Intrinsic ratchets: A Hamiltonian approach

A. V. Plyukhin

Phys. Rev. E 98, 042130 – Published 17 October 2018

Abstract

An asymmetric Brownian particle subjected to an external time-dependent force may acquire a net drift velocity and thus operate as a motor or ratchet, even if the external force is represented by an unbiased time-periodic function or by a zero-centered noise. For an adequate description of such ratchets, a conventional Langevin equation linear in the particle's velocity is insufficient, and one needs to take into account the first nonlinear correction to the dissipation force which emerges beyond the weak coupling limit. We derived microscopically the relevant nonlinear Langevin equation by extending the standard projection operation technique beyond the weak coupling limit. The particle is modeled as a rigid cluster of atoms and its asymmetry may be geometrical, compositional (when a cluster is composed of atoms of different types), or due to a combination of both factors. The drift velocity is quadratic in the external force's amplitude and increases with decreasing the force's frequency (for a periodic force) and inverse correlation time (for a fluctuating force). The maximum value of the drift velocity is independent on the particle's mass and achieved in the adiabatic limit, i.e., for an infinitely slow change of the external field.

Received 20 June 2018

DOI:https://doi.org/10.1103/PhysRevE.98.042130

Physics Subject Headings (PhySH)

Brownian motion Nonequilibrium statistical mechanics

Statistical Physics & Thermodynamics

Authors & Affiliations

A. V. Plyukhin^*

Department of Mathematics, Saint Anselm College, Manchester, New Hampshire 03102, USA

^*aplyukhin@anselm.edu

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Vol. 98, Iss. 4 — October 2018

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Images

Figure 1
Simplest atomic clusters with different types of asymmetry. Left: a dimer made of two different atoms (structural asymmetry). Right: a trimer made of three identical atoms (geometrical asymmetry).
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Figure 2
Two dimer configurations with the same radii of the atom-molecule interaction spheres, $σ_{1} = 2$ and $σ_{2} = 1$ , and different lengths $d$ . Configuration $A$ with $d = 2$ (on the left) is characterized by a significant overlapping of the interaction spheres and is found to have a larger value of the net drift velocity.
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Figure 3
Main plot: The average displacement $〈 X (t) 〉$ of a dimer of configuration $A$ (left on Fig. 2) activated by the external harmonic force $F_{ex} (t) = F_{0} sin ω t$ (acting on each atom) with $F_{0} = 2$ and $ω = 0.1$ . For units and values of other parameters see Eq. (85) and the text above it. The slope of tangent lines (dashed lines) to minima and maxima gives the net velocity of the dimer $V_{net} \approx 0.02$ . Inset: The corresponding average velocity $〈 V (t) 〉$ . The amplitude of positive peaks of the curve $〈 V (t) 〉$ is slightly larger than that of negative ones, which results in the net motion to the right. The net velocity $V_{net}$ can be also determined as a time average of $〈 V (t) 〉$ .
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Figure 4
Main plot: The average displacement $〈 X (t) 〉$ of a dimer of configuration $A$ (left on Fig. 2) activated by the external telegraph noise $F_{ex} (t)$ for three values of the correlation time $τ_{c}$ . Inset: A specific realization of $F_{ex} (t)$ with $τ_{c} = 1$ . The average $〈 X (t) 〉$ is calculated over about $10^{5}$ simulation runs.
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Figure 5
Two equilateral trimer configurations (in equilibrium) with the same radius $σ$ of bath-atom interaction spheres and different side lengths $2 a$ . The simulation shows that the net drift velocity of configuration $A$ with $2 a = σ$ (on the left) is more than three times higher than that of configuration $B$ with $a = σ$ (on the right); see Fig. 6.
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Figure 6
The average displacement (solid lines) of two trimer configurations $A$ and $B$ shown in Fig. 5 activated by a harmonic external force. The slopes of tangent lines to minima (dashed) give the net velocity $V_{net}$ of a cluster. We found $V_{net} \approx 0.07$ for configuration $A$ and $V_{net} \approx 0.02$ for configuration $B$ . The simulation parameters are: the molecule-atom mass ratio $m / M_{a} = 0.075$ (molecule-trimer mass ratio $m / M = 0.025$ ), external force frequency $ω = 0.1$ and amplitude $F_{0} = 2$ , squared harmonic bond frequency $Ω^{2} = k / M_{a} = 4.5$ , bath density $ρ = 0.2$ .
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