10. 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 10.3 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 10.1:

and similar for (t). Subtracting from etc., we have

(10.4) | |||

(10.5) |

Defining

(10.6) |

(10.7) | |||

(10.8) |

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

(10.11) | |||

(10.12) | |||

(10.13) |

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 10.9-10.10.

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 10.1 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 10.14 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]).

F. J. Vesely / University of Vienna