It sounds like you're trying to solve the Langevin Equation. This is a model of Brownian motion where the particle experiences stochastic kicks at discrete time intervals. Your force, in this case, is a random variable you draw from a distribution each time step (instead of being given by an explicit formula).
For the simplest case, the "kicks" are generated by thermal noise, and have a gaussian distribution.
From the Wikipedia article linked above, the equation of motion is:
$$
m \frac{d^2 \vec{x}}{dt^2} = - \lambda \frac{d\vec{x}}{dt} + \vec{f(t)}
$$
Here the $\lambda$ term is from viscous friction with the fluid, and $\vec{f}$ is the random force. For a particle of radius $a$ in a fluid of dynamic viscosity $\eta$, then Stokes' Law says $\lambda = 6 \pi \eta a$.
If time is continuous, the correlation function for the kicks is:
$$
\langle f_i(t) f_j(t') \rangle = 2 \lambda k_B T \delta_{ij}\delta(t-t')
$$
Where $T$ is the temperature and $f_i$ is the $i$'th component of the force. To generate a force $\vec{f}(t)$ obeying this correlation function you may want to investigate gaussian processes.
Finally, there IS a relation between these (microphysical) constants and the diffusion coefficient $D$. It is called the Stokes-Einstein relation and was the first example of the Fluctuation-Dissipation Theorem. For the Langevin equation above, the relation is:
$$
D = \frac{k_B T}{\lambda} \ .
$$
Note: the Langevin equation is NOT an ordinary differential equation. It is a Stochastic differential equation, so the usual (e.g. Runge-Kutta) numerical methods won't apply.
Best Answer
The colloidal particels of a colloidal solution when viewed through a ultramicroscope show a constant zig-zag motion known as Brownian movement.
$Size\space of\space colloidal\space particles :1nm - 100nm$
Brownian movement is a characteristic property of colloidal particles. This motion is independent of the nature of the colloid but depends in the size of particles and the viscosity of the solution. Smaller the size and lesser the viscosity, faster the motion. The motion becomes intense at higher temperature.
The Brownian movement is due to the unbalanced bombardment of the particles by the molecules of the dispersion medium. As the size of particles increases, the probability of uneven bombardment decreases and the Brownian movement becomes slow. (Here particles refer to colloidal particles)
The Brownian movement has a string effect which does not permit the particles to settle and thus, is responsible for the stability of sols.