Ratchet effect on a relativistic particle driven by external forces

TL;DR

The paper analyzes ratchet transport of a damped relativistic particle driven by zero-average periodic forces, focusing on bi-harmonic and time-periodic piecewise-constant drives. By transforming to a linear momentum equation, it obtains an exact solution and expresses the ratchet velocity as a nonlinear functional v[f], enabling a functional Taylor expansion that clarifies the limitations of the traditional moment method. The authors derive the leading cubic term for general forcing and a specific cubic expression for piecewise-constant forcing, including a damping-induced ratchet with a non-monotonic dependence on the damping parameter. They demonstrate that the moment method is generally invalid except in the overdamped limit, highlighting the necessity of the functional approach to predict current shapes across parameter regimes and revealing how damping can enable ratchet currents under time-reversible forcing.

Abstract

We study the ratchet effect of a damped relativistic particle driven by both asymmetric temporal bi-harmonic and time-periodic piecewise constant forces. This system can be formally solved for any external force, providing the ratchet velocity as a non-linear functional of the driving force. This allows us to explicitly illustrate the functional Taylor expansion formalism recently proposed for this kind of systems. The Taylor expansion reveals particularly useful to obtain the shape of the current when the force is periodic, piecewise constant. We also illustrate the somewhat counterintuitive effect that introducing damping may induce a ratchet effect. When the force is symmetric under time-reversal and the system is undamped, under symmetry principles no ratchet effect is possible. In this situation increasing damping generates a ratchet current which, upon increasing the damping coefficient eventually reaches a maximum and decreases toward zero. We argue that this effect is not specific of this example and should appear in any ratchet system with tunable damping driven by a time-reversible external force.

Figures (1)

• Figure 1: Plot of the current velocity $v$, in units of $\epsilon_1^2\epsilon_2 /(M\omega)^3$, vs. the damping coefficient $\beta$, in units of the frequency, $\omega$, induced by a biharmonic force like (\ref{['eq-f']}) with $\phi_1=\phi_2=0$. Notice that this force is time reversible, i.e., $f(-t)=f(t)$.