TY - JOUR

T1 - Fluctuation-Dissipation Relation for Systems with Spatially Varying Friction

AU - Farago, Oded

AU - Grønbech-Jensen, Niels

N1 - Funding Information:
This work was supported by the US Department of Energy Project DE-NE0000536 000.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - When a particle diffuses in a medium with spatially dependent friction coefficient α(r) at constant temperature T, it drifts toward the low friction end of the system even in the absence of any real physical force f. This phenomenon, which has been previously studied in the context of non-inertial Brownian dynamics, is termed "spurious drift", although the drift is real and stems from an inertial effect taking place at the short temporal scales. Here, we study the diffusion of particles in inhomogeneous media within the framework of the inertial Langevin equation. We demonstrate that the quantity which characterizes the dynamics with non-uniform α(r) is not the displacement of the particle Δr = r - r0 (where r0 is the initial position), but rather ΔA(r) = A(r) - A(r0), where A(r) is the primitive function of α(r). We derive expressions relating the mean and variance of ΔA to f, T, and the duration of the dynamics Δt. For a constant friction coefficient α(r) = α, these expressions reduce to the well known forms of the force-drift and fluctuation-dissipation relations. We introduce a very accurate method for Langevin dynamics simulations in systems with spatially varying α(r), and use the method to validate the newly derived expressions.

AB - When a particle diffuses in a medium with spatially dependent friction coefficient α(r) at constant temperature T, it drifts toward the low friction end of the system even in the absence of any real physical force f. This phenomenon, which has been previously studied in the context of non-inertial Brownian dynamics, is termed "spurious drift", although the drift is real and stems from an inertial effect taking place at the short temporal scales. Here, we study the diffusion of particles in inhomogeneous media within the framework of the inertial Langevin equation. We demonstrate that the quantity which characterizes the dynamics with non-uniform α(r) is not the displacement of the particle Δr = r - r0 (where r0 is the initial position), but rather ΔA(r) = A(r) - A(r0), where A(r) is the primitive function of α(r). We derive expressions relating the mean and variance of ΔA to f, T, and the duration of the dynamics Δt. For a constant friction coefficient α(r) = α, these expressions reduce to the well known forms of the force-drift and fluctuation-dissipation relations. We introduce a very accurate method for Langevin dynamics simulations in systems with spatially varying α(r), and use the method to validate the newly derived expressions.

KW - Computer simulations

KW - Diffusion

KW - Fick's law

KW - Fluctuation-dissipation theorem

KW - Itô-Stratonovich Dilemma

KW - Langevin dynamics

UR - http://www.scopus.com/inward/record.url?scp=84905573050&partnerID=8YFLogxK

U2 - 10.1007/s10955-014-1045-4

DO - 10.1007/s10955-014-1045-4

M3 - Article

AN - SCOPUS:84905573050

VL - 156

SP - 1093

EP - 1110

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 6

ER -