| 1379 |
|
The Hamiltonian of rigid body can be separated in terms of kinetic |
| 1380 |
|
energy and potential energy, $H = T(p,\pi ) + V(q,Q)$. The equations |
| 1381 |
|
of motion corresponding to potential energy and kinetic energy are |
| 1382 |
< |
listed in Table~\ref{introTable:rbEquations} |
| 1382 |
> |
listed in Table~\ref{introTable:rbEquations}. |
| 1383 |
|
\begin{table} |
| 1384 |
|
\caption{EQUATIONS OF MOTION DUE TO POTENTIAL AND KINETIC ENERGIES} |
| 1385 |
|
\label{introTable:rbEquations} |
| 1439 |
|
has been applied in a variety of studies. This section will review |
| 1440 |
|
the theory of Langevin dynamics. A brief derivation of generalized |
| 1441 |
|
Langevin equation will be given first. Following that, we will |
| 1442 |
< |
discuss the physical meaning of the terms appearing in the equation |
| 1443 |
< |
as well as the calculation of friction tensor from hydrodynamics |
| 1444 |
< |
theory. |
| 1442 |
> |
discuss the physical meaning of the terms appearing in the equation. |
| 1443 |
|
|
| 1444 |
|
\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation} |
| 1445 |
|
|
| 1588 |
|
One may notice that $R(t)$ depends only on initial conditions, which |
| 1589 |
|
implies it is completely deterministic within the context of a |
| 1590 |
|
harmonic bath. However, it is easy to verify that $R(t)$ is totally |
| 1591 |
< |
uncorrelated to $x$ and $\dot x$,$\left\langle {x(t)R(t)} |
| 1591 |
> |
uncorrelated to $x$ and $\dot x$, $\left\langle {x(t)R(t)} |
| 1592 |
|
\right\rangle = 0, \left\langle {\dot x(t)R(t)} \right\rangle = |
| 1593 |
|
0.$ This property is what we expect from a truly random process. As |
| 1594 |
|
long as the model chosen for $R(t)$ was a gaussian distribution in |
| 1617 |
|
infinitely quickly to motions in the system. Thus, $\xi (t)$ can be |
| 1618 |
|
taken as a $delta$ function in time: |
| 1619 |
|
\[ |
| 1620 |
< |
\xi (t) = 2\xi _0 \delta (t) |
| 1620 |
> |
\xi (t) = 2\xi _0 \delta (t). |
| 1621 |
|
\] |
| 1622 |
|
Hence, the convolution integral becomes |
| 1623 |
|
\[ |
