88 |
|
trajectory, along which a dynamical system may move from one point |
89 |
|
to another within a specified time, is derived by finding the path |
90 |
|
which minimizes the time integral of the difference between the |
91 |
< |
kinetic, $K$, and potential energies, $U$, |
91 |
> |
kinetic $K$, and potential energies $U$, |
92 |
|
\begin{equation} |
93 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0}. |
94 |
|
\label{introEquation:halmitonianPrinciple1} |
243 |
|
Governed by the principles of mechanics, the phase points change |
244 |
|
their locations which would change the density at any time at phase |
245 |
|
space. Hence, the density distribution is also to be taken as a |
246 |
< |
function of the time. |
247 |
< |
|
248 |
< |
The number of systems $\delta N$ at time $t$ can be determined by, |
246 |
> |
function of the time. The number of systems $\delta N$ at time $t$ |
247 |
> |
can be determined by, |
248 |
|
\begin{equation} |
249 |
|
\delta N = \rho (q,p,t)dq_1 \ldots dq_f dp_1 \ldots dp_f. |
250 |
|
\label{introEquation:deltaN} |
277 |
|
\begin{equation} |
278 |
|
\langle A(q , p) \rangle_t = \frac{{\int { \ldots \int {A(q,p)\rho |
279 |
|
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}{{\int { \ldots \int {\rho |
280 |
< |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }} |
280 |
> |
(q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
281 |
|
\label{introEquation:ensembelAverage} |
282 |
|
\end{equation} |
283 |
|
|
291 |
|
\begin{equation} |
292 |
|
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
293 |
|
\end{equation} |
294 |
< |
A canonical ensemble (NVT)is an ensemble of systems, each of which |
294 |
> |
A canonical ensemble (NVT) is an ensemble of systems, each of which |
295 |
|
can share its energy with a large heat reservoir. The distribution |
296 |
|
of the total energy amongst the possible dynamical states is given |
297 |
|
by the partition function, |
409 |
|
q_i }}} \right)}. |
410 |
|
\label{introEquation:poissonBracket} |
411 |
|
\end{equation} |
412 |
< |
Substituting equations of motion in Hamiltonian formalism( |
413 |
< |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
414 |
< |
Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into |
412 |
> |
Substituting equations of motion in Hamiltonian formalism |
413 |
> |
(Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
414 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum}) into |
415 |
|
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
416 |
|
Liouville's theorem using Poisson bracket notion, |
417 |
|
\begin{equation} |
445 |
|
many-body system in Statistical Mechanics. Fortunately, the Ergodic |
446 |
|
Hypothesis makes a connection between time average and the ensemble |
447 |
|
average. It states that the time average and average over the |
448 |
< |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}. |
448 |
> |
statistical ensemble are identical \cite{Frenkel1996, Leach2001}: |
449 |
|
\begin{equation} |
450 |
|
\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty } |
451 |
|
\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma |
459 |
|
a properly weighted statistical average. This allows the researcher |
460 |
|
freedom of choice when deciding how best to measure a given |
461 |
|
observable. In case an ensemble averaged approach sounds most |
462 |
< |
reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be |
462 |
> |
reasonable, the Monte Carlo methods\cite{Metropolis1949} can be |
463 |
|
utilized. Or if the system lends itself to a time averaging |
464 |
|
approach, the Molecular Dynamics techniques in |
465 |
|
Sec.~\ref{introSection:molecularDynamics} will be the best |
509 |
|
\dot x = f(x) |
510 |
|
\end{equation} |
511 |
|
where $x = x(q,p)^T$, this system is a canonical Hamiltonian, if |
512 |
+ |
$f(r) = J\nabla _x H(r)$. Here, $H = H (q, p)$ is Hamiltonian |
513 |
+ |
function and $J$ is the skew-symmetric matrix |
514 |
|
\begin{equation} |
514 |
– |
f(r) = J\nabla _x H(r). |
515 |
– |
\end{equation} |
516 |
– |
$H = H (q, p)$ is Hamiltonian function and $J$ is the skew-symmetric |
517 |
– |
matrix |
518 |
– |
\begin{equation} |
515 |
|
J = \left( {\begin{array}{*{20}c} |
516 |
|
0 & I \\ |
517 |
|
{ - I} & 0 \\ |
521 |
|
where $I$ is an identity matrix. Using this notation, Hamiltonian |
522 |
|
system can be rewritten as, |
523 |
|
\begin{equation} |
524 |
< |
\frac{d}{{dt}}x = J\nabla _x H(x) |
524 |
> |
\frac{d}{{dt}}x = J\nabla _x H(x). |
525 |
|
\label{introEquation:compactHamiltonian} |
526 |
|
\end{equation}In this case, $f$ is |
527 |
|
called a \emph{Hamiltonian vector field}. Another generalization of |
533 |
|
|
534 |
|
\subsection{\label{introSection:exactFlow}Exact Flow} |
535 |
|
|
536 |
< |
Let $x(t)$ be the exact solution of the ODE system, |
537 |
< |
\begin{equation} |
538 |
< |
\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE} |
539 |
< |
\end{equation} |
540 |
< |
The exact flow(solution) $\varphi_\tau$ is defined by |
545 |
< |
\[ |
546 |
< |
x(t+\tau) =\varphi_\tau(x(t)) |
536 |
> |
Let $x(t)$ be the exact solution of the ODE |
537 |
> |
system,$\frac{{dx}}{{dt}} = f(x) \label{introEquation:ODE}$, we can |
538 |
> |
define its exact flow(solution) $\varphi_\tau$ |
539 |
> |
\[ x(t+\tau) |
540 |
> |
=\varphi_\tau(x(t)) |
541 |
|
\] |
542 |
|
where $\tau$ is a fixed time step and $\varphi$ is a map from phase |
543 |
|
space to itself. The flow has the continuous group property, |
559 |
|
}{{\partial x_i }}} } (x) \equiv \exp (\tau f)(x). |
560 |
|
\label{introEquation:exponentialOperator} |
561 |
|
\end{equation} |
568 |
– |
|
562 |
|
In most cases, it is not easy to find the exact flow $\varphi_\tau$. |
563 |
|
Instead, we use an approximate map, $\psi_\tau$, which is usually |
564 |
|
called integrator. The order of an integrator $\psi_\tau$ is $p$, if |
605 |
|
\] |
606 |
|
Using chain rule, one may obtain, |
607 |
|
\[ |
608 |
< |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G, |
608 |
> |
\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \dot \nabla G, |
609 |
|
\] |
610 |
|
which is the condition for conserving \emph{first integral}. For a |
611 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
612 |
|
smooth function $G$ is given by, |
613 |
|
\begin{eqnarray} |
614 |
< |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
615 |
< |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
614 |
> |
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \notag\\ |
615 |
> |
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). |
616 |
|
\label{introEquation:firstIntegral1} |
617 |
|
\end{eqnarray} |
618 |
< |
Using poisson bracket notion, Equation |
619 |
< |
\ref{introEquation:firstIntegral1} can be rewritten as |
618 |
> |
Using poisson bracket notion, Eq.~\ref{introEquation:firstIntegral1} |
619 |
> |
can be rewritten as |
620 |
|
\[ |
621 |
|
\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)). |
622 |
|
\] |
623 |
|
Therefore, the sufficient condition for $G$ to be the \emph{first |
624 |
< |
integral} of a Hamiltonian system is |
632 |
< |
\[ |
633 |
< |
\left\{ {G,H} \right\} = 0. |
634 |
< |
\] |
624 |
> |
integral} of a Hamiltonian system is $\left\{ {G,H} \right\} = 0.$ |
625 |
|
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
626 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
627 |
|
0$. When designing any numerical methods, one should always try to |
640 |
|
\item Runge-Kutta methods |
641 |
|
\item Splitting methods |
642 |
|
\end{enumerate} |
653 |
– |
|
643 |
|
Generating function\cite{Channell1990} tends to lead to methods |
644 |
|
which are cumbersome and difficult to use. In dissipative systems, |
645 |
|
variational methods can capture the decay of energy |
646 |
|
accurately\cite{Kane2000}. Since their geometrically unstable nature |
647 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
648 |
|
methods are not suitable for Hamiltonian system. Recently, various |
649 |
< |
high-order explicit Runge-Kutta methods |
650 |
< |
\cite{Owren1992,Chen2003}have been developed to overcome this |
651 |
< |
instability. However, due to computational penalty involved in |
652 |
< |
implementing the Runge-Kutta methods, they have not attracted much |
653 |
< |
attention from the Molecular Dynamics community. Instead, splitting |
654 |
< |
methods have been widely accepted since they exploit natural |
655 |
< |
decompositions of the system\cite{Tuckerman1992, McLachlan1998}. |
649 |
> |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
650 |
> |
have been developed to overcome this instability. However, due to |
651 |
> |
computational penalty involved in implementing the Runge-Kutta |
652 |
> |
methods, they have not attracted much attention from the Molecular |
653 |
> |
Dynamics community. Instead, splitting methods have been widely |
654 |
> |
accepted since they exploit natural decompositions of the |
655 |
> |
system\cite{Tuckerman1992, McLachlan1998}. |
656 |
|
|
657 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
658 |
|
|
693 |
|
where $\phi$ and $\psi$ both are symplectic maps. Thus operator |
694 |
|
splitting in this context automatically generates a symplectic map. |
695 |
|
|
696 |
< |
The Lie-Trotter splitting(\ref{introEquation:firstOrderSplitting}) |
697 |
< |
introduces local errors proportional to $h^2$, while Strang |
698 |
< |
splitting gives a second-order decomposition, |
696 |
> |
The Lie-Trotter |
697 |
> |
splitting(Eq.~\ref{introEquation:firstOrderSplitting}) introduces |
698 |
> |
local errors proportional to $h^2$, while Strang splitting gives a |
699 |
> |
second-order decomposition, |
700 |
|
\begin{equation} |
701 |
|
\varphi _h = \varphi _{1,h/2} \circ \varphi _{2,h} \circ \varphi |
702 |
|
_{1,h/2} , \label{introEquation:secondOrderSplitting} |
793 |
|
\begin{eqnarray*} |
794 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
795 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
796 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
796 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots |
797 |
> |
). |
798 |
|
\end{eqnarray*} |
799 |
< |
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0,\] the dominant local |
799 |
> |
Since $ [X,Y] + [Y,X] = 0$ and $ [X,X] = 0$, the dominant local |
800 |
|
error of Spring splitting is proportional to $h^3$. The same |
801 |
< |
procedure can be applied to a general splitting, of the form |
801 |
> |
procedure can be applied to a general splitting of the form |
802 |
|
\begin{equation} |
803 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
804 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
943 |
|
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
944 |
|
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
945 |
|
which making large simulations prohibitive in the absence of any |
946 |
< |
algorithmic tricks. |
947 |
< |
|
948 |
< |
A natural approach to avoid system size issues is to represent the |
949 |
< |
bulk behavior by a finite number of the particles. However, this |
950 |
< |
approach will suffer from the surface effect at the edges of the |
951 |
< |
simulation. To offset this, \textit{Periodic boundary conditions} |
952 |
< |
(see Fig.~\ref{introFig:pbc}) is developed to simulate bulk |
953 |
< |
properties with a relatively small number of particles. In this |
954 |
< |
method, the simulation box is replicated throughout space to form an |
955 |
< |
infinite lattice. During the simulation, when a particle moves in |
956 |
< |
the primary cell, its image in other cells move in exactly the same |
957 |
< |
direction with exactly the same orientation. Thus, as a particle |
967 |
< |
leaves the primary cell, one of its images will enter through the |
968 |
< |
opposite face. |
946 |
> |
algorithmic tricks. A natural approach to avoid system size issues |
947 |
> |
is to represent the bulk behavior by a finite number of the |
948 |
> |
particles. However, this approach will suffer from the surface |
949 |
> |
effect at the edges of the simulation. To offset this, |
950 |
> |
\textit{Periodic boundary conditions} (see Fig.~\ref{introFig:pbc}) |
951 |
> |
is developed to simulate bulk properties with a relatively small |
952 |
> |
number of particles. In this method, the simulation box is |
953 |
> |
replicated throughout space to form an infinite lattice. During the |
954 |
> |
simulation, when a particle moves in the primary cell, its image in |
955 |
> |
other cells move in exactly the same direction with exactly the same |
956 |
> |
orientation. Thus, as a particle leaves the primary cell, one of its |
957 |
> |
images will enter through the opposite face. |
958 |
|
\begin{figure} |
959 |
|
\centering |
960 |
|
\includegraphics[width=\linewidth]{pbc.eps} |
1017 |
|
monitor the motions of molecules. Although the dynamics of the |
1018 |
|
system can be described qualitatively from animation, quantitative |
1019 |
|
trajectory analysis are more useful. According to the principles of |
1020 |
< |
Statistical Mechanics, Sec.~\ref{introSection:statisticalMechanics}, |
1021 |
< |
one can compute thermodynamic properties, analyze fluctuations of |
1022 |
< |
structural parameters, and investigate time-dependent processes of |
1023 |
< |
the molecule from the trajectories. |
1020 |
> |
Statistical Mechanics in |
1021 |
> |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1022 |
> |
thermodynamic properties, analyze fluctuations of structural |
1023 |
> |
parameters, and investigate time-dependent processes of the molecule |
1024 |
> |
from the trajectories. |
1025 |
|
|
1026 |
|
\subsubsection{\label{introSection:thermodynamicsProperties}\textbf{Thermodynamic Properties}} |
1027 |
|
|
1061 |
|
The pair distribution functions $g(r)$ gives the probability that a |
1062 |
|
particle $i$ will be located at a distance $r$ from a another |
1063 |
|
particle $j$ in the system |
1064 |
< |
\[ |
1064 |
> |
\begin{equation} |
1065 |
|
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1066 |
|
\ne i} {\delta (r - r_{ij} )} } } \right\rangle = \frac{\rho |
1067 |
|
(r)}{\rho}. |
1068 |
< |
\] |
1068 |
> |
\end{equation} |
1069 |
|
Note that the delta function can be replaced by a histogram in |
1070 |
|
computer simulation. Peaks in $g(r)$ represent solvent shells, and |
1071 |
|
the height of these peaks gradually decreases to 1 as the liquid of |
1102 |
|
Here $u_{tot}$ is the net dipole of the entire system and is given |
1103 |
|
by |
1104 |
|
\[ |
1105 |
< |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1105 |
> |
u_{tot} (t) = \sum\limits_i {u_i (t)}. |
1106 |
|
\] |
1107 |
|
In principle, many time correlation functions can be related with |
1108 |
|
Fourier transforms of the infrared, Raman, and inelastic neutron |
1111 |
|
each frequency using the following relationship: |
1112 |
|
\[ |
1113 |
|
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1114 |
< |
i2\pi vt} dt} |
1114 |
> |
i2\pi vt} dt}. |
1115 |
|
\] |
1116 |
|
|
1117 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1179 |
|
Q^T Q = 1, \label{introEquation:orthogonalConstraint} |
1180 |
|
\end{equation} |
1181 |
|
which is used to ensure rotation matrix's unitarity. Differentiating |
1182 |
< |
\ref{introEquation:orthogonalConstraint} and using Equation |
1183 |
< |
\ref{introEquation:RBMotionMomentum}, one may obtain, |
1182 |
> |
Eq.~\ref{introEquation:orthogonalConstraint} and using |
1183 |
> |
Eq.~\ref{introEquation:RBMotionMomentum}, one may obtain, |
1184 |
|
\begin{equation} |
1185 |
|
Q^T PJ^{ - 1} + J^{ - 1} P^T Q = 0 . \\ |
1186 |
|
\label{introEquation:RBFirstOrderConstraint} |
1189 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
1190 |
|
the equations of motion, |
1191 |
|
\begin{eqnarray} |
1192 |
< |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
1193 |
< |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
1194 |
< |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
1192 |
> |
\frac{{dq}}{{dt}} & = & \frac{p}{m}, \label{introEquation:RBMotionPosition}\\ |
1193 |
> |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q), \label{introEquation:RBMotionMomentum}\\ |
1194 |
> |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1}, \label{introEquation:RBMotionRotation}\\ |
1195 |
|
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
1196 |
|
\end{eqnarray} |
1197 |
|
In general, there are two ways to satisfy the holonomic constraints. |
1247 |
|
\end{array} |
1248 |
|
\label{introEqaution:RBMotionPI} |
1249 |
|
\end{equation} |
1250 |
< |
as well as holonomic constraints, |
1251 |
< |
\[ |
1252 |
< |
\begin{array}{l} |
1263 |
< |
\Pi J^{ - 1} + J^{ - 1} \Pi ^t = 0, \\ |
1264 |
< |
Q^T Q = 1 .\\ |
1265 |
< |
\end{array} |
1266 |
< |
\] |
1267 |
< |
For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in |
1268 |
< |
so(3)^ \star$, the hat-map isomorphism, |
1250 |
> |
as well as holonomic constraints $\Pi J^{ - 1} + J^{ - 1} \Pi ^t = |
1251 |
> |
0$ and $Q^T Q = 1$. For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a |
1252 |
> |
matrix $\hat v \in so(3)^ \star$, the hat-map isomorphism, |
1253 |
|
\begin{equation} |
1254 |
|
v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left( |
1255 |
|
{\begin{array}{*{20}c} |
1267 |
|
Using Eq.~\ref{introEqaution:RBMotionPI}, one can construct a skew |
1268 |
|
matrix, |
1269 |
|
\begin{eqnarray} |
1270 |
< |
(\dot \Pi - \dot \Pi ^T ){\rm{ }} &= &{\rm{ }}(\Pi - \Pi ^T ){\rm{ |
1271 |
< |
}}(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
1272 |
< |
+ \sum\limits_i {[Q^T F_i |
1289 |
< |
(r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - (\Lambda - \Lambda ^T ). |
1290 |
< |
\label{introEquation:skewMatrixPI} |
1270 |
> |
(\dot \Pi - \dot \Pi ^T )&= &(\Pi - \Pi ^T )(J^{ - 1} \Pi + \Pi J^{ - 1} ) \notag \\ |
1271 |
> |
& & + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
1272 |
> |
(\Lambda - \Lambda ^T ). \label{introEquation:skewMatrixPI} |
1273 |
|
\end{eqnarray} |
1274 |
|
Since $\Lambda$ is symmetric, the last term of |
1275 |
|
Eq.~\ref{introEquation:skewMatrixPI} is zero, which implies the |
1314 |
|
\begin{equation} |
1315 |
|
\frac{d}{{dt}}\pi = J(\pi )\nabla _\pi T^r (\pi ). |
1316 |
|
\end{equation} |
1317 |
< |
One may notice that each $T_i^r$ in Equation |
1318 |
< |
\ref{introEquation:rotationalKineticRB} can be solved exactly. For |
1319 |
< |
instance, the equations of motion due to $T_1^r$ are given by |
1317 |
> |
One may notice that each $T_i^r$ in |
1318 |
> |
Eq.~\ref{introEquation:rotationalKineticRB} can be solved exactly. |
1319 |
> |
For instance, the equations of motion due to $T_1^r$ are given by |
1320 |
|
\begin{equation} |
1321 |
|
\frac{d}{{dt}}\pi = R_1 \pi ,\frac{d}{{dt}}Q = QR_1 |
1322 |
|
\label{introEqaution:RBMotionSingleTerm} |
1323 |
|
\end{equation} |
1324 |
< |
where |
1324 |
> |
with |
1325 |
|
\[ R_1 = \left( {\begin{array}{*{20}c} |
1326 |
|
0 & 0 & 0 \\ |
1327 |
|
0 & 0 & {\pi _1 } \\ |
1328 |
|
0 & { - \pi _1 } & 0 \\ |
1329 |
|
\end{array}} \right). |
1330 |
|
\] |
1331 |
< |
The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is |
1331 |
> |
The solutions of Eq.~\ref{introEqaution:RBMotionSingleTerm} is |
1332 |
|
\[ |
1333 |
|
\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) = |
1334 |
|
Q(0)e^{\Delta tR_1 } |
1350 |
|
\] |
1351 |
|
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
1352 |
|
manner. In order to construct a second-order symplectic method, we |
1353 |
< |
split the angular kinetic Hamiltonian function can into five terms |
1353 |
> |
split the angular kinetic Hamiltonian function into five terms |
1354 |
|
\[ |
1355 |
|
T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2 |
1356 |
|
) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r |
1393 |
|
Splitting for Rigid Body} |
1394 |
|
|
1395 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
1396 |
< |
energy and potential energy, |
1397 |
< |
\[ |
1398 |
< |
H = T(p,\pi ) + V(q,Q). |
1417 |
< |
\] |
1418 |
< |
The equations of motion corresponding to potential energy and |
1419 |
< |
kinetic energy are listed in the below table, |
1396 |
> |
energy and potential energy,$H = T(p,\pi ) + V(q,Q)$. The equations |
1397 |
> |
of motion corresponding to potential energy and kinetic energy are |
1398 |
> |
listed in the below table, |
1399 |
|
\begin{table} |
1400 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
1401 |
|
\begin{center} |
1433 |
|
T(p,\pi ) =T^t (p) + T^r (\pi ). |
1434 |
|
\end{equation} |
1435 |
|
where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is |
1436 |
< |
defined by \ref{introEquation:rotationalKineticRB}. Therefore, the |
1437 |
< |
corresponding propagators are given by |
1436 |
> |
defined by Eq.~\ref{introEquation:rotationalKineticRB}. Therefore, |
1437 |
> |
the corresponding propagators are given by |
1438 |
|
\[ |
1439 |
|
\varphi _{\Delta t,T} = \varphi _{\Delta t,T^t } \circ \varphi |
1440 |
|
_{\Delta t,T^r }. |
1441 |
|
\] |
1442 |
|
Finally, we obtain the overall symplectic propagators for freely |
1443 |
|
moving rigid bodies |
1444 |
< |
\begin{eqnarray*} |
1445 |
< |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \\ |
1446 |
< |
& & \circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \\ |
1444 |
> |
\begin{eqnarray} |
1445 |
> |
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
1446 |
> |
& & \circ \varphi _{\Delta t,T^t } \circ \varphi _{\Delta t/2,\pi _1 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t,\pi _3 } \circ \varphi _{\Delta t/2,\pi _2 } \circ \varphi _{\Delta t/2,\pi _1 } \notag\\ |
1447 |
|
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
1448 |
|
\label{introEquation:overallRBFlowMaps} |
1449 |
< |
\end{eqnarray*} |
1449 |
> |
\end{eqnarray} |
1450 |
|
|
1451 |
|
\section{\label{introSection:langevinDynamics}Langevin Dynamics} |
1452 |
|
As an alternative to newtonian dynamics, Langevin dynamics, which |
1524 |
|
solved easily. Then, by applying the inverse Laplace transform, also |
1525 |
|
known as the Bromwich integral, we can retrieve the solutions of the |
1526 |
|
original problems. Let $f(t)$ be a function defined on $ [0,\infty ) |
1527 |
< |
$. The Laplace transform of f(t) is a new function defined as |
1527 |
> |
$, the Laplace transform of $f(t)$ is a new function defined as |
1528 |
|
\[ |
1529 |
|
L(f(t)) \equiv F(p) = \int_0^\infty {f(t)e^{ - pt} dt} |
1530 |
|
\] |
1539 |
|
\end{eqnarray*} |
1540 |
|
Applying the Laplace transform to the bath coordinates, we obtain |
1541 |
|
\begin{eqnarray*} |
1542 |
< |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x) \\ |
1543 |
< |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }} \\ |
1542 |
> |
p^2 L(x_\alpha ) - px_\alpha (0) - \dot x_\alpha (0) & = & - \omega _\alpha ^2 L(x_\alpha ) + \frac{{g_\alpha }}{{\omega _\alpha }}L(x), \\ |
1543 |
> |
L(x_\alpha ) & = & \frac{{\frac{{g_\alpha }}{{\omega _\alpha }}L(x) + px_\alpha (0) + \dot x_\alpha (0)}}{{p^2 + \omega _\alpha ^2 }}. \\ |
1544 |
|
\end{eqnarray*} |
1545 |
|
By the same way, the system coordinates become |
1546 |
|
\begin{eqnarray*} |
1547 |
|
mL(\ddot x) & = & |
1548 |
|
- \sum\limits_{\alpha = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha \omega _\alpha ^2 }}\frac{p}{{p^2 + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2 + \omega _\alpha ^2 }}g_\alpha x_\alpha (0) - \frac{1}{{p^2 + \omega _\alpha ^2 }}g_\alpha \dot x_\alpha (0)} \right\}} \\ |
1549 |
< |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} |
1549 |
> |
& & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}}. |
1550 |
|
\end{eqnarray*} |
1551 |
|
With the help of some relatively important inverse Laplace |
1552 |
|
transformations: |
1605 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
1606 |
|
implies it is completely deterministic within the context of a |
1607 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
1608 |
< |
uncorrelated to $x$ and $\dot x$, |
1609 |
< |
\[ |
1610 |
< |
\begin{array}{l} |
1611 |
< |
\left\langle {x(t)R(t)} \right\rangle = 0, \\ |
1633 |
< |
\left\langle {\dot x(t)R(t)} \right\rangle = 0. \\ |
1634 |
< |
\end{array} |
1635 |
< |
\] |
1636 |
< |
This property is what we expect from a truly random process. As long |
1637 |
< |
as the model chosen for $R(t)$ was a gaussian distribution in |
1608 |
> |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
1609 |
> |
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
1610 |
> |
0.$ This property is what we expect from a truly random process. As |
1611 |
> |
long as the model chosen for $R(t)$ was a gaussian distribution in |
1612 |
|
general, the stochastic nature of the GLE still remains. |
1639 |
– |
|
1613 |
|
%dynamic friction kernel |
1614 |
|
The convolution integral |
1615 |
|
\[ |
1654 |
|
|
1655 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}\textbf{The Second Fluctuation Dissipation Theorem}} |
1656 |
|
|
1657 |
< |
Defining a new set of coordinates, |
1657 |
> |
Defining a new set of coordinates |
1658 |
|
\[ |
1659 |
|
q_\alpha (t) = x_\alpha (t) - \frac{1}{{m_\alpha \omega _\alpha |
1660 |
< |
^2 }}x(0) |
1661 |
< |
\], |
1660 |
> |
^2 }}x(0), |
1661 |
> |
\] |
1662 |
|
we can rewrite $R(T)$ as |
1663 |
|
\[ |
1664 |
|
R(t) = \sum\limits_{\alpha = 1}^N {g_\alpha q_\alpha (t)}. |
1675 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
1676 |
|
\begin{equation} |
1677 |
|
\xi (t) = \left\langle {R(t)R(0)} \right\rangle |
1678 |
< |
\label{introEquation:secondFluctuationDissipation}. |
1678 |
> |
\label{introEquation:secondFluctuationDissipation}, |
1679 |
|
\end{equation} |
1680 |
< |
In effect, it acts as a constraint on the possible ways in which one |
1681 |
< |
can model the random force and friction kernel. |
1680 |
> |
which acts as a constraint on the possible ways in which one can |
1681 |
> |
model the random force and friction kernel. |