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Isothermal Langevin dynamics in systems with power-law spatially dependent friction


  Shaked Regev [1]  ,  Niels Gronbech Jensen [2,3]  ,  Oded Farago [1,4]  
 [1] Department of Biomedical Engineering, Ben Gurion University of the Negev
[2] Department of Mechanical and Aerospace Engineering, University of California Davis
[3] Department of Mathematics, University of California Davis
[4] Ilse Katz Institute for Nanoscale Science and Technology, Ben-Gurion University of the Negev

We study the dynamics of Brownian particles in a heterogeneous one-dimensional medium with

a spatially-dependent diffusion coefficient of the form D(x) ∼ |x|c, at constant temperature. The

particle’s probability distribution function (PDF) is calculated both analytically, by solving Fick’s

diffusion equation, and from numerical simulations of the underdamped Langevin equation. At large

times, the PDFs calculated by both approaches yield identical results, corresponding to subdiffusion

for c < 0, and superdiffusion for 0 < c < 1. For c > 1, the diffusion equation predicts that the

particles accelerate. Here, we show that this phenomenon, previously considered in several works as

an illustration for the possible dramatic effects of spatially-dependent thermal noise, is unphysical.

We argue that in an isothermal medium, the motion cannot exceed the ballistic limit (<x2>∼ t2).

The ballistic limit is reached when the friction coefficient drops sufficiently fast at large distances

from the origin, and is correctly captured by Langevin’s equation.