A multiple-timing analysis of temporal ratcheting Aref Hashemi12 Edward T. Gilman3and Aditya S. Khair4 1Department of Applied and Computational Mathematics and

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A multiple-timing analysis of temporal ratcheting

Aref Hashemi1,2*, Edward T. Gilman3and Aditya S. Khair4*

1*Department of Applied and Computational Mathematics and

Statistics, University of Notre Dame, Notre Dame, IN, United States.

2Courant Institute, New York University, New York, NY, United States.

3Department of Mathematics, Rice University, Houston, TX, United

States.

4*Department of Chemical Engineering, Carnegie Mellon University,

Pittsburgh, PA, United States.

*Corresponding author(s). E-mail(s): aref.hashemi@nd.edu;

akhair@andrew.cmu.edu;

Abstract

We develop a two-timing perturbation analysis to provide quantitative insights

on the existence of temporal ratchets in an exemplary system of a particle moving

in a tank of ﬂuid in response to an external vibration of the tank. We consider

two-mode vibrations with angular frequencies ωand αω, where αis a rational

number. If αis a ratio of odd and even integers (e.g., 2

1,3

2,4

3), the system yields

a net response: here, a nonzero time-average particle velocity. Our ﬁrst-order

perturbation solution predicts the existence of temporal ratchets for α= 2.

Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting

eﬀect for α=3

2and 4

3appears at the third-order perturbation solution. More

importantly, we ﬁnd closed form formulas for the magnitude and direction of the

induced net velocities for these αvalues. On a broader scale, our methodology

oﬀers a new mathematical approach to study the complicated nature of temporal

ratchets in physical systems.

1 Introduction

A zero time-average oscillatory excitation with certain broken time symmetries can

induce a net response, i.e., a nonzero time-average solution, in nonlinear dynami-

cal systems [1–7]. In particular, periodic excitations that are not shift-symmetric (or

arXiv:2210.07555v2 [physics.app-ph] 4 Apr 2024

f(t)

u(t)

f(t)

x(t) =Zt

0hu(s)−˙

f(s)ids

Fig. 1 Schematic diagram of the problem. A particle moves at a velocity u(t) in a large tank that

is fully ﬁlled with liquid and vibrates vertically at a periodic velocity given by ˙

f(t). x(t) denotes

the vertical location of the particle relative to the tank. The tank vibrates up and down with a

displacement f(t) with respect to the initial location of the particle.

antiperiodic) can induce a net drift when acting on nonlinear systems [6,8]. Unlike

the classical Feynman-Smoluchowski ratchet [9], where the symmetry is broken in the

physical domain, these “temporal ratchets” are caused by time asymmetries. Previous

theoretical and experimental studies have established the existence of temporal ratch-

ets in point particles [1,2,10–14], optical [15–20] and quantum [21,22] lattices, and

mechanical and microﬂuidic systems [23–28]. However, the underlying mechanism by

which the temporal ratcheting is induced has remained relatively obscure. Notably, it

is unclear as to what determines or, rather, how to predict the sign and magnitude of

the net response, e.g., the direction and speed of net motion. Typically, the sign and

magnitude of the net response are determined by a direct numerical integration of the

governing equation. Here, in this study, we introduce multiple time-scale theory as a

method to analyze temporal ratchets. Our approach shows a new way to obtain an

analytical approximation to the ratcheting behavior and, thereby, provides a theoret-

ical understanding of their origin. Importantly, we provide a new methodology to ﬁnd

closed-form formula for the magnitude and direction of the net motion in temporal

ratchets.

2 Physical Problem

As a model system, consider a particle immersed in a large (compared to the size of

the particle) enclosed tank, which is ﬁlled with a liquid and vertically placed on top

of a platform (see Fig. 1). The platform vibrates up and down with a periodic wave

form f(t) = ℓ˜

f(t), where ℓis the nominal amplitude of oscillation, and tis time. The

Newton’s second law for the particle motion can be expressed as

m˙u=−a(u−˙

f)−bsgn(u−˙

f)(u−˙

f)2−V g∆ρ, (1)

where mand uare the particle mass and velocity, respectively; sgn(x) is the sign func-

tion; aand bare positive constants; V g∆ρis the gravitational force on the particle; and

overdot denotes diﬀerentiation with respect to time. The terms on the right hand side

(RHS) of (1) correspond to linear drag/friction, a nonlinear drag, and gravitational

forces, respectively.

Using ω−1and ℓω as, respectively, the time and velocity scales, and changing the

dependent variable to v=u−˙

f(with vas the particle velocity relative to the tank),

we ﬁnd the dimensionless form of (1),

d˜v

d˜

t=−d2˜

d˜

t2−k˜v−ϵsgn(˜v)˜v2−w, (2)

where k=a/(mω), ϵ=bℓ/m, and w=V g∆ρ/(mℓω2). To make further progress,

we approximate the sgn(˜v) function with Taylor series expansion of a smooth sigmoid

function tanh(˜v),

d˜v

d˜

t=−d2˜

d˜

t2−k˜v−ϵ˜v−1

3˜v3+···˜v2−w. (3)

While the approximation sgn(˜v)≈tanh(˜v) might be quantitatively inaccurate, it does

capture the most important qualitative characteristics of the resistance term, such

as being nonlinear, odd in v, and monotonic increasing. Finally, for |˜v|≪ 1, which

physically implies that the particle moves at only a slightly diﬀerent velocity than the

tank, we truncate the series to obtain

dt =−d2f

dt2−kv −ϵv3−w, (4)

where we dropped the ∼decoration for simplicity of the presentations. To close the

problem, we assume the initial condition v(t= 0) = 0, i.e., the relative velocity is

initially zero.

To proceed, we assume a speciﬁc two-mode excitation

f(t) = 1

2[sin(t) + csin(αt)] ,(5)

with αas a rational number, and 2τ= 2π/gcd(1, α), where gcd(1, α) denotes the

greatest common divisor of 1 and α. Without any loss of generality, we assume α≥1;

any two-mode waveform with α < 1 can be expressed as another waveform with

α:= 1/α by choosing (αω)−1as the time scale. In (5), cis assumed to be an O(1)

constant that represents the relative importance of the two modes of excitation.

Fig. 2 Representative numerical solution to (4). Moving average (on time intervals of length 8π) of

the relative position of the particle, x(t) = Rt

0v(s)ds, versus time for diﬀerent αvalues with w= 0.001

(a) and w= 0 (b). Parameters: c= 1, k= 1, ϵ= 0.25.

Fig. 2 shows representative numerical results for the particle position (relative

to the tank), i.e., x(t) = Rt

0v(s)ds, versus time for diﬀerent values of α, and with

and without gravitational forces. (Note that we are not interested in cases where the

system behavior is largely dominated by gravitational forces; so we present results for

w= 0.001 and w= 0 only.) When w̸= 0 (Fig. 2(a)), one would expect the particle

to follow, on average, the direction of the gravitational force. This is the case for

α= 1,3

2,5

3,19

9; the particle moves on average downward despite oscillating. Perhaps

the simplest intuition is that in the absence of the nonlinear term, one could simply

ﬁnd the average velocity as ⟨v⟩=−w/k. (Here, ⟨χ⟩denotes a time-average over the

period 2τ: namely, ⟨χ⟩=1

2τRt+2τ

tχds.) However, the system behaves qualitatively

diﬀerent for α= 2 and the particle moves upward on average, against gravity.

Further analysis indicates that there are two diﬀerent phenomena governing the

average motion of the particle. Fig. 2(b) shows the same results but for w= 0. Here,

the particle does not move, on average, for α= 1,5

3,19

9(the particle experiences an

initial drift but then remains, on average, in the same location) but drifts downward

and upward with a constant velocity for α=3

2and α= 2, respectively. Note that the

total resistance term in (4) is nonlinear and odd in v. The latter indicates that the

system has no direction bias when w= 0. Under such conditions, it can be proved that

an antiperiodic f(t), for which f(t+τ) = −f(t), invariably yields a zero net response,

i.e., ⟨v⟩= 0. However, if such a symmetry is broken, the system will, potentially,

behave as a ratchet; see Hashemi et al. [27] for details. It is straightforward to show

that if αcan be expressed as a ratio of two odd integers (e.g., 1

1,5

3,19

9. . . ), f(t) is

antiperiodic, and hence, yields no net response [27]. This is not necessarily true when

f(t) is non-antiperiodic, e.g., for α=3

2and 2; as shown in Fig. 2(b), the particle

experiences a constant drift for these αvalues. In short, the two phenomena governing

the net motion of the particle are (i) spatial ratcheting eﬀect due to the presence of

gravitational forces, and (ii) temporal ratcheting due to non-antiperiodic excitations.

In fact, one can see the system behavior as a superposition of these two ratcheting

eﬀects. Notably, in Fig. 2(a) the temporal ratcheting eﬀect when α= 2 is strong

enough to reverse the direction of motion from downward to upward.

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摘要：

Amultiple-timinganalysisoftemporalratchetingArefHashemi1,2*,EdwardT.Gilman3andAdityaS.Khair4*1*DepartmentofAppliedandComputationalMathematicsandStatistics,UniversityofNotreDame,NotreDame,IN,UnitedStates.2CourantInstitute,NewYorkUniversity,NewYork,NY,UnitedStates.3DepartmentofMathematics,RiceUniversi...

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A multiple-timing analysis of temporal ratcheting Aref Hashemi12 Edward T. Gilman3and Aditya S. Khair4 1Department of Applied and Computational Mathematics and

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