Brownian motion itself can be described approximately by a Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on the Brownian particle. On long timescales, the mathematical Brownian motion is well described by a Langevin equation. On small timescales, inertial effects are prevalent in the Langevin equation. However the mathematical "Brownian motion" is exempt of such inertial effects. Note that inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular. so that simply removing the inertia term from this equation would not yield an exact description,
Brownian motion the particle. The displacement of a particle undergoing Brownian motion is obtained by solving the diffusion equation under appropriate boundary conditions and finding the rms of the solution. This shows that the displacement varies as the square root of the time (not linearly), which explains why previous experimental results concerning the velocity of Brownian particles gave nonsensical results. A linear time dependence was incorrectly assumed. At very short time scales, however, the motion of a particle is dominated by its inertia and its displacement will be linearly dependent on time: Δ"x" = "v"Δ"t". So the instantaneous velocity of the Brownian