A note on the fluctuation–dissipation relation for the generalized Langevin equation with hydrodynamic backflow

• The generalized Langevin equation aimed to describe the Brownian motion in pure and visco-elastic fluids is solved.
• The time correlation function of the thermal force driving the Brownian particles is found in a simple way.
• For pure liquids, the decay of this correlation function is slower than it has been believed so far.
• The reason for this discrepancy is explained.

This paper is devoted to finding the fluctuation–dissipation relation (FDR) for the generalized Langevin equation (GLE) with the Boussinesq–Basset (BB) force in which the Stokes friction is generalized to a convolution of a memory kernel with the velocity of a Brownian particle. First, the solution of such GLE with hydrodynamic backflow is obtained. Using this solution, we find in a simple and easily controllable way the time correlation function of the thermal force driving the particles. If the GLE is used with the original BB force for pure liquids, the FDR known from the literature is corrected. It is shown that in this case the FDR contains, in addition to the known term ∼t−3/2∼t−3/2, a more slowly decaying contribution ∼t−1/2∼t−1/2.