Ratchet universality in coupled dissipative oscillators without external bias

Pedro J. Martínez and Ricardo Chacón

Phys. Rev. E 104, 024224 – Published 30 August 2021

Abstract

Directed ratchet transport is generally observed in nonautonomous systems as a result of the interplay of nonlinearity, symmetry breaking, and nonequilibrium fluctuations. Here we demonstrate that ratchet dynamics can appear in significant transporting degrees of freedom of dissipative coupled systems without external bias due to unidirectional coupling of oscillatory degrees of freedom (which are also nonbiasing in any direction), while optimal enhancement of directed ratchet transport occurs when the initial conditions and parameters of such ratcheting degrees of freedom are suitably chosen as predicted by the theory of ratchet universality. The simple case of linear oscillatory degrees of freedom is discussed in detail, and numerical experiments are described which confirm all the theoretical predictions, including the dependence of current (velocity) reversals on the initial conditions and the ratcheting degrees-of-freedom parameters. This autonomous ratchet scenario could be exploited technologically, for instance, in the context of noncontact, rack-and-pinion type, nanoscale setups with coupling from the lateral Casimir force, and is relevant for studies of molecular motors in the biological realm.

Received 1 July 2021
Accepted 16 August 2021

DOI:https://doi.org/10.1103/PhysRevE.104.024224

Physics Subject Headings (PhySH)

Brownian motion

Statistical Physics & Thermodynamics

Authors & Affiliations

Pedro J. Martínez ^1,2 and Ricardo Chacón ^3,4

¹Instituto de Nanociencia y Materiales de Aragón, CSIC-Universidad de Zaragoza, E-50009 Zaragoza, Spain
²Departamento de Física Aplicada, E.I.N.A., Universidad de Zaragoza, E-50018 Zaragoza, Spain
³Departamento de Física Aplicada, E.I.I., Universidad de Extremadura, Apartado Postal 382, E-06006 Badajoz, Spain
⁴Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06006 Badajoz, Spain

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Issue

Vol. 104, Iss. 2 — August 2021

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Images

Figure 1
(a) Optimal value of the amplitude factor $η_{opt}$ [cf. Eq. (4)] vs eigenfrequency $ω$ for $λ_{z 0} = λ_{y 0} = 1$ . (b) Contour plot of $η_{opt}$ [cf. Eq. (4)] vs $λ_{y 0}$ and $λ_{z 0}$ for $ω = 1$ . Fixed parameter: $| y_{0} / z_{0} | = 1$ .
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Figure 2
Hyperbolas and straight lines in the $λ_{y 0} - λ_{z 0}$ plane which are associated with the optimal breaking of the shift symmetry of the effective ratcheting signal [cf. Eqs. (5) and (6), respectively] for $| y_{0} / z_{0} | = 1.4$ (solid lines) and $| y_{0} / z_{0} | = 2.5$ (dashed lines). Fixed parameters: $η = 0.8, ω = 0.25$ .
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Figure 3
(a) Loci in the $λ_{y 0} - λ_{z 0}$ plane for which the effective initial phase difference between the two harmonics of the effective ratcheting signal, $Ψ_{eff}$ , takes the optimal values $Ψ_{eff}^{opt} = {0, π}$ [cf. Eq. (7)] and three values of the eigenfrequency: $ω = 0.8$ (solid lines), $ω = 0.5$ (dotted lines), $ω = 0.25$ (dashed lines). The curves that pass through the origin correspond to $Ψ_{eff}^{opt} = 0$ , while the others correspond to $Ψ_{eff}^{opt} = π$ . (b) Loci in the $λ_{y 0} - λ_{z 0}$ plane for which the effective initial phase difference between the two harmonics of the effective ratcheting signal, $Ψ_{eff}$ , takes the least favorable values $Ψ_{eff}^{unfav} = {π / 2, 3 π / 2}$ [cf. Eq. (8)] and three values of the eigenfrequency: $ω = 0.8$ (solid lines), $ω = 0.5$ (dotted lines), $ω = 0.25$ (dashed lines). The curves with $λ_{y 0} > 0 (< 0)$ correspond to $Ψ_{eff}^{unfav} = π / 2 (3 π / 2)$ .
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Figure 4
Average velocity $〈 〈 \dot{x} 〉 〉$ [cf. Eqs. (1) and (13)] in the $λ_{y 0} - λ_{z 0}$ plane for (a) $η = 0.4, ω = 0.25, y_{0} = z_{0} = 1$ , (b) $η = 0.8, ω = 0.25, y_{0} = z_{0} = 1$ , (c) $η = 8 / 9, ω = 0.25, y_{0} = z_{0} = 1$ , and (d) $η = 0.35, ω = 2, y_{0} = 4, z_{0} = 2$ . Also plotted are the theoretical predictions for the optimal breaking of the shift symmetry [cf. Eqs. (5) and (6); dashed white (hyperbola and straight) lines, respectively], the optimal breaking of the time reversal symmetry [cf. Eq. (7); solid white lines], and the case with no breaking of the time reversal symmetry [cf. Eq. (8); solid black lines] of the effective ratcheting signal. Fixed parameter: $σ = 2$ .
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Figure 5
Average velocity $〈 〈 \dot{x} 〉 〉$ [cf. Eqs. (1) and (13)] in the $ω - η$ plane for $λ_{z 0} = λ_{y 0} = | y_{0} / z_{0} | = 1, σ = 1$ . Also plotted is the predicted optimal value of the amplitude factor $η_{opt}$ vs eigenfrequency $ω$ [cf. Eq. (4); solid line].
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Figure 6
Average velocity $〈 〈 \dot{x} 〉 〉$ [cf. Eqs. (1) and (13)] vs (a) $λ_{z 0}$ and (b) $λ_{y 0}$ for $λ_{y 0} = 1.5$ and $λ_{z 0} = 3$ , respectively, and two values of the noise strength: $σ = 0$ (dashed line, left scale) and $σ = 2$ (solid line, right scale). The vertical dotted lines $C_{1, 2}$ indicate the values at which the respective arguments intersect the corresponding hyperbolas [cf. Eq. (5)] in each case. Fixed parameters: $η = 0.8, ω = 0.25, | z_{0} / y_{0} | = 1$ .
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