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 of Brownian motion in any linear, isotropic viscoelastic material is proportional to the real part of the complex dynamic fluidity 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 intrance with mass mR. The synthesis of this 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.

💡 Research Summary

The manuscript presents a unified framework that links the frequency‑domain characteristics of Brownian motion to a simple rheological analogue. Starting from the classical Langevin equation for a spherical probe of mass m and radius R in a Newtonian fluid, the author derives the well‑known exponential velocity autocorrelation and the corresponding Lorentzian power spectral density (PSD). By introducing the generalized Langevin equation with a memory kernel ζ(t), the analysis is extended to arbitrary linear, isotropic viscoelastic media. The key insight is that the complex dynamic modulus G(ω) of the surrounding material, when placed in parallel with an “inerter” (a two‑terminal element whose force is proportional to the relative acceleration of its terminals, with inertance m_R = m/6πR), yields an effective complex dynamic fluidity φ(ω)=iω G(ω). The PSD of the probe’s velocity is then simply
 S(ω)=2 Re{VAC(ω)} = (N k_B T)/(3πR) Re{φ(ω)}.
Thus the real part of the fluidity—i.e., the real part of iω G(ω)—directly gives the observable power spectrum. For a purely viscous fluid (η) the fluidity reduces to 1/η·1/(1+iωτ) with τ=m/6πRη, reproducing the classic Lorentzian shape. When the host medium follows a Maxwell model, Jeffreys model, or a fractional (sub‑diffusive) law, the complex modulus acquires additional poles and branch cuts, and the PSD accordingly displays multiple Lorentzian components or power‑law tails. The author demonstrates that increasing the elastic spring stiffness in the Maxwell analogue drives the spectrum toward the Newtonian limit, while a fractional exponent α (0<α<1) yields S(ω)∝ω^{−(1−α)} at low frequencies, matching experimental observations of long‑time memory in crowded or polymeric fluids.

The paper also revisits the correspondence principle originally proposed by Mason and Weitz, showing that the mean‑square displacement ⟨Δr²(t)⟩ can be expressed as (N k_B T)/(3πR) γ(t), where γ(t) is the creep compliance of the parallel network of the material and the inerter. This provides a direct route from microrheological measurements (MSD) to the rheological function of the medium without solving integral equations.

Strengths of the work include: (i) a clear physical interpretation of the inertial term as an inerter, which bridges mechanical network theory and stochastic thermodynamics; (ii) compact analytical expressions for PSDs across a wide class of viscoelastic models; (iii) the ability to incorporate hydrodynamic memory (through the inerter) without resorting to cumbersome Basset‑type kernels.

Potential limitations are: the analysis assumes linear response and neglects non‑Newtonian shear‑thinning or active forces; the inertial element is treated as distributed uniformly over the particle surface, which may be inaccurate for non‑spherical probes; and the derivation presumes thermal equilibrium, so extensions to active or driven systems would require additional terms.

Overall, the study offers a powerful and elegant tool for microrheology: by measuring the power spectrum of a probe’s Brownian motion, one can directly infer the real part of the complex fluidity of the surrounding medium, and the rheological analogue (dashpot + inerter) provides an intuitive circuit‑like picture that simplifies both theoretical calculations and experimental data interpretation.