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$\Longrightarrow$ Solution of (1):
$
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})$ 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')
\\
R_{n}\equiv\int \limits_{0}^{\Delta t} f(\Delta t-t') a(t_{n}+t')
\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} $
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 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 (3) 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:
where
$ \begin{eqnarray}
\langle v(0) a(t) \rangle &=& 0
\;\;\; 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:
vesely may-06
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