Stationary and transient work-fluctuation theorems for a dragged Brownian particle

Abstract

Recently Wang et al. carried out a laboratory experiment, where a Brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center. This experiment confirmed a theoretically predicted work related integrated transient fluctuation theorem (ITFT), which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state. The corresponding integrated stationary state fluctuation theorem (ISSFT) was not observed. Using an overdamped Langevin equation and an arbitrary motion for the center of the harmonic force, all quantities of interest for these theorems and the corresponding nonintegrated ones (TFT and SSFT, respectively) are theoretically explicitly obtained in this paper. While the TFT and the ITFT are satisfied for all times, the SSFT and the ISSFT only hold asymptotically in time. Suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the ISSFT could be observed, are given. In addition, a nontrivial long-time relation between the ITFT and the ISSFT was discovered, which could be observed experimentally, especially in the case of a resonant circular motion of the center of the harmonic force.

Fluctuation relation for a Lévy particle

“…The study of fluctuations in systems driven in nonequilibrium steady states has concentrated lately on a special symmetry property referred to as the fluctuation theorem or fluctuation relation. The first studies of this symmetry focused on the entropy production of nonequilibrium systems [1,2], but it was soon realized that the same symmetry holds for other quantities of interest, such as the work W τ performed on a nonequilibrium system during a time τ [3]. In the context of this quantity, we define a fluctuation relation [4] as follows.…”

“…For now we leave these considerations aside, and attempt to calculate the probability density P (W τ ) of the work W τ performed on the Lévy particle by the harmonic drag force during a time τ [3]:…”

Fluctuation theorem for an optically trapped tracer in dense colloids. A simulation study

“…where ǫ(τ) = τ r 1 − e −τ/τ r /τ [21,22]. As the time interval τ diverges ǫ(τ → ∞) → 0, and the above expression tends to the more conventional form of the FT:…”

“…where M S (τ) = γ 0 v 2 trap τ is the average work, and V S (τ) = 2γ 0 k B T v 2 trap τ − τ r 1 − e −τ/τ r , with τ r = γ 0 /k [21,22]. In this model, this work is dissipated to the medium directly.…”

“…For an isolated particle in a medium with friction and neglecting the inertial term, this work is Gaussian distributed, and in the stationary state reads [21,22]:…”