Comment on the paper by R. Maniar and A. Bhattacharyay, [Random walk model for coordinate-dependent diffusion in a force field, Physica A 584 (2021) 126348].

Paper (Maniar and Bhattacharyay, 2021) introduces a discrete random-walk (RW) model with a site dependent waiting time and an external force i.e. a site dependent drift. The RW is compared with a Smoluchowski equation in continuous space and time with space dependent diffusivity and space dependent damping. Using numerical simulations in Maniar and Bhattacharyay (2021) it is shown that the equilibrium distribution satisfies locally the Stokes–Einstein relation i.e. there is a constant temperature in the whole system. Here we show that the result can be recovered analytically with a continuous limit in the RW master equation. Moreover, we evidence that the RW model should be carefully defined in order to satisfy locally Stokes–Einstein relation, indeed a different definition of the discrete RW process describes a system that satisfies the Smoluchowski equation with constant damping, where the temperature is not uniform. [ABSTRACT FROM AUTHOR]