9 that U ( x, t ) could as well be incorporated into the drift term instead of the diffusion term. However, differing from [34, 35] our cases (ii) and (iii) do not incorporate an external potential. The absence of a spatial ratchet potential is compensated by strongly nonlin- ear prefactors of the diffusion and the drift term that result in mean flow even for harmonic (i.e., time-symmetric driving). In the context of ’particle ratchets’ related systems are studied [36, 37] where a mean flux is created without spatial ratchet potential but through the response of two different (weakly) coupled degrees of freedom to time-periodic driving. The de- grees of freedom are related to two different interacting par- ticle species in [36], and to pancake vortices and Josephson vortices in layered superconductors in [37]. However, in con- trast to our case in [36, 37] the driving is time-asymmetric. Time-symmetric driving can as well induce mean transport in a ’particle ratchet’ without spatially varying potential if the coupling of the different degrees of freedom is sufficiently nonlinear. The only example we are aware of is the “vortex diode” discussed in [38]. Note finally, that similar 4th orde r terms as the capillarity term in Eq. (7) may be included in higher order Fokker-Planck equations. pare the relevant vibration frequencies in the model and in the experiment. Finally, we would like to discuss the relation of the dis- cussed ratchet mechanisms to transport a continuous phase to ’particle ratchets’. We do this by pointing out some for- mal analogies between the employed thin film equation and a Fokker-Planck equation for transport of discrete objects in a particle ratchet. The film thickness evolution equation shows similarities to a Fokker-Planck equation for interact- ing particles in an external ratchet potential [34, 35]. Its non- dimensional form is with ∂$_{t}$ h = − ∂$_{x}$ { Q ( h ) ∂$_{x}$ [ ∂$_{xx}$h + P ( h, x, t )] + Q ( h ) F ( t ) } . (7) ∂$_{t}$ W = − ∂$_{x}$ { W ∂$_{x}$P ( W, x, t ) + W F ( t ) } (8) P ( W, x, t ) = − T ( t ) log W − ˜ gW − U ( x, t ) (9) Acknowledgments This work was supported by the European Union under grant PITN-GA-2008-214919 (MULTIFLOW). In the particular case, it is an evolution equation for the one-particle distribution function W ( x, t ) , where U ( x, t ) is a flashing ratchet potential (note that the one used in [34, 35 ] is static) ’rocked’ periodically by F ( t ) . The function T ( t ) stands for a periodic modulation of the non-dimensional tem- perature, and ˜ g is a non-dimensional interaction parameter. One notes that both, the first order drift or transport term an d the second order diffusion term are modulated in time. Note doi:10.1103/PhysRevLett.96.154502 [10] G. Lagubeau, M. Le Merrer, C. Clanet, D. Qu´er´e, Leiden frost on a ratchet, Preprint. 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