From hydrodynamics we know that the friction constant \( \xi \) is given by

$$ \xi =6\pi \eta a/m $$

where \( \eta \) is the viscosity of the solvent and a is the radius of the particle .

Solving the second equation in the previous slide we get

$$ \mathbf{v}(t)=\mathbf{v}_{0}e^{-\xi t}+\int_{0}^{t}d\tau e^{-\xi (t-\tau )}\mathbf{F }(\tau ). $$