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 noise it has more energy at lower frequencies, even more so than pink noise. It decreases in power by 6 dB per octave (20 dB per decade) and, when heard, has a "damped" or "soft" quality compared to white and pink noise. The sound is a low roar resembling a waterfall or heavy rainfall. See also violet noise, which is a 6 dB "increase" per octave. Strictly, Brownian motion has a Gaussian probability distribution, but "red noise" could apply to any signal with the 1/"f" frequency spectrum. A Brownian motion, also called a Wiener process, is obtained as the integral of