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Next: 7. Quantum Mechanical Simulation Up: 6. Simulation and Statistical Previous: 6.5.2 Particle-Mesh Methods (PM

6.6 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} (6.19)

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$}(t) \rangle$ $\textstyle =$ $\displaystyle 0 \hspace{60pt}
{\rm for} \;\;\; t\geq 0$ (6.20)
$\displaystyle \langle \mbox{$\bf a$}(0) \cdot \mbox{$\bf a$}(t) \rangle$ $\textstyle =$ $\displaystyle 3 \frac{2\eta kT}{m} \, \delta (t)$ (6.21)


$\Longrightarrow$Solution of 6.19:

\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$}(t_{n}+t')\, dt'$ (6.22)
$\displaystyle \mbox{$\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$}(t_{n}+t')\, dt'$ (6.23)

e(t)\equiv e^{\textstyle -\eta t}\,,\;\;\;\;\;
f(t)\equiv \frac{1-e^{\textstyle -\eta t}}{\eta}
\end{displaymath} (6.24)

$\displaystyle \mbox{$\bf V$}_{n}\equiv\int \limits_{0}^{\Delta t} e(\Delta t-t') \,\mbox{$\bf a$}(t_{n}+t')$     (6.25)
$\displaystyle \mbox{$\bf R$}_{n}\equiv\int \limits_{0}^{\Delta t} f(\Delta t-t') \,\mbox{$\bf a$}(t_{n}+t')$     (6.26)

we may write the stepwise solution

$\displaystyle \mbox{$\bf v$}_{n+1}$ $\textstyle =$ $\displaystyle \mbox{$\bf v$}_{n}\, e(\Delta t) + \mbox{$\bf V$}_{n}$ (6.27)
$\displaystyle \mbox{$\bf r$}_{n+1}$ $\textstyle =$ $\displaystyle \mbox{$\bf r$}_{n}+\mbox{$\bf v$}_{n}\, f(\Delta t) + \mbox{$\bf R$}_{n}$ (6.28)

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

$\displaystyle \langle V_{n}^{2}\rangle$ $\textstyle =$ $\displaystyle \frac{kT}{m}\left[ 1-e^{2}(\Delta t)\right]$ (6.29)
$\displaystyle \langle R_{n}^{2}\rangle$ $\textstyle =$ $\displaystyle \frac{kT}{m \eta^{2}}
\left[ 2 \eta \Delta t - 3 + 4e(\Delta t) - e^{2}(\Delta t)\right]$ (6.30)
$\displaystyle \langle V_{n}R_{n}\rangle$ $\textstyle =$ $\displaystyle \frac{kT\eta}{m} \, f^{2}(\Delta t)$ (6.31)

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 6.27-6.28.

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} (6.32)

$\displaystyle \langle v(0) a(t) \rangle$ $\textstyle =$ $\displaystyle 0 \hspace{60pt}
\mbox{\rm for} \;\;\; t\geq 0$ (6.33)
$\displaystyle \langle a(0)a(t) \rangle$ $\textstyle =$ $\displaystyle \frac{kT}{m} \, M(t)$ (6.34)

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:
next up previous
Next: 7. Quantum Mechanical Simulation Up: 6. Simulation and Statistical Previous: 6.5.2 Particle-Mesh Methods (PM
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001