Simulating a Brownian type of motion: The rescaling-velocity approach versus the Langevin approach

The dynamics of a classical test particle that evolves deterministically in a potential field and whose velocity is then randomized at regular intervals of time is discussed. A limiting procedure for this type of Brownian type of motion is given, which results in a motion like that described by the Langevin equation. Exact analytical solutions for free and harmonically bound particle, are obtained. It is shown that if the time intervals between randomizing events coincide with the period of harmonic oscillations, thermal equilibrium is not reached. For anharmonic background potentials such as a resonant behavior also shows up as a slowing down of the relaxation.