Consider a free particle of mass m with equation of motion described by m d v d t = − − v μ μ + η η ( t ) , where v = d r / d t is the particle velocity, μ μ is the particle mobility, and η η ( t ) = m a ( t ) is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale t c of particle collisions, i.e. η η ( t ) ¯ ¯ = 0 . The general solution to the equation of motion is v ( t ) = v ( 0 ) e − − t / τ τ + ∫ ∫ 0 t a ( t ′ ) e − − ( t − − t ′ ) / τ τ d t ′ , where τ τ = m μ μ is the correlation time of the noise term. It can also be shown that the autocorrelation function of the particle velocity v is given by