Spectral analysis of Brownian motion with its rheological analogues

The power spectrum of the Brownian motion of probe microparticles with mass m and radius R immersed in a viscoelastic material reveals valuable information about repetitive patterns and correlation structures that manifest in the frequency domain. In this paper, we employ a viscous–viscoelastic correspondence principle for Brownian motion and we show that the power spectrum (power spectral density) of Brownian motion in any linear, isotropic viscoelastic material is proportional to the real part of the complex dynamic fluidity (complex mobility) of a linear rheological network that is a parallel connection of the linear viscoelastic material within which the Brownian particles are immersed and an inerter, with distributed inertance mR=m6πR. The synthesis of this deterministic rheological analogue simplifies appreciably the calculation of the power spectrum for Brownian motion within viscoelastic materials such as Maxwell fluids, Jeffreys fluids, subdiffusive materials, or in dense viscous fluids that give rise to hydrodynamic memory.