# What is meant with overdamped motion?

I'm learning about Brownian motion. I use the approximation of overdamped motion. I read that the average acceleration is $$0$$ then, but I don't really understand the concept. So, what does overdamped exactly mean, especially in the context of Brownian motion, and why is the acceleration $$0$$? Thanks!

• Jan 29, 2023 at 14:40

Overdamped means that viscosity forces are much more "relevant" than inertia. When this is the case, essentially any movement will very quickly reach terminal velocity, so the acceleration will be $$0$$.
As a toy model, imagine trying to push an object through a viscous medium. If we apply some constant force $$F$$, then Newton's second law gives us the following differential equation: $$m\ddot x=F-b\dot x$$
It is easy to see that terminal velocity is $$v_T=F/b$$, and the relevant time scale to relax to terminal velocity here is $$\tau=m/b$$. If the system is overdamped, then $$b$$ is very large, which makes $$\tau$$ very small. In other words, the more damping you have, the faster you reach terminal velocity where the acceleration is $$0$$.
Iin this regime you can create a "trick Newton's second law" where the velocity is proportional to the applied force: $$F=b\dot x$$ Don't show introductory physics students this ;)