Particles dispersed in a dilute gas: Limits of validity of the Langevin equation

The problem of the limits of validity of the Langevin equation is considered in detail in the case of (microscopic) test-particles in very dilute gases. It is shown that, in this case, the current Langevin equation follows from the Newton’s law in an exact way only in the Maxwell test-particle–gas-particle interaction model, and in an approximate way only in the Rayleigh-gas limit and in the low-velocity limit, while in any other interaction model, or limit, only a Langevin-like equation with speed-dependent friction coefficient and speed-dependent fluctuating force can be written. Such a circumstance, although probably limited to the particular physical situation considered in this paper, suggests that, in general, some preliminary, specific check of the validity of the Langevin equation should be performed before using the said equation to interpret laboratory experiments.