| 376 |
|
bead $i$ and origin $O$, the elements of resistance tensor at |
| 377 |
|
arbitrary origin $O$ can be written as |
| 378 |
|
\begin{eqnarray} |
| 379 |
+ |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 380 |
|
\Xi _{}^{tt} & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 381 |
|
\Xi _{}^{tr} & = & \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 382 |
|
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } |
| 383 |
|
U_j + 6 \eta V {\bf I}. \notag |
| 383 |
– |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 384 |
|
\end{eqnarray} |
| 385 |
|
The final term in the expression for $\Xi^{rr}$ is correction that |
| 386 |
|
accounts for errors in the rotational motion of certain kinds of bead |
| 469 |
|
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a |
| 470 |
|
system force $\mathbf{F}_{s}$, a frictional or dissipative force |
| 471 |
|
$\mathbf{F}_{f}$ and stochastic force $\mathbf{F}_{r}$. While the |
| 472 |
< |
evolution of the system in Newtownian mechanics is typically done in the |
| 473 |
< |
lab-fixed frame, it is convenient to handle the rotation of rigid |
| 474 |
< |
bodies in the body-fixed frame. Thus the friction and random forces are |
| 475 |
< |
calculated in body-fixed frame and converted back to lab-fixed frame |
| 476 |
< |
using the rigid body's rotation matrix ($Q$): |
| 472 |
> |
evolution of the system in Newtonian mechanics is typically done in |
| 473 |
> |
the lab-fixed frame, it is convenient to handle the dynamics of rigid |
| 474 |
> |
bodies in the body-fixed frame. Thus the friction and random forces |
| 475 |
> |
are calculated in body-fixed frame and may be converted back to |
| 476 |
> |
lab-fixed frame using the rigid body's rotation matrix ($Q$): |
| 477 |
|
\begin{equation} |
| 478 |
< |
\begin{array}{l} |
| 479 |
< |
\mathbf{F}_{f}(t) = Q^{T} \mathbf{F}_{f}^b (t), \\ |
| 480 |
< |
\mathbf{F}_{r}(t) = Q^{T} \mathbf{F}_{r}^b (t). \\ |
| 481 |
< |
\end{array} |
| 478 |
> |
\mathbf{F}_{f,r} = |
| 479 |
> |
\left( \begin{array}{c} |
| 480 |
> |
\mathbf{f}_{f,r} \\ |
| 481 |
> |
\mathbf{\tau}_{f,r} |
| 482 |
> |
\end{array} \right) |
| 483 |
> |
= |
| 484 |
> |
\left( \begin{array}{c} |
| 485 |
> |
Q^{T} \mathbf{f}^{b}_{f,r} \\ |
| 486 |
> |
Q^{T} \mathbf{\tau}^{b}_{f,r} |
| 487 |
> |
\end{array} \right) |
| 488 |
|
\end{equation} |
| 489 |
< |
Here, the body-fixed friction force $\mathbf{F}_{f}^b$ is proportional to |
| 490 |
< |
the body-fixed velocity at the center of resistance $\mathbf{v}_{R}^b$ and |
| 491 |
< |
angular velocity $\mathbf{\omega}$ |
| 489 |
> |
The body-fixed friction force, $\mathbf{F}_{f}^b$, is proportional to |
| 490 |
> |
the velocity at the center of resistance $\mathbf{v}_{R}^b$ (in the |
| 491 |
> |
body-fixed frame) and the angular velocity $\mathbf{\omega}$ |
| 492 |
|
\begin{equation} |
| 493 |
|
\mathbf{F}_{f}^b (t) = \left( \begin{array}{l} |
| 494 |
|
\mathbf{f}_{f}^b (t) \\ |
| 501 |
|
\mathbf{\omega} (t) \\ |
| 502 |
|
\end{array} \right), |
| 503 |
|
\end{equation} |
| 504 |
< |
while the random force $\mathbf{F}_{r}^l$ is a Gaussian stochastic variable |
| 505 |
< |
with zero mean and variance |
| 504 |
> |
while the random force, $\mathbf{F}_{r}$, is a Gaussian stochastic |
| 505 |
> |
variable with zero mean and variance |
| 506 |
|
\begin{equation} |
| 507 |
< |
\left\langle {\mathbf{F}_{r}^l (t) (\mathbf{F}_{r}^l (t'))^T } \right\rangle = |
| 507 |
> |
\left\langle {\mathbf{F}_{r}(t) (\mathbf{F}_{r}(t'))^T } \right\rangle = |
| 508 |
|
\left\langle {\mathbf{F}_{r}^b (t) (\mathbf{F}_{r}^b (t'))^T } \right\rangle = |
| 509 |
|
2 k_B T \Xi_R \delta(t - t'). \label{randomForce} |
| 510 |
|
\end{equation} |
| 511 |
< |
Once the $6\times6$ resistance tensor at the center of resistance |
| 512 |
< |
($\Xi_R$) is known, obtaining a stochastic vector that has the |
| 513 |
< |
properties in Eq. (\ref{eq:randomForce}) can be done efficiently by |
| 514 |
< |
carrying out a one-time Cholesky decomposition to obtain the square |
| 515 |
< |
root matrix of $\Xi_R$.\cite{SchlickBook} Each time a random force |
| 516 |
< |
vector is needed, a gaussian random vector is generated and then the |
| 517 |
< |
square root matrix is multiplied onto this vector. |
| 511 |
> |
$\Xi_R$ is the $6\times6$ resistance tensor at the center of |
| 512 |
> |
resistance. Once this tensor is known for a given rigid body, |
| 513 |
> |
obtaining a stochastic vector that has the properties in |
| 514 |
> |
Eq. (\ref{eq:randomForce}) can be done efficiently by carrying out a |
| 515 |
> |
one-time Cholesky decomposition to obtain the square root matrix of |
| 516 |
> |
the resistance tensor $\Xi_R = \mathbf{S} \mathbf{S}^{T}$, where |
| 517 |
> |
$\mathbf{S}$ is a lower triangular matrix.\cite{SchlickBook} A vector |
| 518 |
> |
with the statistics required for the random force can then be obtained |
| 519 |
> |
by multiplying $\mathbf{S}$ onto a 6-vector $Z$ which has elements |
| 520 |
> |
chosen from a Gaussian distribution, such that: |
| 521 |
> |
\begin{equation} |
| 522 |
> |
\langle Z_i \rangle = 0, \hspace{1in} \langle Z_i \cdot Z_j \rangle = \frac{2 k_B |
| 523 |
> |
T}{\delta t} \delta_{ij}. |
| 524 |
> |
\end{equation} |
| 525 |
> |
The random force, $F_{r}^{b} = \mathbf{S} Z$, can be shown to have the |
| 526 |
> |
correct ohmic |
| 527 |
|
|
| 528 |
+ |
|
| 529 |
+ |
Each |
| 530 |
+ |
time a random force vector is needed, a gaussian random vector is |
| 531 |
+ |
generated and then the square root matrix is multiplied onto this |
| 532 |
+ |
vector. |
| 533 |
+ |
|
| 534 |
|
The equation of motion for $\mathbf{v}$ can be written as |
| 535 |
|
\begin{equation} |
| 536 |
< |
m \dot{\mathbf{v}} (t) = \mathbf{f}_{s} (t) + \mathbf{f}_{f}^l (t) + |
| 536 |
> |
m \dot{\mathbf{v}} (t) = \mathbf{f}_{s}^l (t) + \mathbf{f}_{f}^l (t) + |
| 537 |
|
\mathbf{f}_{r}^l (t) |
| 538 |
|
\end{equation} |
| 539 |
|
Since the frictional force is applied at the center of resistance |
| 542 |
|
frictional torque at the center of mass, $\tau_{f}^b (t)$, is |
| 543 |
|
given by |
| 544 |
|
\begin{equation} |
| 545 |
< |
\tau_{f}^b \leftarrow \tau_{f}^b + \mathbf{r}_{MR} \times \mathbf{f}_{r}^b |
| 545 |
> |
\tau_{f}^b \leftarrow \tau_{f}^b + \mathbf{r}_{MR} \times \mathbf{f}_{f}^b |
| 546 |
|
\end{equation} |
| 547 |
|
where $r_{MR}$ is the vector from the center of mass to the center |
| 548 |
|
of the resistance. Instead of integrating the angular velocity in |
| 549 |
|
lab-fixed frame, we consider the equation of angular momentum in |
| 550 |
|
body-fixed frame |
| 551 |
|
\begin{equation} |
| 552 |
< |
\dot j(t) = \tau_{s} (t) + \tau_{f}^b (t) + \tau_{r}^b(t) |
| 552 |
> |
\dot j(t) = \tau_{s}^b (t) + \tau_{f}^b (t) + \tau_{r}^b(t) |
| 553 |
|
\end{equation} |
| 554 |
|
Embedding the friction terms into force and torque, one can integrate |
| 555 |
|
the Langevin equations of motion for rigid body of arbitrary shape in |
