| 6 |
|
Closely related to Classical Mechanics, Molecular Dynamics |
| 7 |
|
simulations are carried out by integrating the equations of motion |
| 8 |
|
for a given system of particles. There are three fundamental ideas |
| 9 |
< |
behind classical mechanics. Firstly, One can determine the state of |
| 9 |
> |
behind classical mechanics. Firstly, one can determine the state of |
| 10 |
|
a mechanical system at any time of interest; Secondly, all the |
| 11 |
|
mechanical properties of the system at that time can be determined |
| 12 |
|
by combining the knowledge of the properties of the system with the |
| 17 |
|
\subsection{\label{introSection:newtonian}Newtonian Mechanics} |
| 18 |
|
The discovery of Newton's three laws of mechanics which govern the |
| 19 |
|
motion of particles is the foundation of the classical mechanics. |
| 20 |
< |
Newton¡¯s first law defines a class of inertial frames. Inertial |
| 20 |
> |
Newton's first law defines a class of inertial frames. Inertial |
| 21 |
|
frames are reference frames where a particle not interacting with |
| 22 |
|
other bodies will move with constant speed in the same direction. |
| 23 |
< |
With respect to inertial frames Newton¡¯s second law has the form |
| 23 |
> |
With respect to inertial frames, Newton's second law has the form |
| 24 |
|
\begin{equation} |
| 25 |
< |
F = \frac {dp}{dt} = \frac {mv}{dt} |
| 25 |
> |
F = \frac {dp}{dt} = \frac {mdv}{dt} |
| 26 |
|
\label{introEquation:newtonSecondLaw} |
| 27 |
|
\end{equation} |
| 28 |
|
A point mass interacting with other bodies moves with the |
| 29 |
|
acceleration along the direction of the force acting on it. Let |
| 30 |
|
$F_{ij}$ be the force that particle $i$ exerts on particle $j$, and |
| 31 |
|
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
| 32 |
< |
Newton¡¯s third law states that |
| 32 |
> |
Newton's third law states that |
| 33 |
|
\begin{equation} |
| 34 |
|
F_{ij} = -F_{ji} |
| 35 |
|
\label{introEquation:newtonThirdLaw} |
| 46 |
|
\end{equation} |
| 47 |
|
The torque $\tau$ with respect to the same origin is defined to be |
| 48 |
|
\begin{equation} |
| 49 |
< |
N \equiv r \times F \label{introEquation:torqueDefinition} |
| 49 |
> |
\tau \equiv r \times F \label{introEquation:torqueDefinition} |
| 50 |
|
\end{equation} |
| 51 |
|
Differentiating Eq.~\ref{introEquation:angularMomentumDefinition}, |
| 52 |
|
\[ |
| 59 |
|
\] |
| 60 |
|
thus, |
| 61 |
|
\begin{equation} |
| 62 |
< |
\dot L = r \times \dot p = N |
| 62 |
> |
\dot L = r \times \dot p = \tau |
| 63 |
|
\end{equation} |
| 64 |
|
If there are no external torques acting on a body, the angular |
| 65 |
|
momentum of it is conserved. The last conservation theorem state |
| 68 |
|
\end{equation} |
| 69 |
|
is conserved. All of these conserved quantities are |
| 70 |
|
important factors to determine the quality of numerical integration |
| 71 |
< |
scheme for rigid body \cite{Dullweber1997}. |
| 71 |
> |
schemes for rigid bodies \cite{Dullweber1997}. |
| 72 |
|
|
| 73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 74 |
|
|
| 75 |
< |
Newtonian Mechanics suffers from two important limitations: it |
| 76 |
< |
describes their motion in special cartesian coordinate systems. |
| 77 |
< |
Another limitation of Newtonian mechanics becomes obvious when we |
| 78 |
< |
try to describe systems with large numbers of particles. It becomes |
| 79 |
< |
very difficult to predict the properties of the system by carrying |
| 80 |
< |
out calculations involving the each individual interaction between |
| 81 |
< |
all the particles, even if we know all of the details of the |
| 82 |
< |
interaction. In order to overcome some of the practical difficulties |
| 83 |
< |
which arise in attempts to apply Newton's equation to complex |
| 84 |
< |
system, alternative procedures may be developed. |
| 75 |
> |
Newtonian Mechanics suffers from two important limitations: motions |
| 76 |
> |
can only be described in cartesian coordinate systems. Moreover, It |
| 77 |
> |
become impossible to predict analytically the properties of the |
| 78 |
> |
system even if we know all of the details of the interaction. In |
| 79 |
> |
order to overcome some of the practical difficulties which arise in |
| 80 |
> |
attempts to apply Newton's equation to complex system, approximate |
| 81 |
> |
numerical procedures may be developed. |
| 82 |
|
|
| 83 |
< |
\subsubsection{\label{introSection:halmiltonPrinciple}Hamilton's |
| 84 |
< |
Principle} |
| 83 |
> |
\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
| 84 |
> |
Principle}} |
| 85 |
|
|
| 86 |
|
Hamilton introduced the dynamical principle upon which it is |
| 87 |
< |
possible to base all of mechanics and, indeed, most of classical |
| 88 |
< |
physics. Hamilton's Principle may be stated as follow, |
| 87 |
> |
possible to base all of mechanics and most of classical physics. |
| 88 |
> |
Hamilton's Principle may be stated as follows, |
| 89 |
|
|
| 90 |
|
The actual trajectory, along which a dynamical system may move from |
| 91 |
|
one point to another within a specified time, is derived by finding |
| 92 |
|
the path which minimizes the time integral of the difference between |
| 93 |
< |
the kinetic, $K$, and potential energies, $U$ \cite{tolman79}. |
| 93 |
> |
the kinetic, $K$, and potential energies, $U$. |
| 94 |
|
\begin{equation} |
| 95 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
| 96 |
|
\label{introEquation:halmitonianPrinciple1} |
| 97 |
|
\end{equation} |
| 98 |
|
|
| 99 |
|
For simple mechanical systems, where the forces acting on the |
| 100 |
< |
different part are derivable from a potential and the velocities are |
| 101 |
< |
small compared with that of light, the Lagrangian function $L$ can |
| 102 |
< |
be define as the difference between the kinetic energy of the system |
| 106 |
< |
and its potential energy, |
| 100 |
> |
different parts are derivable from a potential, the Lagrangian |
| 101 |
> |
function $L$ can be defined as the difference between the kinetic |
| 102 |
> |
energy of the system and its potential energy, |
| 103 |
|
\begin{equation} |
| 104 |
|
L \equiv K - U = L(q_i ,\dot q_i ) , |
| 105 |
|
\label{introEquation:lagrangianDef} |
| 110 |
|
\label{introEquation:halmitonianPrinciple2} |
| 111 |
|
\end{equation} |
| 112 |
|
|
| 113 |
< |
\subsubsection{\label{introSection:equationOfMotionLagrangian}The |
| 114 |
< |
Equations of Motion in Lagrangian Mechanics} |
| 113 |
> |
\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The |
| 114 |
> |
Equations of Motion in Lagrangian Mechanics}} |
| 115 |
|
|
| 116 |
< |
For a holonomic system of $f$ degrees of freedom, the equations of |
| 117 |
< |
motion in the Lagrangian form is |
| 116 |
> |
For a system of $f$ degrees of freedom, the equations of motion in |
| 117 |
> |
the Lagrangian form is |
| 118 |
|
\begin{equation} |
| 119 |
|
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - |
| 120 |
|
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f |
| 128 |
|
Arising from Lagrangian Mechanics, Hamiltonian Mechanics was |
| 129 |
|
introduced by William Rowan Hamilton in 1833 as a re-formulation of |
| 130 |
|
classical mechanics. If the potential energy of a system is |
| 131 |
< |
independent of generalized velocities, the generalized momenta can |
| 136 |
< |
be defined as |
| 131 |
> |
independent of velocities, the momenta can be defined as |
| 132 |
|
\begin{equation} |
| 133 |
|
p_i = \frac{\partial L}{\partial \dot q_i} |
| 134 |
|
\label{introEquation:generalizedMomenta} |
| 167 |
|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
| 168 |
|
find |
| 169 |
|
\begin{equation} |
| 170 |
< |
\frac{{\partial H}}{{\partial p_k }} = q_k |
| 170 |
> |
\frac{{\partial H}}{{\partial p_k }} = \dot {q_k} |
| 171 |
|
\label{introEquation:motionHamiltonianCoordinate} |
| 172 |
|
\end{equation} |
| 173 |
|
\begin{equation} |
| 174 |
< |
\frac{{\partial H}}{{\partial q_k }} = - p_k |
| 174 |
> |
\frac{{\partial H}}{{\partial q_k }} = - \dot {p_k} |
| 175 |
|
\label{introEquation:motionHamiltonianMomentum} |
| 176 |
|
\end{equation} |
| 177 |
|
and |
| 184 |
|
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 185 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
| 186 |
|
equation of motion. Due to their symmetrical formula, they are also |
| 187 |
< |
known as the canonical equations of motions \cite{Goldstein01}. |
| 187 |
> |
known as the canonical equations of motions \cite{Goldstein2001}. |
| 188 |
|
|
| 189 |
|
An important difference between Lagrangian approach and the |
| 190 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
| 191 |
< |
function of the generalized velocities $\dot q_i$ and the |
| 192 |
< |
generalized coordinates $q_i$, while the Hamiltonian is considered |
| 193 |
< |
to be a function of the generalized momenta $p_i$ and the conjugate |
| 194 |
< |
generalized coordinate $q_i$. Hamiltonian Mechanics is more |
| 195 |
< |
appropriate for application to statistical mechanics and quantum |
| 196 |
< |
mechanics, since it treats the coordinate and its time derivative as |
| 197 |
< |
independent variables and it only works with 1st-order differential |
| 203 |
< |
equations\cite{Marion90}. |
| 191 |
> |
function of the generalized velocities $\dot q_i$ and coordinates |
| 192 |
> |
$q_i$, while the Hamiltonian is considered to be a function of the |
| 193 |
> |
generalized momenta $p_i$ and the conjugate coordinates $q_i$. |
| 194 |
> |
Hamiltonian Mechanics is more appropriate for application to |
| 195 |
> |
statistical mechanics and quantum mechanics, since it treats the |
| 196 |
> |
coordinate and its time derivative as independent variables and it |
| 197 |
> |
only works with 1st-order differential equations\cite{Marion1990}. |
| 198 |
|
|
| 199 |
|
In Newtonian Mechanics, a system described by conservative forces |
| 200 |
|
conserves the total energy \ref{introEquation:energyConservation}. |
| 224 |
|
possible states. Each possible state of the system corresponds to |
| 225 |
|
one unique point in the phase space. For mechanical systems, the |
| 226 |
|
phase space usually consists of all possible values of position and |
| 227 |
< |
momentum variables. Consider a dynamic system in a cartesian space, |
| 228 |
< |
where each of the $6f$ coordinates and momenta is assigned to one of |
| 229 |
< |
$6f$ mutually orthogonal axes, the phase space of this system is a |
| 230 |
< |
$6f$ dimensional space. A point, $x = (q_1 , \ldots ,q_f ,p_1 , |
| 231 |
< |
\ldots ,p_f )$, with a unique set of values of $6f$ coordinates and |
| 232 |
< |
momenta is a phase space vector. |
| 227 |
> |
momentum variables. Consider a dynamic system of $f$ particles in a |
| 228 |
> |
cartesian space, where each of the $6f$ coordinates and momenta is |
| 229 |
> |
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
| 230 |
> |
this system is a $6f$ dimensional space. A point, $x = (q_1 , \ldots |
| 231 |
> |
,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$ |
| 232 |
> |
coordinates and momenta is a phase space vector. |
| 233 |
|
|
| 234 |
+ |
%%%fix me |
| 235 |
|
A microscopic state or microstate of a classical system is |
| 236 |
|
specification of the complete phase space vector of a system at any |
| 237 |
|
instant in time. An ensemble is defined as a collection of systems |
| 252 |
|
regions of the phase space. The condition of an ensemble at any time |
| 253 |
|
can be regarded as appropriately specified by the density $\rho$ |
| 254 |
|
with which representative points are distributed over the phase |
| 255 |
< |
space. The density of distribution for an ensemble with $f$ degrees |
| 256 |
< |
of freedom is defined as, |
| 255 |
> |
space. The density distribution for an ensemble with $f$ degrees of |
| 256 |
> |
freedom is defined as, |
| 257 |
|
\begin{equation} |
| 258 |
|
\rho = \rho (q_1 , \ldots ,q_f ,p_1 , \ldots ,p_f ,t). |
| 259 |
|
\label{introEquation:densityDistribution} |
| 260 |
|
\end{equation} |
| 261 |
|
Governed by the principles of mechanics, the phase points change |
| 262 |
< |
their value which would change the density at any time at phase |
| 263 |
< |
space. Hence, the density of distribution is also to be taken as a |
| 262 |
> |
their locations which would change the density at any time at phase |
| 263 |
> |
space. Hence, the density distribution is also to be taken as a |
| 264 |
|
function of the time. |
| 265 |
|
|
| 266 |
|
The number of systems $\delta N$ at time $t$ can be determined by, |
| 268 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
| 269 |
|
\label{introEquation:deltaN} |
| 270 |
|
\end{equation} |
| 271 |
< |
Assuming a large enough population of systems are exploited, we can |
| 272 |
< |
sufficiently approximate $\delta N$ without introducing |
| 273 |
< |
discontinuity when we go from one region in the phase space to |
| 274 |
< |
another. By integrating over the whole phase space, |
| 271 |
> |
Assuming a large enough population of systems, we can sufficiently |
| 272 |
> |
approximate $\delta N$ without introducing discontinuity when we go |
| 273 |
> |
from one region in the phase space to another. By integrating over |
| 274 |
> |
the whole phase space, |
| 275 |
|
\begin{equation} |
| 276 |
|
N = \int { \ldots \int {\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f |
| 277 |
|
\label{introEquation:totalNumberSystem} |
| 283 |
|
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 284 |
|
\label{introEquation:unitProbability} |
| 285 |
|
\end{equation} |
| 286 |
< |
With the help of Equation(\ref{introEquation:unitProbability}) and |
| 287 |
< |
the knowledge of the system, it is possible to calculate the average |
| 286 |
> |
With the help of Eq.~\ref{introEquation:unitProbability} and the |
| 287 |
> |
knowledge of the system, it is possible to calculate the average |
| 288 |
|
value of any desired quantity which depends on the coordinates and |
| 289 |
|
momenta of the system. Even when the dynamics of the real system is |
| 290 |
|
complex, or stochastic, or even discontinuous, the average |
| 291 |
< |
properties of the ensemble of possibilities as a whole may still |
| 292 |
< |
remain well defined. For a classical system in thermal equilibrium |
| 293 |
< |
with its environment, the ensemble average of a mechanical quantity, |
| 294 |
< |
$\langle A(q , p) \rangle_t$, takes the form of an integral over the |
| 295 |
< |
phase space of the system, |
| 291 |
> |
properties of the ensemble of possibilities as a whole remaining |
| 292 |
> |
well defined. For a classical system in thermal equilibrium with its |
| 293 |
> |
environment, the ensemble average of a mechanical quantity, $\langle |
| 294 |
> |
A(q , p) \rangle_t$, takes the form of an integral over the phase |
| 295 |
> |
space of the system, |
| 296 |
|
\begin{equation} |
| 297 |
|
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
| 298 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
| 302 |
|
|
| 303 |
|
There are several different types of ensembles with different |
| 304 |
|
statistical characteristics. As a function of macroscopic |
| 305 |
< |
parameters, such as temperature \textit{etc}, partition function can |
| 306 |
< |
be used to describe the statistical properties of a system in |
| 305 |
> |
parameters, such as temperature \textit{etc}, the partition function |
| 306 |
> |
can be used to describe the statistical properties of a system in |
| 307 |
|
thermodynamic equilibrium. |
| 308 |
|
|
| 309 |
|
As an ensemble of systems, each of which is known to be thermally |
| 310 |
< |
isolated and conserve energy, Microcanonical ensemble(NVE) has a |
| 311 |
< |
partition function like, |
| 310 |
> |
isolated and conserve energy, the Microcanonical ensemble (NVE) has |
| 311 |
> |
a partition function like, |
| 312 |
|
\begin{equation} |
| 313 |
|
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 314 |
|
\end{equation} |
| 315 |
< |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
| 315 |
> |
A canonical ensemble (NVT)is an ensemble of systems, each of which |
| 316 |
|
can share its energy with a large heat reservoir. The distribution |
| 317 |
|
of the total energy amongst the possible dynamical states is given |
| 318 |
|
by the partition function, |
| 321 |
|
\label{introEquation:NVTPartition} |
| 322 |
|
\end{equation} |
| 323 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 324 |
< |
TS$. Since most experiment are carried out under constant pressure |
| 325 |
< |
condition, isothermal-isobaric ensemble(NPT) play a very important |
| 326 |
< |
role in molecular simulation. The isothermal-isobaric ensemble allow |
| 327 |
< |
the system to exchange energy with a heat bath of temperature $T$ |
| 328 |
< |
and to change the volume as well. Its partition function is given as |
| 324 |
> |
TS$. Since most experiments are carried out under constant pressure |
| 325 |
> |
condition, the isothermal-isobaric ensemble (NPT) plays a very |
| 326 |
> |
important role in molecular simulations. The isothermal-isobaric |
| 327 |
> |
ensemble allow the system to exchange energy with a heat bath of |
| 328 |
> |
temperature $T$ and to change the volume as well. Its partition |
| 329 |
> |
function is given as |
| 330 |
|
\begin{equation} |
| 331 |
|
\Delta (N,P,T) = - e^{\beta G}. |
| 332 |
|
\label{introEquation:NPTPartition} |
| 335 |
|
|
| 336 |
|
\subsection{\label{introSection:liouville}Liouville's theorem} |
| 337 |
|
|
| 338 |
< |
The Liouville's theorem is the foundation on which statistical |
| 339 |
< |
mechanics rests. It describes the time evolution of phase space |
| 338 |
> |
Liouville's theorem is the foundation on which statistical mechanics |
| 339 |
> |
rests. It describes the time evolution of the phase space |
| 340 |
|
distribution function. In order to calculate the rate of change of |
| 341 |
< |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
| 342 |
< |
consider the two faces perpendicular to the $q_1$ axis, which are |
| 343 |
< |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
| 344 |
< |
leaving the opposite face is given by the expression, |
| 341 |
> |
$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider |
| 342 |
> |
the two faces perpendicular to the $q_1$ axis, which are located at |
| 343 |
> |
$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the |
| 344 |
> |
opposite face is given by the expression, |
| 345 |
|
\begin{equation} |
| 346 |
|
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
| 347 |
|
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
| 365 |
|
+ \frac{{\partial \dot p_i }}{{\partial p_i }}} \right)} = 0 , |
| 366 |
|
\end{equation} |
| 367 |
|
which cancels the first terms of the right hand side. Furthermore, |
| 368 |
< |
divining $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
| 368 |
> |
dividing $ \delta q_1 \ldots \delta q_f \delta p_1 \ldots \delta |
| 369 |
|
p_f $ in both sides, we can write out Liouville's theorem in a |
| 370 |
|
simple form, |
| 371 |
|
\begin{equation} |
| 377 |
|
|
| 378 |
|
Liouville's theorem states that the distribution function is |
| 379 |
|
constant along any trajectory in phase space. In classical |
| 380 |
< |
statistical mechanics, since the number of particles in the system |
| 381 |
< |
is huge, we may be able to believe the system is stationary, |
| 380 |
> |
statistical mechanics, since the number of members in an ensemble is |
| 381 |
> |
huge and constant, we can assume the local density has no reason |
| 382 |
> |
(other than classical mechanics) to change, |
| 383 |
|
\begin{equation} |
| 384 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
| 385 |
|
\label{introEquation:stationary} |
| 392 |
|
\label{introEquation:densityAndHamiltonian} |
| 393 |
|
\end{equation} |
| 394 |
|
|
| 395 |
< |
\subsubsection{\label{introSection:phaseSpaceConservation}Conservation of Phase Space} |
| 395 |
> |
\subsubsection{\label{introSection:phaseSpaceConservation}\textbf{Conservation of Phase Space}} |
| 396 |
|
Lets consider a region in the phase space, |
| 397 |
|
\begin{equation} |
| 398 |
|
\delta v = \int { \ldots \int {dq_1 } ...dq_f dp_1 } ..dp_f . |
| 399 |
|
\end{equation} |
| 400 |
|
If this region is small enough, the density $\rho$ can be regarded |
| 401 |
< |
as uniform over the whole phase space. Thus, the number of phase |
| 402 |
< |
points inside this region is given by, |
| 401 |
> |
as uniform over the whole integral. Thus, the number of phase points |
| 402 |
> |
inside this region is given by, |
| 403 |
|
\begin{equation} |
| 404 |
|
\delta N = \rho \delta v = \rho \int { \ldots \int {dq_1 } ...dq_f |
| 405 |
|
dp_1 } ..dp_f. |
| 411 |
|
\end{equation} |
| 412 |
|
With the help of stationary assumption |
| 413 |
|
(\ref{introEquation:stationary}), we obtain the principle of the |
| 414 |
< |
\emph{conservation of extension in phase space}, |
| 414 |
> |
\emph{conservation of volume in phase space}, |
| 415 |
|
\begin{equation} |
| 416 |
|
\frac{d}{{dt}}(\delta v) = \frac{d}{{dt}}\int { \ldots \int {dq_1 } |
| 417 |
|
...dq_f dp_1 } ..dp_f = 0. |
| 418 |
|
\label{introEquation:volumePreserving} |
| 419 |
|
\end{equation} |
| 420 |
|
|
| 421 |
< |
\subsubsection{\label{introSection:liouvilleInOtherForms}Liouville's Theorem in Other Forms} |
| 421 |
> |
\subsubsection{\label{introSection:liouvilleInOtherForms}\textbf{Liouville's Theorem in Other Forms}} |
| 422 |
|
|
| 423 |
|
Liouville's theorem can be expresses in a variety of different forms |
| 424 |
|
which are convenient within different contexts. For any two function |
| 432 |
|
\label{introEquation:poissonBracket} |
| 433 |
|
\end{equation} |
| 434 |
|
Substituting equations of motion in Hamiltonian formalism( |
| 435 |
< |
\ref{introEquation:motionHamiltonianCoordinate} , |
| 436 |
< |
\ref{introEquation:motionHamiltonianMomentum} ) into |
| 437 |
< |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
| 438 |
< |
theorem using Poisson bracket notion, |
| 435 |
> |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
| 436 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into |
| 437 |
> |
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
| 438 |
> |
Liouville's theorem using Poisson bracket notion, |
| 439 |
|
\begin{equation} |
| 440 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
| 441 |
|
{\rho ,H} \right\}. |
| 460 |
|
Various thermodynamic properties can be calculated from Molecular |
| 461 |
|
Dynamics simulation. By comparing experimental values with the |
| 462 |
|
calculated properties, one can determine the accuracy of the |
| 463 |
< |
simulation and the quality of the underlying model. However, both of |
| 464 |
< |
experiment and computer simulation are usually performed during a |
| 463 |
> |
simulation and the quality of the underlying model. However, both |
| 464 |
> |
experiments and computer simulations are usually performed during a |
| 465 |
|
certain time interval and the measurements are averaged over a |
| 466 |
|
period of them which is different from the average behavior of |
| 467 |
< |
many-body system in Statistical Mechanics. Fortunately, Ergodic |
| 468 |
< |
Hypothesis is proposed to make a connection between time average and |
| 469 |
< |
ensemble average. It states that time average and average over the |
| 470 |
< |
statistical ensemble are identical \cite{Frenkel1996, leach01:mm}. |
| 467 |
> |
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 468 |
> |
Hypothesis makes a connection between time average and the ensemble |
| 469 |
> |
average. It states that the time average and average over the |
| 470 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
| 471 |
|
\begin{equation} |
| 472 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 473 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 481 |
|
a properly weighted statistical average. This allows the researcher |
| 482 |
|
freedom of choice when deciding how best to measure a given |
| 483 |
|
observable. In case an ensemble averaged approach sounds most |
| 484 |
< |
reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be |
| 484 |
> |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
| 485 |
|
utilized. Or if the system lends itself to a time averaging |
| 486 |
|
approach, the Molecular Dynamics techniques in |
| 487 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 488 |
|
choice\cite{Frenkel1996}. |
| 489 |
|
|
| 490 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 491 |
< |
A variety of numerical integrators were proposed to simulate the |
| 492 |
< |
motions. They usually begin with an initial conditionals and move |
| 493 |
< |
the objects in the direction governed by the differential equations. |
| 494 |
< |
However, most of them ignore the hidden physical law contained |
| 495 |
< |
within the equations. Since 1990, geometric integrators, which |
| 496 |
< |
preserve various phase-flow invariants such as symplectic structure, |
| 497 |
< |
volume and time reversal symmetry, are developed to address this |
| 498 |
< |
issue. The velocity verlet method, which happens to be a simple |
| 499 |
< |
example of symplectic integrator, continues to gain its popularity |
| 500 |
< |
in molecular dynamics community. This fact can be partly explained |
| 501 |
< |
by its geometric nature. |
| 491 |
> |
A variety of numerical integrators have been proposed to simulate |
| 492 |
> |
the motions of atoms in MD simulation. They usually begin with |
| 493 |
> |
initial conditionals and move the objects in the direction governed |
| 494 |
> |
by the differential equations. However, most of them ignore the |
| 495 |
> |
hidden physical laws contained within the equations. Since 1990, |
| 496 |
> |
geometric integrators, which preserve various phase-flow invariants |
| 497 |
> |
such as symplectic structure, volume and time reversal symmetry, are |
| 498 |
> |
developed to address this issue\cite{Dullweber1997, McLachlan1998, |
| 499 |
> |
Leimkuhler1999}. The velocity verlet method, which happens to be a |
| 500 |
> |
simple example of symplectic integrator, continues to gain |
| 501 |
> |
popularity in the molecular dynamics community. This fact can be |
| 502 |
> |
partly explained by its geometric nature. |
| 503 |
|
|
| 504 |
< |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifold} |
| 505 |
< |
A \emph{manifold} is an abstract mathematical space. It locally |
| 506 |
< |
looks like Euclidean space, but when viewed globally, it may have |
| 507 |
< |
more complicate structure. A good example of manifold is the surface |
| 508 |
< |
of Earth. It seems to be flat locally, but it is round if viewed as |
| 509 |
< |
a whole. A \emph{differentiable manifold} (also known as |
| 510 |
< |
\emph{smooth manifold}) is a manifold with an open cover in which |
| 511 |
< |
the covering neighborhoods are all smoothly isomorphic to one |
| 512 |
< |
another. In other words,it is possible to apply calculus on |
| 515 |
< |
\emph{differentiable manifold}. A \emph{symplectic manifold} is |
| 516 |
< |
defined as a pair $(M, \omega)$ which consisting of a |
| 504 |
> |
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 505 |
> |
A \emph{manifold} is an abstract mathematical space. It looks |
| 506 |
> |
locally like Euclidean space, but when viewed globally, it may have |
| 507 |
> |
more complicated structure. A good example of manifold is the |
| 508 |
> |
surface of Earth. It seems to be flat locally, but it is round if |
| 509 |
> |
viewed as a whole. A \emph{differentiable manifold} (also known as |
| 510 |
> |
\emph{smooth manifold}) is a manifold on which it is possible to |
| 511 |
> |
apply calculus on \emph{differentiable manifold}. A \emph{symplectic |
| 512 |
> |
manifold} is defined as a pair $(M, \omega)$ which consists of a |
| 513 |
|
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 514 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 515 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
| 516 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 517 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 518 |
< |
$\omega(x, x) = 0$. Cross product operation in vector field is an |
| 519 |
< |
example of symplectic form. |
| 518 |
> |
$\omega(x, x) = 0$. The cross product operation in vector field is |
| 519 |
> |
an example of symplectic form. |
| 520 |
|
|
| 521 |
< |
One of the motivations to study \emph{symplectic manifold} in |
| 521 |
> |
One of the motivations to study \emph{symplectic manifolds} in |
| 522 |
|
Hamiltonian Mechanics is that a symplectic manifold can represent |
| 523 |
|
all possible configurations of the system and the phase space of the |
| 524 |
|
system can be described by it's cotangent bundle. Every symplectic |
| 525 |
|
manifold is even dimensional. For instance, in Hamilton equations, |
| 526 |
|
coordinate and momentum always appear in pairs. |
| 527 |
|
|
| 532 |
– |
Let $(M,\omega)$ and $(N, \eta)$ be symplectic manifolds. A map |
| 533 |
– |
\[ |
| 534 |
– |
f : M \rightarrow N |
| 535 |
– |
\] |
| 536 |
– |
is a \emph{symplectomorphism} if it is a \emph{diffeomorphims} and |
| 537 |
– |
the \emph{pullback} of $\eta$ under f is equal to $\omega$. |
| 538 |
– |
Canonical transformation is an example of symplectomorphism in |
| 539 |
– |
classical mechanics. |
| 540 |
– |
|
| 528 |
|
\subsection{\label{introSection:ODE}Ordinary Differential Equations} |
| 529 |
|
|
| 530 |
< |
For a ordinary differential system defined as |
| 530 |
> |
For an ordinary differential system defined as |
| 531 |
|
\begin{equation} |
| 532 |
|
\dot x = f(x) |
| 533 |
|
\end{equation} |
| 534 |
< |
where $x = x(q,p)^T$, this system is canonical Hamiltonian, if |
| 534 |
> |
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
| 535 |
|
\begin{equation} |
| 536 |
|
f(r) = J\nabla _x H(r). |
| 537 |
|
\end{equation} |
| 552 |
|
\end{equation}In this case, $f$ is |
| 553 |
|
called a \emph{Hamiltonian vector field}. |
| 554 |
|
|
| 555 |
< |
Another generalization of Hamiltonian dynamics is Poisson Dynamics, |
| 555 |
> |
Another generalization of Hamiltonian dynamics is Poisson |
| 556 |
> |
Dynamics\cite{Olver1986}, |
| 557 |
|
\begin{equation} |
| 558 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 559 |
|
\end{equation} |
| 600 |
|
|
| 601 |
|
\subsection{\label{introSection:geometricProperties}Geometric Properties} |
| 602 |
|
|
| 603 |
< |
The hidden geometric properties of ODE and its flow play important |
| 604 |
< |
roles in numerical studies. Many of them can be found in systems |
| 605 |
< |
which occur naturally in applications. |
| 603 |
> |
The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE |
| 604 |
> |
and its flow play important roles in numerical studies. Many of them |
| 605 |
> |
can be found in systems which occur naturally in applications. |
| 606 |
|
|
| 607 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 608 |
|
a \emph{symplectic} flow if it satisfies, |
| 646 |
|
which is the condition for conserving \emph{first integral}. For a |
| 647 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
| 648 |
|
smooth function $G$ is given by, |
| 649 |
< |
\begin{equation} |
| 650 |
< |
\begin{array}{c} |
| 651 |
< |
\frac{{dG(x(t))}}{{dt}} = [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 652 |
< |
= [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 665 |
< |
\end{array} |
| 649 |
> |
|
| 650 |
> |
\begin{eqnarray} |
| 651 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 652 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 653 |
|
\label{introEquation:firstIntegral1} |
| 654 |
< |
\end{equation} |
| 654 |
> |
\end{eqnarray} |
| 655 |
> |
|
| 656 |
> |
|
| 657 |
|
Using poisson bracket notion, Equation |
| 658 |
|
\ref{introEquation:firstIntegral1} can be rewritten as |
| 659 |
|
\[ |
| 668 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 669 |
|
0$. |
| 670 |
|
|
| 671 |
< |
|
| 683 |
< |
When designing any numerical methods, one should always try to |
| 671 |
> |
When designing any numerical methods, one should always try to |
| 672 |
|
preserve the structural properties of the original ODE and its flow. |
| 673 |
|
|
| 674 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 675 |
|
A lot of well established and very effective numerical methods have |
| 676 |
|
been successful precisely because of their symplecticities even |
| 677 |
|
though this fact was not recognized when they were first |
| 678 |
< |
constructed. The most famous example is leapfrog methods in |
| 679 |
< |
molecular dynamics. In general, symplectic integrators can be |
| 678 |
> |
constructed. The most famous example is the Verlet-leapfrog methods |
| 679 |
> |
in molecular dynamics. In general, symplectic integrators can be |
| 680 |
|
constructed using one of four different methods. |
| 681 |
|
\begin{enumerate} |
| 682 |
|
\item Generating functions |
| 685 |
|
\item Splitting methods |
| 686 |
|
\end{enumerate} |
| 687 |
|
|
| 688 |
< |
Generating function tends to lead to methods which are cumbersome |
| 689 |
< |
and difficult to use. In dissipative systems, variational methods |
| 690 |
< |
can capture the decay of energy accurately. Since their |
| 691 |
< |
geometrically unstable nature against non-Hamiltonian perturbations, |
| 692 |
< |
ordinary implicit Runge-Kutta methods are not suitable for |
| 693 |
< |
Hamiltonian system. Recently, various high-order explicit |
| 694 |
< |
Runge--Kutta methods have been developed to overcome this |
| 688 |
> |
Generating function\cite{Channell1990} tends to lead to methods |
| 689 |
> |
which are cumbersome and difficult to use. In dissipative systems, |
| 690 |
> |
variational methods can capture the decay of energy |
| 691 |
> |
accurately\cite{Kane2000}. Since their geometrically unstable nature |
| 692 |
> |
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 693 |
> |
methods are not suitable for Hamiltonian system. Recently, various |
| 694 |
> |
high-order explicit Runge-Kutta methods |
| 695 |
> |
\cite{Owren1992,Chen2003}have been developed to overcome this |
| 696 |
|
instability. However, due to computational penalty involved in |
| 697 |
< |
implementing the Runge-Kutta methods, they do not attract too much |
| 698 |
< |
attention from Molecular Dynamics community. Instead, splitting have |
| 699 |
< |
been widely accepted since they exploit natural decompositions of |
| 700 |
< |
the system\cite{Tuckerman92}. |
| 697 |
> |
implementing the Runge-Kutta methods, they have not attracted much |
| 698 |
> |
attention from the Molecular Dynamics community. Instead, splitting |
| 699 |
> |
methods have been widely accepted since they exploit natural |
| 700 |
> |
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
| 701 |
|
|
| 702 |
< |
\subsubsection{\label{introSection:splittingMethod}Splitting Method} |
| 702 |
> |
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 703 |
|
|
| 704 |
|
The main idea behind splitting methods is to decompose the discrete |
| 705 |
|
$\varphi_h$ as a composition of simpler flows, |
| 720 |
|
energy respectively, which is a natural decomposition of the |
| 721 |
|
problem. If $H_1$ and $H_2$ can be integrated using exact flows |
| 722 |
|
$\varphi_1(t)$ and $\varphi_2(t)$, respectively, a simple first |
| 723 |
< |
order is then given by the Lie-Trotter formula |
| 723 |
> |
order expression is then given by the Lie-Trotter formula |
| 724 |
|
\begin{equation} |
| 725 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 726 |
|
\label{introEquation:firstOrderSplitting} |
| 746 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 747 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 748 |
|
\end{equation} |
| 749 |
< |
which has a local error proportional to $h^3$. Sprang splitting's |
| 750 |
< |
popularity in molecular simulation community attribute to its |
| 751 |
< |
symmetric property, |
| 749 |
> |
which has a local error proportional to $h^3$. The Sprang |
| 750 |
> |
splitting's popularity in molecular simulation community attribute |
| 751 |
> |
to its symmetric property, |
| 752 |
|
\begin{equation} |
| 753 |
|
\varphi _h^{ - 1} = \varphi _{ - h}. |
| 754 |
|
\label{introEquation:timeReversible} |
| 755 |
< |
\end{equation} |
| 755 |
> |
\end{equation},appendixFig:architecture |
| 756 |
|
|
| 757 |
< |
\subsubsection{\label{introSection:exampleSplittingMethod}Example of Splitting Method} |
| 757 |
> |
\subsubsection{\label{introSection:exampleSplittingMethod}\textbf{Example of Splitting Method}} |
| 758 |
|
The classical equation for a system consisting of interacting |
| 759 |
|
particles can be written in Hamiltonian form, |
| 760 |
|
\[ |
| 814 |
|
\label{introEquation:positionVerlet2} |
| 815 |
|
\end{align} |
| 816 |
|
|
| 817 |
< |
\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods} |
| 817 |
> |
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 818 |
|
|
| 819 |
|
Baker-Campbell-Hausdorff formula can be used to determine the local |
| 820 |
|
error of splitting method in terms of commutator of the |
| 833 |
|
\[ |
| 834 |
|
[X,Y] = XY - YX . |
| 835 |
|
\] |
| 836 |
< |
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 837 |
< |
can obtain |
| 838 |
< |
\begin{equation} |
| 839 |
< |
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 |
| 840 |
< |
[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 841 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} + |
| 842 |
< |
\ldots ) |
| 854 |
< |
\end{equation} |
| 836 |
> |
Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to |
| 837 |
> |
Sprang splitting, we can obtain |
| 838 |
> |
\begin{eqnarray*} |
| 839 |
> |
\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 \\ |
| 840 |
> |
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 841 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
| 842 |
> |
\end{eqnarray*} |
| 843 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 844 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 845 |
|
procedure can be applied to general splitting, of the form |
| 847 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 848 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 849 |
|
\end{equation} |
| 850 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
| 850 |
> |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 851 |
|
order method. Yoshida proposed an elegant way to compose higher |
| 852 |
< |
order methods based on symmetric splitting. Given a symmetric second |
| 853 |
< |
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
| 854 |
< |
method can be constructed by composing, |
| 852 |
> |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 853 |
> |
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 854 |
> |
fourth-order symmetric method can be constructed by composing, |
| 855 |
|
\[ |
| 856 |
|
\varphi _h^{(4)} = \varphi _{\alpha h}^{(2)} \circ \varphi _{\beta |
| 857 |
|
h}^{(2)} \circ \varphi _{\alpha h}^{(2)} |
| 900 |
|
\end{enumerate} |
| 901 |
|
These three individual steps will be covered in the following |
| 902 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 903 |
< |
initialization of a simulation. Sec.~\ref{introSec:production} will |
| 904 |
< |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
| 905 |
< |
provides the theoretical tools for trajectory analysis. |
| 903 |
> |
initialization of a simulation. Sec.~\ref{introSection:production} |
| 904 |
> |
will discusses issues in production run. |
| 905 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 906 |
> |
trajectory analysis. |
| 907 |
|
|
| 908 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 909 |
|
|
| 910 |
< |
\subsubsection{Preliminary preparation} |
| 910 |
> |
\subsubsection{\textbf{Preliminary preparation}} |
| 911 |
|
|
| 912 |
|
When selecting the starting structure of a molecule for molecular |
| 913 |
|
simulation, one may retrieve its Cartesian coordinates from public |
| 925 |
|
instead of placing lipids randomly in solvent, since we are not |
| 926 |
|
interested in self-aggregation and it takes a long time to happen. |
| 927 |
|
|
| 928 |
< |
\subsubsection{Minimization} |
| 928 |
> |
\subsubsection{\textbf{Minimization}} |
| 929 |
|
|
| 930 |
|
It is quite possible that some of molecules in the system from |
| 931 |
|
preliminary preparation may be overlapped with each other. This |
| 947 |
|
matrix and insufficient storage capacity to store them, most |
| 948 |
|
Newton-Raphson methods can not be used with very large models. |
| 949 |
|
|
| 950 |
< |
\subsubsection{Heating} |
| 950 |
> |
\subsubsection{\textbf{Heating}} |
| 951 |
|
|
| 952 |
|
Typically, Heating is performed by assigning random velocities |
| 953 |
|
according to a Gaussian distribution for a temperature. Beginning at |
| 959 |
|
net linear momentum and angular momentum of the system should be |
| 960 |
|
shifted to zero. |
| 961 |
|
|
| 962 |
< |
\subsubsection{Equilibration} |
| 962 |
> |
\subsubsection{\textbf{Equilibration}} |
| 963 |
|
|
| 964 |
|
The purpose of equilibration is to allow the system to evolve |
| 965 |
|
spontaneously for a period of time and reach equilibrium. The |
| 973 |
|
|
| 974 |
|
\subsection{\label{introSection:production}Production} |
| 975 |
|
|
| 976 |
< |
Production run is the most important steps of the simulation, in |
| 976 |
> |
Production run is the most important step of the simulation, in |
| 977 |
|
which the equilibrated structure is used as a starting point and the |
| 978 |
|
motions of the molecules are collected for later analysis. In order |
| 979 |
|
to capture the macroscopic properties of the system, the molecular |
| 989 |
|
A natural approach to avoid system size issue is to represent the |
| 990 |
|
bulk behavior by a finite number of the particles. However, this |
| 991 |
|
approach will suffer from the surface effect. To offset this, |
| 992 |
< |
\textit{Periodic boundary condition} is developed to simulate bulk |
| 993 |
< |
properties with a relatively small number of particles. In this |
| 994 |
< |
method, the simulation box is replicated throughout space to form an |
| 995 |
< |
infinite lattice. During the simulation, when a particle moves in |
| 996 |
< |
the primary cell, its image in other cells move in exactly the same |
| 997 |
< |
direction with exactly the same orientation. Thus, as a particle |
| 998 |
< |
leaves the primary cell, one of its images will enter through the |
| 999 |
< |
opposite face. |
| 1000 |
< |
%\begin{figure} |
| 1001 |
< |
%\centering |
| 1002 |
< |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
| 1003 |
< |
%\caption[An illustration of periodic boundary conditions]{A 2-D |
| 1004 |
< |
%illustration of periodic boundary conditions. As one particle leaves |
| 1005 |
< |
%the right of the simulation box, an image of it enters the left.} |
| 1006 |
< |
%\label{introFig:pbc} |
| 1007 |
< |
%\end{figure} |
| 992 |
> |
\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc}) |
| 993 |
> |
is developed to simulate bulk properties with a relatively small |
| 994 |
> |
number of particles. In this method, the simulation box is |
| 995 |
> |
replicated throughout space to form an infinite lattice. During the |
| 996 |
> |
simulation, when a particle moves in the primary cell, its image in |
| 997 |
> |
other cells move in exactly the same direction with exactly the same |
| 998 |
> |
orientation. Thus, as a particle leaves the primary cell, one of its |
| 999 |
> |
images will enter through the opposite face. |
| 1000 |
> |
\begin{figure} |
| 1001 |
> |
\centering |
| 1002 |
> |
\includegraphics[width=\linewidth]{pbc.eps} |
| 1003 |
> |
\caption[An illustration of periodic boundary conditions]{A 2-D |
| 1004 |
> |
illustration of periodic boundary conditions. As one particle leaves |
| 1005 |
> |
the left of the simulation box, an image of it enters the right.} |
| 1006 |
> |
\label{introFig:pbc} |
| 1007 |
> |
\end{figure} |
| 1008 |
|
|
| 1009 |
|
%cutoff and minimum image convention |
| 1010 |
|
Another important technique to improve the efficiency of force |
| 1022 |
|
reciprocal sums with rapid and absolute convergence, has proved to |
| 1023 |
|
minimize the periodicity artifacts in liquid simulations. Taking the |
| 1024 |
|
advantages of the fast Fourier transform (FFT) for calculating |
| 1025 |
< |
discrete Fourier transforms, the particle mesh-based methods are |
| 1026 |
< |
accelerated from $O(N^{3/2})$ to $O(N logN)$. An alternative |
| 1027 |
< |
approach is \emph{fast multipole method}, which treats Coulombic |
| 1028 |
< |
interaction exactly at short range, and approximate the potential at |
| 1029 |
< |
long range through multipolar expansion. In spite of their wide |
| 1030 |
< |
acceptances at the molecular simulation community, these two methods |
| 1031 |
< |
are hard to be implemented correctly and efficiently. Instead, we |
| 1032 |
< |
use a damped and charge-neutralized Coulomb potential method |
| 1033 |
< |
developed by Wolf and his coworkers. The shifted Coulomb potential |
| 1034 |
< |
for particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 1025 |
> |
discrete Fourier transforms, the particle mesh-based |
| 1026 |
> |
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 1027 |
> |
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast |
| 1028 |
> |
multipole method}\cite{Greengard1987, Greengard1994}, which treats |
| 1029 |
> |
Coulombic interaction exactly at short range, and approximate the |
| 1030 |
> |
potential at long range through multipolar expansion. In spite of |
| 1031 |
> |
their wide acceptances at the molecular simulation community, these |
| 1032 |
> |
two methods are hard to be implemented correctly and efficiently. |
| 1033 |
> |
Instead, we use a damped and charge-neutralized Coulomb potential |
| 1034 |
> |
method developed by Wolf and his coworkers\cite{Wolf1999}. The |
| 1035 |
> |
shifted Coulomb potential for particle $i$ and particle $j$ at |
| 1036 |
> |
distance $r_{rj}$ is given by: |
| 1037 |
|
\begin{equation} |
| 1038 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 1039 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 1043 |
|
where $\alpha$ is the convergence parameter. Due to the lack of |
| 1044 |
|
inherent periodicity and rapid convergence,this method is extremely |
| 1045 |
|
efficient and easy to implement. |
| 1046 |
< |
%\begin{figure} |
| 1047 |
< |
%\centering |
| 1048 |
< |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
| 1049 |
< |
%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} |
| 1050 |
< |
%\label{introFigure:shiftedCoulomb} |
| 1051 |
< |
%\end{figure} |
| 1046 |
> |
\begin{figure} |
| 1047 |
> |
\centering |
| 1048 |
> |
\includegraphics[width=\linewidth]{shifted_coulomb.eps} |
| 1049 |
> |
\caption[An illustration of shifted Coulomb potential]{An |
| 1050 |
> |
illustration of shifted Coulomb potential.} |
| 1051 |
> |
\label{introFigure:shiftedCoulomb} |
| 1052 |
> |
\end{figure} |
| 1053 |
|
|
| 1054 |
|
%multiple time step |
| 1055 |
|
|
| 1065 |
|
parameters, and investigate time-dependent processes of the molecule |
| 1066 |
|
from the trajectories. |
| 1067 |
|
|
| 1068 |
< |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
| 1068 |
> |
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamics Properties}} |
| 1069 |
|
|
| 1070 |
|
Thermodynamics properties, which can be expressed in terms of some |
| 1071 |
|
function of the coordinates and momenta of all particles in the |
| 1087 |
|
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
| 1088 |
|
\end{equation} |
| 1089 |
|
|
| 1090 |
< |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
| 1090 |
> |
\subsubsection{\label{introSection:structuralProperties}\textbf{Structural Properties}} |
| 1091 |
|
|
| 1092 |
|
Structural Properties of a simple fluid can be described by a set of |
| 1093 |
|
distribution functions. Among these functions,\emph{pair |
| 1095 |
|
function}, is of most fundamental importance to liquid-state theory. |
| 1096 |
|
Pair distribution function can be gathered by Fourier transforming |
| 1097 |
|
raw data from a series of neutron diffraction experiments and |
| 1098 |
< |
integrating over the surface factor \cite{Powles73}. The experiment |
| 1099 |
< |
result can serve as a criterion to justify the correctness of the |
| 1100 |
< |
theory. Moreover, various equilibrium thermodynamic and structural |
| 1101 |
< |
properties can also be expressed in terms of radial distribution |
| 1102 |
< |
function \cite{allen87:csl}. |
| 1098 |
> |
integrating over the surface factor \cite{Powles1973}. The |
| 1099 |
> |
experiment result can serve as a criterion to justify the |
| 1100 |
> |
correctness of the theory. Moreover, various equilibrium |
| 1101 |
> |
thermodynamic and structural properties can also be expressed in |
| 1102 |
> |
terms of radial distribution function \cite{Allen1987}. |
| 1103 |
|
|
| 1104 |
|
A pair distribution functions $g(r)$ gives the probability that a |
| 1105 |
|
particle $i$ will be located at a distance $r$ from a another |
| 1127 |
|
%\label{introFigure:pairDistributionFunction} |
| 1128 |
|
%\end{figure} |
| 1129 |
|
|
| 1130 |
< |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
| 1131 |
< |
Properties} |
| 1130 |
> |
\subsubsection{\label{introSection:timeDependentProperties}\textbf{Time-dependent |
| 1131 |
> |
Properties}} |
| 1132 |
|
|
| 1133 |
|
Time-dependent properties are usually calculated using \emph{time |
| 1134 |
|
correlation function}, which correlates random variables $A$ and $B$ |
| 1177 |
|
movement of the objects in 3D gaming engine or other physics |
| 1178 |
|
simulator is governed by the rigid body dynamics. In molecular |
| 1179 |
|
simulation, rigid body is used to simplify the model in |
| 1180 |
< |
protein-protein docking study{\cite{Gray03}}. |
| 1180 |
> |
protein-protein docking study\cite{Gray2003}. |
| 1181 |
|
|
| 1182 |
|
It is very important to develop stable and efficient methods to |
| 1183 |
|
integrate the equations of motion of orientational degrees of |
| 1185 |
|
rotational degrees of freedom. However, due to its singularity, the |
| 1186 |
|
numerical integration of corresponding equations of motion is very |
| 1187 |
|
inefficient and inaccurate. Although an alternative integrator using |
| 1188 |
< |
different sets of Euler angles can overcome this difficulty\cite{}, |
| 1189 |
< |
the computational penalty and the lost of angular momentum |
| 1190 |
< |
conservation still remain. A singularity free representation |
| 1191 |
< |
utilizing quaternions was developed by Evans in 1977. Unfortunately, |
| 1192 |
< |
this approach suffer from the nonseparable Hamiltonian resulted from |
| 1193 |
< |
quaternion representation, which prevents the symplectic algorithm |
| 1194 |
< |
to be utilized. Another different approach is to apply holonomic |
| 1195 |
< |
constraints to the atoms belonging to the rigid body. Each atom |
| 1196 |
< |
moves independently under the normal forces deriving from potential |
| 1197 |
< |
energy and constraint forces which are used to guarantee the |
| 1198 |
< |
rigidness. However, due to their iterative nature, SHAKE and Rattle |
| 1199 |
< |
algorithm converge very slowly when the number of constraint |
| 1200 |
< |
increases. |
| 1188 |
> |
different sets of Euler angles can overcome this |
| 1189 |
> |
difficulty\cite{Barojas1973}, the computational penalty and the lost |
| 1190 |
> |
of angular momentum conservation still remain. A singularity free |
| 1191 |
> |
representation utilizing quaternions was developed by Evans in |
| 1192 |
> |
1977\cite{Evans1977}. Unfortunately, this approach suffer from the |
| 1193 |
> |
nonseparable Hamiltonian resulted from quaternion representation, |
| 1194 |
> |
which prevents the symplectic algorithm to be utilized. Another |
| 1195 |
> |
different approach is to apply holonomic constraints to the atoms |
| 1196 |
> |
belonging to the rigid body. Each atom moves independently under the |
| 1197 |
> |
normal forces deriving from potential energy and constraint forces |
| 1198 |
> |
which are used to guarantee the rigidness. However, due to their |
| 1199 |
> |
iterative nature, SHAKE and Rattle algorithm converge very slowly |
| 1200 |
> |
when the number of constraint increases\cite{Ryckaert1977, |
| 1201 |
> |
Andersen1983}. |
| 1202 |
|
|
| 1203 |
|
The break through in geometric literature suggests that, in order to |
| 1204 |
|
develop a long-term integration scheme, one should preserve the |
| 1205 |
|
symplectic structure of the flow. Introducing conjugate momentum to |
| 1206 |
|
rotation matrix $Q$ and re-formulating Hamiltonian's equation, a |
| 1207 |
< |
symplectic integrator, RSHAKE, was proposed to evolve the |
| 1208 |
< |
Hamiltonian system in a constraint manifold by iteratively |
| 1207 |
> |
symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve |
| 1208 |
> |
the Hamiltonian system in a constraint manifold by iteratively |
| 1209 |
|
satisfying the orthogonality constraint $Q_T Q = 1$. An alternative |
| 1210 |
< |
method using quaternion representation was developed by Omelyan. |
| 1211 |
< |
However, both of these methods are iterative and inefficient. In |
| 1212 |
< |
this section, we will present a symplectic Lie-Poisson integrator |
| 1213 |
< |
for rigid body developed by Dullweber and his |
| 1214 |
< |
coworkers\cite{Dullweber1997} in depth. |
| 1210 |
> |
method using quaternion representation was developed by |
| 1211 |
> |
Omelyan\cite{Omelyan1998}. However, both of these methods are |
| 1212 |
> |
iterative and inefficient. In this section, we will present a |
| 1213 |
> |
symplectic Lie-Poisson integrator for rigid body developed by |
| 1214 |
> |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1215 |
|
|
| 1216 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body} |
| 1217 |
|
The motion of the rigid body is Hamiltonian with the Hamiltonian |
| 1243 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1244 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1245 |
|
the equations of motion, |
| 1246 |
< |
\[ |
| 1247 |
< |
\begin{array}{c} |
| 1248 |
< |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1249 |
< |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1250 |
< |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1251 |
< |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 1252 |
< |
\end{array} |
| 1260 |
< |
\] |
| 1246 |
> |
|
| 1247 |
> |
\begin{eqnarray} |
| 1248 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1249 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1250 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1251 |
> |
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1252 |
> |
\end{eqnarray} |
| 1253 |
|
|
| 1254 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1255 |
|
We can use constraint force provided by lagrange multiplier on the |
| 1330 |
|
\[ |
| 1331 |
|
\hat vu = v \times u |
| 1332 |
|
\] |
| 1341 |
– |
|
| 1333 |
|
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1334 |
|
matrix, |
| 1335 |
|
\begin{equation} |
| 1336 |
< |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1336 |
> |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
| 1337 |
|
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1338 |
|
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1339 |
|
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1342 |
|
\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange |
| 1343 |
|
multiplier $\Lambda$ is absent from the equations of motion. This |
| 1344 |
|
unique property eliminate the requirement of iterations which can |
| 1345 |
< |
not be avoided in other methods\cite{}. |
| 1345 |
> |
not be avoided in other methods\cite{Kol1997, Omelyan1998}. |
| 1346 |
|
|
| 1347 |
|
Applying hat-map isomorphism, we obtain the equation of motion for |
| 1348 |
|
angular momentum on body frame |
| 1362 |
|
|
| 1363 |
|
If there is not external forces exerted on the rigid body, the only |
| 1364 |
|
contribution to the rotational is from the kinetic potential (the |
| 1365 |
< |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
| 1366 |
< |
rigid body is an example of Lie-Poisson system with Hamiltonian |
| 1376 |
< |
function |
| 1365 |
> |
first term of \ref{introEquation:bodyAngularMotion}). The free rigid |
| 1366 |
> |
body is an example of Lie-Poisson system with Hamiltonian function |
| 1367 |
|
\begin{equation} |
| 1368 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1369 |
|
\label{introEquation:rotationalKineticRB} |
| 1608 |
|
\] |
| 1609 |
|
where $p$ is real and $L$ is called the Laplace Transform |
| 1610 |
|
Operator. Below are some important properties of Laplace transform |
| 1611 |
< |
\begin{equation} |
| 1612 |
< |
\begin{array}{c} |
| 1613 |
< |
L(x + y) = L(x) + L(y) \\ |
| 1614 |
< |
L(ax) = aL(x) \\ |
| 1615 |
< |
L(\dot x) = pL(x) - px(0) \\ |
| 1616 |
< |
L(\ddot x) = p^2 L(x) - px(0) - \dot x(0) \\ |
| 1617 |
< |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p) \\ |
| 1618 |
< |
\end{array} |
| 1629 |
< |
\end{equation} |
| 1611 |
> |
|
| 1612 |
> |
\begin{eqnarray*} |
| 1613 |
> |
L(x + y) & = & L(x) + L(y) \\ |
| 1614 |
> |
L(ax) & = & aL(x) \\ |
| 1615 |
> |
L(\dot x) & = & pL(x) - px(0) \\ |
| 1616 |
> |
L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\ |
| 1617 |
> |
L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\ |
| 1618 |
> |
\end{eqnarray*} |
| 1619 |
|
|
| 1620 |
+ |
|
| 1621 |
|
Applying Laplace transform to the bath coordinates, we obtain |
| 1622 |
< |
\[ |
| 1623 |
< |
\begin{array}{c} |
| 1624 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) = - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
| 1625 |
< |
L(x_\alpha ) = \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1626 |
< |
\end{array} |
| 1637 |
< |
\] |
| 1622 |
> |
\begin{eqnarray*} |
| 1623 |
> |
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) \\ |
| 1624 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
| 1625 |
> |
\end{eqnarray*} |
| 1626 |
> |
|
| 1627 |
|
By the same way, the system coordinates become |
| 1628 |
< |
\[ |
| 1629 |
< |
\begin{array}{c} |
| 1630 |
< |
mL(\ddot x) = - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1631 |
< |
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
| 1643 |
< |
\end{array} |
| 1644 |
< |
\] |
| 1628 |
> |
\begin{eqnarray*} |
| 1629 |
> |
mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\ |
| 1630 |
> |
& & \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\}} \\ |
| 1631 |
> |
\end{eqnarray*} |
| 1632 |
|
|
| 1633 |
|
With the help of some relatively important inverse Laplace |
| 1634 |
|
transformations: |
| 1640 |
|
\end{array} |
| 1641 |
|
\] |
| 1642 |
|
, we obtain |
| 1643 |
< |
\begin{align} |
| 1644 |
< |
m\ddot x &= - \frac{{\partial W(x)}}{{\partial x}} - |
| 1643 |
> |
\begin{eqnarray*} |
| 1644 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
| 1645 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1646 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
| 1647 |
< |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
| 1648 |
< |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
| 1649 |
< |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 1650 |
< |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
| 1651 |
< |
% |
| 1652 |
< |
&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1647 |
> |
_\alpha t)\dot x(t - \tau )d\tau } } \right\}} \\ |
| 1648 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1649 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1650 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1651 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1652 |
> |
\end{eqnarray*} |
| 1653 |
> |
\begin{eqnarray*} |
| 1654 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1655 |
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1656 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 1657 |
< |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
| 1658 |
< |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
| 1659 |
< |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
| 1660 |
< |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
| 1661 |
< |
(\omega _\alpha t)} \right\}} |
| 1662 |
< |
\end{align} |
| 1674 |
< |
|
| 1657 |
> |
t)\dot x(t - \tau )d} \tau } \\ |
| 1658 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1659 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1660 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1661 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1662 |
> |
\end{eqnarray*} |
| 1663 |
|
Introducing a \emph{dynamic friction kernel} |
| 1664 |
|
\begin{equation} |
| 1665 |
|
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1682 |
|
\end{equation} |
| 1683 |
|
which is known as the \emph{generalized Langevin equation}. |
| 1684 |
|
|
| 1685 |
< |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel} |
| 1685 |
> |
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
| 1686 |
|
|
| 1687 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
| 1688 |
|
implies it is completely deterministic within the context of a |
| 1737 |
|
\end{equation} |
| 1738 |
|
which is known as the Langevin equation. The static friction |
| 1739 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1740 |
< |
or be determined by Stokes' law for regular shaped particles.A |
| 1740 |
> |
or be determined by Stokes' law for regular shaped particles. A |
| 1741 |
|
briefly review on calculating friction tensor for arbitrary shaped |
| 1742 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1743 |
|
|
| 1744 |
< |
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 1744 |
> |
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1745 |
|
|
| 1746 |
|
Defining a new set of coordinates, |
| 1747 |
|
\[ |
| 1753 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1754 |
|
\] |
| 1755 |
|
And since the $q$ coordinates are harmonic oscillators, |
| 1756 |
< |
\[ |
| 1757 |
< |
\begin{array}{c} |
| 1758 |
< |
\left\langle {q_\alpha ^2 } \right\rangle = \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1759 |
< |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1760 |
< |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle = \delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1761 |
< |
\left\langle {R(t)R(0)} \right\rangle = \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
| 1762 |
< |
= \sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1763 |
< |
= kT\xi (t) \\ |
| 1764 |
< |
\end{array} |
| 1765 |
< |
\] |
| 1756 |
> |
|
| 1757 |
> |
\begin{eqnarray*} |
| 1758 |
> |
\left\langle {q_\alpha ^2 } \right\rangle & = & \frac{{kT}}{{m_\alpha \omega _\alpha ^2 }} \\ |
| 1759 |
> |
\left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t) \\ |
| 1760 |
> |
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1761 |
> |
\left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha {\sum\limits_\beta {g_\alpha g_\beta \left\langle {q_\alpha (t)q_\beta (0)} \right\rangle } } \\ |
| 1762 |
> |
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1763 |
> |
& = &kT\xi (t) \\ |
| 1764 |
> |
\end{eqnarray*} |
| 1765 |
> |
|
| 1766 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1767 |
|
\begin{equation} |
| 1768 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
| 1770 |
|
\end{equation} |
| 1771 |
|
In effect, it acts as a constraint on the possible ways in which one |
| 1772 |
|
can model the random force and friction kernel. |
| 1785 |
– |
|
| 1786 |
– |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1787 |
– |
Theoretically, the friction kernel can be determined using velocity |
| 1788 |
– |
autocorrelation function. However, this approach become impractical |
| 1789 |
– |
when the system become more and more complicate. Instead, various |
| 1790 |
– |
approaches based on hydrodynamics have been developed to calculate |
| 1791 |
– |
the friction coefficients. The friction effect is isotropic in |
| 1792 |
– |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 1793 |
– |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 1794 |
– |
\[ |
| 1795 |
– |
\Xi = \left( {\begin{array}{*{20}c} |
| 1796 |
– |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1797 |
– |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 1798 |
– |
\end{array}} \right). |
| 1799 |
– |
\] |
| 1800 |
– |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 1801 |
– |
tensor and rotational resistance (friction) tensor respectively, |
| 1802 |
– |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 1803 |
– |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 1804 |
– |
particle moves in a fluid, it may experience friction force or |
| 1805 |
– |
torque along the opposite direction of the velocity or angular |
| 1806 |
– |
velocity, |
| 1807 |
– |
\[ |
| 1808 |
– |
\left( \begin{array}{l} |
| 1809 |
– |
F_R \\ |
| 1810 |
– |
\tau _R \\ |
| 1811 |
– |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1812 |
– |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1813 |
– |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1814 |
– |
\end{array}} \right)\left( \begin{array}{l} |
| 1815 |
– |
v \\ |
| 1816 |
– |
w \\ |
| 1817 |
– |
\end{array} \right) |
| 1818 |
– |
\] |
| 1819 |
– |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1820 |
– |
toque. |
| 1821 |
– |
|
| 1822 |
– |
\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape} |
| 1823 |
– |
|
| 1824 |
– |
For a spherical particle, the translational and rotational friction |
| 1825 |
– |
constant can be calculated from Stoke's law, |
| 1826 |
– |
\[ |
| 1827 |
– |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1828 |
– |
{6\pi \eta R} & 0 & 0 \\ |
| 1829 |
– |
0 & {6\pi \eta R} & 0 \\ |
| 1830 |
– |
0 & 0 & {6\pi \eta R} \\ |
| 1831 |
– |
\end{array}} \right) |
| 1832 |
– |
\] |
| 1833 |
– |
and |
| 1834 |
– |
\[ |
| 1835 |
– |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1836 |
– |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1837 |
– |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1838 |
– |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1839 |
– |
\end{array}} \right) |
| 1840 |
– |
\] |
| 1841 |
– |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1842 |
– |
hydrodynamics radius. |
| 1843 |
– |
|
| 1844 |
– |
Other non-spherical shape, such as cylinder and ellipsoid |
| 1845 |
– |
\textit{etc}, are widely used as reference for developing new |
| 1846 |
– |
hydrodynamics theory, because their properties can be calculated |
| 1847 |
– |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 1848 |
– |
also called a triaxial ellipsoid, which is given in Cartesian |
| 1849 |
– |
coordinates by |
| 1850 |
– |
\[ |
| 1851 |
– |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 1852 |
– |
}} = 1 |
| 1853 |
– |
\] |
| 1854 |
– |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 1855 |
– |
due to the complexity of the elliptic integral, only the ellipsoid |
| 1856 |
– |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 1857 |
– |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 1858 |
– |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 1859 |
– |
\[ |
| 1860 |
– |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1861 |
– |
} }}{b}, |
| 1862 |
– |
\] |
| 1863 |
– |
and oblate, |
| 1864 |
– |
\[ |
| 1865 |
– |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1866 |
– |
}}{a} |
| 1867 |
– |
\], |
| 1868 |
– |
one can write down the translational and rotational resistance |
| 1869 |
– |
tensors |
| 1870 |
– |
\[ |
| 1871 |
– |
\begin{array}{l} |
| 1872 |
– |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1873 |
– |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1874 |
– |
\end{array}, |
| 1875 |
– |
\] |
| 1876 |
– |
and |
| 1877 |
– |
\[ |
| 1878 |
– |
\begin{array}{l} |
| 1879 |
– |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1880 |
– |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1881 |
– |
\end{array}. |
| 1882 |
– |
\] |
| 1883 |
– |
|
| 1884 |
– |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape} |
| 1885 |
– |
|
| 1886 |
– |
Unlike spherical and other regular shaped molecules, there is not |
| 1887 |
– |
analytical solution for friction tensor of any arbitrary shaped |
| 1888 |
– |
rigid molecules. The ellipsoid of revolution model and general |
| 1889 |
– |
triaxial ellipsoid model have been used to approximate the |
| 1890 |
– |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1891 |
– |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1892 |
– |
combination of rotational diffusion coefficients, $D$-space is not |
| 1893 |
– |
unique\cite{Wegener79} as well as the intrinsic coupling between |
| 1894 |
– |
translational and rotational motion of rigid body\cite{}, general |
| 1895 |
– |
ellipsoid is not always suitable for modeling arbitrarily shaped |
| 1896 |
– |
rigid molecule. A number of studies have been devoted to determine |
| 1897 |
– |
the friction tensor for irregularly shaped rigid bodies using more |
| 1898 |
– |
advanced method\cite{} where the molecule of interest was modeled by |
| 1899 |
– |
combinations of spheres(beads)\cite{} and the hydrodynamics |
| 1900 |
– |
properties of the molecule can be calculated using the hydrodynamic |
| 1901 |
– |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 1902 |
– |
immersed in a continuous medium. Due to hydrodynamics interaction, |
| 1903 |
– |
the ``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 1904 |
– |
unperturbed velocity $v_i$, |
| 1905 |
– |
\[ |
| 1906 |
– |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1907 |
– |
\] |
| 1908 |
– |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1909 |
– |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1910 |
– |
proportional to its ``net'' velocity |
| 1911 |
– |
\begin{equation} |
| 1912 |
– |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1913 |
– |
\label{introEquation:tensorExpression} |
| 1914 |
– |
\end{equation} |
| 1915 |
– |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1916 |
– |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1917 |
– |
\ref{introEquation:tensorExpression} |
| 1918 |
– |
\begin{equation} |
| 1919 |
– |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1920 |
– |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1921 |
– |
\label{introEquation:oseenTensor} |
| 1922 |
– |
\end{equation} |
| 1923 |
– |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1924 |
– |
A second order expression for element of different size was |
| 1925 |
– |
introduced by Rotne and Prager\cite{} and improved by Garc\'{i}a de |
| 1926 |
– |
la Torre and Bloomfield, |
| 1927 |
– |
\begin{equation} |
| 1928 |
– |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1929 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1930 |
– |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1931 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1932 |
– |
\label{introEquation:RPTensorNonOverlapped} |
| 1933 |
– |
\end{equation} |
| 1934 |
– |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1935 |
– |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1936 |
– |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1937 |
– |
overlapping beads with the same radius, $\sigma$, is given by |
| 1938 |
– |
\begin{equation} |
| 1939 |
– |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1940 |
– |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1941 |
– |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1942 |
– |
\label{introEquation:RPTensorOverlapped} |
| 1943 |
– |
\end{equation} |
| 1944 |
– |
|
| 1945 |
– |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1946 |
– |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1947 |
– |
$B_{ij}$ blocks |
| 1948 |
– |
\begin{equation} |
| 1949 |
– |
B = \left( {\begin{array}{*{20}c} |
| 1950 |
– |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1951 |
– |
\vdots & \ddots & \vdots \\ |
| 1952 |
– |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1953 |
– |
\end{array}} \right), |
| 1954 |
– |
\end{equation} |
| 1955 |
– |
where $B_{ij}$ is given by |
| 1956 |
– |
\[ |
| 1957 |
– |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1958 |
– |
)T_{ij} |
| 1959 |
– |
\] |
| 1960 |
– |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 1961 |
– |
$B$, we obtain |
| 1962 |
– |
|
| 1963 |
– |
\[ |
| 1964 |
– |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1965 |
– |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1966 |
– |
\vdots & \ddots & \vdots \\ |
| 1967 |
– |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1968 |
– |
\end{array}} \right) |
| 1969 |
– |
\] |
| 1970 |
– |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1971 |
– |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1972 |
– |
\[ |
| 1973 |
– |
U_i = \left( {\begin{array}{*{20}c} |
| 1974 |
– |
0 & { - z_i } & {y_i } \\ |
| 1975 |
– |
{z_i } & 0 & { - x_i } \\ |
| 1976 |
– |
{ - y_i } & {x_i } & 0 \\ |
| 1977 |
– |
\end{array}} \right) |
| 1978 |
– |
\] |
| 1979 |
– |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1980 |
– |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1981 |
– |
arbitrary origin $O$ can be written as |
| 1982 |
– |
\begin{equation} |
| 1983 |
– |
\begin{array}{l} |
| 1984 |
– |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1985 |
– |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1986 |
– |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1987 |
– |
\end{array} |
| 1988 |
– |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1989 |
– |
\end{equation} |
| 1990 |
– |
|
| 1991 |
– |
The resistance tensor depends on the origin to which they refer. The |
| 1992 |
– |
proper location for applying friction force is the center of |
| 1993 |
– |
resistance (reaction), at which the trace of rotational resistance |
| 1994 |
– |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1995 |
– |
resistance is defined as an unique point of the rigid body at which |
| 1996 |
– |
the translation-rotation coupling tensor are symmetric, |
| 1997 |
– |
\begin{equation} |
| 1998 |
– |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1999 |
– |
\label{introEquation:definitionCR} |
| 2000 |
– |
\end{equation} |
| 2001 |
– |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 2002 |
– |
we can easily find out that the translational resistance tensor is |
| 2003 |
– |
origin independent, while the rotational resistance tensor and |
| 2004 |
– |
translation-rotation coupling resistance tensor depend on the |
| 2005 |
– |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 2006 |
– |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 2007 |
– |
obtain the resistance tensor at $P$ by |
| 2008 |
– |
\begin{equation} |
| 2009 |
– |
\begin{array}{l} |
| 2010 |
– |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 2011 |
– |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 2012 |
– |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
| 2013 |
– |
\end{array} |
| 2014 |
– |
\label{introEquation:resistanceTensorTransformation} |
| 2015 |
– |
\end{equation} |
| 2016 |
– |
where |
| 2017 |
– |
\[ |
| 2018 |
– |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 2019 |
– |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 2020 |
– |
{z_i } & 0 & { - x_{OP} } \\ |
| 2021 |
– |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 2022 |
– |
\end{array}} \right) |
| 2023 |
– |
\] |
| 2024 |
– |
Using Equations \ref{introEquation:definitionCR} and |
| 2025 |
– |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 2026 |
– |
the position of center of resistance, |
| 2027 |
– |
\[ |
| 2028 |
– |
\left( \begin{array}{l} |
| 2029 |
– |
x_{OR} \\ |
| 2030 |
– |
y_{OR} \\ |
| 2031 |
– |
z_{OR} \\ |
| 2032 |
– |
\end{array} \right) = \left( {\begin{array}{*{20}c} |
| 2033 |
– |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 2034 |
– |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 2035 |
– |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 2036 |
– |
\end{array}} \right)^{ - 1} \left( \begin{array}{l} |
| 2037 |
– |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 2038 |
– |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 2039 |
– |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 2040 |
– |
\end{array} \right). |
| 2041 |
– |
\] |
| 2042 |
– |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 2043 |
– |
joining center of resistance $R$ and origin $O$. |
