Experimental Verification of a Modified Fluctuation-Dissipation Relation for a Micron-Sized Particle in a Nonequilibrium Steady State

Abstract

A modified fluctuation-dissipation-theorem (MFDT) for a non-equilibrium steady state (NESS) is experimentally checked by studying the position fluctuations of a colloidal particle whose motion is confined in a toroidal optical trap. The NESS is generated by means of a rotating laser beam which exerts on the particle a sinusoidal conservative force plus a constant non-conservative one. The MFDT is shown to be perfectly verified by the experimental data. It can be interpreted as an equilibrium-like fluctuation-dissipation relation in the Lagrangian frame of the mean local velocity of the particle.The validity of the fluctuation-dissipation theorem (FDT) in systems out of thermal equilibrium has been the subject of intensive study during the last years. We recall that for a system in equilibrium with a thermal bath at temperature T the FDT establishes a simple relation between the 2-time correlation function C(t − s) of a given observable and the linear response function R(t − s) of this observable to a weak external perturbationHowever, Eq. (1) is not necessarily fulfilled out of equilibrium and violations are observed in a variety of systems such as glassy materials [1,2,3,4,5], granular matter [6], and biophysical systems [7]. This motivated a theoretical work devoted to a search of a general framework describing FD relations, see the review [8] or [9,10,11,12,13,14] for recent attempts in simple stochastic systems. In the same spirit, a modified fluctuation-dissipation theorem (MFDT) has been recently formulated for a non-equilibrium steady dynamics governed by the Langevin equation with nonconservative forces [15]. In particular, this MFDT holds for the overdamped motion of a particle on a circle, with angular position θ, in the presence of a periodic potential H(θ) = H(θ + 2π) and a constant non-conservative force Fθwhere ζ is a white noise term of mean ζ t = 0 and covariance ζ t ζ s = 2Dδ(t − s), with D the (bare) diffusivity. This is a system that may exhibit an increase in the effective diffusivity [16,17]. Here, we shall study the dynamical non-equilibrium steady state (NESS) reached for observables that depend only on the particle position on the circle so are periodic functions of the angle θ. Such a state corresponds to a constant non-vanishing probability current j along the circle and a periodic invariant probability density function ρ 0 (θ) that allow us to define a mean local velocity v 0 (θ) = j/ρ 0 (θ). This is the average velocity of the particle at θ. For a stochastic system in NESS evolving according to Eq. (2), the MFDT reads for t ≥ swhere the 2-time correlation of a given observable O(θ) is defined byand the linear response function to a δ-perturbation of the conjugated variable h t is given by the functional derivativeIn Eq. (5), ... h denotes the average in the perturbed time-dependent state obtained from the NESS by replacing H(θ) in Eq. (2) by H(θ) − h t O(θ). It reduces for h = 0 to the NESS average ... 0 . In Eq. (3), the correlation b(t − s) is given byThis new term takes into accoun...

Large deviations of the current for driven periodic diffusions

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Modified fluctuation-dissipation theorem for non-equilibrium steady states and applications to molecular motors

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