| 67 |
|
\begin{equation}E = T + V. \label{introEquation:energyConservation} |
| 68 |
|
\end{equation} |
| 69 |
|
All of these conserved quantities are important factors to determine |
| 70 |
< |
the quality of numerical integration schemes for rigid bodies |
| 71 |
< |
\cite{Dullweber1997}. |
| 70 |
> |
the quality of numerical integration schemes for rigid |
| 71 |
> |
bodies.\cite{Dullweber1997} |
| 72 |
|
|
| 73 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 74 |
|
|
| 178 |
|
where Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 179 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
| 180 |
|
equation of motion. Due to their symmetrical formula, they are also |
| 181 |
< |
known as the canonical equations of motions \cite{Goldstein2001}. |
| 181 |
> |
known as the canonical equations of motions.\cite{Goldstein2001} |
| 182 |
|
|
| 183 |
|
An important difference between Lagrangian approach and the |
| 184 |
|
Hamiltonian approach is that the Lagrangian is considered to be a |
| 188 |
|
Hamiltonian Mechanics is more appropriate for application to |
| 189 |
|
statistical mechanics and quantum mechanics, since it treats the |
| 190 |
|
coordinate and its time derivative as independent variables and it |
| 191 |
< |
only works with 1st-order differential equations\cite{Marion1990}. |
| 191 |
> |
only works with 1st-order differential equations.\cite{Marion1990} |
| 192 |
|
In Newtonian Mechanics, a system described by conservative forces |
| 193 |
|
conserves the total energy |
| 194 |
|
(Eq.~\ref{introEquation:energyConservation}). It follows that |
| 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} |
| 281 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
| 282 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 283 |
|
\label{introEquation:ensembelAverage} |
| 284 |
– |
\end{equation} |
| 285 |
– |
|
| 286 |
– |
There are several different types of ensembles with different |
| 287 |
– |
statistical characteristics. As a function of macroscopic |
| 288 |
– |
parameters, such as temperature \textit{etc}, the partition function |
| 289 |
– |
can be used to describe the statistical properties of a system in |
| 290 |
– |
thermodynamic equilibrium. As an ensemble of systems, each of which |
| 291 |
– |
is known to be thermally isolated and conserve energy, the |
| 292 |
– |
Microcanonical ensemble (NVE) has a partition function like, |
| 293 |
– |
\begin{equation} |
| 294 |
– |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
| 295 |
– |
\end{equation} |
| 296 |
– |
A canonical ensemble (NVT) is an ensemble of systems, each of which |
| 297 |
– |
can share its energy with a large heat reservoir. The distribution |
| 298 |
– |
of the total energy amongst the possible dynamical states is given |
| 299 |
– |
by the partition function, |
| 300 |
– |
\begin{equation} |
| 301 |
– |
\Omega (N,V,T) = e^{ - \beta A}. |
| 302 |
– |
\label{introEquation:NVTPartition} |
| 284 |
|
\end{equation} |
| 304 |
– |
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 305 |
– |
TS$. Since most experiments are carried out under constant pressure |
| 306 |
– |
condition, the isothermal-isobaric ensemble (NPT) plays a very |
| 307 |
– |
important role in molecular simulations. The isothermal-isobaric |
| 308 |
– |
ensemble allow the system to exchange energy with a heat bath of |
| 309 |
– |
temperature $T$ and to change the volume as well. Its partition |
| 310 |
– |
function is given as |
| 311 |
– |
\begin{equation} |
| 312 |
– |
\Delta (N,P,T) = - e^{\beta G}. |
| 313 |
– |
\label{introEquation:NPTPartition} |
| 314 |
– |
\end{equation} |
| 315 |
– |
Here, $G = U - TS + PV$, and $G$ is called Gibbs free energy. |
| 285 |
|
|
| 286 |
|
\subsection{\label{introSection:liouville}Liouville's theorem} |
| 287 |
|
|
| 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 }} - |
| 416 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
| 417 |
|
Hypothesis makes a connection between time average and the ensemble |
| 418 |
|
average. It states that the time average and average over the |
| 419 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
| 419 |
> |
statistical ensemble are identical:\cite{Frenkel1996, Leach2001} |
| 420 |
|
\begin{equation} |
| 421 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
| 422 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
| 434 |
|
utilized. Or if the system lends itself to a time averaging |
| 435 |
|
approach, the Molecular Dynamics techniques in |
| 436 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
| 437 |
< |
choice\cite{Frenkel1996}. |
| 437 |
> |
choice.\cite{Frenkel1996} |
| 438 |
|
|
| 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 |
| 443 |
< |
by the differential equations. However, most of them ignore the |
| 444 |
< |
hidden physical laws contained within the equations. Since 1990, |
| 445 |
< |
geometric integrators, which preserve various phase-flow invariants |
| 446 |
< |
such as symplectic structure, volume and time reversal symmetry, |
| 447 |
< |
were developed to address this issue\cite{Dullweber1997, |
| 448 |
< |
McLachlan1998, Leimkuhler1999}. The velocity Verlet method, which |
| 449 |
< |
happens to be a simple example of symplectic integrator, continues |
| 450 |
< |
to gain popularity in the molecular dynamics community. This fact |
| 451 |
< |
can be partly explained by its geometric nature. |
| 442 |
> |
initial conditions and move the objects in the direction governed by |
| 443 |
> |
the differential equations. However, most of them ignore the hidden |
| 444 |
> |
physical laws contained within the equations. Since 1990, geometric |
| 445 |
> |
integrators, which preserve various phase-flow invariants such as |
| 446 |
> |
symplectic structure, volume and time reversal symmetry, were |
| 447 |
> |
developed to address this issue.\cite{Dullweber1997, McLachlan1998, |
| 448 |
> |
Leimkuhler1999} The velocity Verlet method, which happens to be a |
| 449 |
> |
simple example of symplectic integrator, continues to gain |
| 450 |
> |
popularity in the molecular dynamics community. This fact can be |
| 451 |
> |
partly explained by its geometric nature. |
| 452 |
|
|
| 453 |
|
\subsection{\label{introSection:symplecticManifold}Symplectic Manifolds} |
| 454 |
|
A \emph{manifold} is an abstract mathematical space. It looks |
| 457 |
|
surface of Earth. It seems to be flat locally, but it is round if |
| 458 |
|
viewed as a whole. A \emph{differentiable manifold} (also known as |
| 459 |
|
\emph{smooth manifold}) is a manifold on which it is possible to |
| 460 |
< |
apply calculus\cite{Hirsch1997}. A \emph{symplectic manifold} is |
| 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 |
|
$\omega(\lambda_1x_1+\lambda_2x_2, y) = \lambda_1\omega(x_1, y)+ |
| 466 |
|
\lambda_2\omega(x_2, y)$, $\omega(x, y) = - \omega(y, x)$ and |
| 467 |
< |
$\omega(x, x) = 0$\cite{McDuff1998}. The cross product operation in |
| 467 |
> |
$\omega(x, x) = 0$.\cite{McDuff1998} The cross product operation in |
| 468 |
|
vector field is an example of symplectic form. One of the |
| 469 |
|
motivations to study \emph{symplectic manifolds} in Hamiltonian |
| 470 |
|
Mechanics is that a symplectic manifold can represent all possible |
| 471 |
|
configurations of the system and the phase space of the system can |
| 472 |
< |
be described by it's cotangent bundle\cite{Jost2002}. Every |
| 472 |
> |
be described by it's cotangent bundle.\cite{Jost2002} Every |
| 473 |
|
symplectic manifold is even dimensional. For instance, in Hamilton |
| 474 |
|
equations, coordinate and momentum always appear in pairs. |
| 475 |
|
|
| 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} |
| 496 |
|
\label{introEquation:compactHamiltonian} |
| 497 |
|
\end{equation}In this case, $f$ is |
| 498 |
|
called a \emph{Hamiltonian vector field}. Another generalization of |
| 499 |
< |
Hamiltonian dynamics is Poisson Dynamics\cite{Olver1986}, |
| 499 |
> |
Hamiltonian dynamics is Poisson Dynamics,\cite{Olver1986} |
| 500 |
|
\begin{equation} |
| 501 |
|
\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian} |
| 502 |
|
\end{equation} |
| 503 |
< |
The most obvious change being that matrix $J$ now depends on $x$. |
| 503 |
> |
where the most obvious change being that matrix $J$ now depends on |
| 504 |
> |
$x$. |
| 505 |
|
|
| 506 |
|
\subsection{\label{introSection:exactFlow}Exact Propagator} |
| 507 |
|
|
| 508 |
|
Let $x(t)$ be the exact solution of the ODE |
| 509 |
< |
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
| 510 |
< |
define its exact propagator(solution) $\varphi_\tau$ |
| 509 |
> |
system, |
| 510 |
> |
\begin{equation} |
| 511 |
> |
\frac{{dx}}{{dt}} = f(x), \label{introEquation:ODE} |
| 512 |
> |
\end{equation} we can |
| 513 |
> |
define its exact propagator $\varphi_\tau$: |
| 514 |
|
\[ x(t+\tau) |
| 515 |
|
=\varphi_\tau(x(t)) |
| 516 |
|
\] |
| 528 |
|
\begin{equation} |
| 529 |
|
\varphi _\tau = \varphi _{ - \tau }^{ - 1}. |
| 530 |
|
\end{equation} |
| 531 |
< |
The exact propagator can also be written in terms of operator, |
| 531 |
> |
The exact propagator can also be written as an operator, |
| 532 |
|
\begin{equation} |
| 533 |
|
\varphi _\tau (x) = e^{\tau \sum\limits_i {f_i (x)\frac{\partial |
| 534 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
| 582 |
|
\] |
| 583 |
|
Using the chain rule, one may obtain, |
| 584 |
|
\[ |
| 585 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
| 585 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \cdot \nabla G, |
| 586 |
|
\] |
| 587 |
|
which is the condition for conserved quantities. For a canonical |
| 588 |
|
Hamiltonian system, the time evolution of an arbitrary smooth |
| 621 |
|
Generating functions\cite{Channell1990} tend to lead to methods |
| 622 |
|
which are cumbersome and difficult to use. In dissipative systems, |
| 623 |
|
variational methods can capture the decay of energy |
| 624 |
< |
accurately\cite{Kane2000}. Since they are geometrically unstable |
| 624 |
> |
accurately.\cite{Kane2000} Since they are geometrically unstable |
| 625 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
| 626 |
< |
methods are not suitable for Hamiltonian system. Recently, various |
| 627 |
< |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
| 628 |
< |
have been developed to overcome this instability. However, due to |
| 629 |
< |
computational penalty involved in implementing the Runge-Kutta |
| 630 |
< |
methods, they have not attracted much attention from the Molecular |
| 631 |
< |
Dynamics community. Instead, splitting methods have been widely |
| 632 |
< |
accepted since they exploit natural decompositions of the |
| 633 |
< |
system\cite{Tuckerman1992, McLachlan1998}. |
| 626 |
> |
methods are not suitable for Hamiltonian |
| 627 |
> |
system.\cite{Cartwright1992} Recently, various high-order explicit |
| 628 |
> |
Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to |
| 629 |
> |
overcome this instability. However, due to computational penalty |
| 630 |
> |
involved in implementing the Runge-Kutta methods, they have not |
| 631 |
> |
attracted much attention from the Molecular Dynamics community. |
| 632 |
> |
Instead, splitting methods have been widely accepted since they |
| 633 |
> |
exploit natural decompositions of the system.\cite{McLachlan1998, |
| 634 |
> |
Tuckerman1992} |
| 635 |
|
|
| 636 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
| 637 |
|
|
| 654 |
|
problem. If $H_1$ and $H_2$ can be integrated using exact |
| 655 |
|
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
| 656 |
|
simple first order expression is then given by the Lie-Trotter |
| 657 |
< |
formula |
| 657 |
> |
formula\cite{Trotter1959} |
| 658 |
|
\begin{equation} |
| 659 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
| 660 |
|
\label{introEquation:firstOrderSplitting} |
| 675 |
|
The Lie-Trotter |
| 676 |
|
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
| 677 |
|
local errors proportional to $h^2$, while the Strang splitting gives |
| 678 |
< |
a second-order decomposition, |
| 678 |
> |
a second-order decomposition,\cite{Strang1968} |
| 679 |
|
\begin{equation} |
| 680 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
| 681 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
| 708 |
|
\end{align} |
| 709 |
|
where $F(t)$ is the force at time $t$. This integration scheme is |
| 710 |
|
known as \emph{velocity verlet} which is |
| 711 |
< |
symplectic(\ref{introEquation:SymplecticFlowComposition}), |
| 712 |
< |
time-reversible(\ref{introEquation:timeReversible}) and |
| 713 |
< |
volume-preserving (\ref{introEquation:volumePreserving}). These |
| 711 |
> |
symplectic(Eq.~\ref{introEquation:SymplecticFlowComposition}), |
| 712 |
> |
time-reversible(Eq.~\ref{introEquation:timeReversible}) and |
| 713 |
> |
volume-preserving (Eq.~\ref{introEquation:volumePreserving}). These |
| 714 |
|
geometric properties attribute to its long-time stability and its |
| 715 |
|
popularity in the community. However, the most commonly used |
| 716 |
|
velocity verlet integration scheme is written as below, |
| 731 |
|
|
| 732 |
|
\item Use the half step velocities to move positions one whole step, $\Delta t$. |
| 733 |
|
|
| 734 |
< |
\item Evaluate the forces at the new positions, $\mathbf{q}(\Delta t)$, and use the new forces to complete the velocity move. |
| 734 |
> |
\item Evaluate the forces at the new positions, $q(\Delta t)$, and use the new forces to complete the velocity move. |
| 735 |
|
|
| 736 |
|
\item Repeat from step 1 with the new position, velocities, and forces assuming the roles of the initial values. |
| 737 |
|
\end{enumerate} |
| 750 |
|
|
| 751 |
|
\subsubsection{\label{introSection:errorAnalysis}\textbf{Error Analysis and Higher Order Methods}} |
| 752 |
|
|
| 753 |
< |
The Baker-Campbell-Hausdorff formula can be used to determine the |
| 754 |
< |
local error of a splitting method in terms of the commutator of the |
| 755 |
< |
operators(\ref{introEquation:exponentialOperator}) associated with |
| 756 |
< |
the sub-propagator. For operators $hX$ and $hY$ which are associated |
| 757 |
< |
with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have |
| 753 |
> |
The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used |
| 754 |
> |
to determine the local error of a splitting method in terms of the |
| 755 |
> |
commutator of the |
| 756 |
> |
operators(Eq.~\ref{introEquation:exponentialOperator}) associated |
| 757 |
> |
with the sub-propagator. For operators $hX$ and $hY$ which are |
| 758 |
> |
associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we |
| 759 |
> |
have |
| 760 |
|
\begin{equation} |
| 761 |
|
\exp (hX + hY) = \exp (hZ) |
| 762 |
|
\end{equation} |
| 786 |
|
\end{equation} |
| 787 |
|
A careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 788 |
|
order methods. Yoshida proposed an elegant way to compose higher |
| 789 |
< |
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 789 |
> |
order methods based on symmetric splitting.\cite{Yoshida1990} Given |
| 790 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 791 |
|
fourth-order symmetric method can be constructed by composing, |
| 792 |
|
\[ |
| 838 |
|
These three individual steps will be covered in the following |
| 839 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 840 |
|
initialization of a simulation. Sec.~\ref{introSection:production} |
| 841 |
< |
will discuss issues of production runs. |
| 841 |
> |
discusses issues of production runs. |
| 842 |
|
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 843 |
|
analysis of trajectories. |
| 844 |
|
|
| 872 |
|
minimization to find a more reasonable conformation. Several energy |
| 873 |
|
minimization methods have been developed to exploit the energy |
| 874 |
|
surface and to locate the local minimum. While converging slowly |
| 875 |
< |
near the minimum, steepest descent method is extremely robust when |
| 875 |
> |
near the minimum, the steepest descent method is extremely robust when |
| 876 |
|
systems are strongly anharmonic. Thus, it is often used to refine |
| 877 |
|
structures from crystallographic data. Relying on the Hessian, |
| 878 |
|
advanced methods like Newton-Raphson converge rapidly to a local |
| 891 |
|
temperature. Beginning at a lower temperature and gradually |
| 892 |
|
increasing the temperature by assigning larger random velocities, we |
| 893 |
|
end up setting the temperature of the system to a final temperature |
| 894 |
< |
at which the simulation will be conducted. In heating phase, we |
| 894 |
> |
at which the simulation will be conducted. In the heating phase, we |
| 895 |
|
should also keep the system from drifting or rotating as a whole. To |
| 896 |
|
do this, the net linear momentum and angular momentum of the system |
| 897 |
|
is shifted to zero after each resampling from the Maxwell -Boltzman |
| 906 |
|
properties \textit{etc}, become independent of time. Strictly |
| 907 |
|
speaking, minimization and heating are not necessary, provided the |
| 908 |
|
equilibration process is long enough. However, these steps can serve |
| 909 |
< |
as a means to arrive at an equilibrated structure in an effective |
| 909 |
> |
as a mean to arrive at an equilibrated structure in an effective |
| 910 |
|
way. |
| 911 |
|
|
| 912 |
|
\subsection{\label{introSection:production}Production} |
| 947 |
|
%cutoff and minimum image convention |
| 948 |
|
Another important technique to improve the efficiency of force |
| 949 |
|
evaluation is to apply spherical cutoffs where particles farther |
| 950 |
< |
than a predetermined distance are not included in the calculation |
| 951 |
< |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
| 952 |
< |
discontinuity in the potential energy curve. Fortunately, one can |
| 950 |
> |
than a predetermined distance are not included in the |
| 951 |
> |
calculation.\cite{Frenkel1996} The use of a cutoff radius will cause |
| 952 |
> |
a discontinuity in the potential energy curve. Fortunately, one can |
| 953 |
|
shift a simple radial potential to ensure the potential curve go |
| 954 |
|
smoothly to zero at the cutoff radius. The cutoff strategy works |
| 955 |
|
well for Lennard-Jones interaction because of its short range |
| 958 |
|
in simulations. The Ewald summation, in which the slowly decaying |
| 959 |
|
Coulomb potential is transformed into direct and reciprocal sums |
| 960 |
|
with rapid and absolute convergence, has proved to minimize the |
| 961 |
< |
periodicity artifacts in liquid simulations. Taking the advantages |
| 962 |
< |
of the fast Fourier transform (FFT) for calculating discrete Fourier |
| 963 |
< |
transforms, the particle mesh-based |
| 961 |
> |
periodicity artifacts in liquid simulations. Taking advantage of |
| 962 |
> |
fast Fourier transform (FFT) techniques for calculating discrete |
| 963 |
> |
Fourier transforms, the particle mesh-based |
| 964 |
|
methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from |
| 965 |
|
$O(N^{3/2})$ to $O(N logN)$. An alternative approach is the |
| 966 |
|
\emph{fast multipole method}\cite{Greengard1987, Greengard1994}, |
| 970 |
|
simulation community, these two methods are difficult to implement |
| 971 |
|
correctly and efficiently. Instead, we use a damped and |
| 972 |
|
charge-neutralized Coulomb potential method developed by Wolf and |
| 973 |
< |
his coworkers\cite{Wolf1999}. The shifted Coulomb potential for |
| 973 |
> |
his coworkers.\cite{Wolf1999} The shifted Coulomb potential for |
| 974 |
|
particle $i$ and particle $j$ at distance $r_{rj}$ is given by: |
| 975 |
|
\begin{equation} |
| 976 |
|
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
| 977 |
|
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
| 978 |
|
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
| 979 |
< |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
| 979 |
> |
r_{ij})}{r_{ij}}\right\}, \label{introEquation:shiftedCoulomb} |
| 980 |
|
\end{equation} |
| 981 |
|
where $\alpha$ is the convergence parameter. Due to the lack of |
| 982 |
|
inherent periodicity and rapid convergence,this method is extremely |
| 993 |
|
|
| 994 |
|
\subsection{\label{introSection:Analysis} Analysis} |
| 995 |
|
|
| 996 |
< |
Recently, advanced visualization technique have become applied to |
| 996 |
> |
Recently, advanced visualization techniques have been applied to |
| 997 |
|
monitor the motions of molecules. Although the dynamics of the |
| 998 |
|
system can be described qualitatively from animation, quantitative |
| 999 |
|
trajectory analysis is more useful. According to the principles of |
| 1033 |
|
function}, is of most fundamental importance to liquid theory. |
| 1034 |
|
Experimentally, pair distribution functions can be gathered by |
| 1035 |
|
Fourier transforming raw data from a series of neutron diffraction |
| 1036 |
< |
experiments and integrating over the surface factor |
| 1037 |
< |
\cite{Powles1973}. The experimental results can serve as a criterion |
| 1038 |
< |
to justify the correctness of a liquid model. Moreover, various |
| 1039 |
< |
equilibrium thermodynamic and structural properties can also be |
| 1040 |
< |
expressed in terms of the radial distribution function |
| 1041 |
< |
\cite{Allen1987}. The pair distribution functions $g(r)$ gives the |
| 1042 |
< |
probability that a particle $i$ will be located at a distance $r$ |
| 1043 |
< |
from a another particle $j$ in the system |
| 1036 |
> |
experiments and integrating over the surface |
| 1037 |
> |
factor.\cite{Powles1973} The experimental results can serve as a |
| 1038 |
> |
criterion to justify the correctness of a liquid model. Moreover, |
| 1039 |
> |
various equilibrium thermodynamic and structural properties can also |
| 1040 |
> |
be expressed in terms of the radial distribution |
| 1041 |
> |
function.\cite{Allen1987} The pair distribution functions $g(r)$ |
| 1042 |
> |
gives the probability that a particle $i$ will be located at a |
| 1043 |
> |
distance $r$ from a another particle $j$ in the system |
| 1044 |
|
\begin{equation} |
| 1045 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1046 |
|
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
| 1063 |
|
\label{introEquation:timeCorrelationFunction} |
| 1064 |
|
\end{equation} |
| 1065 |
|
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1066 |
< |
function is called an \emph{autocorrelation function}. One example |
| 1091 |
< |
of an auto correlation function is the velocity auto-correlation |
| 1066 |
> |
functions are called \emph{autocorrelation functions}. One typical example is the velocity autocorrelation |
| 1067 |
|
function which is directly related to transport properties of |
| 1068 |
|
molecular liquids: |
| 1069 |
< |
\[ |
| 1069 |
> |
\begin{equation} |
| 1070 |
|
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
| 1071 |
|
\right\rangle } dt |
| 1072 |
< |
\] |
| 1072 |
> |
\end{equation} |
| 1073 |
|
where $D$ is diffusion constant. Unlike the velocity autocorrelation |
| 1074 |
|
function, which is averaged over time origins and over all the |
| 1075 |
|
atoms, the dipole autocorrelation functions is calculated for the |
| 1076 |
|
entire system. The dipole autocorrelation function is given by: |
| 1077 |
< |
\[ |
| 1077 |
> |
\begin{equation} |
| 1078 |
|
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
| 1079 |
|
\right\rangle |
| 1080 |
< |
\] |
| 1080 |
> |
\end{equation} |
| 1081 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
| 1082 |
|
by |
| 1083 |
< |
\[ |
| 1083 |
> |
\begin{equation} |
| 1084 |
|
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
| 1085 |
< |
\] |
| 1085 |
> |
\end{equation} |
| 1086 |
|
In principle, many time correlation functions can be related to |
| 1087 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
| 1088 |
|
scattering spectra of molecular liquids. In practice, one can |
| 1089 |
|
extract the IR spectrum from the intensity of the molecular dipole |
| 1090 |
|
fluctuation at each frequency using the following relationship: |
| 1091 |
< |
\[ |
| 1091 |
> |
\begin{equation} |
| 1092 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
| 1093 |
|
i2\pi vt} dt}. |
| 1094 |
< |
\] |
| 1094 |
> |
\end{equation} |
| 1095 |
|
|
| 1096 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1097 |
|
|
| 1098 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1099 |
< |
areas, from engineering, physics, to chemistry. For example, |
| 1099 |
> |
areas, including engineering, physics and chemistry. For example, |
| 1100 |
|
missiles and vehicles are usually modeled by rigid bodies. The |
| 1101 |
|
movement of the objects in 3D gaming engines or other physics |
| 1102 |
|
simulators is governed by rigid body dynamics. In molecular |
| 1103 |
|
simulations, rigid bodies are used to simplify protein-protein |
| 1104 |
< |
docking studies\cite{Gray2003}. |
| 1104 |
> |
docking studies.\cite{Gray2003} |
| 1105 |
|
|
| 1106 |
|
It is very important to develop stable and efficient methods to |
| 1107 |
|
integrate the equations of motion for orientational degrees of |
| 1113 |
|
angles can overcome this difficulty\cite{Barojas1973}, the |
| 1114 |
|
computational penalty and the loss of angular momentum conservation |
| 1115 |
|
still remain. A singularity-free representation utilizing |
| 1116 |
< |
quaternions was developed by Evans in 1977\cite{Evans1977}. |
| 1117 |
< |
Unfortunately, this approach uses a nonseparable Hamiltonian |
| 1118 |
< |
resulting from the quaternion representation, which prevents the |
| 1116 |
> |
quaternions was developed by Evans in 1977.\cite{Evans1977} |
| 1117 |
> |
Unfortunately, this approach used a nonseparable Hamiltonian |
| 1118 |
> |
resulting from the quaternion representation, which prevented the |
| 1119 |
|
symplectic algorithm from being utilized. Another different approach |
| 1120 |
|
is to apply holonomic constraints to the atoms belonging to the |
| 1121 |
|
rigid body. Each atom moves independently under the normal forces |
| 1122 |
|
deriving from potential energy and constraint forces which are used |
| 1123 |
|
to guarantee the rigidness. However, due to their iterative nature, |
| 1124 |
|
the SHAKE and Rattle algorithms also converge very slowly when the |
| 1125 |
< |
number of constraints increases\cite{Ryckaert1977, Andersen1983}. |
| 1125 |
> |
number of constraints increases.\cite{Ryckaert1977, Andersen1983} |
| 1126 |
|
|
| 1127 |
|
A break-through in geometric literature suggests that, in order to |
| 1128 |
|
develop a long-term integration scheme, one should preserve the |
| 1132 |
|
proposed to evolve the Hamiltonian system in a constraint manifold |
| 1133 |
|
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 1134 |
|
An alternative method using the quaternion representation was |
| 1135 |
< |
developed by Omelyan\cite{Omelyan1998}. However, both of these |
| 1135 |
> |
developed by Omelyan.\cite{Omelyan1998} However, both of these |
| 1136 |
|
methods are iterative and inefficient. In this section, we descibe a |
| 1137 |
|
symplectic Lie-Poisson integrator for rigid bodies developed by |
| 1138 |
|
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 1139 |
|
|
| 1140 |
|
\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Bodies} |
| 1141 |
< |
The motion of a rigid body is Hamiltonian with the Hamiltonian |
| 1167 |
< |
function |
| 1141 |
> |
The Hamiltonian of a rigid body is given by |
| 1142 |
|
\begin{equation} |
| 1143 |
|
H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) + |
| 1144 |
|
V(q,Q) + \frac{1}{2}tr[(QQ^T - 1)\Lambda ]. |
| 1145 |
|
\label{introEquation:RBHamiltonian} |
| 1146 |
|
\end{equation} |
| 1147 |
< |
Here, $q$ and $Q$ are the position and rotation matrix for the |
| 1148 |
< |
rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , and |
| 1149 |
< |
$J$, a diagonal matrix, is defined by |
| 1147 |
> |
Here, $q$ and $Q$ are the position vector and rotation matrix for |
| 1148 |
> |
the rigid-body, $p$ and $P$ are conjugate momenta to $q$ and $Q$ , |
| 1149 |
> |
and $J$, a diagonal matrix, is defined by |
| 1150 |
|
\[ |
| 1151 |
|
I_{ii}^{ - 1} = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} } |
| 1152 |
|
\] |
| 1156 |
|
\begin{equation} |
| 1157 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
| 1158 |
|
\end{equation} |
| 1159 |
< |
which is used to ensure rotation matrix's unitarity. Differentiating |
| 1160 |
< |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
| 1161 |
< |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1188 |
< |
\begin{equation} |
| 1189 |
< |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1190 |
< |
\label{introEquation:RBFirstOrderConstraint} |
| 1191 |
< |
\end{equation} |
| 1192 |
< |
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1193 |
< |
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1159 |
> |
which is used to ensure the rotation matrix's unitarity. Using |
| 1160 |
> |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and Eq.~ |
| 1161 |
> |
\ref{introEquation:motionHamiltonianMomentum}, one can write down |
| 1162 |
|
the equations of motion, |
| 1163 |
|
\begin{eqnarray} |
| 1164 |
|
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
| 1166 |
|
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
| 1167 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1168 |
|
\end{eqnarray} |
| 1169 |
+ |
Differentiating Eq.~\ref{introEquation:orthogonalConstraint} and |
| 1170 |
+ |
using Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
| 1171 |
+ |
\begin{equation} |
| 1172 |
+ |
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
| 1173 |
+ |
\label{introEquation:RBFirstOrderConstraint} |
| 1174 |
+ |
\end{equation} |
| 1175 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1176 |
|
We can use a constraint force provided by a Lagrange multiplier on |
| 1177 |
< |
the normal manifold to keep the motion on constraint space. Or we |
| 1178 |
< |
can simply evolve the system on the constraint manifold. These two |
| 1179 |
< |
methods have been proved to be equivalent. The holonomic constraint |
| 1180 |
< |
and equations of motions define a constraint manifold for rigid |
| 1181 |
< |
bodies |
| 1177 |
> |
the normal manifold to keep the motion on the constraint space. Or |
| 1178 |
> |
we can simply evolve the system on the constraint manifold. These |
| 1179 |
> |
two methods have been proved to be equivalent. The holonomic |
| 1180 |
> |
constraint and equations of motions define a constraint manifold for |
| 1181 |
> |
rigid bodies |
| 1182 |
|
\[ |
| 1183 |
|
M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0} |
| 1184 |
|
\right\}. |
| 1185 |
|
\] |
| 1186 |
< |
Unfortunately, this constraint manifold is not the cotangent bundle |
| 1187 |
< |
$T^* SO(3)$ which can be consider as a symplectic manifold on Lie |
| 1188 |
< |
rotation group $SO(3)$. However, it turns out that under symplectic |
| 1189 |
< |
transformation, the cotangent space and the phase space are |
| 1216 |
< |
diffeomorphic. By introducing |
| 1186 |
> |
Unfortunately, this constraint manifold is not $T^* SO(3)$ which is |
| 1187 |
> |
a symplectic manifold on Lie rotation group $SO(3)$. However, it |
| 1188 |
> |
turns out that under symplectic transformation, the cotangent space |
| 1189 |
> |
and the phase space are diffeomorphic. By introducing |
| 1190 |
|
\[ |
| 1191 |
|
\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right), |
| 1192 |
|
\] |
| 1193 |
< |
the mechanical system subject to a holonomic constraint manifold $M$ |
| 1193 |
> |
the mechanical system subjected to a holonomic constraint manifold $M$ |
| 1194 |
|
can be re-formulated as a Hamiltonian system on the cotangent space |
| 1195 |
|
\[ |
| 1196 |
|
T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q = |
| 1252 |
|
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
| 1253 |
|
Lagrange multiplier $\Lambda$ is absent from the equations of |
| 1254 |
|
motion. This unique property eliminates the requirement of |
| 1255 |
< |
iterations which can not be avoided in other methods\cite{Kol1997, |
| 1256 |
< |
Omelyan1998}. Applying the hat-map isomorphism, we obtain the |
| 1257 |
< |
equation of motion for angular momentum on body frame |
| 1255 |
> |
iterations which can not be avoided in other methods.\cite{Kol1997, |
| 1256 |
> |
Omelyan1998} Applying the hat-map isomorphism, we obtain the |
| 1257 |
> |
equation of motion for angular momentum in the body frame |
| 1258 |
|
\begin{equation} |
| 1259 |
|
\dot \pi = \pi \times I^{ - 1} \pi + \sum\limits_i {\left( {Q^T |
| 1260 |
|
F_i (r,Q)} \right) \times X_i }. |
| 1267 |
|
\] |
| 1268 |
|
|
| 1269 |
|
\subsection{\label{introSection:SymplecticFreeRB}Symplectic |
| 1270 |
< |
Lie-Poisson Integrator for Free Rigid Body} |
| 1270 |
> |
Lie-Poisson Integrator for Free Rigid Bodies} |
| 1271 |
|
|
| 1272 |
|
If there are no external forces exerted on the rigid body, the only |
| 1273 |
|
contribution to the rotational motion is from the kinetic energy |
| 1319 |
|
\end{array}} \right),\theta _1 = \frac{{\pi _1 }}{{I_1 }}\Delta t. |
| 1320 |
|
\] |
| 1321 |
|
To reduce the cost of computing expensive functions in $e^{\Delta |
| 1322 |
< |
tR_1 }$, we can use Cayley transformation to obtain a single-aixs |
| 1323 |
< |
propagator, |
| 1324 |
< |
\[ |
| 1325 |
< |
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1326 |
< |
). |
| 1327 |
< |
\] |
| 1328 |
< |
The propagator maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1322 |
> |
tR_1 }$, we can use the Cayley transformation to obtain a |
| 1323 |
> |
single-aixs propagator, |
| 1324 |
> |
\begin{eqnarray*} |
| 1325 |
> |
e^{\Delta tR_1 } & \approx & (1 - \Delta tR_1 )^{ - 1} (1 + \Delta |
| 1326 |
> |
tR_1 ) \\ |
| 1327 |
> |
% |
| 1328 |
> |
& \approx & \left( \begin{array}{ccc} |
| 1329 |
> |
1 & 0 & 0 \\ |
| 1330 |
> |
0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
| 1331 |
> |
\theta^2 / 4} \\ |
| 1332 |
> |
0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
| 1333 |
> |
\theta^2 / 4} |
| 1334 |
> |
\end{array} |
| 1335 |
> |
\right). |
| 1336 |
> |
\end{eqnarray*} |
| 1337 |
> |
The propagators for $T_2^r$ and $T_3^r$ can be found in the same |
| 1338 |
|
manner. In order to construct a second-order symplectic method, we |
| 1339 |
|
split the angular kinetic Hamiltonian function into five terms |
| 1340 |
|
\[ |
| 1350 |
|
\circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi |
| 1351 |
|
_1 }. |
| 1352 |
|
\] |
| 1353 |
< |
The non-canonical Lie-Poisson bracket ${F, G}$ of two function |
| 1372 |
< |
$F(\pi )$ and $G(\pi )$ is defined by |
| 1353 |
> |
The non-canonical Lie-Poisson bracket $\{F, G\}$ of two functions $F(\pi )$ and $G(\pi )$ is defined by |
| 1354 |
|
\[ |
| 1355 |
|
\{ F,G\} (\pi ) = [\nabla _\pi F(\pi )]^T J(\pi )\nabla _\pi G(\pi |
| 1356 |
|
). |
| 1359 |
|
function $G$ is zero, $F$ is a \emph{Casimir}, which is the |
| 1360 |
|
conserved quantity in Poisson system. We can easily verify that the |
| 1361 |
|
norm of the angular momentum, $\parallel \pi |
| 1362 |
< |
\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel |
| 1362 |
> |
\parallel$, is a \emph{Casimir}.\cite{McLachlan1993} Let $F(\pi ) = S(\frac{{\parallel |
| 1363 |
|
\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ , |
| 1364 |
|
then by the chain rule |
| 1365 |
|
\[ |
| 1378 |
|
Splitting for Rigid Body} |
| 1379 |
|
|
| 1380 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1381 |
< |
energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations |
| 1381 |
> |
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations |
| 1382 |
|
of motion corresponding to potential energy and kinetic energy are |
| 1383 |
< |
listed in the below table, |
| 1383 |
> |
listed in Table~\ref{introTable:rbEquations}. |
| 1384 |
|
\begin{table} |
| 1385 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1386 |
+ |
\label{introTable:rbEquations} |
| 1387 |
|
\begin{center} |
| 1388 |
|
\begin{tabular}{|l|l|} |
| 1389 |
|
\hline |
| 1430 |
|
\begin{eqnarray} |
| 1431 |
|
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
| 1432 |
|
& & \circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \notag\\ |
| 1433 |
< |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1433 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} . |
| 1434 |
|
\label{introEquation:overallRBFlowMaps} |
| 1435 |
|
\end{eqnarray} |
| 1436 |
|
|
| 1438 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
| 1439 |
|
mimics a simple heat bath with stochastic and dissipative forces, |
| 1440 |
|
has been applied in a variety of studies. This section will review |
| 1441 |
< |
the theory of Langevin dynamics. A brief derivation of generalized |
| 1441 |
> |
the theory of Langevin dynamics. A brief derivation of the generalized |
| 1442 |
|
Langevin equation will be given first. Following that, we will |
| 1443 |
< |
discuss the physical meaning of the terms appearing in the equation |
| 1462 |
< |
as well as the calculation of friction tensor from hydrodynamics |
| 1463 |
< |
theory. |
| 1443 |
> |
discuss the physical meaning of the terms appearing in the equation. |
| 1444 |
|
|
| 1445 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1446 |
|
|
| 1449 |
|
environment, has been widely used in quantum chemistry and |
| 1450 |
|
statistical mechanics. One of the successful applications of |
| 1451 |
|
Harmonic bath model is the derivation of the Generalized Langevin |
| 1452 |
< |
Dynamics (GLE). Lets consider a system, in which the degree of |
| 1452 |
> |
Dynamics (GLE). Consider a system, in which the degree of |
| 1453 |
|
freedom $x$ is assumed to couple to the bath linearly, giving a |
| 1454 |
|
Hamiltonian of the form |
| 1455 |
|
\begin{equation} |
| 1460 |
|
with this degree of freedom, $H_B$ is a harmonic bath Hamiltonian, |
| 1461 |
|
\[ |
| 1462 |
|
H_B = \sum\limits_{\alpha = 1}^N {\left\{ {\frac{{p_\alpha ^2 |
| 1463 |
< |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha \omega _\alpha ^2 } |
| 1463 |
> |
}}{{2m_\alpha }} + \frac{1}{2}m_\alpha x_\alpha ^2 } |
| 1464 |
|
\right\}} |
| 1465 |
|
\] |
| 1466 |
|
where the index $\alpha$ runs over all the bath degrees of freedom, |
| 1505 |
|
differential equations,the Laplace transform is the appropriate tool |
| 1506 |
|
to solve this problem. The basic idea is to transform the difficult |
| 1507 |
|
differential equations into simple algebra problems which can be |
| 1508 |
< |
solved easily. Then, by applying the inverse Laplace transform, also |
| 1509 |
< |
known as the Bromwich integral, we can retrieve the solutions of the |
| 1510 |
< |
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
| 1511 |
< |
$, the Laplace transform of $f(t)$ is a new function defined as |
| 1508 |
> |
solved easily. Then, by applying the inverse Laplace transform, we |
| 1509 |
> |
can retrieve the solutions of the original problems. Let $f(t)$ be a |
| 1510 |
> |
function defined on $ [0,\infty ) $, the Laplace transform of $f(t)$ |
| 1511 |
> |
is a new function defined as |
| 1512 |
|
\[ |
| 1513 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
| 1514 |
|
\] |
| 1515 |
|
where $p$ is real and $L$ is called the Laplace Transform |
| 1516 |
< |
Operator. Below are some important properties of Laplace transform |
| 1516 |
> |
Operator. Below are some important properties of the Laplace transform |
| 1517 |
|
\begin{eqnarray*} |
| 1518 |
|
L(x + y) & = & L(x) + L(y) \\ |
| 1519 |
|
L(ax) & = & aL(x) \\ |
| 1526 |
|
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), \\ |
| 1527 |
|
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
| 1528 |
|
\end{eqnarray*} |
| 1529 |
< |
By the same way, the system coordinates become |
| 1529 |
> |
In the same way, the system coordinates become |
| 1530 |
|
\begin{eqnarray*} |
| 1531 |
|
mL(\ddot x) & = & |
| 1532 |
|
- \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\}} \\ |
| 1550 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1551 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1552 |
|
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1553 |
< |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1554 |
< |
\end{eqnarray*} |
| 1555 |
< |
\begin{eqnarray*} |
| 1556 |
< |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1557 |
< |
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1558 |
< |
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 1553 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}}\\ |
| 1554 |
> |
% |
| 1555 |
> |
& = & - |
| 1556 |
> |
\frac{{\partial W(x)}}{{\partial x}} - \int_0^t {\sum\limits_{\alpha |
| 1557 |
> |
= 1}^N {\left( { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha |
| 1558 |
> |
^2 }}} \right)\cos (\omega _\alpha |
| 1559 |
|
t)\dot x(t - \tau )d} \tau } \\ |
| 1560 |
|
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1561 |
|
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1582 |
|
(t)\dot x(t - \tau )d\tau } + R(t) |
| 1583 |
|
\label{introEuqation:GeneralizedLangevinDynamics} |
| 1584 |
|
\end{equation} |
| 1585 |
< |
which is known as the \emph{generalized Langevin equation}. |
| 1585 |
> |
which is known as the \emph{generalized Langevin equation} (GLE). |
| 1586 |
|
|
| 1587 |
|
\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}\textbf{Random Force and Dynamic Friction Kernel}} |
| 1588 |
|
|
| 1589 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
| 1590 |
|
implies it is completely deterministic within the context of a |
| 1591 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1592 |
< |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
| 1592 |
> |
uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)} |
| 1593 |
|
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
| 1594 |
|
0.$ This property is what we expect from a truly random process. As |
| 1595 |
|
long as the model chosen for $R(t)$ was a gaussian distribution in |
| 1618 |
|
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
| 1619 |
|
taken as a $delta$ function in time: |
| 1620 |
|
\[ |
| 1621 |
< |
\xi (t) = 2\xi _0 \delta (t) |
| 1621 |
> |
\xi (t) = 2\xi _0 \delta (t). |
| 1622 |
|
\] |
| 1623 |
|
Hence, the convolution integral becomes |
| 1624 |
|
\[ |
| 1633 |
|
which is known as the Langevin equation. The static friction |
| 1634 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1635 |
|
or be determined by Stokes' law for regular shaped particles. A |
| 1636 |
< |
briefly review on calculating friction tensor for arbitrary shaped |
| 1636 |
> |
brief review on calculating friction tensors for arbitrary shaped |
| 1637 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1638 |
|
|
| 1639 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
| 1643 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
| 1644 |
|
^2 }}x(0), |
| 1645 |
|
\] |
| 1646 |
< |
we can rewrite $R(T)$ as |
| 1646 |
> |
we can rewrite $R(t)$ as |
| 1647 |
|
\[ |
| 1648 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
| 1649 |
|
\] |
| 1654 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1655 |
|
\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 } } \\ |
| 1656 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1657 |
< |
& = &kT\xi (t) \\ |
| 1657 |
> |
& = &kT\xi (t) |
| 1658 |
|
\end{eqnarray*} |
| 1659 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1660 |
|
\begin{equation} |
