The Langevin equation - nature of the forces The Langevin equation is given by: 
$$m\ddot x=-\gamma \dot x+f(t)$$
where $f(t)$ is some stochastic force. I know that the first term on the RHS is to do with viscous drag and the second term is to do with random fluctuations in the number particle collisions. I have also heard it said  it is due to random density fluctuations - is this the same? Lastly what is the main difference between the origin of the two forces, since they are both fundamentally due to particle collisions?
 A: The first term on the RHS arises from the assumption that the particle is moving through a viscous fluid (in this case, with no net flow velocity).  The collisions from the front deliver a higher impulse than collisions from the rear, on average, which leads to a coherent net force directed opposite the particle velocity.
The second term arises from the fact that the viscous fluid is actually made up of a huge number of tiny particles which randomly kick the particle from all directions.  In addition to the coherent drag mentioned above, these discrete kicks can also make the particle randomly jiggle in various directions, which results in an easily observable diffusion effect.
In essence, the first term encapsulates the coherent influence of the tiny fluid molecules which tends to reduce the average particle velocity to zero, while the second term represents the incoherent influence of the tiny fluid molecules which causes the particle to bounce around and diffuse rather than just sit perfectly still.
