Time-Shift Invariance Determines the Functional Shape of the Current in Dissipative Rocking Ratchets

Cuesta, José A.; Quintero, Niurka R.; Álvarez-Nodarse, R.

doi:10.1103/physrevx.3.041014

Cited by 20 publications

(47 citation statements)

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“…1 that the numerical results are consistent with the η opt = 2/3 prediction of the theories in the limit of weak external forces, we can actually go further and show that a recent extension of the theory developed in Ref. [9], valid for arbitrarily large forces [11], fits perfectly with the results presented in [1,2]. For the case of harmonic mixing represented by Eq.…”

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confidence: 83%

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

2013

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A recent paper [P. J. Martínez and R. Chacón, Phys. Rev. E 87, 062114 (2013)] presents numerical simulations on a system exhibiting directed ratchet transport of a driven overdamped Brownian particle subjected to a spatially periodic, symmetric potential. The authors claim that their simulations prove the existence of a universal waveform of the external force that optimally enhances directed transport, hence confirmin the validity of a previous conjecture put forth by one of them in the limit of vanishing noise intensity. With minor corrections due to noise, the conjecture holds even in the presence of noise, according to the authors. On the basis of their results the authors claim that all previous theories, which predict a different optimal force waveform, are incorrect. In this Comment we provide sufficien numerical evidence showing that there is no such universal force waveform and that the evidence obtained by the authors otherwise is due to their particular choice of parameters. Our simulations also suggest that previous theories correctly predict the shape of the optimal waveform within their validity regime, namely, when the forcing is weak. On the contrary, the aforementioned conjecture does not hold. The authors of Ref.[1] (see also the erratum [2]) simulate the equatioṅwhere γ is the global amplitude of the force; 0 η 1 and φ account for the relative amplitude and initial phase difference of the two harmonics, respectively; ξ (t) is a Gaussian white noise with zero mean and ξ (t)ξ (t + s) = δ(s); and σ is proportional to the temperature of the system. This system exhibits ratchet transport if the external force breaks both a time-shift symmetry, namely, if F bihar (t) = −F bihar (t + T /2) (T being the period of F bihar ), and a time-reversal symmetry, i.e., F bihar (t) = −F bihar (−t). This happens for all 0 < η < 1 and all φ = 0,π. If initially the particle starts at x(t 0 ) = x 0 , the ratchet current can be obtained aswhere · represents an ensemble average over all trajectories satisfying the same initial condition. Obviously the ratchet current v will be a function of the parameters of the system, in particular of those that defin the external force. Since for η = 0,1 or φ = 0,π the force breaks neither the time-shift symmetry nor the time-reversal symmetry (hence v = 0), it is easily foreseen that for a certain combination of the parameters of the force v must be maximal (in absolute value). . An argument based on an affin transformation of the force leads the authors to conclude that this optimal shape of the force will hold even for σ > 0, albeit some deviations are to be expected.This result is universal in the sense that is independent of γ and φ. Figure 1(a) of [2] confirm that this is an accurate prediction even for the high intensities of the noise they use in their simulations (σ = 2,3,4). The other parameters are set to ω = 0.08π, φ = π/2, and γ = 2 throughout their paper.They go on to claim that since all previous theories [4-9] predict a form of the ratchet current given by [2] t...

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confidence: 83%

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

2013

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“…In the special case when Γ is invariant under time-shift, i.e., Γ[f(t + τ)] = Γ[f(t)] for all 0 < τ < T, we recover the results in [2] …”

Section: Theoremsupporting

confidence: 62%

Functionals of Harmonics Functions

Quintero

Cuesta

Álvarez-Nodarse

2018

The First International Conference on Symmetry

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Let C s T , with T > 0, be the set of continuous, T-periodic functions f : R → R s , and let Γ : C s T → R be a real functional on C s T . If Γ is n times Fréchet differentiable on C s T , then it has an n-th order Taylor expansion around 0 (see e.g., [1]). Such a Taylor expansion can be obtained as the n-th order truncation of the serieswhere n = (n 1 , . . . , n s ) and we have introduced the notationThe kernels c n 1 ,...,n s (t 11 , . . . , t sn s ) are all real, T-periodic, and symmetric in all their arguments. In this contribution we will prove the following theorem.Theorem 1. Let Γ be a functional with Taylor series (1), and takewhere q ≡ (q 1 , . . . , q s ) ∈ N s is such that gcd(q 1 , . . . , q s ) = 1 and ω = 2π/T. Then,where, φ ≡ (φ 1 , . . . , φ s ), ≡ ( 1 , . . . , s ), and functions C x ( ) and θ x ( ) do not depend on φ and are even in each i , i = 1, . . . , s, for every x ∈ S + . x ∈ S + is the set of vectors x whose leftmost nonzero component is positive.In the special case when Γ is invariant under time-shift, i.e., Γ[f(t + τ)] = Γ[f(t)] for all 0 < τ < T, we recover the results in [2]

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“…This implies, for example, that current reversals can be induced by solely changing the relative phase between f 1 and f 2 . Furthermore, in [5], for a non-small amplitude limit, two interesting effects have been theoretically predicted: a deviation from the sinusoidal shape of v as a function of the phase; and the dependence of θ 0 and A 0 on the amplitudes of the forces. This latter fact leads to an unexpected phenomenon related with the appearance of current reversals by changing the amplitudes of the harmonics.…”

Section: Introductionmentioning

confidence: 97%

Nonsinusoidal current and current reversals in a gating ratchet

Dinís

Quintero

2015

Phys. Rev. E

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In this work, the ratchet dynamics of Brownian particles driven by an external sinusoidal (harmonic) force is investigated. The gating ratchet effect is observed when another harmonic is used to modulate the spatially symmetric potential in which the particles move. For small amplitudes of the harmonics, it is shown that the current (average velocity) of particles exhibits a sinusoidal shape as a function of a precise combination of the phases of both harmonics. By increasing the amplitudes of the harmonics beyond the small-limit regime, departures from the sinusoidal behavior are observed and current reversals can also be induced. These current reversals persist even for the overdamped dynamics of the particles.

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Time-Shift Invariance Determines the Functional Shape of the Current in Dissipative Rocking Ratchets

Cited by 20 publications

References 49 publications

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

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

Functionals of Harmonics Functions

Nonsinusoidal current and current reversals in a gating ratchet

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