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10. Stochastic Dynamics

In many applications we encounter widely varying time scales. In such cases the ``fast''_ degrees of freedom dominate the choice of the time step, although they may be of lesser interest.
Example: A few slow-moving heavy ions in a bath of many light water molecules.

Strategy: Mimick the effect of the secondary particles by suitably sampled stochastic forces $\bf S$.

LANGEVIN'S equation of motion for a single ion in a viscous solvent:

\dot{\mbox{$\bf v$}}(t)=-\eta \mbox{$\bf v$}(t)+\mbox{$\bf a$}(t)
\end{displaymath} (10.1)

where the statistical properties of the stochastic acceleration $\mbox{$\bf a$}\equiv \mbox{$\bf S$}/m$ are
$\displaystyle \langle \mbox{$\bf v$}(0) \cdot \mbox{$\bf a$}(t) \rangle$ $\textstyle =$ $\displaystyle 0 \hspace{60pt}
{\rm for} \;\;\; t\geq 0$ (10.2)
$\displaystyle \langle \mbox{$\bf a$}(0) \cdot \mbox{$\bf a$}(t) \rangle$ $\textstyle =$ $\displaystyle 3 \frac{2\eta kT}{m}   \delta (t)$ (10.3)


$\Longrightarrow$Solution of 10.1:

\mbox{$\bf v$}(t)=\mbox{$\bf v$}(0)e^{-\eta t}+\int \limits_{0}^{t}
e^{\textstyle-\eta(t-t')} \mbox{$\bf a$}(t')  dt'

and similar for $\bf r$(t). Subtracting $\mbox{$\bf v$}(t_{n})$ from $\mbox{$\bf v$}(t_{n})$ etc., we have
$\displaystyle \mbox{$\bf v$}_{n+1}$ $\textstyle =$ $\displaystyle \mbox{$\bf v$}_{n}e^{\textstyle -\eta \Delta t}
+\int \limits_{0}^{\Delta t}
e^{\textstyle -\eta(\Delta t-t')} \mbox{$\bf a$}(t_{n}+t')  dt'$ (10.4)
$\displaystyle \mbox{$\bf r$}_{n+1}$ $\textstyle =$ $\displaystyle \mbox{$\bf r$}_{n}+\mbox{$\bf v$}_{n}
\frac{1-e^{\textstyle -\eta...
...rac{1-e^{\textstyle -\eta(\Delta t-t')}}{\eta}  \mbox{$\bf a$}(t_{n}+t')  dt'$ (10.5)

e(t)\equiv e^{\textstyle -\eta t} ,\;\;\;\;\;
f(t)\equiv \frac{1-e^{\textstyle -\eta t}}{\eta}
\end{displaymath} (10.6)

$\displaystyle \mbox{$\bf V$}_{n}\equiv\int \limits_{0}^{\Delta t} e(\Delta t-t')  \mbox{$\bf a$}(t_{n}+t')$     (10.7)
$\displaystyle \mbox{$\bf R$}_{n}\equiv\int \limits_{0}^{\Delta t} f(\Delta t-t')  \mbox{$\bf a$}(t_{n}+t')$     (10.8)

we may write the stepwise solution

$\displaystyle \mbox{$\bf v$}_{n+1}$ $\textstyle =$ $\displaystyle \mbox{$\bf v$}_{n}  e(\Delta t) + \mbox{$\bf V$}_{n}$ (10.9)
$\displaystyle \mbox{$\bf r$}_{n+1}$ $\textstyle =$ $\displaystyle \mbox{$\bf r$}_{n}+\mbox{$\bf v$}_{n}  f(\Delta t) + \mbox{$\bf R$}_{n}$ (10.10)

The components of the stochastic vectors $\mbox{$\bf V$}_{n}, \mbox{$\bf R$}_{n}$ are time integrals of the function $\mbox{$\bf a$}(t)$ whose statistical properties are given. $\Longrightarrow$ $\mbox{$\bf V$}_{n}, \mbox{$\bf R$}_{n}$ are themselves random variates with known statistics: $\langle V_{n}\rangle=\langle R_{n}\rangle=0$, $\langle V_{n}V_{n+1}\rangle=\langle R_{n} R_{n+1}\rangle=0$, and

$\displaystyle \langle V_{n}^{2}\rangle$ $\textstyle =$ $\displaystyle \frac{kT}{m}\left[ 1-e^{2}(\Delta t)\right]$ (10.11)
$\displaystyle \langle R_{n}^{2}\rangle$ $\textstyle =$ $\displaystyle \frac{kT}{m \eta^{2}}
\left[ 2 \eta \Delta t - 3 + 4e(\Delta t) - e^{2}(\Delta t)\right]$ (10.12)
$\displaystyle \langle V_{n}R_{n}\rangle$ $\textstyle =$ $\displaystyle \frac{kT\eta}{m}   f^{2}(\Delta t)$ (10.13)

In the chapter about stochastics we described a method to produce pairs of correlated Gaussian variates. We may apply this here to generate $V_{n}, R_{n}$ and insert these in 10.9-10.10.

The stochastic force need not be $\delta$-correlated. If the solvent particles have a mass that is comparable with that of the solute, they will also move with similar speeds. In such cases the generalized Langevin equation applies:
\dot{v}(t) = - \int_{0}^{t} M(t-t')  v(t')  dt' + a(t)
\end{displaymath} (10.14)

$\displaystyle \langle v(0) a(t) \rangle$ $\textstyle =$ $\displaystyle 0 \hspace{60pt}
\mbox{\rm for} \;\;\; t\geq 0$ (10.15)
$\displaystyle \langle a(0)a(t) \rangle$ $\textstyle =$ $\displaystyle \frac{kT}{m}   M(t)$ (10.16)

This is a stochastic integrodifferential equation involving the ``history'' of the solute particle's motion in the form of the memory function $M(t)$ (see [MORI 65]). Still, we may assume that $M(t)$ decays fast.

Two approaches:
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F. J. Vesely / University of Vienna