Ratchet universality in the presence of thermal noise

Abstract

We show that directed ratchet transport of a driven overdamped Brownian particle subjected to a spatially periodic and symmetric potential can be reliably controlled by tailoring a biharmonic temporal force, in coherence with the degree-of-symmetry-breaking mechanism. We demonstrate that the effect of finite temperature on the purely deterministic ratchet scenario can be understood as an effective noise-induced change of the potential barrier which is in turn controlled by the degreeof-symmetry-breaking mechanism. Remarkably, we find that the same universal scenario holds for any symmetric periodic potential, while optimal directed ratchet transport occurs when the impulse transmitted (spatial integral over a half-period) by the symmetric spatial force is maximum.PACS numbers: 05.40.-a Directed transport without any net external force, the ratchet effect [1][2][3], has been an intensely studied interdisciplinary subject over the last few decades owing to its relevance in biology where ratchet mechanisms are found to underlie the working principles of molecular motors [4,5], and to its wide range of potential technological applications including micro-and nano-technologies. Directed ratchet transport (DRT) is today understood to be a result of the interplay of symmetry breaking [6], nonlinearity, and non-equilibrium fluctuations, in which these fluctuations include temporal noise [2], spatial disorder [7], and quenched temporal disorder [8]. In extremely small systems, including many of those occurring in biological and liquid environments as well as many nanoscale devices, DRT is often suitably described by overdamped ratchets, in which inertial effects are negligible in comparison with friction effects [2,[9][10][11]. Here, we show how ratchet universality [12] works subtly in the context of noisy overdamped ratchets by studying the dynamics of a universal model -a Brownian particle moving on a periodic substrate subjected to a biharmonic excitation [2,3],where γ is an amplitude factor, and the parameters η ∈ [0, 1] and ϕ account for the relative amplitude and initial phase difference of the two harmonics, respectively, while ξ (t) is a Gaussian white noise with zero mean and ξ (t) ξ (t + s) = δ (s), and σ = 2k b T with k b and T being the Boltzmann constant and temperature, respectively. For deterministic ratchets, this has been shown to also be the case for topological solitons [8] and matterwave solitons [13]. It is worth noting that, in spite of the abundance of numerical findings, the theoretical understanding of the directed transport phenomena represented by Eq. (1) remains far from being satisfactory [14] even about half a century after the earliest studies [15][16][17]. Indeed, all the earlier theoretical predictions (cf. Refs. [3,6,16,17,18]) indicate that the dependence of DRT velocity on the amplitudes of the two harmonics should scale asNote that this expression presents, as a function of η, a single maximum at η = 2/3, irrespective of the particular value of temperature, including the limiting val...

Comment on “Ratchet universality in the presence of thermal noise”

“…The predicted universal value η opt = 4/5 is reached at no value of γ and the closest it gets to it is at γ ≈ 2, precisely the value used in the simulations of Refs. [1,2]. We need to make clear at this point that by setting γ = 2 in our simulations our results reproduce accurately the plots of Fig.…”

“…Accordingly, we set φ = π/2 and fit the simulation results of v vs η to a function of the form −η 2 (1 − η)A(η), where we take for A(η) a (2,2) Padé approximant. 1 The result is plotted in Fig. 3(b) to show that this fit is a very accurate description of v(η) and therefore correctly predicts the deviation of η opt from its weak force approximation η opt = 2/3.…”

“…In this regime, only a very specific dependence of the coefficients A n with γ , which does not occur in the case of Eq. (1), would yield the universality claimed in [1][2][3].…”

“…1. Simulations of the stochastic differential equation (1) have been performed using the second-order weak predictor-corrector method [10] with time step t = 0.01, initial condition x(0) = 0, and a final integration time t f = 200π/ω. The ratchet velocity v has been computed using formula (2) averaging over 5000 realizations of the noise.…”

Inertial frictional ratchets and their load bearing efficiencies

“…For example, a 'square-wave' drive with a large constant force for a small duration and a small constant force in the opposite direction for a longer duration [24] gives an appropriate asymmetric drive. A biharmonic drive being an another example [25,26] of asymmetric drive used to obtain ratchet current in overdamped as well as underdamped symmetric periodic potential systems. In most of the cases, in order to obtain ratchet current the medium (substrate) is considered spatially uniform so that the friction coefficient is constant in space.…”