| 1448 |
|
\begin{eqnarray} |
| 1449 |
|
\varphi _{\Delta t} &=& \varphi _{\Delta t/2,F} \circ \varphi _{\Delta t/2,\tau } \notag\\ |
| 1450 |
|
& & \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\\ |
| 1451 |
< |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} .\\ |
| 1451 |
> |
& & \circ \varphi _{\Delta t/2,\tau } \circ \varphi _{\Delta t/2,F} . |
| 1452 |
|
\label{introEquation:overallRBFlowMaps} |
| 1453 |
|
\end{eqnarray} |
| 1454 |
|
|
| 1674 |
|
\left\langle {q_\alpha (t)q_\beta (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha (t)q_\alpha (0)} \right\rangle \\ |
| 1675 |
|
\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 } } \\ |
| 1676 |
|
& = &\sum\limits_\alpha {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha t)} \\ |
| 1677 |
< |
& = &kT\xi (t) \\ |
| 1677 |
> |
& = &kT\xi (t) |
| 1678 |
|
\end{eqnarray*} |
| 1679 |
|
Thus, we recover the \emph{second fluctuation dissipation theorem} |
| 1680 |
|
\begin{equation} |
