6.6 Stochastic Dynamics

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

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

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

(6.22) | |||

(6.23) |

Defining

(6.24) |

(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

(6.29) | |||

(6.30) | |||

(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 and insert these in 6.27-6.28.

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

where

This is a stochastic

- 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]).

See also: