| 208 |
|
The thermodynamic behaviors and properties of Molecular Dynamics |
| 209 |
|
simulation are governed by the principle of Statistical Mechanics. |
| 210 |
|
The following section will give a brief introduction to some of the |
| 211 |
< |
Statistical Mechanics concepts and theorem presented in this |
| 211 |
> |
Statistical Mechanics concepts and theorems presented in this |
| 212 |
|
dissertation. |
| 213 |
|
|
| 214 |
|
\subsection{\label{introSection:ensemble}Phase Space and Ensemble} |
| 372 |
|
Liouville's theorem can be expressed in a variety of different forms |
| 373 |
|
which are convenient within different contexts. For any two function |
| 374 |
|
$F$ and $G$ of the coordinates and momenta of a system, the Poisson |
| 375 |
< |
bracket ${F, G}$ is defined as |
| 375 |
> |
bracket $\{F,G\}$ is defined as |
| 376 |
|
\begin{equation} |
| 377 |
|
\left\{ {F,G} \right\} = \sum\limits_i {\left( {\frac{{\partial |
| 378 |
|
F}}{{\partial q_i }}\frac{{\partial G}}{{\partial p_i }} - |
| 439 |
|
\section{\label{introSection:geometricIntegratos}Geometric Integrators} |
| 440 |
|
A variety of numerical integrators have been proposed to simulate |
| 441 |
|
the motions of atoms in MD simulation. They usually begin with |
| 442 |
< |
initial conditionals and move the objects in the direction governed |
| 442 |
> |
initial conditions and move the objects in the direction governed |
| 443 |
|
by the differential equations. However, most of them ignore the |
| 444 |
|
hidden physical laws contained within the equations. Since 1990, |
| 445 |
|
geometric integrators, which preserve various phase-flow invariants |
| 459 |
|
\emph{smooth manifold}) is a manifold on which it is possible to |
| 460 |
|
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
| 461 |
|
defined as a pair $(M, \omega)$ which consists of a |
| 462 |
< |
\emph{differentiable manifold} $M$ and a close, non-degenerated, |
| 462 |
> |
\emph{differentiable manifold} $M$ and a close, non-degenerate, |
| 463 |
|
bilinear symplectic form, $\omega$. A symplectic form on a vector |
| 464 |
|
space $V$ is a function $\omega(x, y)$ which satisfies |
| 465 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 479 |
|
\begin{equation} |
| 480 |
|
\dot x = f(x) |
| 481 |
|
\end{equation} |
| 482 |
< |
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
| 482 |
> |
where $x = x(q,p)$, this system is a canonical Hamiltonian, if |
| 483 |
|
$f(x) = J\nabla _x H(x)$. Here, $H = H (q, p)$ is Hamiltonian |
| 484 |
|
function and $J$ is the skew-symmetric matrix |
| 485 |
|
\begin{equation} |
| 505 |
|
\subsection{\label{introSection:exactFlow}Exact Propagator} |
| 506 |
|
|
| 507 |
|
Let $x(t)$ be the exact solution of the ODE |
| 508 |
< |
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
| 509 |
< |
define its exact propagator(solution) $\varphi_\tau$ |
| 508 |
> |
system, |
| 509 |
> |
\begin{equation} |
| 510 |
> |
\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE} |
| 511 |
> |
\end{equation} we can |
| 512 |
> |
define its exact propagator $\varphi_\tau$: |
| 513 |
|
\[ x(t+\tau) |
| 514 |
|
=\varphi_\tau(x(t)) |
| 515 |
|
\] |
| 527 |
|
\begin{equation} |
| 528 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 529 |
|
\end{equation} |
| 530 |
< |
The exact propagator can also be written in terms of operator, |
| 530 |
> |
The exact propagator can also be written as an operator, |
| 531 |
|
\begin{equation} |
| 532 |
|
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 533 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 581 |
|
\] |
| 582 |
|
Using the chain rule, one may obtain, |
| 583 |
|
\[ |
| 584 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
| 584 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G, |
| 585 |
|
\] |
| 586 |
|
which is the condition for conserved quantities. For a canonical |
| 587 |
|
Hamiltonian system, the time evolution of an arbitrary smooth |
| 706 |
|
\end{align} |
| 707 |
|
where $F(t)$ is the force at time $t$. This integration scheme is |
| 708 |
|
known as \emph{velocity verlet} which is |
| 709 |
< |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
| 710 |
< |
time-reversible(\ref{introEquation:timeReversible}) and |
| 711 |
< |
volume-preserving (\ref{introEquation:volumePreserving}). These |
| 709 |
> |
symplectic(Eq.~\ref{introEquation:SymplecticFlowComposition}), |
| 710 |
> |
time-reversible(Eq.~\ref{introEquation:timeReversible}) and |
| 711 |
> |
volume-preserving (Eq.~\ref{introEquation:volumePreserving}). These |
| 712 |
|
geometric properties attribute to its long-time stability and its |
| 713 |
|
popularity in the community. However, the most commonly used |
| 714 |
|
velocity verlet integration scheme is written as below, |
| 729 |
|
|
| 730 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 731 |
|
|
| 732 |
< |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
| 732 |
> |
\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move. |
| 733 |
|
|
| 734 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 735 |
|
\end{enumerate} |
| 750 |
|
|
| 751 |
|
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 752 |
|
local error of a splitting method in terms of the commutator of the |
| 753 |
< |
operators(\ref{introEquation:exponentialOperator}) associated with |
| 753 |
> |
operators(Eq.~\ref{introEquation:exponentialOperator}) associated with |
| 754 |
|
the sub-propagator. For operators $hX$ and $hY$ which are associated |
| 755 |
|
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 756 |
|
\begin{equation} |
| 834 |
|
These three individual steps will be covered in the following |
| 835 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 836 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
| 837 |
< |
will discuss issues of production runs. |
| 837 |
> |
discusses issues of production runs. |
| 838 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 839 |
|
analysis of trajectories. |
| 840 |
|
|
| 868 |
|
minimization to find a more reasonable conformation. Several energy |
| 869 |
|
minimization methods have been developed to exploit the energy |
| 870 |
|
surface and to locate the local minimum. While converging slowly |
| 871 |
< |
near the minimum, steepest descent method is extremely robust when |
| 871 |
> |
near the minimum, the steepest descent method is extremely robust when |
| 872 |
|
systems are strongly anharmonic. Thus, it is often used to refine |
| 873 |
|
structures from crystallographic data. Relying on the Hessian, |
| 874 |
|
advanced methods like Newton-Raphson converge rapidly to a local |
| 887 |
|
temperature. Beginning at a lower temperature and gradually |
| 888 |
|
increasing the temperature by assigning larger random velocities, we |
| 889 |
|
end up setting the temperature of the system to a final temperature |
| 890 |
< |
at which the simulation will be conducted. In heating phase, we |
| 890 |
> |
at which the simulation will be conducted. In the heating phase, we |
| 891 |
|
should also keep the system from drifting or rotating as a whole. To |
| 892 |
|
do this, the net linear momentum and angular momentum of the system |
| 893 |
|
is shifted to zero after each resampling from the Maxwell -Boltzman |
| 902 |
|
properties \textit{etc}, become independent of time. Strictly |
| 903 |
|
speaking, minimization and heating are not necessary, provided the |
| 904 |
|
equilibration process is long enough. However, these steps can serve |
| 905 |
< |
as a means to arrive at an equilibrated structure in an effective |
| 905 |
> |
as a mean to arrive at an equilibrated structure in an effective |
| 906 |
|
way. |
| 907 |
|
|
| 908 |
|
\subsection{\label{introSection:production}Production} |
| 954 |
|
in simulations. The Ewald summation, in which the slowly decaying |
| 955 |
|
Coulomb potential is transformed into direct and reciprocal sums |
| 956 |
|
with rapid and absolute convergence, has proved to minimize the |
| 957 |
< |
periodicity artifacts in liquid simulations. Taking the advantages |
| 958 |
< |
of the fast Fourier transform (FFT) for calculating discrete Fourier |
| 957 |
> |
periodicity artifacts in liquid simulations. Taking advantage |
| 958 |
> |
of fast Fourier transform (FFT) techniques for calculating discrete Fourier |
| 959 |
|
transforms, the particle mesh-based |
| 960 |
|
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 961 |
|
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
| 972 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 973 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 974 |
|
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
| 975 |
< |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
| 975 |
> |
r_{ij})}{r_{ij}}\right\}, \label{introEquation:shiftedCoulomb} |
| 976 |
|
\end{equation} |
| 977 |
|
where $\alpha$ is the convergence parameter. Due to the lack of |
| 978 |
|
inherent periodicity and rapid convergence,this method is extremely |
| 989 |
|
|
| 990 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 991 |
|
|
| 992 |
< |
Recently, advanced visualization technique have become applied to |
| 992 |
> |
Recently, advanced visualization techniques have been applied to |
| 993 |
|
monitor the motions of molecules. Although the dynamics of the |
| 994 |
|
system can be described qualitatively from animation, quantitative |
| 995 |
|
trajectory analysis is more useful. According to the principles of |
| 1059 |
|
\label{introEquation:timeCorrelationFunction} |
| 1060 |
|
\end{equation} |
| 1061 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1062 |
< |
function is called an \emph{autocorrelation function}. One example |
| 1060 |
< |
of an auto correlation function is the velocity auto-correlation |
| 1062 |
> |
functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation |
| 1063 |
|
function which is directly related to transport properties of |
| 1064 |
|
molecular liquids: |
| 1065 |
< |
\[ |
| 1065 |
> |
\begin{equation} |
| 1066 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
| 1067 |
|
\right\rangle } dt |
| 1068 |
< |
\] |
| 1068 |
> |
\end{equation} |
| 1069 |
|
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1070 |
|
function, which is averaged over time origins and over all the |
| 1071 |
|
atoms, the dipole autocorrelation functions is calculated for the |
| 1072 |
|
entire system. The dipole autocorrelation function is given by: |
| 1073 |
< |
\[ |
| 1073 |
> |
\begin{equation} |
| 1074 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1075 |
|
\right\rangle |
| 1076 |
< |
\] |
| 1076 |
> |
\end{equation} |
| 1077 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
| 1078 |
|
by |
| 1079 |
< |
\[ |
| 1079 |
> |
\begin{equation} |
| 1080 |
|
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
| 1081 |
< |
\] |
| 1081 |
> |
\end{equation} |
| 1082 |
|
In principle, many time correlation functions can be related to |
| 1083 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1084 |
|
scattering spectra of molecular liquids. In practice, one can |
| 1085 |
|
extract the IR spectrum from the intensity of the molecular dipole |
| 1086 |
|
fluctuation at each frequency using the following relationship: |
| 1087 |
< |
\[ |
| 1087 |
> |
\begin{equation} |
| 1088 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1089 |
|
i2\pi vt} dt}. |
| 1090 |
< |
\] |
| 1090 |
> |
\end{equation} |
| 1091 |
|
|
| 1092 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1093 |
|
|
| 1094 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1095 |
< |
areas, from engineering, physics, to chemistry. For example, |
| 1095 |
> |
areas, including engineering, physics and chemistry. For example, |
| 1096 |
|
missiles and vehicles are usually modeled by rigid bodies. The |
| 1097 |
|
movement of the objects in 3D gaming engines or other physics |
| 1098 |
|
simulators is governed by rigid body dynamics. In molecular |
| 1110 |
|
computational penalty and the loss of angular momentum conservation |
| 1111 |
|
still remain. A singularity-free representation utilizing |
| 1112 |
|
quaternions was developed by Evans in 1977\cite{Evans1977}. |
| 1113 |
< |
Unfortunately, this approach uses a nonseparable Hamiltonian |
| 1114 |
< |
resulting from the quaternion representation, which prevents the |
| 1113 |
> |
Unfortunately, this approach used a nonseparable Hamiltonian |
| 1114 |
> |
resulting from the quaternion representation, which prevented the |
| 1115 |
|
symplectic algorithm from being utilized. Another different approach |
| 1116 |
|
is to apply holonomic constraints to the atoms belonging to the |
| 1117 |
|
rigid body. Each atom moves independently under the normal forces |
| 1134 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1135 |
|
|
| 1136 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1137 |
< |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
| 1136 |
< |
function |
| 1137 |
> |
The Hamiltonian of a rigid body is given by |
| 1138 |
|
\begin{equation} |
| 1139 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1140 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1153 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1154 |
|
\end{equation} |
| 1155 |
|
which is used to ensure the rotation matrix's unitarity. Using |
| 1156 |
< |
Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1157 |
< |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1156 |
> |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~ |
| 1157 |
> |
\ref{introEquation:motionHamiltonianMomentum}, one can write down |
| 1158 |
|
the equations of motion, |
| 1159 |
|
\begin{eqnarray} |
| 1160 |
|
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
| 1186 |
|
\[ |
| 1187 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1188 |
|
\] |
| 1189 |
< |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 1189 |
> |
the mechanical system subjected to a holonomic constraint manifold $M$ |
| 1190 |
|
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1191 |
|
\[ |
| 1192 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1346 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1347 |
|
_1 }. |
| 1348 |
|
\] |
| 1349 |
< |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1349 |
< |
$F(\pi )$ and $G(\pi )$ is defined by |
| 1349 |
> |
The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by |
| 1350 |
|
\[ |
| 1351 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1352 |
|
). |
| 1355 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1356 |
|
conserved quantity in Poisson system. We can easily verify that the |
| 1357 |
|
norm of the angular momentum, $\parallel \pi |
| 1358 |
< |
\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1358 |
> |
\parallel$, is a \emph{Casimir}\cite{McLachlan1993}. Let $F(\pi ) = S(\frac{{\parallel |
| 1359 |
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1360 |
|
then by the chain rule |
| 1361 |
|
\[ |
| 1374 |
|
Splitting for Rigid Body} |
| 1375 |
|
|
| 1376 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1377 |
< |
energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations |
| 1377 |
> |
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations |
| 1378 |
|
of motion corresponding to potential energy and kinetic energy are |
| 1379 |
< |
listed in the below table, |
| 1379 |
> |
listed in Table~\ref{introTable:rbEquations}. |
| 1380 |
|
\begin{table} |
| 1381 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1382 |
+ |
\label{introTable:rbEquations} |
| 1383 |
|
\begin{center} |
| 1384 |
|
\begin{tabular}{|l|l|} |
| 1385 |
|
\hline |
| 1434 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1435 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
| 1436 |
|
has been applied in a variety of studies. This section will review |
| 1437 |
< |
the theory of Langevin dynamics. A brief derivation of generalized |
| 1437 |
> |
the theory of Langevin dynamics. A brief derivation of the generalized |
| 1438 |
|
Langevin equation will be given first. Following that, we will |
| 1439 |
< |
discuss the physical meaning of the terms appearing in the equation |
| 1439 |
< |
as well as the calculation of friction tensor from hydrodynamics |
| 1440 |
< |
theory. |
| 1439 |
> |
discuss the physical meaning of the terms appearing in the equation. |
| 1440 |
|
|
| 1441 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1442 |
|
|
| 1445 |
|
environment, has been widely used in quantum chemistry and |
| 1446 |
|
statistical mechanics. One of the successful applications of |
| 1447 |
|
Harmonic bath model is the derivation of the Generalized Langevin |
| 1448 |
< |
Dynamics (GLE). Lets consider a system, in which the degree of |
| 1448 |
> |
Dynamics (GLE). Consider a system, in which the degree of |
| 1449 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1450 |
|
Hamiltonian of the form |
| 1451 |
|
\begin{equation} |
| 1456 |
|
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
| 1457 |
|
\[ |
| 1458 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1459 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1459 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha x_\alpha ^2 } |
| 1460 |
|
\right\}} |
| 1461 |
|
\] |
| 1462 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1509 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1510 |
|
\] |
| 1511 |
|
where $p$ is real and $L$ is called the Laplace Transform |
| 1512 |
< |
Operator. Below are some important properties of Laplace transform |
| 1512 |
> |
Operator. Below are some important properties of the Laplace transform |
| 1513 |
|
\begin{eqnarray*} |
| 1514 |
|
L(x + y) & = & L(x) + L(y) \\ |
| 1515 |
|
L(ax) & = & aL(x) \\ |
| 1578 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1579 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1580 |
|
\end{equation} |
| 1581 |
< |
which is known as the \emph{generalized Langevin equation}. |
| 1581 |
> |
which is known as the \emph{generalized Langevin equation} (GLE). |
| 1582 |
|
|
| 1583 |
|
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
| 1584 |
|
|
| 1585 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
| 1586 |
|
implies it is completely deterministic within the context of a |
| 1587 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1588 |
< |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
| 1588 |
> |
uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)} |
| 1589 |
|
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
| 1590 |
|
0.$ This property is what we expect from a truly random process. As |
| 1591 |
|
long as the model chosen for $R(t)$ was a gaussian distribution in |
| 1614 |
|
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
| 1615 |
|
taken as a $delta$ function in time: |
| 1616 |
|
\[ |
| 1617 |
< |
\xi (t) = 2\xi _0 \delta (t) |
| 1617 |
> |
\xi (t) = 2\xi _0 \delta (t). |
| 1618 |
|
\] |
| 1619 |
|
Hence, the convolution integral becomes |
| 1620 |
|
\[ |
| 1639 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1640 |
|
^2 }}x(0), |
| 1641 |
|
\] |
| 1642 |
< |
we can rewrite $R(T)$ as |
| 1642 |
> |
we can rewrite $R(t)$ as |
| 1643 |
|
\[ |
| 1644 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1645 |
|
\] |
