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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 .
LANGEVIN'S equation of motion for a single ion in a viscous solvent:
|
(6.19) |
where the statistical properties of the stochastic
acceleration
are
Explanation:
-
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
,
we have yet to specify its statistical distribution; the usual choice
is a Gauss distribution for the components of
Solution of 6.19:
and similar for (t).
Subtracting
from
etc.,
we have
Defining
|
(6.24) |
and
|
|
|
(6.25) |
|
|
|
(6.26) |
we may write the stepwise solution
The components of the stochastic vectors
are time integrals of the function
whose statistical properties
are given.
are themselves
random variates with known statistics:
,
, and
In the chapter about stochastics we described a method to produce
pairs of correlated Gaussian variates. We may apply this here to
generate and insert these in 6.27-6.28.
Generalization:
The stochastic force need not be -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:
|
(6.32) |
where
This is a stochastic integrodifferential equation involving
the ``history'' of the solute particle's motion in the form
of the memory function (see [MORI 65]).
Still, we may assume that decays fast.
Two approaches:
- Approximate the memory function by a suitable class of functions:
assuming that the Laplace transform
may be represented
by a truncated chain fraction in the variable ,
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 may be neglected after
time steps.
Using a tabulated autocorrelation function 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 time steps, one arrives at a stepwise procedure
to produce
and
; see [SMITH 90], and also
[NILSSON 90]).
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