Reply to “Comment on ‘Ratchet universality in the presence of thermal noise’ ”

Abstract

The Comment by Quintero et al. does not dispute the central result of our paper [Phys. Rev. E 87, 062114 (2013)] which is a theory explaining the interplay between thermal noise and symmetry breaking in the ratchet transport of a Brownian particle moving on a periodic substrate subjected to a temporal biharmonic excitation γ [η sin (ωt) + α (1 − η) sin (2ωt + ϕ)]. In the Comment, the authors claim, on the sole basis of their numerical simulations for the particular case α = 2, 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." Here we demonstrate by means of theoretical arguments and additional numerical simulations that all the conclusions of our original article are preserved. The foregoing Comment by Quintero et al.[1] offers some criticisms on certain particular (numerical) aspects of our previous paper [2], in which we studied theoretically and numerically a universal model -a Brownian particle moving on a periodic substrate subjected to a biharmonic excitation,where γ is an amplitude factor, and the parameters (η ∈ [0, 1] , α > 0) 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

Ratchet universality in the bidirectional escape from a symmetric potential well

“…For deterministic ratchets, the effectiveness of the theory of ratchet universality has been demonstrated in diverse physical contexts in which the driving forces are chosen to be biharmonic, such as in the cases of cold atoms in optical lattices [27,28], topological solitons [29], Bose-Einstein condensates exposed to a sawtoothlike optical lattice potential [30], matter-wave solitons [31], and one-dimensional granular chains [32]. Also, the interplay between thermal noise and symmetry breaking in the directed ratchet transport (DRT) of a Brownian particle moving on a periodic substrate subjected to a homogeneous temporal biharmonic force [33][34][35] as well as the cases of a driven Brownian particle subjected to a vibrating periodic potential [26], a driven Brownian particle in the presence of non-Gaussian noise [36], and coupled Brownian motors with stochastic interactions in the crowded environment [37] have been explained quantitatively in coherence with the degree-of-symmetry-breaking (DSB) mechanism, as predicted by the theory of ratchet universality [24,25].…”

Controlling directed ratchet transport of driven overdamped Brownian particles subjected to a vibrating periodic potential: ratchet universality versus harmonic-mixing perturbation theory

“…At first sight, this aspect of controllability should be easier to investigate in non-chaotic physical contexts such as those of certain extremely small systems, including many nanoscale devices and systems occurring in biological and liquid environments, in which DRT is often suitably described by overdamped ratchets [2,[14][15][16]. Thus, the interplay between thermal noise and symmetry breaking in the DRT of a Brownian particle moving on a periodic substrate subjected to a homogeneous temporal biharmonic excitation has been explained quantitatively in coherence with the degree-of-symmetry-breaking (DSB) mechanism [17][18][19], as predicted by the theory of ratchet universality (RU) [20,21]. For deterministic ratchets subjected to biharmonic forces, it has been shown [20,21] that there exists a universal force waveform which optimally enhances directed transport by symmetry breaking, i.e., once one has identified the relevant spatio-temporal symmetries to be broken in a given equation of motion, this optimal waveform maximally enhances the ratchet effect in that any other waveform yields a lower ratchet effect while the remaining equation parameters are held constant.…”

“…Examples are cold atoms in optical lattices [22,23], topological solitons [11], Bose-Einstein condensates exposed to a sawtooth-like optical lattice potential [24], matter-wave solitons [13], one-dimensional granular chains [25], and Bose-Einstein condensates under an unbiased periodic driving potential [26]. Additionally, it has been demonstrated through the example of a driven overdamped Brownian particle 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 DSB mechanism [17][18][19].…”

“…For the sake of clarity, we shall confine ourselves to the regime where the DSB mechanism dominates over the thermal inter-well activation mechanism [17][18][19]. Leaving aside the effect of noise (an effective change of the potential barrier which is in turn controlled by the DSB mechanism [17][18][19]), RU predicts (for σ = 0) that the optimal value of the relative amplitude η comes from the condition that the amplitude of sin (ωt) must be twice as large as that of cos (2ωt + ϕ) in Eq. (3) with F (t) given by Eq.…”

Comment on "Directed motion of spheres by unbiased driving forces in viscous fluids beyond the Stokes' law regime"

“…This is incorrect. It has been demonstrated that optimal enhancement of directed motion is achieved when maximal effective (i.e., critical) symmetry breaking occurs [4,5], while the effect of fi-nite temperature on the purely deterministic criticality scenario can be understood as an effective noise-induced change of the potential barrier which is in turn controlled by the degree-of-symmetry-breaking mechanism [8][9][10].…”