Hamiltonian formalism( |
| 384 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
| 385 |
< |
Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into |
| 383 |
> |
Substituting equations of motion in Hamiltonian formalism |
| 384 |
> |
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
| 385 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into |
| 386 |
|
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
| 387 |
|
Liouville's theorem using Poisson bracket notion, |
| 388 |
|
\begin{equation} |
| 403 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - iL\rho |
| 404 |
|
\label{introEquation:liouvilleTheoremInOperator} |
| 405 |
|
\end{equation} |
| 406 |
< |
|
| 406 |
> |
which can help define a propagator $\rho (t) = e^{-iLt} \rho (0)$. |
| 407 |
|
\subsection{\label{introSection:ergodic}The Ergodic Hypothesis} |
| 408 |
|
|
| 409 |
|
Various thermodynamic properties can be calculated from Molecular |
| 412 |
|
simulation and the quality of the underlying model. However, both |
| 413 |
|
experiments and computer simulations are usually performed during a |
| 414 |
|
certain time interval and the measurements are averaged over a |
| 415 |
< |
period of them which is different from the average behavior of |
| 415 |
> |
period of time which is different from the average behavior of |
| 416 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 417 |
|
Hypothesis makes a connection between time average and the ensemble |
| 418 |
|
average. It states that the time average and average over the |
| 419 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
| 419 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
| 420 |
|
\begin{equation} |
| 421 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 422 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 425 |
|
where $\langle A(q , p) \rangle_t$ is an equilibrium value of a |
| 426 |
|
physical quantity and $\rho (p(t), q(t))$ is the equilibrium |
| 427 |
|
distribution function. If an observation is averaged over a |
| 428 |
< |
sufficiently long time (longer than relaxation time), all accessible |
| 429 |
< |
microstates in phase space are assumed to be equally probed, giving |
| 430 |
< |
a properly weighted statistical average. This allows the researcher |
| 431 |
< |
freedom of choice when deciding how best to measure a given |
| 432 |
< |
observable. In case an ensemble averaged approach sounds most |
| 433 |
< |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
| 428 |
> |
sufficiently long time (longer than the relaxation time), all |
| 429 |
> |
accessible microstates in phase space are assumed to be equally |
| 430 |
> |
probed, giving a properly weighted statistical average. This allows |
| 431 |
> |
the researcher freedom of choice when deciding how best to measure a |
| 432 |
> |
given observable. In case an ensemble averaged approach sounds most |
| 433 |
> |
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
| 434 |
|
utilized. Or if the system lends itself to a time averaging |
| 435 |
|
approach, the Molecular Dynamics techniques in |
| 436 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 443 |
|
by the differential equations. However, most of them ignore the |
| 444 |
|
hidden physical laws contained within the equations. Since 1990, |
| 445 |
|
geometric integrators, which preserve various phase-flow invariants |
| 446 |
< |
such as symplectic structure, volume and time reversal symmetry, are |
| 447 |
< |
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
| 448 |
< |
Leimkuhler1999}. The velocity Verlet method, which happens to be a |
| 449 |
< |
simple example of symplectic integrator, continues to gain |
| 450 |
< |
popularity in the molecular dynamics community. This fact can be |
| 451 |
< |
partly explained by its geometric nature. |
| 446 |
> |
such as symplectic structure, volume and time reversal symmetry, |
| 447 |
> |
were developed to address this issue\cite{Dullweber1997, |
| 448 |
> |
McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which |
| 449 |
> |
happens to be a simple example of symplectic integrator, continues |
| 450 |
> |
to gain popularity in the molecular dynamics community. This fact |
| 451 |
> |
can be partly explained by its geometric nature. |
| 452 |
|
|
| 453 |
|
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 454 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
| 457 |
|
surface of Earth. It seems to be flat locally, but it is round if |
| 458 |
|
viewed as a whole. A \emph{differentiable manifold} (also known as |
| 459 |
|
\emph{smooth manifold}) is a manifold on which it is possible to |
| 460 |
< |
apply calculus on \emph{differentiable manifold}. A \emph{symplectic |
| 461 |
< |
manifold} is defined as a pair $(M, \omega)$ which consists of a |
| 460 |
> |
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
| 461 |
> |
defined as a pair $(M, \omega)$ which consists of a |
| 462 |
|
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 463 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 464 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
| 465 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 466 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 467 |
< |
$\omega(x, x) = 0$. The cross product operation in vector field is |
| 468 |
< |
an example of symplectic form. |
| 469 |
< |
|
| 470 |
< |
One of the motivations to study \emph{symplectic manifolds} in |
| 471 |
< |
Hamiltonian Mechanics is that a symplectic manifold can represent |
| 472 |
< |
all possible configurations of the system and the phase space of the |
| 473 |
< |
system can be described by it's cotangent bundle. Every symplectic |
| 474 |
< |
manifold is even dimensional. For instance, in Hamilton equations, |
| 510 |
< |
coordinate and momentum always appear in pairs. |
| 467 |
> |
$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in |
| 468 |
> |
vector field is an example of symplectic form. One of the |
| 469 |
> |
motivations to study \emph{symplectic manifolds} in Hamiltonian |
| 470 |
> |
Mechanics is that a symplectic manifold can represent all possible |
| 471 |
> |
configurations of the system and the phase space of the system can |
| 472 |
> |
be described by it's cotangent bundle\cite{Jost2002}. Every |
| 473 |
> |
symplectic manifold is even dimensional. For instance, in Hamilton |
| 474 |
> |
equations, coordinate and momentum always appear in pairs. |
| 475 |
|
|
| 476 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 477 |
|
|
| 480 |
|
\dot x = f(x) |
| 481 |
|
\end{equation} |
| 482 |
|
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
| 483 |
+ |
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian |
| 484 |
+ |
function and $J$ is the skew-symmetric matrix |
| 485 |
|
\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} |
| 486 |
|
J = \left( {\begin{array}{*{20}c} |
| 487 |
|
0 & I \\ |
| 488 |
|
{ - I} & 0 \\ |
| 492 |
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
| 493 |
|
system can be rewritten as, |
| 494 |
|
\begin{equation} |
| 495 |
< |
\frac{d}{{dt}}x = J\nabla _x H(x) |
| 495 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x). |
| 496 |
|
\label{introEquation:compactHamiltonian} |
| 497 |
|
\end{equation}In this case, $f$ is |
| 498 |
< |
called a \emph{Hamiltonian vector field}. |
| 499 |
< |
|
| 539 |
< |
Another generalization of Hamiltonian dynamics is Poisson |
| 540 |
< |
Dynamics\cite{Olver1986}, |
| 498 |
> |
called a \emph{Hamiltonian vector field}. Another generalization of |
| 499 |
> |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
| 500 |
|
\begin{equation} |
| 501 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 502 |
|
\end{equation} |
| 503 |
|
The most obvious change being that matrix $J$ now depends on $x$. |
| 504 |
|
|
| 505 |
< |
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 505 |
> |
\subsection{\label{introSection:exactFlow}Exact Propagator} |
| 506 |
|
|
| 507 |
< |
Let $x(t)$ be the exact solution of the ODE system, |
| 507 |
> |
Let $x(t)$ be the exact solution of the ODE |
| 508 |
> |
system, |
| 509 |
|
\begin{equation} |
| 510 |
< |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
| 511 |
< |
\end{equation} |
| 512 |
< |
The exact flow(solution) $\varphi_\tau$ is defined by |
| 513 |
< |
\[ |
| 514 |
< |
x(t+\tau) =\varphi_\tau(x(t)) |
| 510 |
> |
\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE} |
| 511 |
> |
\end{equation} we can |
| 512 |
> |
define its exact propagator $\varphi_\tau$: |
| 513 |
> |
\[ x(t+\tau) |
| 514 |
> |
=\varphi_\tau(x(t)) |
| 515 |
|
\] |
| 516 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 517 |
< |
space to itself. The flow has the continuous group property, |
| 517 |
> |
space to itself. The propagator has the continuous group property, |
| 518 |
|
\begin{equation} |
| 519 |
|
\varphi _{\tau _1 } \circ \varphi _{\tau _2 } = \varphi _{\tau _1 |
| 520 |
|
+ \tau _2 } . |
| 523 |
|
\begin{equation} |
| 524 |
|
\varphi _\tau \circ \varphi _{ - \tau } = I |
| 525 |
|
\end{equation} |
| 526 |
< |
Therefore, the exact flow is self-adjoint, |
| 526 |
> |
Therefore, the exact propagator is self-adjoint, |
| 527 |
|
\begin{equation} |
| 528 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 529 |
|
\end{equation} |
| 530 |
< |
The exact flow can also be written in terms of the of an operator, |
| 530 |
> |
The exact propagator can also be written in terms of operator, |
| 531 |
|
\begin{equation} |
| 532 |
|
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 533 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 534 |
|
\label{introEquation:exponentialOperator} |
| 535 |
|
\end{equation} |
| 536 |
< |
|
| 537 |
< |
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
| 538 |
< |
Instead, we use an approximate map, $\psi_\tau$, which is usually |
| 539 |
< |
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
| 540 |
< |
the Taylor series of $\psi_\tau$ agree to order $p$, |
| 536 |
> |
In most cases, it is not easy to find the exact propagator |
| 537 |
> |
$\varphi_\tau$. Instead, we use an approximate map, $\psi_\tau$, |
| 538 |
> |
which is usually called an integrator. The order of an integrator |
| 539 |
> |
$\psi_\tau$ is $p$, if the Taylor series of $\psi_\tau$ agree to |
| 540 |
> |
order $p$, |
| 541 |
|
\begin{equation} |
| 542 |
|
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 543 |
|
\end{equation} |
| 545 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 546 |
|
|
| 547 |
|
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
| 548 |
< |
ODE and its flow play important roles in numerical studies. Many of |
| 549 |
< |
them can be found in systems which occur naturally in applications. |
| 550 |
< |
|
| 551 |
< |
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 592 |
< |
a \emph{symplectic} flow if it satisfies, |
| 548 |
> |
ODE and its propagator play important roles in numerical studies. |
| 549 |
> |
Many of them can be found in systems which occur naturally in |
| 550 |
> |
applications. Let $\varphi$ be the propagator of Hamiltonian vector |
| 551 |
> |
field, $\varphi$ is a \emph{symplectic} propagator if it satisfies, |
| 552 |
|
\begin{equation} |
| 553 |
|
{\varphi '}^T J \varphi ' = J. |
| 554 |
|
\end{equation} |
| 555 |
|
According to Liouville's theorem, the symplectic volume is invariant |
| 556 |
< |
under a Hamiltonian flow, which is the basis for classical |
| 557 |
< |
statistical mechanics. Furthermore, the flow of a Hamiltonian vector |
| 558 |
< |
field on a symplectic manifold can be shown to be a |
| 556 |
> |
under a Hamiltonian propagator, which is the basis for classical |
| 557 |
> |
statistical mechanics. Furthermore, the propagator of a Hamiltonian |
| 558 |
> |
vector field on a symplectic manifold can be shown to be a |
| 559 |
|
symplectomorphism. As to the Poisson system, |
| 560 |
|
\begin{equation} |
| 561 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 562 |
|
\end{equation} |
| 563 |
< |
is the property that must be preserved by the integrator. |
| 564 |
< |
|
| 565 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
| 566 |
< |
source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ |
| 567 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
| 568 |
< |
be volume-preserving. |
| 569 |
< |
|
| 611 |
< |
Changing the variables $y = h(x)$ in an ODE |
| 612 |
< |
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
| 563 |
> |
is the property that must be preserved by the integrator. It is |
| 564 |
> |
possible to construct a \emph{volume-preserving} propagator for a |
| 565 |
> |
source free ODE ($ \nabla \cdot f = 0 $), if the propagator |
| 566 |
> |
satisfies $ \det d\varphi = 1$. One can show easily that a |
| 567 |
> |
symplectic propagator will be volume-preserving. Changing the |
| 568 |
> |
variables $y = h(x)$ in an ODE (Eq.~\ref{introEquation:ODE}) will |
| 569 |
> |
result in a new system, |
| 570 |
|
\[ |
| 571 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 572 |
|
\] |
| 573 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 574 |
< |
In other words, the flow of this vector field is reversible if and |
| 575 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 576 |
< |
|
| 577 |
< |
A \emph{first integral}, or conserved quantity of a general |
| 578 |
< |
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 622 |
< |
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 574 |
> |
In other words, the propagator of this vector field is reversible if |
| 575 |
> |
and only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
| 576 |
> |
conserved quantity of a general differential function is a function |
| 577 |
> |
$ G:R^{2d} \to R^d $ which is constant for all solutions of the ODE |
| 578 |
> |
$\frac{{dx}}{{dt}} = f(x)$ , |
| 579 |
|
\[ |
| 580 |
|
\frac{{dG(x(t))}}{{dt}} = 0. |
| 581 |
|
\] |
| 582 |
< |
Using chain rule, one may obtain, |
| 582 |
> |
Using the chain rule, one may obtain, |
| 583 |
|
\[ |
| 584 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
| 584 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G, |
| 585 |
|
\] |
| 586 |
< |
which is the condition for conserving \emph{first integral}. For a |
| 587 |
< |
canonical Hamiltonian system, the time evolution of an arbitrary |
| 588 |
< |
smooth function $G$ is given by, |
| 633 |
< |
|
| 586 |
> |
which is the condition for conserved quantities. For a canonical |
| 587 |
> |
Hamiltonian system, the time evolution of an arbitrary smooth |
| 588 |
> |
function $G$ is given by, |
| 589 |
|
\begin{eqnarray} |
| 590 |
< |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 591 |
< |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 590 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
| 591 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
| 592 |
|
\label{introEquation:firstIntegral1} |
| 593 |
|
\end{eqnarray} |
| 594 |
< |
|
| 595 |
< |
|
| 641 |
< |
Using poisson bracket notion, Equation |
| 642 |
< |
\ref{introEquation:firstIntegral1} can be rewritten as |
| 594 |
> |
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1} |
| 595 |
> |
can be rewritten as |
| 596 |
|
\[ |
| 597 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 598 |
|
\] |
| 599 |
< |
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 600 |
< |
integral} of a Hamiltonian system is |
| 601 |
< |
\[ |
| 602 |
< |
\left\{ {G,H} \right\} = 0. |
| 650 |
< |
\] |
| 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 |
< |
|
| 599 |
> |
Therefore, the sufficient condition for $G$ to be a conserved |
| 600 |
> |
quantity of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ As |
| 601 |
> |
is well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 602 |
> |
is a conserved quantity, which is due to the fact $\{ H,H\} = 0$. |
| 603 |
|
When designing any numerical methods, one should always try to |
| 604 |
< |
preserve the structural properties of the original ODE and its flow. |
| 604 |
> |
preserve the structural properties of the original ODE and its |
| 605 |
> |
propagator. |
| 606 |
|
|
| 607 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 608 |
|
A lot of well established and very effective numerical methods have |
| 609 |
< |
been successful precisely because of their symplecticities even |
| 609 |
> |
been successful precisely because of their symplectic nature even |
| 610 |
|
though this fact was not recognized when they were first |
| 611 |
|
constructed. The most famous example is the Verlet-leapfrog method |
| 612 |
|
in molecular dynamics. In general, symplectic integrators can be |
| 617 |
|
\item Runge-Kutta methods |
| 618 |
|
\item Splitting methods |
| 619 |
|
\end{enumerate} |
| 620 |
< |
|
| 672 |
< |
Generating function\cite{Channell1990} tends to lead to methods |
| 620 |
> |
Generating functions\cite{Channell1990} tend to lead to methods |
| 621 |
|
which are cumbersome and difficult to use. In dissipative systems, |
| 622 |
|
variational methods can capture the decay of energy |
| 623 |
< |
accurately\cite{Kane2000}. Since their geometrically unstable nature |
| 623 |
> |
accurately\cite{Kane2000}. Since they are geometrically unstable |
| 624 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 625 |
|
methods are not suitable for Hamiltonian system. Recently, various |
| 626 |
< |
high-order explicit Runge-Kutta methods |
| 627 |
< |
\cite{Owren1992,Chen2003}have been developed to overcome this |
| 628 |
< |
instability. However, due to computational penalty involved in |
| 629 |
< |
implementing the Runge-Kutta methods, they have not attracted much |
| 630 |
< |
attention from the Molecular Dynamics community. Instead, splitting |
| 631 |
< |
methods have been widely accepted since they exploit natural |
| 632 |
< |
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
| 626 |
> |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
| 627 |
> |
have been developed to overcome this instability. However, due to |
| 628 |
> |
computational penalty involved in implementing the Runge-Kutta |
| 629 |
> |
methods, they have not attracted much attention from the Molecular |
| 630 |
> |
Dynamics community. Instead, splitting methods have been widely |
| 631 |
> |
accepted since they exploit natural decompositions of the |
| 632 |
> |
system\cite{Tuckerman1992, McLachlan1998}. |
| 633 |
|
|
| 634 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 635 |
|
|
| 636 |
|
The main idea behind splitting methods is to decompose the discrete |
| 637 |
< |
$\varphi_h$ as a composition of simpler flows, |
| 637 |
> |
$\varphi_h$ as a composition of simpler propagators, |
| 638 |
|
\begin{equation} |
| 639 |
|
\varphi _h = \varphi _{h_1 } \circ \varphi _{h_2 } \ldots \circ |
| 640 |
|
\varphi _{h_n } |
| 641 |
|
\label{introEquation:FlowDecomposition} |
| 642 |
|
\end{equation} |
| 643 |
< |
where each of the sub-flow is chosen such that each represent a |
| 644 |
< |
simpler integration of the system. |
| 645 |
< |
|
| 698 |
< |
Suppose that a Hamiltonian system takes the form, |
| 643 |
> |
where each of the sub-propagator is chosen such that each represent |
| 644 |
> |
a simpler integration of the system. Suppose that a Hamiltonian |
| 645 |
> |
system takes the form, |
| 646 |
|
\[ |
| 647 |
|
H = H_1 + H_2. |
| 648 |
|
\] |
| 649 |
|
Here, $H_1$ and $H_2$ may represent different physical processes of |
| 650 |
|
the system. For instance, they may relate to kinetic and potential |
| 651 |
|
energy respectively, which is a natural decomposition of the |
| 652 |
< |
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
| 653 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
| 654 |
< |
order expression is then given by the Lie-Trotter formula |
| 652 |
> |
problem. If $H_1$ and $H_2$ can be integrated using exact |
| 653 |
> |
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
| 654 |
> |
simple first order expression is then given by the Lie-Trotter |
| 655 |
> |
formula |
| 656 |
|
\begin{equation} |
| 657 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 658 |
|
\label{introEquation:firstOrderSplitting} |
| 661 |
|
continuous $\varphi _i$ over a time $h$. By definition, as |
| 662 |
|
$\varphi_i(t)$ is the exact solution of a Hamiltonian system, it |
| 663 |
|
must follow that each operator $\varphi_i(t)$ is a symplectic map. |
| 664 |
< |
It is easy to show that any composition of symplectic flows yields a |
| 665 |
< |
symplectic map, |
| 664 |
> |
It is easy to show that any composition of symplectic propagators |
| 665 |
> |
yields a symplectic map, |
| 666 |
|
\begin{equation} |
| 667 |
|
(\varphi '\phi ')^T J\varphi '\phi ' = \phi '^T \varphi '^T J\varphi |
| 668 |
|
'\phi ' = \phi '^T J\phi ' = J, |
| 670 |
|
\end{equation} |
| 671 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
| 672 |
|
splitting in this context automatically generates a symplectic map. |
| 673 |
< |
|
| 674 |
< |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
| 675 |
< |
introduces local errors proportional to $h^2$, while Strang |
| 676 |
< |
splitting gives a second-order decomposition, |
| 673 |
> |
The Lie-Trotter |
| 674 |
> |
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
| 675 |
> |
local errors proportional to $h^2$, while the Strang splitting gives |
| 676 |
> |
a second-order decomposition, |
| 677 |
|
\begin{equation} |
| 678 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 679 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 680 |
|
\end{equation} |
| 681 |
< |
which has a local error proportional to $h^3$. The Sprang |
| 681 |
> |
which has a local error proportional to $h^3$. The Strang |
| 682 |
|
splitting's popularity in molecular simulation community attribute |
| 683 |
|
to its symmetric property, |
| 684 |
|
\begin{equation} |
| 706 |
|
\end{align} |
| 707 |
|
where $F(t)$ is the force at time $t$. This integration scheme is |
| 708 |
|
known as \emph{velocity verlet} which is |
| 709 |
< |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
| 710 |
< |
time-reversible(\ref{introEquation:timeReversible}) and |
| 711 |
< |
volume-preserving (\ref{introEquation:volumePreserving}). These |
| 709 |
> |
symplectic(Eq.~\ref{introEquation:SymplecticFlowComposition}), |
| 710 |
> |
time-reversible(Eq.~\ref{introEquation:timeReversible}) and |
| 711 |
> |
volume-preserving (Eq.~\ref{introEquation:volumePreserving}). These |
| 712 |
|
geometric properties attribute to its long-time stability and its |
| 713 |
|
popularity in the community. However, the most commonly used |
| 714 |
|
velocity verlet integration scheme is written as below, |
| 733 |
|
|
| 734 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 735 |
|
\end{enumerate} |
| 788 |
– |
|
| 736 |
|
By simply switching the order of the propagators in the splitting |
| 737 |
|
and composing a new integrator, the \emph{position verlet} |
| 738 |
|
integrator, can be generated, |
| 749 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 750 |
|
|
| 751 |
|
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 752 |
< |
local error of splitting method in terms of the commutator of the |
| 753 |
< |
operators(\ref{introEquation:exponentialOperator}) associated with |
| 754 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associated with |
| 755 |
< |
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 752 |
> |
local error of a splitting method in terms of the commutator of the |
| 753 |
> |
operators(Eq.~\ref{introEquation:exponentialOperator}) associated with |
| 754 |
> |
the sub-propagator. For operators $hX$ and $hY$ which are associated |
| 755 |
> |
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 756 |
|
\begin{equation} |
| 757 |
|
\exp (hX + hY) = \exp (hZ) |
| 758 |
|
\end{equation} |
| 761 |
|
hZ = hX + hY + \frac{{h^2 }}{2}[X,Y] + \frac{{h^3 }}{2}\left( |
| 762 |
|
{[X,[X,Y]] + [Y,[Y,X]]} \right) + \ldots . |
| 763 |
|
\end{equation} |
| 764 |
< |
Here, $[X,Y]$ is the commutators of operator $X$ and $Y$ given by |
| 764 |
> |
Here, $[X,Y]$ is the commutator of operator $X$ and $Y$ given by |
| 765 |
|
\[ |
| 766 |
|
[X,Y] = XY - YX . |
| 767 |
|
\] |
| 768 |
|
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
| 769 |
< |
to the Sprang splitting, we can obtain |
| 769 |
> |
to the Strang splitting, we can obtain |
| 770 |
|
\begin{eqnarray*} |
| 771 |
|
\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 \\ |
| 772 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 773 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
| 773 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots |
| 774 |
> |
). |
| 775 |
|
\end{eqnarray*} |
| 776 |
< |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0,\] the dominant local |
| 777 |
< |
error of Spring splitting is proportional to $h^3$. The same |
| 778 |
< |
procedure can be applied to a general splitting, of the form |
| 776 |
> |
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
| 777 |
> |
error of Strang splitting is proportional to $h^3$. The same |
| 778 |
> |
procedure can be applied to a general splitting of the form |
| 779 |
|
\begin{equation} |
| 780 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 781 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 810 |
|
dynamical information. The basic idea of molecular dynamics is that |
| 811 |
|
macroscopic properties are related to microscopic behavior and |
| 812 |
|
microscopic behavior can be calculated from the trajectories in |
| 813 |
< |
simulations. For instance, instantaneous temperature of an |
| 814 |
< |
Hamiltonian system of $N$ particle can be measured by |
| 813 |
> |
simulations. For instance, instantaneous temperature of a |
| 814 |
> |
Hamiltonian system of $N$ particles can be measured by |
| 815 |
|
\[ |
| 816 |
|
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
| 817 |
|
\] |
| 818 |
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
| 819 |
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
| 820 |
< |
the boltzman constant. |
| 820 |
> |
the Boltzman constant. |
| 821 |
|
|
| 822 |
|
A typical molecular dynamics run consists of three essential steps: |
| 823 |
|
\begin{enumerate} |
| 834 |
|
These three individual steps will be covered in the following |
| 835 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 836 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
| 837 |
< |
will discusse issues in production run. |
| 837 |
> |
discusses issues of production runs. |
| 838 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 839 |
< |
trajectory analysis. |
| 839 |
> |
analysis of trajectories. |
| 840 |
|
|
| 841 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 842 |
|
|
| 848 |
|
thousands of crystal structures of molecules are discovered every |
| 849 |
|
year, many more remain unknown due to the difficulties of |
| 850 |
|
purification and crystallization. Even for molecules with known |
| 851 |
< |
structure, some important information is missing. For example, a |
| 851 |
> |
structures, some important information is missing. For example, a |
| 852 |
|
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 853 |
< |
be added. Moreover, in order to include electrostatic interaction, |
| 853 |
> |
be added. Moreover, in order to include electrostatic interactions, |
| 854 |
|
one may need to specify the partial charges for individual atoms. |
| 855 |
|
Under some circumstances, we may even need to prepare the system in |
| 856 |
|
a special configuration. For instance, when studying transport |
| 870 |
|
surface and to locate the local minimum. While converging slowly |
| 871 |
|
near the minimum, steepest descent method is extremely robust when |
| 872 |
|
systems are strongly anharmonic. Thus, it is often used to refine |
| 873 |
< |
structure from crystallographic data. Relied on the gradient or |
| 874 |
< |
hessian, advanced methods like Newton-Raphson converge rapidly to a |
| 875 |
< |
local minimum, but become unstable if the energy surface is far from |
| 873 |
> |
structures from crystallographic data. Relying on the Hessian, |
| 874 |
> |
advanced methods like Newton-Raphson converge rapidly to a local |
| 875 |
> |
minimum, but become unstable if the energy surface is far from |
| 876 |
|
quadratic. Another factor that must be taken into account, when |
| 877 |
|
choosing energy minimization method, is the size of the system. |
| 878 |
|
Steepest descent and conjugate gradient can deal with models of any |
| 879 |
|
size. Because of the limits on computer memory to store the hessian |
| 880 |
< |
matrix and the computing power needed to diagonalized these |
| 881 |
< |
matrices, most Newton-Raphson methods can not be used with very |
| 934 |
< |
large systems. |
| 880 |
> |
matrix and the computing power needed to diagonalize these matrices, |
| 881 |
> |
most Newton-Raphson methods can not be used with very large systems. |
| 882 |
|
|
| 883 |
|
\subsubsection{\textbf{Heating}} |
| 884 |
|
|
| 885 |
< |
Typically, Heating is performed by assigning random velocities |
| 885 |
> |
Typically, heating is performed by assigning random velocities |
| 886 |
|
according to a Maxwell-Boltzman distribution for a desired |
| 887 |
|
temperature. Beginning at a lower temperature and gradually |
| 888 |
|
increasing the temperature by assigning larger random velocities, we |
| 889 |
< |
end up with setting the temperature of the system to a final |
| 890 |
< |
temperature at which the simulation will be conducted. In heating |
| 891 |
< |
phase, we should also keep the system from drifting or rotating as a |
| 892 |
< |
whole. To do this, the net linear momentum and angular momentum of |
| 893 |
< |
the system is shifted to zero after each resampling from the Maxwell |
| 894 |
< |
-Boltzman distribution. |
| 889 |
> |
end up setting the temperature of the system to a final temperature |
| 890 |
> |
at which the simulation will be conducted. In heating phase, we |
| 891 |
> |
should also keep the system from drifting or rotating as a whole. To |
| 892 |
> |
do this, the net linear momentum and angular momentum of the system |
| 893 |
> |
is shifted to zero after each resampling from the Maxwell -Boltzman |
| 894 |
> |
distribution. |
| 895 |
|
|
| 896 |
|
\subsubsection{\textbf{Equilibration}} |
| 897 |
|
|
| 902 |
|
properties \textit{etc}, become independent of time. Strictly |
| 903 |
|
speaking, minimization and heating are not necessary, provided the |
| 904 |
|
equilibration process is long enough. However, these steps can serve |
| 905 |
< |
as a means to arrive at an equilibrated structure in an effective |
| 905 |
> |
as a mean to arrive at an equilibrated structure in an effective |
| 906 |
|
way. |
| 907 |
|
|
| 908 |
|
\subsection{\label{introSection:production}Production} |
| 918 |
|
calculation of non-bonded forces, such as van der Waals force and |
| 919 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
| 920 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
| 921 |
< |
which making large simulations prohibitive in the absence of any |
| 922 |
< |
algorithmic tricks. |
| 923 |
< |
|
| 924 |
< |
A natural approach to avoid system size issues is to represent the |
| 925 |
< |
bulk behavior by a finite number of the particles. However, this |
| 926 |
< |
approach will suffer from the surface effect at the edges of the |
| 927 |
< |
simulation. To offset this, \textit{Periodic boundary conditions} |
| 928 |
< |
(see Fig.~\ref{introFig:pbc}) is developed to simulate bulk |
| 929 |
< |
properties with a relatively small number of particles. In this |
| 930 |
< |
method, the simulation box is replicated throughout space to form an |
| 931 |
< |
infinite lattice. During the simulation, when a particle moves in |
| 932 |
< |
the primary cell, its image in other cells move in exactly the same |
| 933 |
< |
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. |
| 921 |
> |
which makes large simulations prohibitive in the absence of any |
| 922 |
> |
algorithmic tricks. A natural approach to avoid system size issues |
| 923 |
> |
is to represent the bulk behavior by a finite number of the |
| 924 |
> |
particles. However, this approach will suffer from surface effects |
| 925 |
> |
at the edges of the simulation. To offset this, \textit{Periodic |
| 926 |
> |
boundary conditions} (see Fig.~\ref{introFig:pbc}) were developed to |
| 927 |
> |
simulate bulk properties with a relatively small number of |
| 928 |
> |
particles. In this method, the simulation box is replicated |
| 929 |
> |
throughout space to form an infinite lattice. During the simulation, |
| 930 |
> |
when a particle moves in the primary cell, its image in other cells |
| 931 |
> |
move in exactly the same direction with exactly the same |
| 932 |
> |
orientation. Thus, as a particle leaves the primary cell, one of its |
| 933 |
> |
images will enter through the opposite face. |
| 934 |
|
\begin{figure} |
| 935 |
|
\centering |
| 936 |
|
\includegraphics[width=\linewidth]{pbc.eps} |
| 942 |
|
|
| 943 |
|
%cutoff and minimum image convention |
| 944 |
|
Another important technique to improve the efficiency of force |
| 945 |
< |
evaluation is to apply spherical cutoff where particles farther than |
| 946 |
< |
a predetermined distance are not included in the calculation |
| 945 |
> |
evaluation is to apply spherical cutoffs where particles farther |
| 946 |
> |
than a predetermined distance are not included in the calculation |
| 947 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 948 |
|
discontinuity in the potential energy curve. Fortunately, one can |
| 949 |
< |
shift simple radial potential to ensure the potential curve go |
| 949 |
> |
shift a simple radial potential to ensure the potential curve go |
| 950 |
|
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 951 |
|
well for Lennard-Jones interaction because of its short range |
| 952 |
|
nature. However, simply truncating the electrostatic interaction |
| 972 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 973 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 974 |
|
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
| 975 |
< |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
| 975 |
> |
r_{ij})}{r_{ij}}\right\}, \label{introEquation:shiftedCoulomb} |
| 976 |
|
\end{equation} |
| 977 |
|
where $\alpha$ is the convergence parameter. Due to the lack of |
| 978 |
|
inherent periodicity and rapid convergence,this method is extremely |
| 989 |
|
|
| 990 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 991 |
|
|
| 992 |
< |
Recently, advanced visualization technique have become applied to |
| 992 |
> |
Recently, advanced visualization techniques have been applied to |
| 993 |
|
monitor the motions of molecules. Although the dynamics of the |
| 994 |
|
system can be described qualitatively from animation, quantitative |
| 995 |
< |
trajectory analysis are more useful. According to the principles of |
| 996 |
< |
Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics}, |
| 997 |
< |
one can compute thermodynamic properties, analyze fluctuations of |
| 998 |
< |
structural parameters, and investigate time-dependent processes of |
| 999 |
< |
the molecule from the trajectories. |
| 995 |
> |
trajectory analysis is more useful. According to the principles of |
| 996 |
> |
Statistical Mechanics in |
| 997 |
> |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 998 |
> |
thermodynamic properties, analyze fluctuations of structural |
| 999 |
> |
parameters, and investigate time-dependent processes of the molecule |
| 1000 |
> |
from the trajectories. |
| 1001 |
|
|
| 1002 |
|
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
| 1003 |
|
|
| 1027 |
|
distribution functions. Among these functions,the \emph{pair |
| 1028 |
|
distribution function}, also known as \emph{radial distribution |
| 1029 |
|
function}, is of most fundamental importance to liquid theory. |
| 1030 |
< |
Experimentally, pair distribution function can be gathered by |
| 1030 |
> |
Experimentally, pair distribution functions can be gathered by |
| 1031 |
|
Fourier transforming raw data from a series of neutron diffraction |
| 1032 |
|
experiments and integrating over the surface factor |
| 1033 |
|
\cite{Powles1973}. The experimental results can serve as a criterion |
| 1034 |
|
to justify the correctness of a liquid model. Moreover, various |
| 1035 |
|
equilibrium thermodynamic and structural properties can also be |
| 1036 |
< |
expressed in terms of radial distribution function \cite{Allen1987}. |
| 1037 |
< |
|
| 1038 |
< |
The pair distribution functions $g(r)$ gives the probability that a |
| 1039 |
< |
particle $i$ will be located at a distance $r$ from a another |
| 1040 |
< |
particle $j$ in the system |
| 1095 |
< |
\[ |
| 1036 |
> |
expressed in terms of the radial distribution function |
| 1037 |
> |
\cite{Allen1987}. The pair distribution functions $g(r)$ gives the |
| 1038 |
> |
probability that a particle $i$ will be located at a distance $r$ |
| 1039 |
> |
from a another particle $j$ in the system |
| 1040 |
> |
\begin{equation} |
| 1041 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1042 |
|
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
| 1043 |
|
(r)}{\rho}. |
| 1044 |
< |
\] |
| 1044 |
> |
\end{equation} |
| 1045 |
|
Note that the delta function can be replaced by a histogram in |
| 1046 |
|
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
| 1047 |
|
the height of these peaks gradually decreases to 1 as the liquid of |
| 1059 |
|
\label{introEquation:timeCorrelationFunction} |
| 1060 |
|
\end{equation} |
| 1061 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1062 |
< |
function is called an \emph{autocorrelation function}. One example |
| 1063 |
< |
of an auto correlation function is the velocity auto-correlation |
| 1062 |
> |
functions are called \emph{autocorrelation functions}. One example |
| 1063 |
> |
of auto correlation function is the velocity auto-correlation |
| 1064 |
|
function which is directly related to transport properties of |
| 1065 |
|
molecular liquids: |
| 1066 |
|
\[ |
| 1068 |
|
\right\rangle } dt |
| 1069 |
|
\] |
| 1070 |
|
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1071 |
< |
function, which is averaging over time origins and over all the |
| 1072 |
< |
atoms, the dipole autocorrelation functions are calculated for the |
| 1071 |
> |
function, which is averaged over time origins and over all the |
| 1072 |
> |
atoms, the dipole autocorrelation functions is calculated for the |
| 1073 |
|
entire system. The dipole autocorrelation function is given by: |
| 1074 |
|
\[ |
| 1075 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1078 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
| 1079 |
|
by |
| 1080 |
|
\[ |
| 1081 |
< |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
| 1081 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
| 1082 |
|
\] |
| 1083 |
< |
In principle, many time correlation functions can be related with |
| 1083 |
> |
In principle, many time correlation functions can be related to |
| 1084 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1085 |
|
scattering spectra of molecular liquids. In practice, one can |
| 1086 |
< |
extract the IR spectrum from the intensity of dipole fluctuation at |
| 1087 |
< |
each frequency using the following relationship: |
| 1086 |
> |
extract the IR spectrum from the intensity of the molecular dipole |
| 1087 |
> |
fluctuation at each frequency using the following relationship: |
| 1088 |
|
\[ |
| 1089 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1090 |
< |
i2\pi vt} dt} |
| 1090 |
> |
i2\pi vt} dt}. |
| 1091 |
|
\] |
| 1092 |
|
|
| 1093 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1094 |
|
|
| 1095 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1096 |
|
areas, from engineering, physics, to chemistry. For example, |
| 1097 |
< |
missiles and vehicle are usually modeled by rigid bodies. The |
| 1098 |
< |
movement of the objects in 3D gaming engine or other physics |
| 1099 |
< |
simulator is governed by rigid body dynamics. In molecular |
| 1097 |
> |
missiles and vehicles are usually modeled by rigid bodies. The |
| 1098 |
> |
movement of the objects in 3D gaming engines or other physics |
| 1099 |
> |
simulators is governed by rigid body dynamics. In molecular |
| 1100 |
|
simulations, rigid bodies are used to simplify protein-protein |
| 1101 |
|
docking studies\cite{Gray2003}. |
| 1102 |
|
|
| 1105 |
|
freedom. Euler angles are the natural choice to describe the |
| 1106 |
|
rotational degrees of freedom. However, due to $\frac {1}{sin |
| 1107 |
|
\theta}$ singularities, the numerical integration of corresponding |
| 1108 |
< |
equations of motion is very inefficient and inaccurate. Although an |
| 1109 |
< |
alternative integrator using multiple sets of Euler angles can |
| 1110 |
< |
overcome this difficulty\cite{Barojas1973}, the computational |
| 1111 |
< |
penalty and the loss of angular momentum conservation still remain. |
| 1112 |
< |
A singularity-free representation utilizing quaternions was |
| 1113 |
< |
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
| 1114 |
< |
approach uses a nonseparable Hamiltonian resulting from the |
| 1115 |
< |
quaternion representation, which prevents the symplectic algorithm |
| 1116 |
< |
to be utilized. Another different approach is to apply holonomic |
| 1117 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
| 1118 |
< |
moves independently under the normal forces deriving from potential |
| 1119 |
< |
energy and constraint forces which are used to guarantee the |
| 1120 |
< |
rigidness. However, due to their iterative nature, the SHAKE and |
| 1121 |
< |
Rattle algorithms also converge very slowly when the number of |
| 1122 |
< |
constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1108 |
> |
equations of these motion is very inefficient and inaccurate. |
| 1109 |
> |
Although an alternative integrator using multiple sets of Euler |
| 1110 |
> |
angles can overcome this difficulty\cite{Barojas1973}, the |
| 1111 |
> |
computational penalty and the loss of angular momentum conservation |
| 1112 |
> |
still remain. A singularity-free representation utilizing |
| 1113 |
> |
quaternions was developed by Evans in 1977\cite{Evans1977}. |
| 1114 |
> |
Unfortunately, this approach used a nonseparable Hamiltonian |
| 1115 |
> |
resulting from the quaternion representation, which prevented the |
| 1116 |
> |
symplectic algorithm from being utilized. Another different approach |
| 1117 |
> |
is to apply holonomic constraints to the atoms belonging to the |
| 1118 |
> |
rigid body. Each atom moves independently under the normal forces |
| 1119 |
> |
deriving from potential energy and constraint forces which are used |
| 1120 |
> |
to guarantee the rigidness. However, due to their iterative nature, |
| 1121 |
> |
the SHAKE and Rattle algorithms also converge very slowly when the |
| 1122 |
> |
number of constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1123 |
|
|
| 1124 |
|
A break-through in geometric literature suggests that, in order to |
| 1125 |
|
develop a long-term integration scheme, one should preserve the |
| 1126 |
< |
symplectic structure of the flow. By introducing a conjugate |
| 1126 |
> |
symplectic structure of the propagator. By introducing a conjugate |
| 1127 |
|
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
| 1128 |
|
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
| 1129 |
|
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1131 |
|
An alternative method using the quaternion representation was |
| 1132 |
|
developed by Omelyan\cite{Omelyan1998}. However, both of these |
| 1133 |
|
methods are iterative and inefficient. In this section, we descibe a |
| 1134 |
< |
symplectic Lie-Poisson integrator for rigid body developed by |
| 1134 |
> |
symplectic Lie-Poisson integrator for rigid bodies developed by |
| 1135 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1136 |
|
|
| 1137 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1142 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1143 |
|
\label{introEquation:RBHamiltonian} |
| 1144 |
|
\end{equation} |
| 1145 |
< |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 1146 |
< |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 1147 |
< |
$J$, a diagonal matrix, is defined by |
| 1145 |
> |
Here, $q$ and $Q$ are the position vector and rotation matrix for |
| 1146 |
> |
the rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , |
| 1147 |
> |
and $J$, a diagonal matrix, is defined by |
| 1148 |
|
\[ |
| 1149 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1150 |
|
\] |
| 1154 |
|
\begin{equation} |
| 1155 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1156 |
|
\end{equation} |
| 1157 |
< |
which is used to ensure rotation matrix's unitarity. Differentiating |
| 1158 |
< |
\ref{introEquation:orthogonalConstraint} and using Equation |
| 1159 |
< |
\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1215 |
< |
\begin{equation} |
| 1216 |
< |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1217 |
< |
\label{introEquation:RBFirstOrderConstraint} |
| 1218 |
< |
\end{equation} |
| 1219 |
< |
|
| 1220 |
< |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1221 |
< |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1157 |
> |
which is used to ensure the rotation matrix's unitarity. Using |
| 1158 |
> |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~ |
| 1159 |
> |
\ref{introEquation:motionHamiltonianMomentum}, one can write down |
| 1160 |
|
the equations of motion, |
| 1223 |
– |
|
| 1161 |
|
\begin{eqnarray} |
| 1162 |
< |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1163 |
< |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1164 |
< |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1162 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
| 1163 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\ |
| 1164 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
| 1165 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1166 |
|
\end{eqnarray} |
| 1167 |
< |
|
| 1167 |
> |
Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and |
| 1168 |
> |
using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1169 |
> |
\begin{equation} |
| 1170 |
> |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1171 |
> |
\label{introEquation:RBFirstOrderConstraint} |
| 1172 |
> |
\end{equation} |
| 1173 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1174 |
|
We can use a constraint force provided by a Lagrange multiplier on |
| 1175 |
< |
the normal manifold to keep the motion on constraint space. Or we |
| 1176 |
< |
can simply evolve the system on the constraint manifold. These two |
| 1177 |
< |
methods have been proved to be equivalent. The holonomic constraint |
| 1178 |
< |
and equations of motions define a constraint manifold for rigid |
| 1179 |
< |
bodies |
| 1175 |
> |
the normal manifold to keep the motion on the constraint space. Or |
| 1176 |
> |
we can simply evolve the system on the constraint manifold. These |
| 1177 |
> |
two methods have been proved to be equivalent. The holonomic |
| 1178 |
> |
constraint and equations of motions define a constraint manifold for |
| 1179 |
> |
rigid bodies |
| 1180 |
|
\[ |
| 1181 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1182 |
|
\right\}. |
| 1183 |
|
\] |
| 1184 |
< |
|
| 1185 |
< |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1186 |
< |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
| 1187 |
< |
rotation group $SO(3)$. However, it turns out that under symplectic |
| 1246 |
< |
transformation, the cotangent space and the phase space are |
| 1247 |
< |
diffeomorphic. By introducing |
| 1184 |
> |
Unfortunately, this constraint manifold is not $T^* SO(3)$ which is |
| 1185 |
> |
a symplectic manifold on Lie rotation group $SO(3)$. However, it |
| 1186 |
> |
turns out that under symplectic transformation, the cotangent space |
| 1187 |
> |
and the phase space are diffeomorphic. By introducing |
| 1188 |
|
\[ |
| 1189 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1190 |
|
\] |
| 1191 |
< |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 1191 |
> |
the mechanical system subjected to a holonomic constraint manifold $M$ |
| 1192 |
|
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1193 |
|
\[ |
| 1194 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1195 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1196 |
|
\] |
| 1257 |
– |
|
| 1197 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1198 |
|
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1199 |
|
given as |
| 1212 |
|
\[ |
| 1213 |
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1214 |
|
\] |
| 1215 |
< |
respectively. |
| 1216 |
< |
|
| 1217 |
< |
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, |
| 1215 |
> |
respectively. As a common choice to describe the rotation dynamics |
| 1216 |
> |
of the rigid body, the angular momentum on the body fixed frame $\Pi |
| 1217 |
> |
= Q^t P$ is introduced to rewrite the equations of motion, |
| 1218 |
|
\begin{equation} |
| 1219 |
|
\begin{array}{l} |
| 1220 |
< |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1221 |
< |
\dot Q = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1220 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda, \\ |
| 1221 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1}, \\ |
| 1222 |
|
\end{array} |
| 1223 |
|
\label{introEqaution:RBMotionPI} |
| 1224 |
|
\end{equation} |
| 1225 |
< |
, as well as holonomic constraints, |
| 1226 |
< |
\[ |
| 1227 |
< |
\begin{array}{l} |
| 1291 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1292 |
< |
Q^T Q = 1 \\ |
| 1293 |
< |
\end{array} |
| 1294 |
< |
\] |
| 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, |
| 1225 |
> |
as well as holonomic constraints $\Pi J^{ - 1} + J^{ - 1} \Pi ^t = |
| 1226 |
> |
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a |
| 1227 |
> |
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism, |
| 1228 |
|
\begin{equation} |
| 1229 |
|
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1230 |
|
{\begin{array}{*{20}c} |
| 1237 |
|
will let us associate the matrix products with traditional vector |
| 1238 |
|
operations |
| 1239 |
|
\[ |
| 1240 |
< |
\hat vu = v \times u |
| 1240 |
> |
\hat vu = v \times u. |
| 1241 |
|
\] |
| 1242 |
< |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1242 |
> |
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1243 |
|
matrix, |
| 1244 |
< |
|
| 1245 |
< |
\begin{eqnarray*} |
| 1246 |
< |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ |
| 1247 |
< |
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) + \sum\limits_i {[Q^T F_i |
| 1248 |
< |
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
| 1249 |
< |
\label{introEquation:skewMatrixPI} |
| 1250 |
< |
\end{eqnarray*} |
| 1251 |
< |
|
| 1252 |
< |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1253 |
< |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1254 |
< |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1255 |
< |
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 |
| 1244 |
> |
\begin{eqnarray} |
| 1245 |
> |
(\dot \Pi - \dot \Pi ^T )&= &(\Pi - \Pi ^T )(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
| 1246 |
> |
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1247 |
> |
(\Lambda - \Lambda ^T ). \label{introEquation:skewMatrixPI} |
| 1248 |
> |
\end{eqnarray} |
| 1249 |
> |
Since $\Lambda$ is symmetric, the last term of |
| 1250 |
> |
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
| 1251 |
> |
Lagrange multiplier $\Lambda$ is absent from the equations of |
| 1252 |
> |
motion. This unique property eliminates the requirement of |
| 1253 |
> |
iterations which can not be avoided in other methods\cite{Kol1997, |
| 1254 |
> |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
| 1255 |
> |
equation of motion for angular momentum in the body frame |
| 1256 |
|
\begin{equation} |
| 1257 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1258 |
|
F_i (r,Q)} \right) \times X_i }. |
| 1261 |
|
In the same manner, the equation of motion for rotation matrix is |
| 1262 |
|
given by |
| 1263 |
|
\[ |
| 1264 |
< |
\dot Q = Qskew(I^{ - 1} \pi ) |
| 1264 |
> |
\dot Q = Qskew(I^{ - 1} \pi ). |
| 1265 |
|
\] |
| 1266 |
|
|
| 1267 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1268 |
< |
Lie-Poisson Integrator for Free Rigid Body} |
| 1268 |
> |
Lie-Poisson Integrator for Free Rigid Bodies} |
| 1269 |
|
|
| 1270 |
|
If there are no external forces exerted on the rigid body, the only |
| 1271 |
|
contribution to the rotational motion is from the kinetic energy |
| 1283 |
|
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1284 |
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1285 |
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1286 |
< |
\end{array}} \right) |
| 1286 |
> |
\end{array}} \right). |
| 1287 |
|
\end{equation} |
| 1288 |
|
Thus, the dynamics of free rigid body is governed by |
| 1289 |
|
\begin{equation} |
| 1290 |
< |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1290 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
| 1291 |
|
\end{equation} |
| 1292 |
< |
|
| 1293 |
< |
One may notice that each $T_i^r$ in Equation |
| 1294 |
< |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1369 |
< |
instance, the equations of motion due to $T_1^r$ are given by |
| 1292 |
> |
One may notice that each $T_i^r$ in |
| 1293 |
> |
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly. |
| 1294 |
> |
For instance, the equations of motion due to $T_1^r$ are given by |
| 1295 |
|
\begin{equation} |
| 1296 |
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
| 1297 |
|
\label{introEqaution:RBMotionSingleTerm} |
| 1298 |
|
\end{equation} |
| 1299 |
< |
where |
| 1299 |
> |
with |
| 1300 |
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1301 |
|
0 & 0 & 0 \\ |
| 1302 |
|
0 & 0 & {\pi _1 } \\ |
| 1303 |
|
0 & { - \pi _1 } & 0 \\ |
| 1304 |
|
\end{array}} \right). |
| 1305 |
|
\] |
| 1306 |
< |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1306 |
> |
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is |
| 1307 |
|
\[ |
| 1308 |
|
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
| 1309 |
|
Q(0)e^{\Delta tR_1 } |
| 1317 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1318 |
|
\] |
| 1319 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1320 |
< |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
| 1321 |
< |
propagator, |
| 1322 |
< |
\[ |
| 1323 |
< |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1324 |
< |
) |
| 1325 |
< |
\] |
| 1326 |
< |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1320 |
> |
tR_1 }$, we can use the Cayley transformation to obtain a |
| 1321 |
> |
single-aixs propagator, |
| 1322 |
> |
\begin{eqnarray*} |
| 1323 |
> |
e^{\Delta tR_1 } & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta |
| 1324 |
> |
tR_1 ) \\ |
| 1325 |
> |
% |
| 1326 |
> |
& \approx & \left( \begin{array}{ccc} |
| 1327 |
> |
1 & 0 & 0 \\ |
| 1328 |
> |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
| 1329 |
> |
\theta^2 / 4} \\ |
| 1330 |
> |
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
| 1331 |
> |
\theta^2 / 4} |
| 1332 |
> |
\end{array} |
| 1333 |
> |
\right). |
| 1334 |
> |
\end{eqnarray*} |
| 1335 |
> |
The propagators for $T_2^r$ and $T_3^r$ can be found in the same |
| 1336 |
|
manner. In order to construct a second-order symplectic method, we |
| 1337 |
< |
split the angular kinetic Hamiltonian function can into five terms |
| 1337 |
> |
split the angular kinetic Hamiltonian function into five terms |
| 1338 |
|
\[ |
| 1339 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1340 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1348 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1349 |
|
_1 }. |
| 1350 |
|
\] |
| 1417 |
– |
|
| 1351 |
|
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1352 |
|
$F(\pi )$ and $G(\pi )$ is defined by |
| 1353 |
|
\[ |
| 1354 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1355 |
< |
) |
| 1355 |
> |
). |
| 1356 |
|
\] |
| 1357 |
|
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1358 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1359 |
|
conserved quantity in Poisson system. We can easily verify that the |
| 1360 |
|
norm of the angular momentum, $\parallel \pi |
| 1361 |
< |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1361 |
> |
\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1362 |
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1363 |
|
then by the chain rule |
| 1364 |
|
\[ |
| 1365 |
|
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1366 |
< |
}}{2})\pi |
| 1366 |
> |
}}{2})\pi. |
| 1367 |
|
\] |
| 1368 |
< |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1368 |
> |
Thus, $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel |
| 1369 |
> |
\pi |
| 1370 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1371 |
|
Lie-Poisson integrator is found to be both extremely efficient and |
| 1372 |
|
stable. These properties can be explained by the fact the small |
| 1377 |
|
Splitting for Rigid Body} |
| 1378 |
|
|
| 1379 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1380 |
< |
energy and potential energy, |
| 1381 |
< |
\[ |
| 1382 |
< |
H = T(p,\pi ) + V(q,Q) |
| 1449 |
< |
\] |
| 1450 |
< |
The equations of motion corresponding to potential energy and |
| 1451 |
< |
kinetic energy are listed in the below table, |
| 1380 |
> |
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations |
| 1381 |
> |
of motion corresponding to potential energy and kinetic energy are |
| 1382 |
> |
listed in Table~\ref{introTable:rbEquations} |
| 1383 |
|
\begin{table} |
| 1384 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1385 |
+ |
\label{introTable:rbEquations} |
| 1386 |
|
\begin{center} |
| 1387 |
|
\begin{tabular}{|l|l|} |
| 1388 |
|
\hline |
| 1418 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1419 |
|
\end{equation} |
| 1420 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1421 |
< |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1422 |
< |
corresponding propagators are given by |
| 1421 |
> |
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore, |
| 1422 |
> |
the corresponding propagators are given by |
| 1423 |
|
\[ |
| 1424 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1425 |
|
_{\Delta t,T^r }. |
| 1426 |
|
\] |
| 1427 |
|
Finally, we obtain the overall symplectic propagators for freely |
| 1428 |
|
moving rigid bodies |
| 1429 |
< |
\begin{equation} |
| 1430 |
< |
\begin{array}{c} |
| 1431 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1432 |
< |
\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} |
| 1429 |
> |
\begin{eqnarray} |
| 1430 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
| 1431 |
> |
& & \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\\ |
| 1432 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} . |
| 1433 |
|
\label{introEquation:overallRBFlowMaps} |
| 1434 |
< |
\end{equation} |
| 1434 |
> |
\end{eqnarray} |
| 1435 |
|
|
| 1436 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1437 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1477 |
|
\[ |
| 1478 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1479 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1480 |
< |
\] and combining the last two terms in Equation |
| 1481 |
< |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
| 1552 |
< |
Hamiltonian as |
| 1480 |
> |
\] |
| 1481 |
> |
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as |
| 1482 |
|
\[ |
| 1483 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1484 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1485 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1486 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1486 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}}. |
| 1487 |
|
\] |
| 1488 |
|
Since the first two terms of the new Hamiltonian depend only on the |
| 1489 |
|
system coordinates, we can get the equations of motion for |
| 1500 |
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1501 |
|
\label{introEquation:bathMotionGLE} |
| 1502 |
|
\end{equation} |
| 1574 |
– |
|
| 1503 |
|
In order to derive an equation for $x$, the dynamics of the bath |
| 1504 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1505 |
|
transform which is particularly useful in solving linear ordinary |
| 1506 |
|
differential equations,the Laplace transform is the appropriate tool |
| 1507 |
|
to solve this problem. The basic idea is to transform the difficult |
| 1508 |
|
differential equations into simple algebra problems which can be |
| 1509 |
< |
solved easily. Then, by applying the inverse Laplace transform, also |
| 1510 |
< |
known as the Bromwich integral, we can retrieve the solutions of the |
| 1511 |
< |
original problems. |
| 1512 |
< |
|
| 1585 |
< |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1586 |
< |
transform of f(t) is a new function defined as |
| 1509 |
> |
solved easily. Then, by applying the inverse Laplace transform, we |
| 1510 |
> |
can retrieve the solutions of the original problems. Let $f(t)$ be a |
| 1511 |
> |
function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$ |
| 1512 |
> |
is a new function defined as |
| 1513 |
|
\[ |
| 1514 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1515 |
|
\] |
| 1516 |
|
where $p$ is real and $L$ is called the Laplace Transform |
| 1517 |
|
Operator. Below are some important properties of Laplace transform |
| 1592 |
– |
|
| 1518 |
|
\begin{eqnarray*} |
| 1519 |
|
L(x + y) & = & L(x) + L(y) \\ |
| 1520 |
|
L(ax) & = & aL(x) \\ |
| 1522 |
|
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
| 1523 |
|
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
| 1524 |
|
\end{eqnarray*} |
| 1600 |
– |
|
| 1601 |
– |
|
| 1525 |
|
Applying the Laplace transform to the bath coordinates, we obtain |
| 1526 |
|
\begin{eqnarray*} |
| 1527 |
< |
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) \\ |
| 1528 |
< |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1527 |
> |
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), \\ |
| 1528 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
| 1529 |
|
\end{eqnarray*} |
| 1530 |
< |
|
| 1608 |
< |
By the same way, the system coordinates become |
| 1530 |
> |
In the same way, the system coordinates become |
| 1531 |
|
\begin{eqnarray*} |
| 1532 |
< |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1533 |
< |
& & \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\}} \\ |
| 1532 |
> |
mL(\ddot x) & = & |
| 1533 |
> |
- \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\}} \\ |
| 1534 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}. |
| 1535 |
|
\end{eqnarray*} |
| 1613 |
– |
|
| 1536 |
|
With the help of some relatively important inverse Laplace |
| 1537 |
|
transformations: |
| 1538 |
|
\[ |
| 1542 |
|
L(1) = \frac{1}{p} \\ |
| 1543 |
|
\end{array} |
| 1544 |
|
\] |
| 1545 |
< |
, we obtain |
| 1545 |
> |
we obtain |
| 1546 |
|
\begin{eqnarray*} |
| 1547 |
|
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
| 1548 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1551 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1552 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1553 |
|
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1554 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1555 |
< |
\end{eqnarray*} |
| 1556 |
< |
\begin{eqnarray*} |
| 1557 |
< |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1558 |
< |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1559 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 1554 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}}\\ |
| 1555 |
> |
% |
| 1556 |
> |
& = & - |
| 1557 |
> |
\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha |
| 1558 |
> |
= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha |
| 1559 |
> |
^2 }}} \right)\cos (\omega _\alpha |
| 1560 |
|
t)\dot x(t - \tau )d} \tau } \\ |
| 1561 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1562 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1590 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
| 1591 |
|
implies it is completely deterministic within the context of a |
| 1592 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1593 |
< |
uncorrelated to $x$ and $\dot x$, |
| 1594 |
< |
\[ |
| 1595 |
< |
\begin{array}{l} |
| 1596 |
< |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
| 1675 |
< |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
| 1676 |
< |
\end{array} |
| 1677 |
< |
\] |
| 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 |
| 1593 |
> |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
| 1594 |
> |
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
| 1595 |
> |
0.$ This property is what we expect from a truly random process. As |
| 1596 |
> |
long as the model chosen for $R(t)$ was a gaussian distribution in |
| 1597 |
|
general, the stochastic nature of the GLE still remains. |
| 1681 |
– |
|
| 1598 |
|
%dynamic friction kernel |
| 1599 |
|
The convolution integral |
| 1600 |
|
\[ |
| 1609 |
|
\[ |
| 1610 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
| 1611 |
|
\] |
| 1612 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1612 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1613 |
|
\[ |
| 1614 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1615 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1626 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1627 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1628 |
|
\] |
| 1629 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1629 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1630 |
|
\begin{equation} |
| 1631 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1632 |
|
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1634 |
|
which is known as the Langevin equation. The static friction |
| 1635 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1636 |
|
or be determined by Stokes' law for regular shaped particles. A |
| 1637 |
< |
briefly review on calculating friction tensor for arbitrary shaped |
| 1637 |
> |
brief review on calculating friction tensors for arbitrary shaped |
| 1638 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1639 |
|
|
| 1640 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1641 |
|
|
| 1642 |
< |
Defining a new set of coordinates, |
| 1642 |
> |
Defining a new set of coordinates |
| 1643 |
|
\[ |
| 1644 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1645 |
< |
^2 }}x(0) |
| 1646 |
< |
\], |
| 1645 |
> |
^2 }}x(0), |
| 1646 |
> |
\] |
| 1647 |
|
we can rewrite $R(T)$ as |
| 1648 |
|
\[ |
| 1649 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1650 |
|
\] |
| 1651 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1736 |
– |
|
| 1652 |
|
\begin{eqnarray*} |
| 1653 |
|
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1654 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1655 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1656 |
|
\left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
| 1657 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1658 |
< |
& = &kT\xi (t) \\ |
| 1658 |
> |
& = &kT\xi (t) |
| 1659 |
|
\end{eqnarray*} |
| 1745 |
– |
|
| 1660 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1661 |
|
\begin{equation} |
| 1662 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 1663 |
< |
\label{introEquation:secondFluctuationDissipation}. |
| 1663 |
> |
\label{introEquation:secondFluctuationDissipation}, |
| 1664 |
|
\end{equation} |
| 1665 |
< |
In effect, it acts as a constraint on the possible ways in which one |
| 1666 |
< |
can model the random force and friction kernel. |
| 1665 |
> |
which acts as a constraint on the possible ways in which one can |
| 1666 |
> |
model the random force and friction kernel. |
