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Updated: May 2, 2026

The Diffusion of Passive Tracers in Laminar Shear Flow
Published on: May 1, 2018
Analytical solution of the generalized Langevin equation with hydrodynamic interactions: subdiffusion of heavy
Denis S Grebenkov1, Mahsa Vahabi1
1Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France.
Abstract:
We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.
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