Franz J. Vesely > CompPhys Tutorial > Selected Applications > Stochastic Dynamics

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 Part III: Ch. 6

## 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 $S$.

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

$\dot{ v}(t) = - \eta v(t)+ a(t)$     (6.19)

where the statistical properties of the stochastic acceleration $a \equiv S/m$ are

$\begin{eqnarray} \langle v(0) \cdot a(t) \rangle & = & 0 \;\;\;\; {\rm for} \;\;\; t \geq 0 \\ \langle a(0) \cdot a(t) \rangle & = & 3 \frac{\textstyle 2 \eta kT}{\textstyle m} \delta (t) \end{eqnarray}$     (6.20-6.21)

Explanation:
• $a(t)$ is not correlated to previous values of the ion velocity
• Stochastic and frictional forces are mutually related (both are caused by collisions of the ion with solvent molecules)
• Since equation 6.21 gives us only the a.c.f. of $a(t)$, we have yet to specify its statistical distribution; the usual choice is a Gauss distribution for the components of $a(t)$

$\Longrightarrow$ Solution of 6.19:

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

and similar for $r(t)$. Subtracting $v(t_{n+1})$ from $v(t_{n})$ etc. we have

$\begin{eqnarray} v_{n+1} & = & v_{n} e^{\textstyle -\eta \Delta t} + \int \limits_{0}^{\Delta t} e^{\textstyle -\eta(\Delta t-t')} a(t_{n}+t') dt' \\ r_{n+1} & = & r_{n}+ v_{n} \frac{1-e^{\textstyle -\eta \Delta t}}{\eta} + \int \limits_{0}^{\Delta t} \frac{1-e^{\textstyle -\eta(\Delta t-t')}}{\eta} a(t_{n}+t') dt' \end{eqnarray}$

Defining

$e(t) \equiv e^{\textstyle -\eta \, t},\;\;\;\;\; f(t) \equiv \frac{\textstyle 1-e^{\textstyle -\eta \, t}}{\textstyle \eta}$
and

$\begin{eqnarray} V_{n} \equiv \int \limits_{0}^{\Delta t} e(\Delta t-t') a(t_{n}+t') dt' \\ R_{n} \equiv \int \limits_{0}^{\Delta t} f(\Delta t-t') a(t_{n}+t') dt' \end{eqnarray}$

we may write the stepwise solution

$\begin{eqnarray} v_{n+1} & = & v_{n} e(\Delta t) + V_{n} \\ r_{n+1} & = & r_{n}+ v_{n} f(\Delta t) + R_{n} \end{eqnarray}$      (6.27-6.28)

The components of the stochastic vectors $V_{n}, R_{n}$ are time integrals of the function $a(t)$ whose statistical properties are given. $\Longrightarrow$ $V_{n}, R_{n}$ 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

$\begin{eqnarray} \langle V_{n}^{2} \rangle & = & \frac{kT}{m} \left[ 1-e^{2}(\Delta t) \right] \\ \langle R_{n}^{2} \rangle & = & \frac{kT}{m \eta^{2}} \left[ 2 \eta \Delta t - 3 + 4e(\Delta t) - e^{2}(\Delta t) \right] \\ \langle V_{n} R_{n} \rangle & = & \frac{kT\eta}{m} f^{2}(\Delta t) \end{eqnarray}$

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.

Generalization:
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:

$\dot{v}(t) = - \int \limits_{0}^{t} M(t-t') v(t') dt' + a(t)$      (6.32)
where

$\begin{eqnarray} \langle v(0) a(t) \rangle & = & 0 \;\;\; {\rm for} \;\;\; t \geq 0 \\ \langle a(0) a(t) \rangle & = & \frac{kT}{m} M(t) \end{eqnarray}$

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:
• Approximate the memory function by a suitable class of functions: assuming that the Laplace transform $\widehat{M}(s)$ may be represented by a truncated chain fraction in the variable $s$, the integrodifferential equation may be replaced by a set of coupled differential equations. Written in matrix notation these equations have the same form as 6.19 and may be treated accordingly. [VESELY 84]
• Assume that $M(t)$ may be neglected after $K \approx 20-60$ time steps. Using a tabulated autocorrelation function $M(t)$ one may generate an autoregressive process by the method described in the chapter on stochastics. By replacing the integral in 6.32 by a sum over the most recent $K$ time steps, one arrives at a stepwise procedure to produce $v(t)$ and $r(t)$; see [l#SMITHH">SMITH 90], and also [NILSSON 90]).

vesely 2006

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