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Brownian motion is the fluctuating motion of a suspended particle produced by thermal molecular collisions. Using the standard Brownian displacement conventions, in a uniform fluid without external drift, mean displacement is zero but mean-square displacement grows with time. For overdamped diffusion, $⟨\Delta x^2⟩=2Dt$ in one coordinate and $⟨|\Delta \mathbf{r}{|}^2⟩=6Dt$ in three dimensions, with $D=k_{B}T/\zeta$ and Stokes friction $\zeta =6\pi \eta a$ for a no-slip sphere.

Brownian transport brings nanoparticles into encounters and allows them to explore configurations during self-assembly. Reversible interactions permit rearrangement toward low-free-energy structures; interactions that are too strong can instead trap disordered aggregates. Thermal fluctuations also oppose localization in shallow traps. The symbol $\zeta$ avoids confusing friction with an unrelated chemical potential.