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!
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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 $0$.
In 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 ;)
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