Directed Current due to Broken Time-Space Symmetry

Abstract

We consider the classical dynamics of a particle in a one-dimensional space-periodic potential U(X) = U(X+2pi) under the influence of a time-periodic space-homogeneous external field E(t) = E(t+T). If E(t) is neither a symmetric function of t nor antisymmetric under time shifts E(t+/-T/2) not equal-E(t), an ensemble of trajectories with zero current at t = 0 yields a nonzero finite current as t-->infinity. We explain this effect using symmetry considerations and perturbation theory. Finally we add dissipation (friction) and demonstrate that the resulting set of attractors keeps the broken symmetry property in the basins of attraction and leads to directed currents as well.

Deterministic inhomogeneous inertia ratchets

“…Owing to the complexity of the dynamics, for example the simultaneous presence of periodic and chaotic attractors, there is a need for proper choice of initial conditions and ensemble averaging over them to obtain realistic averages [14,19,29]. This fact has been exemplified in an (zero mean) ac modulated periodic Hamiltonian system [31].…”

“…[16], has also contributed significantly to the understanding of the subject. These deterministic ratchets, unaided by noise, are shown to yield current in overdamped [17,18], underdamped [14,16,19,20,21,22,23,24,25,27,28], as well as in Hamiltonian [29,30,31] periodic potential systems, and also in overdamped quenched disordered [32,34] systems. In these systems net current results, without the presence of applied nonzero average forcing or asymmetric fluctuations, due to the presence of various regular transporting or chaotic attractors depending on the initial conditions for given system parameter values.…”

“…This effect, called the ratchet effect, has also been obtained in underdamped symmetric periodic (non-ratchetlike) potential systems, by considering either the temperature [6,7,8,9] or the friction coefficient [10,11] to be space dependent. Ratchet current can also result in a symmetric periodic potential when the system is driven by an externally applied zero-mean temporally asymmetric forcing aided by Gaussian white noise [12] or by the explicit application of nonthermal noise [13,14]. The ratchet effect is being studied intensively for about two decades and its applications envisaged in periodic systems that are common in many areas of natural sciences [15].…”

“…In Ref. [14] the symmetry criteria for the realization of ratchet current have been discussed in detail.…”

Controlling directed transport in two-dimensional periodic structures under crossed electric fields

“…For e 2 = 0 and any value of the phase one could in principle observe a ratchet e ect in x, due to the breaking of the temporal symmetry f x (t + T=2) = −f x (t) [41]. Note, however, that in the underdamped limit b → 0 one has to deal also with the symmetry f x (−t + T=2) = f x (t), and the condition, = 0; , may be required for inertial systems with very low friction values.…”