| 31 |
|
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
| 32 |
|
Newton's third law states that |
| 33 |
|
\begin{equation} |
| 34 |
< |
F_{ij} = -F_{ji} |
| 34 |
> |
F_{ij} = -F_{ji}. |
| 35 |
|
\label{introEquation:newtonThirdLaw} |
| 36 |
|
\end{equation} |
| 37 |
– |
|
| 37 |
|
Conservation laws of Newtonian Mechanics play very important roles |
| 38 |
|
in solving mechanics problems. The linear momentum of a particle is |
| 39 |
|
conserved if it is free or it experiences no force. The second |
| 62 |
|
\end{equation} |
| 63 |
|
If there are no external torques acting on a body, the angular |
| 64 |
|
momentum of it is conserved. The last conservation theorem state |
| 65 |
< |
that if all forces are conservative, Energy |
| 66 |
< |
\begin{equation}E = T + V \label{introEquation:energyConservation} |
| 65 |
> |
that if all forces are conservative, energy is conserved, |
| 66 |
> |
\begin{equation}E = T + V. \label{introEquation:energyConservation} |
| 67 |
|
\end{equation} |
| 68 |
< |
is conserved. All of these conserved quantities are |
| 69 |
< |
important factors to determine the quality of numerical integration |
| 70 |
< |
schemes for rigid bodies \cite{Dullweber1997}. |
| 68 |
> |
All of these conserved quantities are important factors to determine |
| 69 |
> |
the quality of numerical integration schemes for rigid bodies |
| 70 |
> |
\cite{Dullweber1997}. |
| 71 |
|
|
| 72 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 73 |
|
|
| 74 |
< |
Newtonian Mechanics suffers from two important limitations: motions |
| 75 |
< |
can only be described in cartesian coordinate systems. Moreover, It |
| 76 |
< |
become impossible to predict analytically the properties of the |
| 77 |
< |
system even if we know all of the details of the interaction. In |
| 78 |
< |
order to overcome some of the practical difficulties which arise in |
| 79 |
< |
attempts to apply Newton's equation to complex system, approximate |
| 80 |
< |
numerical procedures may be developed. |
| 74 |
> |
Newtonian Mechanics suffers from a important limitation: motions can |
| 75 |
> |
only be described in cartesian coordinate systems which make it |
| 76 |
> |
impossible to predict analytically the properties of the system even |
| 77 |
> |
if we know all of the details of the interaction. In order to |
| 78 |
> |
overcome some of the practical difficulties which arise in attempts |
| 79 |
> |
to apply Newton's equation to complex system, approximate numerical |
| 80 |
> |
procedures may be developed. |
| 81 |
|
|
| 82 |
|
\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
| 83 |
|
Principle}} |
| 84 |
|
|
| 85 |
|
Hamilton introduced the dynamical principle upon which it is |
| 86 |
|
possible to base all of mechanics and most of classical physics. |
| 87 |
< |
Hamilton's Principle may be stated as follows, |
| 88 |
< |
|
| 89 |
< |
The actual trajectory, along which a dynamical system may move from |
| 90 |
< |
one point to another within a specified time, is derived by finding |
| 91 |
< |
the path which minimizes the time integral of the difference between |
| 93 |
< |
the kinetic, $K$, and potential energies, $U$. |
| 87 |
> |
Hamilton's Principle may be stated as follows: the actual |
| 88 |
> |
trajectory, along which a dynamical system may move from one point |
| 89 |
> |
to another within a specified time, is derived by finding the path |
| 90 |
> |
which minimizes the time integral of the difference between the |
| 91 |
> |
kinetic $K$, and potential energies $U$, |
| 92 |
|
\begin{equation} |
| 93 |
< |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
| 93 |
> |
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}. |
| 94 |
|
\label{introEquation:halmitonianPrinciple1} |
| 95 |
|
\end{equation} |
| 98 |
– |
|
| 96 |
|
For simple mechanical systems, where the forces acting on the |
| 97 |
|
different parts are derivable from a potential, the Lagrangian |
| 98 |
|
function $L$ can be defined as the difference between the kinetic |
| 99 |
|
energy of the system and its potential energy, |
| 100 |
|
\begin{equation} |
| 101 |
< |
L \equiv K - U = L(q_i ,\dot q_i ) , |
| 101 |
> |
L \equiv K - U = L(q_i ,\dot q_i ). |
| 102 |
|
\label{introEquation:lagrangianDef} |
| 103 |
|
\end{equation} |
| 104 |
< |
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
| 104 |
> |
Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
| 105 |
|
\begin{equation} |
| 106 |
< |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
| 106 |
> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} . |
| 107 |
|
\label{introEquation:halmitonianPrinciple2} |
| 108 |
|
\end{equation} |
| 109 |
|
|
| 110 |
|
\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The |
| 111 |
|
Equations of Motion in Lagrangian Mechanics}} |
| 112 |
|
|
| 113 |
< |
For a holonomic system of $f$ degrees of freedom, the equations of |
| 114 |
< |
motion in the Lagrangian form is |
| 113 |
> |
For a system of $f$ degrees of freedom, the equations of motion in |
| 114 |
> |
the Lagrangian form is |
| 115 |
|
\begin{equation} |
| 116 |
|
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - |
| 117 |
|
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f |
| 135 |
|
p_i = \frac{{\partial L}}{{\partial q_i }} |
| 136 |
|
\label{introEquation:generalizedMomentaDot} |
| 137 |
|
\end{equation} |
| 141 |
– |
|
| 138 |
|
With the help of the generalized momenta, we may now define a new |
| 139 |
|
quantity $H$ by the equation |
| 140 |
|
\begin{equation} |
| 142 |
|
\label{introEquation:hamiltonianDefByLagrangian} |
| 143 |
|
\end{equation} |
| 144 |
|
where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
| 145 |
< |
$L$ is the Lagrangian function for the system. |
| 146 |
< |
|
| 151 |
< |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
| 152 |
< |
one can obtain |
| 145 |
> |
$L$ is the Lagrangian function for the system. Differentiating |
| 146 |
> |
Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain |
| 147 |
|
\begin{equation} |
| 148 |
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
| 149 |
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
| 150 |
|
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
| 151 |
< |
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
| 151 |
> |
L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1} |
| 152 |
|
\end{equation} |
| 153 |
< |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the |
| 154 |
< |
second and fourth terms in the parentheses cancel. Therefore, |
| 153 |
> |
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second |
| 154 |
> |
and fourth terms in the parentheses cancel. Therefore, |
| 155 |
|
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
| 156 |
|
\begin{equation} |
| 157 |
|
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
| 158 |
< |
\right)} - \frac{{\partial L}}{{\partial t}}dt |
| 158 |
> |
\right)} - \frac{{\partial L}}{{\partial t}}dt . |
| 159 |
|
\label{introEquation:diffHamiltonian2} |
| 160 |
|
\end{equation} |
| 161 |
|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
| 174 |
|
t}} |
| 175 |
|
\label{introEquation:motionHamiltonianTime} |
| 176 |
|
\end{equation} |
| 177 |
< |
|
| 184 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 177 |
> |
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 178 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
| 179 |
|
equation of motion. Due to their symmetrical formula, they are also |
| 180 |
|
known as the canonical equations of motions \cite{Goldstein2001}. |
| 188 |
|
statistical mechanics and quantum mechanics, since it treats the |
| 189 |
|
coordinate and its time derivative as independent variables and it |
| 190 |
|
only works with 1st-order differential equations\cite{Marion1990}. |
| 198 |
– |
|
| 191 |
|
In Newtonian Mechanics, a system described by conservative forces |
| 192 |
< |
conserves the total energy \ref{introEquation:energyConservation}. |
| 193 |
< |
It follows that Hamilton's equations of motion conserve the total |
| 194 |
< |
Hamiltonian. |
| 192 |
> |
conserves the total energy |
| 193 |
> |
(Eq.~\ref{introEquation:energyConservation}). It follows that |
| 194 |
> |
Hamilton's equations of motion conserve the total Hamiltonian |
| 195 |
|
\begin{equation} |
| 196 |
|
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
| 197 |
|
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
| 198 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
| 199 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
| 200 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
| 201 |
< |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
| 201 |
> |
q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian} |
| 202 |
|
\end{equation} |
| 203 |
|
|
| 204 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
| 219 |
|
momentum variables. Consider a dynamic system of $f$ particles in a |
| 220 |
|
cartesian space, where each of the $6f$ coordinates and momenta is |
| 221 |
|
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
| 222 |
< |
this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots |
| 223 |
< |
,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$ |
| 224 |
< |
coordinates and momenta is a phase space vector. |
| 222 |
> |
this system is a $6f$ dimensional space. A point, $x = |
| 223 |
> |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 224 |
> |
\over q} _1 , \ldots |
| 225 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 226 |
> |
\over q} _f |
| 227 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 228 |
> |
\over p} _1 \ldots |
| 229 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 230 |
> |
\over p} _f )$ , with a unique set of values of $6f$ coordinates and |
| 231 |
> |
momenta is a phase space vector. |
| 232 |
> |
%%%fix me |
| 233 |
|
|
| 234 |
< |
A microscopic state or microstate of a classical system is |
| 235 |
< |
specification of the complete phase space vector of a system at any |
| 236 |
< |
instant in time. An ensemble is defined as a collection of systems |
| 237 |
< |
sharing one or more macroscopic characteristics but each being in a |
| 238 |
< |
unique microstate. The complete ensemble is specified by giving all |
| 239 |
< |
systems or microstates consistent with the common macroscopic |
| 240 |
< |
characteristics of the ensemble. Although the state of each |
| 241 |
< |
individual system in the ensemble could be precisely described at |
| 242 |
< |
any instance in time by a suitable phase space vector, when using |
| 243 |
< |
ensembles for statistical purposes, there is no need to maintain |
| 244 |
< |
distinctions between individual systems, since the numbers of |
| 245 |
< |
systems at any time in the different states which correspond to |
| 246 |
< |
different regions of the phase space are more interesting. Moreover, |
| 247 |
< |
in the point of view of statistical mechanics, one would prefer to |
| 248 |
< |
use ensembles containing a large enough population of separate |
| 249 |
< |
members so that the numbers of systems in such different states can |
| 250 |
< |
be regarded as changing continuously as we traverse different |
| 251 |
< |
regions of the phase space. The condition of an ensemble at any time |
| 234 |
> |
In statistical mechanics, the condition of an ensemble at any time |
| 235 |
|
can be regarded as appropriately specified by the density $\rho$ |
| 236 |
|
with which representative points are distributed over the phase |
| 237 |
|
space. The density distribution for an ensemble with $f$ degrees of |
| 243 |
|
Governed by the principles of mechanics, the phase points change |
| 244 |
|
their locations which would change the density at any time at phase |
| 245 |
|
space. Hence, the density distribution is also to be taken as a |
| 246 |
< |
function of the time. |
| 247 |
< |
|
| 265 |
< |
The number of systems $\delta N$ at time $t$ can be determined by, |
| 246 |
> |
function of the time. The number of systems $\delta N$ at time $t$ |
| 247 |
> |
can be determined by, |
| 248 |
|
\begin{equation} |
| 249 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
| 250 |
|
\label{introEquation:deltaN} |
| 264 |
|
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 265 |
|
\label{introEquation:unitProbability} |
| 266 |
|
\end{equation} |
| 267 |
< |
With the help of Equation(\ref{introEquation:unitProbability}) and |
| 268 |
< |
the knowledge of the system, it is possible to calculate the average |
| 267 |
> |
With the help of Eq.~\ref{introEquation:unitProbability} and the |
| 268 |
> |
knowledge of the system, it is possible to calculate the average |
| 269 |
|
value of any desired quantity which depends on the coordinates and |
| 270 |
|
momenta of the system. Even when the dynamics of the real system is |
| 271 |
|
complex, or stochastic, or even discontinuous, the average |
| 277 |
|
\begin{equation} |
| 278 |
|
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
| 279 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
| 280 |
< |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
| 280 |
> |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 281 |
|
\label{introEquation:ensembelAverage} |
| 282 |
|
\end{equation} |
| 283 |
|
|
| 285 |
|
statistical characteristics. As a function of macroscopic |
| 286 |
|
parameters, such as temperature \textit{etc}, the partition function |
| 287 |
|
can be used to describe the statistical properties of a system in |
| 288 |
< |
thermodynamic equilibrium. |
| 289 |
< |
|
| 290 |
< |
As an ensemble of systems, each of which is known to be thermally |
| 309 |
< |
isolated and conserve energy, the Microcanonical ensemble(NVE) has a |
| 310 |
< |
partition function like, |
| 288 |
> |
thermodynamic equilibrium. As an ensemble of systems, each of which |
| 289 |
> |
is known to be thermally isolated and conserve energy, the |
| 290 |
> |
Microcanonical ensemble (NVE) has a partition function like, |
| 291 |
|
\begin{equation} |
| 292 |
< |
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 292 |
> |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
| 293 |
|
\end{equation} |
| 294 |
< |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
| 294 |
> |
A canonical ensemble (NVT) is an ensemble of systems, each of which |
| 295 |
|
can share its energy with a large heat reservoir. The distribution |
| 296 |
|
of the total energy amongst the possible dynamical states is given |
| 297 |
|
by the partition function, |
| 298 |
|
\begin{equation} |
| 299 |
< |
\Omega (N,V,T) = e^{ - \beta A} |
| 299 |
> |
\Omega (N,V,T) = e^{ - \beta A}. |
| 300 |
|
\label{introEquation:NVTPartition} |
| 301 |
|
\end{equation} |
| 302 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 303 |
|
TS$. Since most experiments are carried out under constant pressure |
| 304 |
< |
condition, the isothermal-isobaric ensemble(NPT) plays a very |
| 304 |
> |
condition, the isothermal-isobaric ensemble (NPT) plays a very |
| 305 |
|
important role in molecular simulations. The isothermal-isobaric |
| 306 |
|
ensemble allow the system to exchange energy with a heat bath of |
| 307 |
|
temperature $T$ and to change the volume as well. Its partition |
| 317 |
|
Liouville's theorem is the foundation on which statistical mechanics |
| 318 |
|
rests. It describes the time evolution of the phase space |
| 319 |
|
distribution function. In order to calculate the rate of change of |
| 320 |
< |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
| 321 |
< |
consider the two faces perpendicular to the $q_1$ axis, which are |
| 322 |
< |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
| 323 |
< |
leaving the opposite face is given by the expression, |
| 320 |
> |
$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider |
| 321 |
> |
the two faces perpendicular to the $q_1$ axis, which are located at |
| 322 |
> |
$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the |
| 323 |
> |
opposite face is given by the expression, |
| 324 |
|
\begin{equation} |
| 325 |
|
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
| 326 |
|
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
| 353 |
|
\frac{{\partial \rho }}{{\partial p_i }}\dot p_i } \right)} = 0 . |
| 354 |
|
\label{introEquation:liouvilleTheorem} |
| 355 |
|
\end{equation} |
| 376 |
– |
|
| 356 |
|
Liouville's theorem states that the distribution function is |
| 357 |
|
constant along any trajectory in phase space. In classical |
| 358 |
< |
statistical mechanics, since the number of particles in the system |
| 359 |
< |
is huge, we may be able to believe the system is stationary, |
| 358 |
> |
statistical mechanics, since the number of members in an ensemble is |
| 359 |
> |
huge and constant, we can assume the local density has no reason |
| 360 |
> |
(other than classical mechanics) to change, |
| 361 |
|
\begin{equation} |
| 362 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
| 363 |
|
\label{introEquation:stationary} |
| 409 |
|
q_i }}} \right)}. |
| 410 |
|
\label{introEquation:poissonBracket} |
| 411 |
|
\end{equation} |
| 412 |
< |
Substituting equations of motion in Hamiltonian formalism( |
| 413 |
< |
\ref{introEquation:motionHamiltonianCoordinate} , |
| 414 |
< |
\ref{introEquation:motionHamiltonianMomentum} ) into |
| 415 |
< |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
| 416 |
< |
theorem using Poisson bracket notion, |
| 412 |
> |
Substituting equations of motion in Hamiltonian formalism |
| 413 |
> |
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
| 414 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into |
| 415 |
> |
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
| 416 |
> |
Liouville's theorem using Poisson bracket notion, |
| 417 |
|
\begin{equation} |
| 418 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
| 419 |
|
{\rho ,H} \right\}. |
| 445 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 446 |
|
Hypothesis makes a connection between time average and the ensemble |
| 447 |
|
average. It states that the time average and average over the |
| 448 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
| 448 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
| 449 |
|
\begin{equation} |
| 450 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 451 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 459 |
|
a properly weighted statistical average. This allows the researcher |
| 460 |
|
freedom of choice when deciding how best to measure a given |
| 461 |
|
observable. In case an ensemble averaged approach sounds most |
| 462 |
< |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
| 462 |
> |
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
| 463 |
|
utilized. Or if the system lends itself to a time averaging |
| 464 |
|
approach, the Molecular Dynamics techniques in |
| 465 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 474 |
|
geometric integrators, which preserve various phase-flow invariants |
| 475 |
|
such as symplectic structure, volume and time reversal symmetry, are |
| 476 |
|
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
| 477 |
< |
Leimkuhler1999}. The velocity verlet method, which happens to be a |
| 477 |
> |
Leimkuhler1999}. The velocity Verlet method, which happens to be a |
| 478 |
|
simple example of symplectic integrator, continues to gain |
| 479 |
|
popularity in the molecular dynamics community. This fact can be |
| 480 |
|
partly explained by its geometric nature. |
| 494 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 495 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 496 |
|
$\omega(x, x) = 0$. The cross product operation in vector field is |
| 497 |
< |
an example of symplectic form. |
| 497 |
> |
an example of symplectic form. One of the motivations to study |
| 498 |
> |
\emph{symplectic manifolds} in Hamiltonian Mechanics is that a |
| 499 |
> |
symplectic manifold can represent all possible configurations of the |
| 500 |
> |
system and the phase space of the system can be described by it's |
| 501 |
> |
cotangent bundle. Every symplectic manifold is even dimensional. For |
| 502 |
> |
instance, in Hamilton equations, coordinate and momentum always |
| 503 |
> |
appear in pairs. |
| 504 |
|
|
| 519 |
– |
One of the motivations to study \emph{symplectic manifolds} in |
| 520 |
– |
Hamiltonian Mechanics is that a symplectic manifold can represent |
| 521 |
– |
all possible configurations of the system and the phase space of the |
| 522 |
– |
system can be described by it's cotangent bundle. Every symplectic |
| 523 |
– |
manifold is even dimensional. For instance, in Hamilton equations, |
| 524 |
– |
coordinate and momentum always appear in pairs. |
| 525 |
– |
|
| 505 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 506 |
|
|
| 507 |
|
For an ordinary differential system defined as |
| 509 |
|
\dot x = f(x) |
| 510 |
|
\end{equation} |
| 511 |
|
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
| 512 |
+ |
$f(r) = J\nabla _x H(r)$. Here, $H = H (q, p)$ is Hamiltonian |
| 513 |
+ |
function and $J$ is the skew-symmetric matrix |
| 514 |
|
\begin{equation} |
| 534 |
– |
f(r) = J\nabla _x H(r). |
| 535 |
– |
\end{equation} |
| 536 |
– |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
| 537 |
– |
matrix |
| 538 |
– |
\begin{equation} |
| 515 |
|
J = \left( {\begin{array}{*{20}c} |
| 516 |
|
0 & I \\ |
| 517 |
|
{ - I} & 0 \\ |
| 521 |
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
| 522 |
|
system can be rewritten as, |
| 523 |
|
\begin{equation} |
| 524 |
< |
\frac{d}{{dt}}x = J\nabla _x H(x) |
| 524 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x). |
| 525 |
|
\label{introEquation:compactHamiltonian} |
| 526 |
|
\end{equation}In this case, $f$ is |
| 527 |
< |
called a \emph{Hamiltonian vector field}. |
| 528 |
< |
|
| 553 |
< |
Another generalization of Hamiltonian dynamics is Poisson |
| 554 |
< |
Dynamics\cite{Olver1986}, |
| 527 |
> |
called a \emph{Hamiltonian vector field}. Another generalization of |
| 528 |
> |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
| 529 |
|
\begin{equation} |
| 530 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 531 |
|
\end{equation} |
| 533 |
|
|
| 534 |
|
\subsection{\label{introSection:exactFlow}Exact Flow} |
| 535 |
|
|
| 536 |
< |
Let $x(t)$ be the exact solution of the ODE system, |
| 537 |
< |
\begin{equation} |
| 538 |
< |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
| 539 |
< |
\end{equation} |
| 540 |
< |
The exact flow(solution) $\varphi_\tau$ is defined by |
| 567 |
< |
\[ |
| 568 |
< |
x(t+\tau) =\varphi_\tau(x(t)) |
| 536 |
> |
Let $x(t)$ be the exact solution of the ODE |
| 537 |
> |
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
| 538 |
> |
define its exact flow(solution) $\varphi_\tau$ |
| 539 |
> |
\[ x(t+\tau) |
| 540 |
> |
=\varphi_\tau(x(t)) |
| 541 |
|
\] |
| 542 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
| 543 |
|
space to itself. The flow has the continuous group property, |
| 559 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 560 |
|
\label{introEquation:exponentialOperator} |
| 561 |
|
\end{equation} |
| 590 |
– |
|
| 562 |
|
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
| 563 |
< |
Instead, we use a approximate map, $\psi_\tau$, which is usually |
| 563 |
> |
Instead, we use an approximate map, $\psi_\tau$, which is usually |
| 564 |
|
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
| 565 |
|
the Taylor series of $\psi_\tau$ agree to order $p$, |
| 566 |
|
\begin{equation} |
| 567 |
< |
\psi_tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 567 |
> |
\psi_\tau(x) = x + \tau f(x) + O(\tau^{p+1}) |
| 568 |
|
\end{equation} |
| 569 |
|
|
| 570 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 571 |
|
|
| 572 |
< |
The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE |
| 573 |
< |
and its flow play important roles in numerical studies. Many of them |
| 574 |
< |
can be found in systems which occur naturally in applications. |
| 604 |
< |
|
| 572 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
| 573 |
> |
ODE and its flow play important roles in numerical studies. Many of |
| 574 |
> |
them can be found in systems which occur naturally in applications. |
| 575 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 576 |
|
a \emph{symplectic} flow if it satisfies, |
| 577 |
|
\begin{equation} |
| 585 |
|
\begin{equation} |
| 586 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 587 |
|
\end{equation} |
| 588 |
< |
is the property must be preserved by the integrator. |
| 589 |
< |
|
| 590 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
| 591 |
< |
source free($ \nabla \cdot f = 0 $) ODE, if the flow satisfies $ |
| 592 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
| 593 |
< |
be volume-preserving. |
| 624 |
< |
|
| 625 |
< |
Changing the variables $y = h(x)$ in a ODE\ref{introEquation:ODE} |
| 626 |
< |
will result in a new system, |
| 588 |
> |
is the property that must be preserved by the integrator. It is |
| 589 |
> |
possible to construct a \emph{volume-preserving} flow for a source |
| 590 |
> |
free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det |
| 591 |
> |
d\varphi = 1$. One can show easily that a symplectic flow will be |
| 592 |
> |
volume-preserving. Changing the variables $y = h(x)$ in an ODE |
| 593 |
> |
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
| 594 |
|
\[ |
| 595 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 596 |
|
\] |
| 597 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 598 |
|
In other words, the flow of this vector field is reversible if and |
| 599 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 600 |
< |
|
| 634 |
< |
A \emph{first integral}, or conserved quantity of a general |
| 599 |
> |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
| 600 |
> |
\emph{first integral}, or conserved quantity of a general |
| 601 |
|
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 602 |
|
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 603 |
|
\[ |
| 605 |
|
\] |
| 606 |
|
Using chain rule, one may obtain, |
| 607 |
|
\[ |
| 608 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
| 608 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
| 609 |
|
\] |
| 610 |
|
which is the condition for conserving \emph{first integral}. For a |
| 611 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
| 612 |
|
smooth function $G$ is given by, |
| 647 |
– |
|
| 613 |
|
\begin{eqnarray} |
| 614 |
< |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 615 |
< |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 614 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
| 615 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
| 616 |
|
\label{introEquation:firstIntegral1} |
| 617 |
|
\end{eqnarray} |
| 618 |
< |
|
| 619 |
< |
|
| 655 |
< |
Using poisson bracket notion, Equation |
| 656 |
< |
\ref{introEquation:firstIntegral1} can be rewritten as |
| 618 |
> |
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1} |
| 619 |
> |
can be rewritten as |
| 620 |
|
\[ |
| 621 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
| 622 |
|
\] |
| 623 |
|
Therefore, the sufficient condition for $G$ to be the \emph{first |
| 624 |
< |
integral} of a Hamiltonian system is |
| 662 |
< |
\[ |
| 663 |
< |
\left\{ {G,H} \right\} = 0. |
| 664 |
< |
\] |
| 624 |
> |
integral} of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ |
| 625 |
|
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 626 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 627 |
< |
0$. |
| 668 |
< |
|
| 669 |
< |
When designing any numerical methods, one should always try to |
| 627 |
> |
0$. When designing any numerical methods, one should always try to |
| 628 |
|
preserve the structural properties of the original ODE and its flow. |
| 629 |
|
|
| 630 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 631 |
|
A lot of well established and very effective numerical methods have |
| 632 |
|
been successful precisely because of their symplecticities even |
| 633 |
|
though this fact was not recognized when they were first |
| 634 |
< |
constructed. The most famous example is the Verlet-leapfrog methods |
| 634 |
> |
constructed. The most famous example is the Verlet-leapfrog method |
| 635 |
|
in molecular dynamics. In general, symplectic integrators can be |
| 636 |
|
constructed using one of four different methods. |
| 637 |
|
\begin{enumerate} |
| 640 |
|
\item Runge-Kutta methods |
| 641 |
|
\item Splitting methods |
| 642 |
|
\end{enumerate} |
| 685 |
– |
|
| 643 |
|
Generating function\cite{Channell1990} tends to lead to methods |
| 644 |
|
which are cumbersome and difficult to use. In dissipative systems, |
| 645 |
|
variational methods can capture the decay of energy |
| 646 |
|
accurately\cite{Kane2000}. Since their geometrically unstable nature |
| 647 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 648 |
|
methods are not suitable for Hamiltonian system. Recently, various |
| 649 |
< |
high-order explicit Runge-Kutta methods |
| 650 |
< |
\cite{Owren1992,Chen2003}have been developed to overcome this |
| 651 |
< |
instability. However, due to computational penalty involved in |
| 652 |
< |
implementing the Runge-Kutta methods, they have not attracted much |
| 653 |
< |
attention from the Molecular Dynamics community. Instead, splitting |
| 654 |
< |
methods have been widely accepted since they exploit natural |
| 655 |
< |
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
| 649 |
> |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
| 650 |
> |
have been developed to overcome this instability. However, due to |
| 651 |
> |
computational penalty involved in implementing the Runge-Kutta |
| 652 |
> |
methods, they have not attracted much attention from the Molecular |
| 653 |
> |
Dynamics community. Instead, splitting methods have been widely |
| 654 |
> |
accepted since they exploit natural decompositions of the |
| 655 |
> |
system\cite{Tuckerman1992, McLachlan1998}. |
| 656 |
|
|
| 657 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 658 |
|
|
| 664 |
|
\label{introEquation:FlowDecomposition} |
| 665 |
|
\end{equation} |
| 666 |
|
where each of the sub-flow is chosen such that each represent a |
| 667 |
< |
simpler integration of the system. |
| 668 |
< |
|
| 712 |
< |
Suppose that a Hamiltonian system takes the form, |
| 667 |
> |
simpler integration of the system. Suppose that a Hamiltonian system |
| 668 |
> |
takes the form, |
| 669 |
|
\[ |
| 670 |
|
H = H_1 + H_2. |
| 671 |
|
\] |
| 693 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
| 694 |
|
splitting in this context automatically generates a symplectic map. |
| 695 |
|
|
| 696 |
< |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
| 697 |
< |
introduces local errors proportional to $h^2$, while Strang |
| 698 |
< |
splitting gives a second-order decomposition, |
| 696 |
> |
The Lie-Trotter |
| 697 |
> |
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
| 698 |
> |
local errors proportional to $h^2$, while Strang splitting gives a |
| 699 |
> |
second-order decomposition, |
| 700 |
|
\begin{equation} |
| 701 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 702 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 709 |
|
\label{introEquation:timeReversible} |
| 710 |
|
\end{equation} |
| 711 |
|
|
| 712 |
< |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Example of Splitting Method}} |
| 712 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Examples of the Splitting Method}} |
| 713 |
|
The classical equation for a system consisting of interacting |
| 714 |
|
particles can be written in Hamiltonian form, |
| 715 |
|
\[ |
| 716 |
|
H = T + V |
| 717 |
|
\] |
| 718 |
|
where $T$ is the kinetic energy and $V$ is the potential energy. |
| 719 |
< |
Setting $H_1 = T, H_2 = V$ and applying Strang splitting, one |
| 719 |
> |
Setting $H_1 = T, H_2 = V$ and applying the Strang splitting, one |
| 720 |
|
obtains the following: |
| 721 |
|
\begin{align} |
| 722 |
|
q(\Delta t) &= q(0) + \dot{q}(0)\Delta t + |
| 743 |
|
\label{introEquation:Lp9b}\\% |
| 744 |
|
% |
| 745 |
|
\dot{q}(\Delta t) &= \dot{q}\biggl (\frac{\Delta t}{2}\biggr ) + |
| 746 |
< |
\frac{\Delta t}{2m}\, F[q(0)]. \label{introEquation:Lp9c} |
| 746 |
> |
\frac{\Delta t}{2m}\, F[q(t)]. \label{introEquation:Lp9c} |
| 747 |
|
\end{align} |
| 748 |
|
From the preceding splitting, one can see that the integration of |
| 749 |
|
the equations of motion would follow: |
| 752 |
|
|
| 753 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 754 |
|
|
| 755 |
< |
\item Evaluate the forces at the new positions, $\mathbf{r}(\Delta t)$, and use the new forces to complete the velocity move. |
| 755 |
> |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
| 756 |
|
|
| 757 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 758 |
|
\end{enumerate} |
| 759 |
< |
|
| 760 |
< |
Simply switching the order of splitting and composing, a new |
| 761 |
< |
integrator, the \emph{position verlet} integrator, can be generated, |
| 759 |
> |
By simply switching the order of the propagators in the splitting |
| 760 |
> |
and composing a new integrator, the \emph{position verlet} |
| 761 |
> |
integrator, can be generated, |
| 762 |
|
\begin{align} |
| 763 |
|
\dot q(\Delta t) &= \dot q(0) + \Delta tF(q(0))\left[ {q(0) + |
| 764 |
|
\frac{{\Delta t}}{{2m}}\dot q(0)} \right], % |
| 771 |
|
|
| 772 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 773 |
|
|
| 774 |
< |
Baker-Campbell-Hausdorff formula can be used to determine the local |
| 775 |
< |
error of splitting method in terms of commutator of the |
| 774 |
> |
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 775 |
> |
local error of splitting method in terms of the commutator of the |
| 776 |
|
operators(\ref{introEquation:exponentialOperator}) associated with |
| 777 |
< |
the sub-flow. For operators $hX$ and $hY$ which are associate to |
| 777 |
> |
the sub-flow. For operators $hX$ and $hY$ which are associated with |
| 778 |
|
$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 779 |
|
\begin{equation} |
| 780 |
|
\exp (hX + hY) = \exp (hZ) |
| 788 |
|
\[ |
| 789 |
|
[X,Y] = XY - YX . |
| 790 |
|
\] |
| 791 |
< |
Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to |
| 792 |
< |
Sprang splitting, we can obtain |
| 791 |
> |
Applying the Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} |
| 792 |
> |
to the Sprang splitting, we can obtain |
| 793 |
|
\begin{eqnarray*} |
| 794 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
| 795 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 796 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
| 796 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots |
| 797 |
> |
). |
| 798 |
|
\end{eqnarray*} |
| 799 |
< |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 799 |
> |
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
| 800 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 801 |
< |
procedure can be applied to general splitting, of the form |
| 801 |
> |
procedure can be applied to a general splitting of the form |
| 802 |
|
\begin{equation} |
| 803 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 804 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 805 |
|
\end{equation} |
| 806 |
< |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 807 |
< |
order method. Yoshida proposed an elegant way to compose higher |
| 806 |
> |
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 807 |
> |
order methods. Yoshida proposed an elegant way to compose higher |
| 808 |
|
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 809 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 810 |
|
fourth-order symmetric method can be constructed by composing, |
| 817 |
|
integrator $ \varphi _h^{(2n + 2)}$ can be composed by |
| 818 |
|
\begin{equation} |
| 819 |
|
\varphi _h^{(2n + 2)} = \varphi _{\alpha h}^{(2n)} \circ \varphi |
| 820 |
< |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)} |
| 820 |
> |
_{\beta h}^{(2n)} \circ \varphi _{\alpha h}^{(2n)}, |
| 821 |
|
\end{equation} |
| 822 |
< |
, if the weights are chosen as |
| 822 |
> |
if the weights are chosen as |
| 823 |
|
\[ |
| 824 |
|
\alpha = - \frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }},\beta = |
| 825 |
|
\frac{{2^{1/(2n + 1)} }}{{2 - 2^{1/(2n + 1)} }} . |
| 857 |
|
These three individual steps will be covered in the following |
| 858 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 859 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
| 860 |
< |
will discusses issues in production run. |
| 860 |
> |
will discusse issues in production run. |
| 861 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 862 |
|
trajectory analysis. |
| 863 |
|
|
| 870 |
|
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
| 871 |
|
thousands of crystal structures of molecules are discovered every |
| 872 |
|
year, many more remain unknown due to the difficulties of |
| 873 |
< |
purification and crystallization. Even for the molecule with known |
| 874 |
< |
structure, some important information is missing. For example, the |
| 873 |
> |
purification and crystallization. Even for molecules with known |
| 874 |
> |
structure, some important information is missing. For example, a |
| 875 |
|
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 876 |
|
be added. Moreover, in order to include electrostatic interaction, |
| 877 |
|
one may need to specify the partial charges for individual atoms. |
| 878 |
|
Under some circumstances, we may even need to prepare the system in |
| 879 |
< |
a special setup. For instance, when studying transport phenomenon in |
| 880 |
< |
membrane system, we may prepare the lipids in bilayer structure |
| 881 |
< |
instead of placing lipids randomly in solvent, since we are not |
| 882 |
< |
interested in self-aggregation and it takes a long time to happen. |
| 879 |
> |
a special configuration. For instance, when studying transport |
| 880 |
> |
phenomenon in membrane systems, we may prepare the lipids in a |
| 881 |
> |
bilayer structure instead of placing lipids randomly in solvent, |
| 882 |
> |
since we are not interested in the slow self-aggregation process. |
| 883 |
|
|
| 884 |
|
\subsubsection{\textbf{Minimization}} |
| 885 |
|
|
| 886 |
|
It is quite possible that some of molecules in the system from |
| 887 |
< |
preliminary preparation may be overlapped with each other. This |
| 888 |
< |
close proximity leads to high potential energy which consequently |
| 889 |
< |
jeopardizes any molecular dynamics simulations. To remove these |
| 890 |
< |
steric overlaps, one typically performs energy minimization to find |
| 891 |
< |
a more reasonable conformation. Several energy minimization methods |
| 892 |
< |
have been developed to exploit the energy surface and to locate the |
| 893 |
< |
local minimum. While converging slowly near the minimum, steepest |
| 894 |
< |
descent method is extremely robust when systems are far from |
| 895 |
< |
harmonic. Thus, it is often used to refine structure from |
| 896 |
< |
crystallographic data. Relied on the gradient or hessian, advanced |
| 897 |
< |
methods like conjugate gradient and Newton-Raphson converge rapidly |
| 898 |
< |
to a local minimum, while become unstable if the energy surface is |
| 899 |
< |
far from quadratic. Another factor must be taken into account, when |
| 887 |
> |
preliminary preparation may be overlapping with each other. This |
| 888 |
> |
close proximity leads to high initial potential energy which |
| 889 |
> |
consequently jeopardizes any molecular dynamics simulations. To |
| 890 |
> |
remove these steric overlaps, one typically performs energy |
| 891 |
> |
minimization to find a more reasonable conformation. Several energy |
| 892 |
> |
minimization methods have been developed to exploit the energy |
| 893 |
> |
surface and to locate the local minimum. While converging slowly |
| 894 |
> |
near the minimum, steepest descent method is extremely robust when |
| 895 |
> |
systems are strongly anharmonic. Thus, it is often used to refine |
| 896 |
> |
structure from crystallographic data. Relied on the gradient or |
| 897 |
> |
hessian, advanced methods like Newton-Raphson converge rapidly to a |
| 898 |
> |
local minimum, but become unstable if the energy surface is far from |
| 899 |
> |
quadratic. Another factor that must be taken into account, when |
| 900 |
|
choosing energy minimization method, is the size of the system. |
| 901 |
|
Steepest descent and conjugate gradient can deal with models of any |
| 902 |
< |
size. Because of the limit of computation power to calculate hessian |
| 903 |
< |
matrix and insufficient storage capacity to store them, most |
| 904 |
< |
Newton-Raphson methods can not be used with very large models. |
| 902 |
> |
size. Because of the limits on computer memory to store the hessian |
| 903 |
> |
matrix and the computing power needed to diagonalized these |
| 904 |
> |
matrices, most Newton-Raphson methods can not be used with very |
| 905 |
> |
large systems. |
| 906 |
|
|
| 907 |
|
\subsubsection{\textbf{Heating}} |
| 908 |
|
|
| 909 |
|
Typically, Heating is performed by assigning random velocities |
| 910 |
< |
according to a Gaussian distribution for a temperature. Beginning at |
| 911 |
< |
a lower temperature and gradually increasing the temperature by |
| 912 |
< |
assigning greater random velocities, we end up with setting the |
| 913 |
< |
temperature of the system to a final temperature at which the |
| 914 |
< |
simulation will be conducted. In heating phase, we should also keep |
| 915 |
< |
the system from drifting or rotating as a whole. Equivalently, the |
| 916 |
< |
net linear momentum and angular momentum of the system should be |
| 917 |
< |
shifted to zero. |
| 910 |
> |
according to a Maxwell-Boltzman distribution for a desired |
| 911 |
> |
temperature. Beginning at a lower temperature and gradually |
| 912 |
> |
increasing the temperature by assigning larger random velocities, we |
| 913 |
> |
end up with setting the temperature of the system to a final |
| 914 |
> |
temperature at which the simulation will be conducted. In heating |
| 915 |
> |
phase, we should also keep the system from drifting or rotating as a |
| 916 |
> |
whole. To do this, the net linear momentum and angular momentum of |
| 917 |
> |
the system is shifted to zero after each resampling from the Maxwell |
| 918 |
> |
-Boltzman distribution. |
| 919 |
|
|
| 920 |
|
\subsubsection{\textbf{Equilibration}} |
| 921 |
|
|
| 931 |
|
|
| 932 |
|
\subsection{\label{introSection:production}Production} |
| 933 |
|
|
| 934 |
< |
Production run is the most important step of the simulation, in |
| 934 |
> |
The production run is the most important step of the simulation, in |
| 935 |
|
which the equilibrated structure is used as a starting point and the |
| 936 |
|
motions of the molecules are collected for later analysis. In order |
| 937 |
|
to capture the macroscopic properties of the system, the molecular |
| 938 |
< |
dynamics simulation must be performed in correct and efficient way. |
| 938 |
> |
dynamics simulation must be performed by sampling correctly and |
| 939 |
> |
efficiently from the relevant thermodynamic ensemble. |
| 940 |
|
|
| 941 |
|
The most expensive part of a molecular dynamics simulation is the |
| 942 |
|
calculation of non-bonded forces, such as van der Waals force and |
| 943 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
| 944 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
| 945 |
|
which making large simulations prohibitive in the absence of any |
| 946 |
< |
computation saving techniques. |
| 947 |
< |
|
| 948 |
< |
A natural approach to avoid system size issue is to represent the |
| 949 |
< |
bulk behavior by a finite number of the particles. However, this |
| 950 |
< |
approach will suffer from the surface effect. To offset this, |
| 990 |
< |
\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc}) |
| 946 |
> |
algorithmic tricks. A natural approach to avoid system size issues |
| 947 |
> |
is to represent the bulk behavior by a finite number of the |
| 948 |
> |
particles. However, this approach will suffer from the surface |
| 949 |
> |
effect at the edges of the simulation. To offset this, |
| 950 |
> |
\textit{Periodic boundary conditions} (see Fig.~\ref{introFig:pbc}) |
| 951 |
|
is developed to simulate bulk properties with a relatively small |
| 952 |
|
number of particles. In this method, the simulation box is |
| 953 |
|
replicated throughout space to form an infinite lattice. During the |
| 966 |
|
|
| 967 |
|
%cutoff and minimum image convention |
| 968 |
|
Another important technique to improve the efficiency of force |
| 969 |
< |
evaluation is to apply cutoff where particles farther than a |
| 970 |
< |
predetermined distance, are not included in the calculation |
| 969 |
> |
evaluation is to apply spherical cutoff where particles farther than |
| 970 |
> |
a predetermined distance are not included in the calculation |
| 971 |
|
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 972 |
|
discontinuity in the potential energy curve. Fortunately, one can |
| 973 |
< |
shift the potential to ensure the potential curve go smoothly to |
| 974 |
< |
zero at the cutoff radius. Cutoff strategy works pretty well for |
| 975 |
< |
Lennard-Jones interaction because of its short range nature. |
| 976 |
< |
However, simply truncating the electrostatic interaction with the |
| 977 |
< |
use of cutoff has been shown to lead to severe artifacts in |
| 978 |
< |
simulations. Ewald summation, in which the slowly conditionally |
| 979 |
< |
convergent Coulomb potential is transformed into direct and |
| 980 |
< |
reciprocal sums with rapid and absolute convergence, has proved to |
| 981 |
< |
minimize the periodicity artifacts in liquid simulations. Taking the |
| 982 |
< |
advantages of the fast Fourier transform (FFT) for calculating |
| 983 |
< |
discrete Fourier transforms, the particle mesh-based |
| 973 |
> |
shift simple radial potential to ensure the potential curve go |
| 974 |
> |
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 975 |
> |
well for Lennard-Jones interaction because of its short range |
| 976 |
> |
nature. However, simply truncating the electrostatic interaction |
| 977 |
> |
with the use of cutoffs has been shown to lead to severe artifacts |
| 978 |
> |
in simulations. The Ewald summation, in which the slowly decaying |
| 979 |
> |
Coulomb potential is transformed into direct and reciprocal sums |
| 980 |
> |
with rapid and absolute convergence, has proved to minimize the |
| 981 |
> |
periodicity artifacts in liquid simulations. Taking the advantages |
| 982 |
> |
of the fast Fourier transform (FFT) for calculating discrete Fourier |
| 983 |
> |
transforms, the particle mesh-based |
| 984 |
|
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 985 |
< |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast |
| 986 |
< |
multipole method}\cite{Greengard1987, Greengard1994}, which treats |
| 987 |
< |
Coulombic interaction exactly at short range, and approximate the |
| 988 |
< |
potential at long range through multipolar expansion. In spite of |
| 989 |
< |
their wide acceptances at the molecular simulation community, these |
| 990 |
< |
two methods are hard to be implemented correctly and efficiently. |
| 991 |
< |
Instead, we use a damped and charge-neutralized Coulomb potential |
| 992 |
< |
method developed by Wolf and his coworkers\cite{Wolf1999}. The |
| 993 |
< |
shifted Coulomb potential for particle $i$ and particle $j$ at |
| 994 |
< |
distance $r_{rj}$ is given by: |
| 985 |
> |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
| 986 |
> |
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
| 987 |
> |
which treats Coulombic interactions exactly at short range, and |
| 988 |
> |
approximate the potential at long range through multipolar |
| 989 |
> |
expansion. In spite of their wide acceptance at the molecular |
| 990 |
> |
simulation community, these two methods are difficult to implement |
| 991 |
> |
correctly and efficiently. Instead, we use a damped and |
| 992 |
> |
charge-neutralized Coulomb potential method developed by Wolf and |
| 993 |
> |
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
| 994 |
> |
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 995 |
|
\begin{equation} |
| 996 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 997 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 1013 |
|
|
| 1014 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 1015 |
|
|
| 1016 |
< |
Recently, advanced visualization technique are widely applied to |
| 1016 |
> |
Recently, advanced visualization technique have become applied to |
| 1017 |
|
monitor the motions of molecules. Although the dynamics of the |
| 1018 |
|
system can be described qualitatively from animation, quantitative |
| 1019 |
< |
trajectory analysis are more appreciable. According to the |
| 1020 |
< |
principles of Statistical Mechanics, |
| 1019 |
> |
trajectory analysis are more useful. According to the principles of |
| 1020 |
> |
Statistical Mechanics in |
| 1021 |
|
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 1022 |
< |
thermodynamics properties, analyze fluctuations of structural |
| 1022 |
> |
thermodynamic properties, analyze fluctuations of structural |
| 1023 |
|
parameters, and investigate time-dependent processes of the molecule |
| 1024 |
|
from the trajectories. |
| 1025 |
|
|
| 1026 |
< |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamics Properties}} |
| 1026 |
> |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
| 1027 |
|
|
| 1028 |
< |
Thermodynamics properties, which can be expressed in terms of some |
| 1028 |
> |
Thermodynamic properties, which can be expressed in terms of some |
| 1029 |
|
function of the coordinates and momenta of all particles in the |
| 1030 |
|
system, can be directly computed from molecular dynamics. The usual |
| 1031 |
|
way to measure the pressure is based on virial theorem of Clausius |
| 1048 |
|
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
| 1049 |
|
|
| 1050 |
|
Structural Properties of a simple fluid can be described by a set of |
| 1051 |
< |
distribution functions. Among these functions,\emph{pair |
| 1051 |
> |
distribution functions. Among these functions,the \emph{pair |
| 1052 |
|
distribution function}, also known as \emph{radial distribution |
| 1053 |
< |
function}, is of most fundamental importance to liquid-state theory. |
| 1054 |
< |
Pair distribution function can be gathered by Fourier transforming |
| 1055 |
< |
raw data from a series of neutron diffraction experiments and |
| 1056 |
< |
integrating over the surface factor \cite{Powles1973}. The |
| 1057 |
< |
experiment result can serve as a criterion to justify the |
| 1058 |
< |
correctness of the theory. Moreover, various equilibrium |
| 1059 |
< |
thermodynamic and structural properties can also be expressed in |
| 1060 |
< |
terms of radial distribution function \cite{Allen1987}. |
| 1061 |
< |
|
| 1102 |
< |
A pair distribution functions $g(r)$ gives the probability that a |
| 1053 |
> |
function}, is of most fundamental importance to liquid theory. |
| 1054 |
> |
Experimentally, pair distribution function can be gathered by |
| 1055 |
> |
Fourier transforming raw data from a series of neutron diffraction |
| 1056 |
> |
experiments and integrating over the surface factor |
| 1057 |
> |
\cite{Powles1973}. The experimental results can serve as a criterion |
| 1058 |
> |
to justify the correctness of a liquid model. Moreover, various |
| 1059 |
> |
equilibrium thermodynamic and structural properties can also be |
| 1060 |
> |
expressed in terms of radial distribution function \cite{Allen1987}. |
| 1061 |
> |
The pair distribution functions $g(r)$ gives the probability that a |
| 1062 |
|
particle $i$ will be located at a distance $r$ from a another |
| 1063 |
|
particle $j$ in the system |
| 1064 |
< |
\[ |
| 1064 |
> |
\begin{equation} |
| 1065 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1066 |
< |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
| 1067 |
< |
\] |
| 1066 |
> |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
| 1067 |
> |
(r)}{\rho}. |
| 1068 |
> |
\end{equation} |
| 1069 |
|
Note that the delta function can be replaced by a histogram in |
| 1070 |
< |
computer simulation. Figure |
| 1071 |
< |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
| 1072 |
< |
distribution function for the liquid argon system. The occurrence of |
| 1113 |
< |
several peaks in the plot of $g(r)$ suggests that it is more likely |
| 1114 |
< |
to find particles at certain radial values than at others. This is a |
| 1115 |
< |
result of the attractive interaction at such distances. Because of |
| 1116 |
< |
the strong repulsive forces at short distance, the probability of |
| 1117 |
< |
locating particles at distances less than about 2.5{\AA} from each |
| 1118 |
< |
other is essentially zero. |
| 1070 |
> |
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
| 1071 |
> |
the height of these peaks gradually decreases to 1 as the liquid of |
| 1072 |
> |
large distance approaches the bulk density. |
| 1073 |
|
|
| 1120 |
– |
%\begin{figure} |
| 1121 |
– |
%\centering |
| 1122 |
– |
%\includegraphics[width=\linewidth]{pdf.eps} |
| 1123 |
– |
%\caption[Pair distribution function for the liquid argon |
| 1124 |
– |
%]{Pair distribution function for the liquid argon} |
| 1125 |
– |
%\label{introFigure:pairDistributionFunction} |
| 1126 |
– |
%\end{figure} |
| 1074 |
|
|
| 1075 |
|
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
| 1076 |
|
Properties}} |
| 1077 |
|
|
| 1078 |
|
Time-dependent properties are usually calculated using \emph{time |
| 1079 |
< |
correlation function}, which correlates random variables $A$ and $B$ |
| 1080 |
< |
at two different time |
| 1079 |
> |
correlation functions}, which correlate random variables $A$ and $B$ |
| 1080 |
> |
at two different times, |
| 1081 |
|
\begin{equation} |
| 1082 |
|
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
| 1083 |
|
\label{introEquation:timeCorrelationFunction} |
| 1084 |
|
\end{equation} |
| 1085 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1086 |
< |
function is called \emph{auto correlation function}. One example of |
| 1087 |
< |
auto correlation function is velocity auto-correlation function |
| 1088 |
< |
which is directly related to transport properties of molecular |
| 1089 |
< |
liquids: |
| 1086 |
> |
function is called an \emph{autocorrelation function}. One example |
| 1087 |
> |
of an auto correlation function is the velocity auto-correlation |
| 1088 |
> |
function which is directly related to transport properties of |
| 1089 |
> |
molecular liquids: |
| 1090 |
|
\[ |
| 1091 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
| 1092 |
|
\right\rangle } dt |
| 1093 |
|
\] |
| 1094 |
< |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
| 1095 |
< |
function which is averaging over time origins and over all the |
| 1096 |
< |
atoms, dipole autocorrelation are calculated for the entire system. |
| 1097 |
< |
The dipole autocorrelation function is given by: |
| 1094 |
> |
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1095 |
> |
function, which is averaging over time origins and over all the |
| 1096 |
> |
atoms, the dipole autocorrelation functions are calculated for the |
| 1097 |
> |
entire system. The dipole autocorrelation function is given by: |
| 1098 |
|
\[ |
| 1099 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1100 |
|
\right\rangle |
| 1102 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
| 1103 |
|
by |
| 1104 |
|
\[ |
| 1105 |
< |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
| 1105 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
| 1106 |
|
\] |
| 1107 |
|
In principle, many time correlation functions can be related with |
| 1108 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1111 |
|
each frequency using the following relationship: |
| 1112 |
|
\[ |
| 1113 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1114 |
< |
i2\pi vt} dt} |
| 1114 |
> |
i2\pi vt} dt}. |
| 1115 |
|
\] |
| 1116 |
|
|
| 1117 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1120 |
|
areas, from engineering, physics, to chemistry. For example, |
| 1121 |
|
missiles and vehicle are usually modeled by rigid bodies. The |
| 1122 |
|
movement of the objects in 3D gaming engine or other physics |
| 1123 |
< |
simulator is governed by the rigid body dynamics. In molecular |
| 1124 |
< |
simulation, rigid body is used to simplify the model in |
| 1125 |
< |
protein-protein docking study\cite{Gray2003}. |
| 1123 |
> |
simulator is governed by rigid body dynamics. In molecular |
| 1124 |
> |
simulations, rigid bodies are used to simplify protein-protein |
| 1125 |
> |
docking studies\cite{Gray2003}. |
| 1126 |
|
|
| 1127 |
|
It is very important to develop stable and efficient methods to |
| 1128 |
< |
integrate the equations of motion of orientational degrees of |
| 1129 |
< |
freedom. Euler angles are the nature choice to describe the |
| 1130 |
< |
rotational degrees of freedom. However, due to its singularity, the |
| 1131 |
< |
numerical integration of corresponding equations of motion is very |
| 1132 |
< |
inefficient and inaccurate. Although an alternative integrator using |
| 1133 |
< |
different sets of Euler angles can overcome this |
| 1134 |
< |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
| 1135 |
< |
of angular momentum conservation still remain. A singularity free |
| 1136 |
< |
representation utilizing quaternions was developed by Evans in |
| 1137 |
< |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
| 1138 |
< |
nonseparable Hamiltonian resulted from quaternion representation, |
| 1139 |
< |
which prevents the symplectic algorithm to be utilized. Another |
| 1140 |
< |
different approach is to apply holonomic constraints to the atoms |
| 1141 |
< |
belonging to the rigid body. Each atom moves independently under the |
| 1142 |
< |
normal forces deriving from potential energy and constraint forces |
| 1143 |
< |
which are used to guarantee the rigidness. However, due to their |
| 1144 |
< |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
| 1145 |
< |
when the number of constraint increases\cite{Ryckaert1977, |
| 1146 |
< |
Andersen1983}. |
| 1128 |
> |
integrate the equations of motion for orientational degrees of |
| 1129 |
> |
freedom. Euler angles are the natural choice to describe the |
| 1130 |
> |
rotational degrees of freedom. However, due to $\frac {1}{sin |
| 1131 |
> |
\theta}$ singularities, the numerical integration of corresponding |
| 1132 |
> |
equations of motion is very inefficient and inaccurate. Although an |
| 1133 |
> |
alternative integrator using multiple sets of Euler angles can |
| 1134 |
> |
overcome this difficulty\cite{Barojas1973}, the computational |
| 1135 |
> |
penalty and the loss of angular momentum conservation still remain. |
| 1136 |
> |
A singularity-free representation utilizing quaternions was |
| 1137 |
> |
developed by Evans in 1977\cite{Evans1977}. Unfortunately, this |
| 1138 |
> |
approach uses a nonseparable Hamiltonian resulting from the |
| 1139 |
> |
quaternion representation, which prevents the symplectic algorithm |
| 1140 |
> |
to be utilized. Another different approach is to apply holonomic |
| 1141 |
> |
constraints to the atoms belonging to the rigid body. Each atom |
| 1142 |
> |
moves independently under the normal forces deriving from potential |
| 1143 |
> |
energy and constraint forces which are used to guarantee the |
| 1144 |
> |
rigidness. However, due to their iterative nature, the SHAKE and |
| 1145 |
> |
Rattle algorithms also converge very slowly when the number of |
| 1146 |
> |
constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1147 |
|
|
| 1148 |
< |
The break through in geometric literature suggests that, in order to |
| 1148 |
> |
A break-through in geometric literature suggests that, in order to |
| 1149 |
|
develop a long-term integration scheme, one should preserve the |
| 1150 |
< |
symplectic structure of the flow. Introducing conjugate momentum to |
| 1151 |
< |
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
| 1152 |
< |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
| 1153 |
< |
the Hamiltonian system in a constraint manifold by iteratively |
| 1154 |
< |
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
| 1155 |
< |
method using quaternion representation was developed by |
| 1156 |
< |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
| 1157 |
< |
iterative and inefficient. In this section, we will present a |
| 1150 |
> |
symplectic structure of the flow. By introducing a conjugate |
| 1151 |
> |
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
| 1152 |
> |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
| 1153 |
> |
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1154 |
> |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 1155 |
> |
An alternative method using the quaternion representation was |
| 1156 |
> |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
| 1157 |
> |
methods are iterative and inefficient. In this section, we descibe a |
| 1158 |
|
symplectic Lie-Poisson integrator for rigid body developed by |
| 1159 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1160 |
|
|
| 1161 |
< |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 1162 |
< |
The motion of the rigid body is Hamiltonian with the Hamiltonian |
| 1161 |
> |
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1162 |
> |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
| 1163 |
|
function |
| 1164 |
|
\begin{equation} |
| 1165 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1173 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1174 |
|
\] |
| 1175 |
|
where $I_{ii}$ is the diagonal element of the inertia tensor. This |
| 1176 |
< |
constrained Hamiltonian equation subjects to a holonomic constraint, |
| 1176 |
> |
constrained Hamiltonian equation is subjected to a holonomic |
| 1177 |
> |
constraint, |
| 1178 |
|
\begin{equation} |
| 1179 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1180 |
|
\end{equation} |
| 1181 |
< |
which is used to ensure rotation matrix's orthogonality. |
| 1182 |
< |
Differentiating \ref{introEquation:orthogonalConstraint} and using |
| 1183 |
< |
Equation \ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1181 |
> |
which is used to ensure rotation matrix's unitarity. Differentiating |
| 1182 |
> |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
| 1183 |
> |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1184 |
|
\begin{equation} |
| 1185 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1186 |
|
\label{introEquation:RBFirstOrderConstraint} |
| 1187 |
|
\end{equation} |
| 1240 |
– |
|
| 1188 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1189 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1190 |
|
the equations of motion, |
| 1244 |
– |
|
| 1191 |
|
\begin{eqnarray} |
| 1192 |
< |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1193 |
< |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1194 |
< |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1192 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
| 1193 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\ |
| 1194 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
| 1195 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1196 |
|
\end{eqnarray} |
| 1251 |
– |
|
| 1197 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1198 |
< |
We can use constraint force provided by lagrange multiplier on the |
| 1199 |
< |
normal manifold to keep the motion on constraint space. Or we can |
| 1200 |
< |
simply evolve the system in constraint manifold. These two methods |
| 1201 |
< |
are proved to be equivalent. The holonomic constraint and equations |
| 1202 |
< |
of motions define a constraint manifold for rigid body |
| 1198 |
> |
We can use a constraint force provided by a Lagrange multiplier on |
| 1199 |
> |
the normal manifold to keep the motion on constraint space. Or we |
| 1200 |
> |
can simply evolve the system on the constraint manifold. These two |
| 1201 |
> |
methods have been proved to be equivalent. The holonomic constraint |
| 1202 |
> |
and equations of motions define a constraint manifold for rigid |
| 1203 |
> |
bodies |
| 1204 |
|
\[ |
| 1205 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1206 |
|
\right\}. |
| 1207 |
|
\] |
| 1262 |
– |
|
| 1208 |
|
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1209 |
< |
$T_{\star}SO(3)$. However, it turns out that under symplectic |
| 1209 |
> |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
| 1210 |
> |
rotation group $SO(3)$. However, it turns out that under symplectic |
| 1211 |
|
transformation, the cotangent space and the phase space are |
| 1212 |
< |
diffeomorphic. Introducing |
| 1212 |
> |
diffeomorphic. By introducing |
| 1213 |
|
\[ |
| 1214 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1215 |
|
\] |
| 1219 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1220 |
|
1,\tilde Q^T \tilde PJ^{ - 1} + J^{ - 1} P^T \tilde Q = 0} \right\} |
| 1221 |
|
\] |
| 1276 |
– |
|
| 1222 |
|
For a body fixed vector $X_i$ with respect to the center of mass of |
| 1223 |
|
the rigid body, its corresponding lab fixed vector $X_0^{lab}$ is |
| 1224 |
|
given as |
| 1237 |
|
\[ |
| 1238 |
|
\nabla _Q V(q,Q) = F(q,Q)X_i^t |
| 1239 |
|
\] |
| 1240 |
< |
respectively. |
| 1241 |
< |
|
| 1242 |
< |
As a common choice to describe the rotation dynamics of the rigid |
| 1298 |
< |
body, angular momentum on body frame $\Pi = Q^t P$ is introduced to |
| 1299 |
< |
rewrite the equations of motion, |
| 1240 |
> |
respectively. As a common choice to describe the rotation dynamics |
| 1241 |
> |
of the rigid body, the angular momentum on the body fixed frame $\Pi |
| 1242 |
> |
= Q^t P$ is introduced to rewrite the equations of motion, |
| 1243 |
|
\begin{equation} |
| 1244 |
|
\begin{array}{l} |
| 1245 |
< |
\mathop \Pi \limits^ \bullet = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda \\ |
| 1246 |
< |
\mathop Q\limits^{{\rm{ }} \bullet } = Q\Pi {\rm{ }}J^{ - 1} \\ |
| 1245 |
> |
\dot \Pi = J^{ - 1} \Pi ^T \Pi + Q^T \sum\limits_i {F_i (q,Q)X_i^T } - \Lambda, \\ |
| 1246 |
> |
\dot Q = Q\Pi {\rm{ }}J^{ - 1}, \\ |
| 1247 |
|
\end{array} |
| 1248 |
|
\label{introEqaution:RBMotionPI} |
| 1249 |
|
\end{equation} |
| 1250 |
< |
, as well as holonomic constraints, |
| 1251 |
< |
\[ |
| 1252 |
< |
\begin{array}{l} |
| 1310 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0 \\ |
| 1311 |
< |
Q^T Q = 1 \\ |
| 1312 |
< |
\end{array} |
| 1313 |
< |
\] |
| 1314 |
< |
|
| 1315 |
< |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
| 1316 |
< |
so(3)^ \star$, the hat-map isomorphism, |
| 1250 |
> |
as well as holonomic constraints $\Pi J^{ - 1} + J^{ - 1} \Pi ^t = |
| 1251 |
> |
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a |
| 1252 |
> |
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism, |
| 1253 |
|
\begin{equation} |
| 1254 |
|
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
| 1255 |
|
{\begin{array}{*{20}c} |
| 1262 |
|
will let us associate the matrix products with traditional vector |
| 1263 |
|
operations |
| 1264 |
|
\[ |
| 1265 |
< |
\hat vu = v \times u |
| 1265 |
> |
\hat vu = v \times u. |
| 1266 |
|
\] |
| 1267 |
< |
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1267 |
> |
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1268 |
|
matrix, |
| 1269 |
+ |
\begin{eqnarray} |
| 1270 |
+ |
(\dot \Pi - \dot \Pi ^T )&= &(\Pi - \Pi ^T )(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
| 1271 |
+ |
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1272 |
+ |
(\Lambda - \Lambda ^T ). \label{introEquation:skewMatrixPI} |
| 1273 |
+ |
\end{eqnarray} |
| 1274 |
+ |
Since $\Lambda$ is symmetric, the last term of |
| 1275 |
+ |
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
| 1276 |
+ |
Lagrange multiplier $\Lambda$ is absent from the equations of |
| 1277 |
+ |
motion. This unique property eliminates the requirement of |
| 1278 |
+ |
iterations which can not be avoided in other methods\cite{Kol1997, |
| 1279 |
+ |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
| 1280 |
+ |
equation of motion for angular momentum on body frame |
| 1281 |
|
\begin{equation} |
| 1334 |
– |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
| 1335 |
– |
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1336 |
– |
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1337 |
– |
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1338 |
– |
\end{equation} |
| 1339 |
– |
Since $\Lambda$ is symmetric, the last term of Equation |
| 1340 |
– |
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1341 |
– |
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1342 |
– |
unique property eliminate the requirement of iterations which can |
| 1343 |
– |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
| 1344 |
– |
|
| 1345 |
– |
Applying hat-map isomorphism, we obtain the equation of motion for |
| 1346 |
– |
angular momentum on body frame |
| 1347 |
– |
\begin{equation} |
| 1282 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1283 |
|
F_i (r,Q)} \right) \times X_i }. |
| 1284 |
|
\label{introEquation:bodyAngularMotion} |
| 1286 |
|
In the same manner, the equation of motion for rotation matrix is |
| 1287 |
|
given by |
| 1288 |
|
\[ |
| 1289 |
< |
\dot Q = Qskew(I^{ - 1} \pi ) |
| 1289 |
> |
\dot Q = Qskew(I^{ - 1} \pi ). |
| 1290 |
|
\] |
| 1291 |
|
|
| 1292 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1293 |
|
Lie-Poisson Integrator for Free Rigid Body} |
| 1294 |
|
|
| 1295 |
< |
If there is not external forces exerted on the rigid body, the only |
| 1296 |
< |
contribution to the rotational is from the kinetic potential (the |
| 1297 |
< |
first term of \ref{introEquation:bodyAngularMotion}). The free rigid |
| 1298 |
< |
body is an example of Lie-Poisson system with Hamiltonian function |
| 1295 |
> |
If there are no external forces exerted on the rigid body, the only |
| 1296 |
> |
contribution to the rotational motion is from the kinetic energy |
| 1297 |
> |
(the first term of \ref{introEquation:bodyAngularMotion}). The free |
| 1298 |
> |
rigid body is an example of a Lie-Poisson system with Hamiltonian |
| 1299 |
> |
function |
| 1300 |
|
\begin{equation} |
| 1301 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1302 |
|
\label{introEquation:rotationalKineticRB} |
| 1308 |
|
0 & {\pi _3 } & { - \pi _2 } \\ |
| 1309 |
|
{ - \pi _3 } & 0 & {\pi _1 } \\ |
| 1310 |
|
{\pi _2 } & { - \pi _1 } & 0 \\ |
| 1311 |
< |
\end{array}} \right) |
| 1311 |
> |
\end{array}} \right). |
| 1312 |
|
\end{equation} |
| 1313 |
|
Thus, the dynamics of free rigid body is governed by |
| 1314 |
|
\begin{equation} |
| 1315 |
< |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ) |
| 1315 |
> |
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
| 1316 |
|
\end{equation} |
| 1317 |
< |
|
| 1318 |
< |
One may notice that each $T_i^r$ in Equation |
| 1319 |
< |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
| 1385 |
< |
instance, the equations of motion due to $T_1^r$ are given by |
| 1317 |
> |
One may notice that each $T_i^r$ in |
| 1318 |
> |
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly. |
| 1319 |
> |
For instance, the equations of motion due to $T_1^r$ are given by |
| 1320 |
|
\begin{equation} |
| 1321 |
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
| 1322 |
|
\label{introEqaution:RBMotionSingleTerm} |
| 1323 |
|
\end{equation} |
| 1324 |
< |
where |
| 1324 |
> |
with |
| 1325 |
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
| 1326 |
|
0 & 0 & 0 \\ |
| 1327 |
|
0 & 0 & {\pi _1 } \\ |
| 1328 |
|
0 & { - \pi _1 } & 0 \\ |
| 1329 |
|
\end{array}} \right). |
| 1330 |
|
\] |
| 1331 |
< |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
| 1331 |
> |
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is |
| 1332 |
|
\[ |
| 1333 |
|
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
| 1334 |
|
Q(0)e^{\Delta tR_1 } |
| 1342 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1343 |
|
\] |
| 1344 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1345 |
< |
tR_1 }$, we can use Cayley transformation, |
| 1345 |
> |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
| 1346 |
> |
propagator, |
| 1347 |
|
\[ |
| 1348 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1349 |
< |
) |
| 1349 |
> |
). |
| 1350 |
|
\] |
| 1351 |
|
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1352 |
< |
manner. |
| 1353 |
< |
|
| 1419 |
< |
In order to construct a second-order symplectic method, we split the |
| 1420 |
< |
angular kinetic Hamiltonian function can into five terms |
| 1352 |
> |
manner. In order to construct a second-order symplectic method, we |
| 1353 |
> |
split the angular kinetic Hamiltonian function into five terms |
| 1354 |
|
\[ |
| 1355 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
| 1356 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
| 1357 |
< |
(\pi _1 ) |
| 1358 |
< |
\]. |
| 1359 |
< |
Concatenating flows corresponding to these five terms, we can obtain |
| 1360 |
< |
an symplectic integrator, |
| 1357 |
> |
(\pi _1 ). |
| 1358 |
> |
\] |
| 1359 |
> |
By concatenating the propagators corresponding to these five terms, |
| 1360 |
> |
we can obtain an symplectic integrator, |
| 1361 |
|
\[ |
| 1362 |
|
\varphi _{\Delta t,T^r } = \varphi _{\Delta t/2,\pi _1 } \circ |
| 1363 |
|
\varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } |
| 1364 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1365 |
|
_1 }. |
| 1366 |
|
\] |
| 1434 |
– |
|
| 1367 |
|
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1368 |
|
$F(\pi )$ and $G(\pi )$ is defined by |
| 1369 |
|
\[ |
| 1370 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1371 |
< |
) |
| 1371 |
> |
). |
| 1372 |
|
\] |
| 1373 |
|
If the Poisson bracket of a function $F$ with an arbitrary smooth |
| 1374 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1379 |
|
then by the chain rule |
| 1380 |
|
\[ |
| 1381 |
|
\nabla _\pi F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2 |
| 1382 |
< |
}}{2})\pi |
| 1382 |
> |
}}{2})\pi. |
| 1383 |
|
\] |
| 1384 |
< |
Thus $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel \pi |
| 1384 |
> |
Thus, $ [\nabla _\pi F(\pi )]^T J(\pi ) = - S'(\frac{{\parallel |
| 1385 |
> |
\pi |
| 1386 |
|
\parallel ^2 }}{2})\pi \times \pi = 0 $. This explicit |
| 1387 |
< |
Lie-Poisson integrator is found to be extremely efficient and stable |
| 1388 |
< |
which can be explained by the fact the small angle approximation is |
| 1389 |
< |
used and the norm of the angular momentum is conserved. |
| 1387 |
> |
Lie-Poisson integrator is found to be both extremely efficient and |
| 1388 |
> |
stable. These properties can be explained by the fact the small |
| 1389 |
> |
angle approximation is used and the norm of the angular momentum is |
| 1390 |
> |
conserved. |
| 1391 |
|
|
| 1392 |
|
\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian |
| 1393 |
|
Splitting for Rigid Body} |
| 1394 |
|
|
| 1395 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1396 |
< |
energy and potential energy, |
| 1397 |
< |
\[ |
| 1398 |
< |
H = T(p,\pi ) + V(q,Q) |
| 1465 |
< |
\] |
| 1466 |
< |
The equations of motion corresponding to potential energy and |
| 1467 |
< |
kinetic energy are listed in the below table, |
| 1396 |
> |
energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations |
| 1397 |
> |
of motion corresponding to potential energy and kinetic energy are |
| 1398 |
> |
listed in the below table, |
| 1399 |
|
\begin{table} |
| 1400 |
< |
\caption{Equations of motion due to Potential and Kinetic Energies} |
| 1400 |
> |
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1401 |
|
\begin{center} |
| 1402 |
|
\begin{tabular}{|l|l|} |
| 1403 |
|
\hline |
| 1411 |
|
\end{tabular} |
| 1412 |
|
\end{center} |
| 1413 |
|
\end{table} |
| 1414 |
< |
A second-order symplectic method is now obtained by the |
| 1415 |
< |
composition of the flow maps, |
| 1414 |
> |
A second-order symplectic method is now obtained by the composition |
| 1415 |
> |
of the position and velocity propagators, |
| 1416 |
|
\[ |
| 1417 |
|
\varphi _{\Delta t} = \varphi _{\Delta t/2,V} \circ \varphi |
| 1418 |
|
_{\Delta t,T} \circ \varphi _{\Delta t/2,V}. |
| 1419 |
|
\] |
| 1420 |
|
Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two |
| 1421 |
< |
sub-flows which corresponding to force and torque respectively, |
| 1421 |
> |
sub-propagators which corresponding to force and torque |
| 1422 |
> |
respectively, |
| 1423 |
|
\[ |
| 1424 |
|
\varphi _{\Delta t/2,V} = \varphi _{\Delta t/2,F} \circ \varphi |
| 1425 |
|
_{\Delta t/2,\tau }. |
| 1426 |
|
\] |
| 1427 |
|
Since the associated operators of $\varphi _{\Delta t/2,F} $ and |
| 1428 |
< |
$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition |
| 1429 |
< |
order inside $\varphi _{\Delta t/2,V}$ does not matter. |
| 1430 |
< |
|
| 1431 |
< |
Furthermore, kinetic potential can be separated to translational |
| 1500 |
< |
kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1428 |
> |
$\circ \varphi _{\Delta t/2,\tau }$ commute, the composition order |
| 1429 |
> |
inside $\varphi _{\Delta t/2,V}$ does not matter. Furthermore, the |
| 1430 |
> |
kinetic energy can be separated to translational kinetic term, $T^t |
| 1431 |
> |
(p)$, and rotational kinetic term, $T^r (\pi )$, |
| 1432 |
|
\begin{equation} |
| 1433 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
| 1434 |
|
\end{equation} |
| 1435 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
| 1436 |
< |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
| 1437 |
< |
corresponding flow maps are given by |
| 1436 |
> |
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore, |
| 1437 |
> |
the corresponding propagators are given by |
| 1438 |
|
\[ |
| 1439 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
| 1440 |
|
_{\Delta t,T^r }. |
| 1441 |
|
\] |
| 1442 |
< |
Finally, we obtain the overall symplectic flow maps for free moving |
| 1443 |
< |
rigid body |
| 1444 |
< |
\begin{equation} |
| 1445 |
< |
\begin{array}{c} |
| 1446 |
< |
\varphi _{\Delta t} = \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
| 1447 |
< |
\circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \\ |
| 1517 |
< |
\circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1518 |
< |
\end{array} |
| 1442 |
> |
Finally, we obtain the overall symplectic propagators for freely |
| 1443 |
> |
moving rigid bodies |
| 1444 |
> |
\begin{eqnarray} |
| 1445 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
| 1446 |
> |
& & \circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \notag\\ |
| 1447 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1448 |
|
\label{introEquation:overallRBFlowMaps} |
| 1449 |
< |
\end{equation} |
| 1449 |
> |
\end{eqnarray} |
| 1450 |
|
|
| 1451 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
| 1452 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1453 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
| 1454 |
|
has been applied in a variety of studies. This section will review |
| 1455 |
< |
the theory of Langevin dynamics simulation. A brief derivation of |
| 1456 |
< |
generalized Langevin equation will be given first. Follow that, we |
| 1457 |
< |
will discuss the physical meaning of the terms appearing in the |
| 1458 |
< |
equation as well as the calculation of friction tensor from |
| 1459 |
< |
hydrodynamics theory. |
| 1455 |
> |
the theory of Langevin dynamics. A brief derivation of generalized |
| 1456 |
> |
Langevin equation will be given first. Following that, we will |
| 1457 |
> |
discuss the physical meaning of the terms appearing in the equation |
| 1458 |
> |
as well as the calculation of friction tensor from hydrodynamics |
| 1459 |
> |
theory. |
| 1460 |
|
|
| 1461 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1462 |
|
|
| 1463 |
< |
Harmonic bath model, in which an effective set of harmonic |
| 1463 |
> |
A harmonic bath model, in which an effective set of harmonic |
| 1464 |
|
oscillators are used to mimic the effect of a linearly responding |
| 1465 |
|
environment, has been widely used in quantum chemistry and |
| 1466 |
|
statistical mechanics. One of the successful applications of |
| 1467 |
< |
Harmonic bath model is the derivation of Deriving Generalized |
| 1468 |
< |
Langevin Dynamics. Lets consider a system, in which the degree of |
| 1467 |
> |
Harmonic bath model is the derivation of the Generalized Langevin |
| 1468 |
> |
Dynamics (GLE). Lets consider a system, in which the degree of |
| 1469 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1470 |
|
Hamiltonian of the form |
| 1471 |
|
\begin{equation} |
| 1472 |
|
H = \frac{{p^2 }}{{2m}} + U(x) + H_B + \Delta U(x,x_1 , \ldots x_N) |
| 1473 |
|
\label{introEquation:bathGLE}. |
| 1474 |
|
\end{equation} |
| 1475 |
< |
Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated |
| 1476 |
< |
with this degree of freedom, $H_B$ is harmonic bath Hamiltonian, |
| 1475 |
> |
Here $p$ is a momentum conjugate to $x$, $m$ is the mass associated |
| 1476 |
> |
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
| 1477 |
|
\[ |
| 1478 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1479 |
|
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1481 |
|
\] |
| 1482 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1483 |
|
$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are |
| 1484 |
< |
the harmonic bath masses, and $\Delta U$ is bilinear system-bath |
| 1484 |
> |
the harmonic bath masses, and $\Delta U$ is a bilinear system-bath |
| 1485 |
|
coupling, |
| 1486 |
|
\[ |
| 1487 |
|
\Delta U = - \sum\limits_{\alpha = 1}^N {g_\alpha x_\alpha x} |
| 1488 |
|
\] |
| 1489 |
< |
where $g_\alpha$ are the coupling constants between the bath and the |
| 1490 |
< |
coordinate $x$. Introducing |
| 1489 |
> |
where $g_\alpha$ are the coupling constants between the bath |
| 1490 |
> |
coordinates ($x_ \alpha$) and the system coordinate ($x$). |
| 1491 |
> |
Introducing |
| 1492 |
|
\[ |
| 1493 |
|
W(x) = U(x) - \sum\limits_{\alpha = 1}^N {\frac{{g_\alpha ^2 |
| 1494 |
|
}}{{2m_\alpha w_\alpha ^2 }}} x^2 |
| 1495 |
< |
\] and combining the last two terms in Equation |
| 1496 |
< |
\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath |
| 1567 |
< |
Hamiltonian as |
| 1495 |
> |
\] |
| 1496 |
> |
and combining the last two terms in Eq.~\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath Hamiltonian as |
| 1497 |
|
\[ |
| 1498 |
|
H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha = 1}^N |
| 1499 |
|
{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha }} + \frac{1}{2}m_\alpha |
| 1500 |
|
w_\alpha ^2 \left( {x_\alpha - \frac{{g_\alpha }}{{m_\alpha |
| 1501 |
< |
w_\alpha ^2 }}x} \right)^2 } \right\}} |
| 1501 |
> |
w_\alpha ^2 }}x} \right)^2 } \right\}}. |
| 1502 |
|
\] |
| 1503 |
|
Since the first two terms of the new Hamiltonian depend only on the |
| 1504 |
|
system coordinates, we can get the equations of motion for |
| 1505 |
< |
Generalized Langevin Dynamics by Hamilton's equations |
| 1577 |
< |
\ref{introEquation:motionHamiltonianCoordinate, |
| 1578 |
< |
introEquation:motionHamiltonianMomentum}, |
| 1505 |
> |
Generalized Langevin Dynamics by Hamilton's equations, |
| 1506 |
|
\begin{equation} |
| 1507 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1508 |
|
\sum\limits_{\alpha = 1}^N {g_\alpha \left( {x_\alpha - |
| 1515 |
|
\frac{{g_\alpha }}{{m_\alpha w_\alpha ^2 }}x} \right). |
| 1516 |
|
\label{introEquation:bathMotionGLE} |
| 1517 |
|
\end{equation} |
| 1591 |
– |
|
| 1518 |
|
In order to derive an equation for $x$, the dynamics of the bath |
| 1519 |
|
variables $x_\alpha$ must be solved exactly first. As an integral |
| 1520 |
|
transform which is particularly useful in solving linear ordinary |
| 1521 |
< |
differential equations, Laplace transform is the appropriate tool to |
| 1522 |
< |
solve this problem. The basic idea is to transform the difficult |
| 1521 |
> |
differential equations,the Laplace transform is the appropriate tool |
| 1522 |
> |
to solve this problem. The basic idea is to transform the difficult |
| 1523 |
|
differential equations into simple algebra problems which can be |
| 1524 |
< |
solved easily. Then applying inverse Laplace transform, also known |
| 1525 |
< |
as the Bromwich integral, we can retrieve the solutions of the |
| 1526 |
< |
original problems. |
| 1527 |
< |
|
| 1602 |
< |
Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace |
| 1603 |
< |
transform of f(t) is a new function defined as |
| 1524 |
> |
solved easily. Then, by applying the inverse Laplace transform, also |
| 1525 |
> |
known as the Bromwich integral, we can retrieve the solutions of the |
| 1526 |
> |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
| 1527 |
> |
$, the Laplace transform of $f(t)$ is a new function defined as |
| 1528 |
|
\[ |
| 1529 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1530 |
|
\] |
| 1531 |
|
where $p$ is real and $L$ is called the Laplace Transform |
| 1532 |
|
Operator. Below are some important properties of Laplace transform |
| 1609 |
– |
|
| 1533 |
|
\begin{eqnarray*} |
| 1534 |
|
L(x + y) & = & L(x) + L(y) \\ |
| 1535 |
|
L(ax) & = & aL(x) \\ |
| 1537 |
|
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
| 1538 |
|
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
| 1539 |
|
\end{eqnarray*} |
| 1540 |
< |
|
| 1618 |
< |
|
| 1619 |
< |
Applying Laplace transform to the bath coordinates, we obtain |
| 1540 |
> |
Applying the Laplace transform to the bath coordinates, we obtain |
| 1541 |
|
\begin{eqnarray*} |
| 1542 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
| 1543 |
< |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1542 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x), \\ |
| 1543 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
| 1544 |
|
\end{eqnarray*} |
| 1624 |
– |
|
| 1545 |
|
By the same way, the system coordinates become |
| 1546 |
|
\begin{eqnarray*} |
| 1547 |
< |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1548 |
< |
& & \mbox{} - \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1547 |
> |
mL(\ddot x) & = & |
| 1548 |
> |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1549 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}. |
| 1550 |
|
\end{eqnarray*} |
| 1630 |
– |
|
| 1551 |
|
With the help of some relatively important inverse Laplace |
| 1552 |
|
transformations: |
| 1553 |
|
\[ |
| 1557 |
|
L(1) = \frac{1}{p} \\ |
| 1558 |
|
\end{array} |
| 1559 |
|
\] |
| 1560 |
< |
, we obtain |
| 1560 |
> |
we obtain |
| 1561 |
|
\begin{eqnarray*} |
| 1562 |
|
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
| 1563 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1605 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
| 1606 |
|
implies it is completely deterministic within the context of a |
| 1607 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1608 |
< |
uncorrelated to $x$ and $\dot x$, |
| 1609 |
< |
\[ |
| 1610 |
< |
\begin{array}{l} |
| 1611 |
< |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
| 1612 |
< |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
| 1693 |
< |
\end{array} |
| 1694 |
< |
\] |
| 1695 |
< |
This property is what we expect from a truly random process. As long |
| 1696 |
< |
as the model, which is gaussian distribution in general, chosen for |
| 1697 |
< |
$R(t)$ is a truly random process, the stochastic nature of the GLE |
| 1698 |
< |
still remains. |
| 1699 |
< |
|
| 1608 |
> |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
| 1609 |
> |
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
| 1610 |
> |
0.$ This property is what we expect from a truly random process. As |
| 1611 |
> |
long as the model chosen for $R(t)$ was a gaussian distribution in |
| 1612 |
> |
general, the stochastic nature of the GLE still remains. |
| 1613 |
|
%dynamic friction kernel |
| 1614 |
|
The convolution integral |
| 1615 |
|
\[ |
| 1624 |
|
\[ |
| 1625 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = \xi _0 (x(t) - x(0)) |
| 1626 |
|
\] |
| 1627 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1627 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1628 |
|
\[ |
| 1629 |
|
m\ddot x = - \frac{\partial }{{\partial x}}\left( {W(x) + |
| 1630 |
|
\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t), |
| 1631 |
|
\] |
| 1632 |
< |
which can be used to describe dynamic caging effect. The other |
| 1633 |
< |
extreme is the bath that responds infinitely quickly to motions in |
| 1634 |
< |
the system. Thus, $\xi (t)$ can be taken as a $delta$ function in |
| 1635 |
< |
time: |
| 1632 |
> |
which can be used to describe the effect of dynamic caging in |
| 1633 |
> |
viscous solvents. The other extreme is the bath that responds |
| 1634 |
> |
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
| 1635 |
> |
taken as a $delta$ function in time: |
| 1636 |
|
\[ |
| 1637 |
|
\xi (t) = 2\xi _0 \delta (t) |
| 1638 |
|
\] |
| 1641 |
|
\int_0^t {\xi (t)\dot x(t - \tau )d\tau } = 2\xi _0 \int_0^t |
| 1642 |
|
{\delta (t)\dot x(t - \tau )d\tau } = \xi _0 \dot x(t), |
| 1643 |
|
\] |
| 1644 |
< |
and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1644 |
> |
and Eq.~\ref{introEuqation:GeneralizedLangevinDynamics} becomes |
| 1645 |
|
\begin{equation} |
| 1646 |
|
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot |
| 1647 |
|
x(t) + R(t) \label{introEquation:LangevinEquation} |
| 1648 |
|
\end{equation} |
| 1649 |
|
which is known as the Langevin equation. The static friction |
| 1650 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1651 |
< |
or be determined by Stokes' law for regular shaped particles.A |
| 1651 |
> |
or be determined by Stokes' law for regular shaped particles. A |
| 1652 |
|
briefly review on calculating friction tensor for arbitrary shaped |
| 1653 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1654 |
|
|
| 1655 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1656 |
|
|
| 1657 |
< |
Defining a new set of coordinates, |
| 1657 |
> |
Defining a new set of coordinates |
| 1658 |
|
\[ |
| 1659 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1660 |
< |
^2 }}x(0) |
| 1661 |
< |
\], |
| 1660 |
> |
^2 }}x(0), |
| 1661 |
> |
\] |
| 1662 |
|
we can rewrite $R(T)$ as |
| 1663 |
|
\[ |
| 1664 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1665 |
|
\] |
| 1666 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1754 |
– |
|
| 1667 |
|
\begin{eqnarray*} |
| 1668 |
|
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1669 |
|
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1672 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1673 |
|
& = &kT\xi (t) \\ |
| 1674 |
|
\end{eqnarray*} |
| 1763 |
– |
|
| 1675 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1676 |
|
\begin{equation} |
| 1677 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 1678 |
< |
\label{introEquation:secondFluctuationDissipation}. |
| 1678 |
> |
\label{introEquation:secondFluctuationDissipation}, |
| 1679 |
|
\end{equation} |
| 1680 |
< |
In effect, it acts as a constraint on the possible ways in which one |
| 1681 |
< |
can model the random force and friction kernel. |
| 1771 |
< |
|
| 1772 |
< |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1773 |
< |
Theoretically, the friction kernel can be determined using velocity |
| 1774 |
< |
autocorrelation function. However, this approach become impractical |
| 1775 |
< |
when the system become more and more complicate. Instead, various |
| 1776 |
< |
approaches based on hydrodynamics have been developed to calculate |
| 1777 |
< |
the friction coefficients. The friction effect is isotropic in |
| 1778 |
< |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 1779 |
< |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 1780 |
< |
\[ |
| 1781 |
< |
\Xi = \left( {\begin{array}{*{20}c} |
| 1782 |
< |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1783 |
< |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 1784 |
< |
\end{array}} \right). |
| 1785 |
< |
\] |
| 1786 |
< |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 1787 |
< |
tensor and rotational resistance (friction) tensor respectively, |
| 1788 |
< |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 1789 |
< |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 1790 |
< |
particle moves in a fluid, it may experience friction force or |
| 1791 |
< |
torque along the opposite direction of the velocity or angular |
| 1792 |
< |
velocity, |
| 1793 |
< |
\[ |
| 1794 |
< |
\left( \begin{array}{l} |
| 1795 |
< |
F_R \\ |
| 1796 |
< |
\tau _R \\ |
| 1797 |
< |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1798 |
< |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1799 |
< |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1800 |
< |
\end{array}} \right)\left( \begin{array}{l} |
| 1801 |
< |
v \\ |
| 1802 |
< |
w \\ |
| 1803 |
< |
\end{array} \right) |
| 1804 |
< |
\] |
| 1805 |
< |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1806 |
< |
toque. |
| 1807 |
< |
|
| 1808 |
< |
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
| 1809 |
< |
|
| 1810 |
< |
For a spherical particle, the translational and rotational friction |
| 1811 |
< |
constant can be calculated from Stoke's law, |
| 1812 |
< |
\[ |
| 1813 |
< |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1814 |
< |
{6\pi \eta R} & 0 & 0 \\ |
| 1815 |
< |
0 & {6\pi \eta R} & 0 \\ |
| 1816 |
< |
0 & 0 & {6\pi \eta R} \\ |
| 1817 |
< |
\end{array}} \right) |
| 1818 |
< |
\] |
| 1819 |
< |
and |
| 1820 |
< |
\[ |
| 1821 |
< |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1822 |
< |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1823 |
< |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1824 |
< |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1825 |
< |
\end{array}} \right) |
| 1826 |
< |
\] |
| 1827 |
< |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1828 |
< |
hydrodynamics radius. |
| 1829 |
< |
|
| 1830 |
< |
Other non-spherical shape, such as cylinder and ellipsoid |
| 1831 |
< |
\textit{etc}, are widely used as reference for developing new |
| 1832 |
< |
hydrodynamics theory, because their properties can be calculated |
| 1833 |
< |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 1834 |
< |
also called a triaxial ellipsoid, which is given in Cartesian |
| 1835 |
< |
coordinates by\cite{Perrin1934, Perrin1936} |
| 1836 |
< |
\[ |
| 1837 |
< |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 1838 |
< |
}} = 1 |
| 1839 |
< |
\] |
| 1840 |
< |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 1841 |
< |
due to the complexity of the elliptic integral, only the ellipsoid |
| 1842 |
< |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 1843 |
< |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 1844 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 1845 |
< |
\[ |
| 1846 |
< |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1847 |
< |
} }}{b}, |
| 1848 |
< |
\] |
| 1849 |
< |
and oblate, |
| 1850 |
< |
\[ |
| 1851 |
< |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1852 |
< |
}}{a} |
| 1853 |
< |
\], |
| 1854 |
< |
one can write down the translational and rotational resistance |
| 1855 |
< |
tensors |
| 1856 |
< |
\[ |
| 1857 |
< |
\begin{array}{l} |
| 1858 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1859 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1860 |
< |
\end{array}, |
| 1861 |
< |
\] |
| 1862 |
< |
and |
| 1863 |
< |
\[ |
| 1864 |
< |
\begin{array}{l} |
| 1865 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1866 |
< |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1867 |
< |
\end{array}. |
| 1868 |
< |
\] |
| 1869 |
< |
|
| 1870 |
< |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
| 1871 |
< |
|
| 1872 |
< |
Unlike spherical and other regular shaped molecules, there is not |
| 1873 |
< |
analytical solution for friction tensor of any arbitrary shaped |
| 1874 |
< |
rigid molecules. The ellipsoid of revolution model and general |
| 1875 |
< |
triaxial ellipsoid model have been used to approximate the |
| 1876 |
< |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1877 |
< |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1878 |
< |
combination of rotational diffusion coefficients, $D$-space is not |
| 1879 |
< |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 1880 |
< |
translational and rotational motion of rigid body, general ellipsoid |
| 1881 |
< |
is not always suitable for modeling arbitrarily shaped rigid |
| 1882 |
< |
molecule. A number of studies have been devoted to determine the |
| 1883 |
< |
friction tensor for irregularly shaped rigid bodies using more |
| 1884 |
< |
advanced method where the molecule of interest was modeled by |
| 1885 |
< |
combinations of spheres(beads)\cite{Carrasco1999} and the |
| 1886 |
< |
hydrodynamics properties of the molecule can be calculated using the |
| 1887 |
< |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
| 1888 |
< |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
| 1889 |
< |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
| 1890 |
< |
than its unperturbed velocity $v_i$, |
| 1891 |
< |
\[ |
| 1892 |
< |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1893 |
< |
\] |
| 1894 |
< |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1895 |
< |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1896 |
< |
proportional to its ``net'' velocity |
| 1897 |
< |
\begin{equation} |
| 1898 |
< |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1899 |
< |
\label{introEquation:tensorExpression} |
| 1900 |
< |
\end{equation} |
| 1901 |
< |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1902 |
< |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1903 |
< |
\ref{introEquation:tensorExpression} |
| 1904 |
< |
\begin{equation} |
| 1905 |
< |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1906 |
< |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1907 |
< |
\label{introEquation:oseenTensor} |
| 1908 |
< |
\end{equation} |
| 1909 |
< |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1910 |
< |
A second order expression for element of different size was |
| 1911 |
< |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
| 1912 |
< |
Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
| 1913 |
< |
\begin{equation} |
| 1914 |
< |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1915 |
< |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1916 |
< |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1917 |
< |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1918 |
< |
\label{introEquation:RPTensorNonOverlapped} |
| 1919 |
< |
\end{equation} |
| 1920 |
< |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1921 |
< |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1922 |
< |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1923 |
< |
overlapping beads with the same radius, $\sigma$, is given by |
| 1924 |
< |
\begin{equation} |
| 1925 |
< |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1926 |
< |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1927 |
< |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1928 |
< |
\label{introEquation:RPTensorOverlapped} |
| 1929 |
< |
\end{equation} |
| 1930 |
< |
|
| 1931 |
< |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1932 |
< |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1933 |
< |
$B_{ij}$ blocks |
| 1934 |
< |
\begin{equation} |
| 1935 |
< |
B = \left( {\begin{array}{*{20}c} |
| 1936 |
< |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1937 |
< |
\vdots & \ddots & \vdots \\ |
| 1938 |
< |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1939 |
< |
\end{array}} \right), |
| 1940 |
< |
\end{equation} |
| 1941 |
< |
where $B_{ij}$ is given by |
| 1942 |
< |
\[ |
| 1943 |
< |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1944 |
< |
)T_{ij} |
| 1945 |
< |
\] |
| 1946 |
< |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 1947 |
< |
$B$, we obtain |
| 1948 |
< |
|
| 1949 |
< |
\[ |
| 1950 |
< |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1951 |
< |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1952 |
< |
\vdots & \ddots & \vdots \\ |
| 1953 |
< |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1954 |
< |
\end{array}} \right) |
| 1955 |
< |
\] |
| 1956 |
< |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1957 |
< |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1958 |
< |
\[ |
| 1959 |
< |
U_i = \left( {\begin{array}{*{20}c} |
| 1960 |
< |
0 & { - z_i } & {y_i } \\ |
| 1961 |
< |
{z_i } & 0 & { - x_i } \\ |
| 1962 |
< |
{ - y_i } & {x_i } & 0 \\ |
| 1963 |
< |
\end{array}} \right) |
| 1964 |
< |
\] |
| 1965 |
< |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1966 |
< |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1967 |
< |
arbitrary origin $O$ can be written as |
| 1968 |
< |
\begin{equation} |
| 1969 |
< |
\begin{array}{l} |
| 1970 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1971 |
< |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1972 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1973 |
< |
\end{array} |
| 1974 |
< |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1975 |
< |
\end{equation} |
| 1976 |
< |
|
| 1977 |
< |
The resistance tensor depends on the origin to which they refer. The |
| 1978 |
< |
proper location for applying friction force is the center of |
| 1979 |
< |
resistance (reaction), at which the trace of rotational resistance |
| 1980 |
< |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1981 |
< |
resistance is defined as an unique point of the rigid body at which |
| 1982 |
< |
the translation-rotation coupling tensor are symmetric, |
| 1983 |
< |
\begin{equation} |
| 1984 |
< |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1985 |
< |
\label{introEquation:definitionCR} |
| 1986 |
< |
\end{equation} |
| 1987 |
< |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1988 |
< |
we can easily find out that the translational resistance tensor is |
| 1989 |
< |
origin independent, while the rotational resistance tensor and |
| 1990 |
< |
translation-rotation coupling resistance tensor depend on the |
| 1991 |
< |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1992 |
< |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1993 |
< |
obtain the resistance tensor at $P$ by |
| 1994 |
< |
\begin{equation} |
| 1995 |
< |
\begin{array}{l} |
| 1996 |
< |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 1997 |
< |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 1998 |
< |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
| 1999 |
< |
\end{array} |
| 2000 |
< |
\label{introEquation:resistanceTensorTransformation} |
| 2001 |
< |
\end{equation} |
| 2002 |
< |
where |
| 2003 |
< |
\[ |
| 2004 |
< |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 2005 |
< |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 2006 |
< |
{z_i } & 0 & { - x_{OP} } \\ |
| 2007 |
< |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 2008 |
< |
\end{array}} \right) |
| 2009 |
< |
\] |
| 2010 |
< |
Using Equations \ref{introEquation:definitionCR} and |
| 2011 |
< |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 2012 |
< |
the position of center of resistance, |
| 2013 |
< |
\begin{eqnarray*} |
| 2014 |
< |
\left( \begin{array}{l} |
| 2015 |
< |
x_{OR} \\ |
| 2016 |
< |
y_{OR} \\ |
| 2017 |
< |
z_{OR} \\ |
| 2018 |
< |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
| 2019 |
< |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 2020 |
< |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 2021 |
< |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 2022 |
< |
\end{array}} \right)^{ - 1} \\ |
| 2023 |
< |
& & \left( \begin{array}{l} |
| 2024 |
< |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 2025 |
< |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 2026 |
< |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 2027 |
< |
\end{array} \right) \\ |
| 2028 |
< |
\end{eqnarray*} |
| 2029 |
< |
|
| 2030 |
< |
|
| 2031 |
< |
|
| 2032 |
< |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 2033 |
< |
joining center of resistance $R$ and origin $O$. |
| 1680 |
> |
which acts as a constraint on the possible ways in which one can |
| 1681 |
> |
model the random force and friction kernel. |
