Pulsating ratchets: From potential symmetry to symmetry of average velocity

A method for symmetry analysis of a deterministic pulsating ratchet (the so-called phase shift indexing method) is proposed, which provides a relationship between the symmetry parameters of the potential profile of a nanoparticle and the average velocity of the corresponding ratchet. The technique and capabilities of the method are demonstrated by determining the type of symmetry of the dependence of the average ratchet velocity in a general-type additive-multiplicative potential, U(x,t)=u(x)+w(x)σ(t) (σ(t) is a periodic function of time), on the distance (phase shift) between the axes or centers of symmetry of its stationary, u(x), and fluctuating, w(x), periodic contributions. It has been established that if the spatial dependence of u(x) and the time dependence of σ(t) belong to the universal type of symmetry (i.e., have both an axis and a center of symmetry mutually shifted by a quarter-period), then the average velocity as a function of the phase shift is also universally symmetric in the overdamped inertialess motion mode [taking into account the hidden Cubero-Renzoni symmetry, Cubero et al., Phys. Rev. Lett. 116, 010602 (2016)10.1103/PhysRevLett.116.010602], while being only antisymmetric with inertia included. The obtained conclusions are confirmed by the results of calculations of the average velocity of pulsating ratchets in overdamped and inertial motion modes. The effectiveness of the proposed method is also compared for a number of reported cases which address the spatiotemporal dependence of potential energy in overdamped, nondissipative, and inertial-dissipative motion modes.