Question

What is Brownian motion ? Derive Einstein's
formula for the mean square displacement of a
Brownian particle.

Answer 1

Brownian motion - the chaotic motion of microscopic particles (plant spores) suspended in a liquid or gas, discovered by the botanist Brown. Brownian particles are set in motion because the force of impact of molecules from different sides is not the same.

An analysis of the interaction of particles with molecules leads to the expression: $<r^2> = \frac{2kT\tau}{3\pi\alpha\eta},$ and since the diffusion coefficient $D = \frac{kT}{3\pi\alpha\eta},$ then the average squared displacement over time $\tau$ is $<r^2> = 2D\tau$.