Diffusion From Equation of Motion

Here we look at the diffusion from another perspective. We will obtain the diffu-sional behavior, i.e., ([r(t) - r(0)] ) t, from the equation of motion. 

A particle suspended in a liquid receives a random force f(t) when solvent molecules collide with the particle. We assume that the random force has the following statistical properties  

In a viscous solvent the motion of the particle is overdamped. Then, the equation of motion of the particle has the friction term and the force term only  

The mean square displacement in time t is calculated from 

The mean square displacement is proportional to t. The particle makes a diffusional motion on all time scales. We also find that the average displacement is zero  

---