Figure 4.

(a) Directed transport in standard cosine potential, V₁ = 1, V₂ = 0, and in a ratchet potential, V₁ = 1, V₂ = 0.4, in the case of harmonic mixing driving that breaks the time-reversal symmetry, ψ = π/2, with amplitudes A₁ = 5, A₂ = 2, and frequency Ω = 1. Transport ceases at ψ = 0 even for ratchet potential with broken symmetry, when the time-reversal symmetry is restored (dash-dotted line). (b) Influence of a tiny (as compare with the periodic force modulation) constant loading force on transport for the case of a ratchet potential in part (a). The transport ceases after some random time, which depends on f_L and initial conditions, and the particle returns accelerating back. A very similar picture also emerges for a cosine potential with V₂ = 0 (not shown). The stalling force is obviously zero. Any genuine ratchet and motor must be characterized by a non-zero stalling force. A symplectic leapfrog/Verlet integration scheme (where no spurious dissipation is introduced by numerics) was used to obtain these results.