Impulse response function for Brownian motion

Motivated from the central role of the mean-square displacement and its second time-derivative - that is the velocity autocorrelation function < v(0)v(t)> = 1/2 d(2)<Delta r(2)(t)>/dt(2) in the description of Brownian motion and its implications to microrheology, we revisit the physical meaning of the first time-derivative of the mean-square displacement d <Delta r(2)(t)>/dt of Brownian particles. By employing a rheological analogue for Brownian motion, we show that the time-derivative of the mean-square displacement of Brownian microspheres with mass m and radius R immersed in any linear, isotropic viscoelastic material is identical to NKBT/3 pi Rh(t), where h(t) is the impulse response function (strain history gamma(t), due to an impulse stress tau(t) = delta(t - 0)) of a rheological network that is a parallel connection of the linear viscoelastic material with an inerter with distributed inertance m(R) = m/6 pi R. The impulse response function h(t) = 3 pi R/NKBT d <Delta r(2)(t)>/dt of the viscoelastic material-inerter parallel connection derived in this paper at the stress-strain level of the rheological analogue is essentially the response function chi(t) = h(t)/6 pi R of the Brownian particles expressed at the force-displacement level by Nishi et al. after making use of the fluctuation-dissipation theorem. By employing the viscoelastic material-inerter rheological analogue we derive the mean-square displacement and its time-derivatives of Brownian particles immersed in a viscoelastic material described with a Maxwell element connected in parallel with a dashpot and we show that for Brownian motion of microparticles immersed in such fluid-like materials, the impulse response function h(t) maintains a finite constant value in the long term.