The analysis carried out in the previous chapter led to a simple, yet fundamental, conclusion: asymmetry alone is not sufficient for the operation of a Brownian motor, as thermal equilibrium inhibits the rectification of fluctuations. In order to obtain average motion, the system has to be taken out of equilibrium. In the examples considered so far, the Brillouin paradox and Feynman's ratchet, this is done by keeping two parts of the system at different temperatures: both metals in the diode of the Brillouin paradox; and the vanes and the wheel in Feynman's ratchet. Note that in these ratchet systems both parts are macroscopic objects, and, thus, keeping them at different temperatures is not technically difficult. It can become more complicated, however, if the whole ratchet system is at the molecular or microscopic level. These are the systems which are most interesting for the realization of Brownian ratchets. Indeed, in these small systems, thermal fluctuations are usually not negligible at all.

This chapter examines different mechanisms to keep the system away from thermal equilibrium, thus introducing three fundamental models of ratchets: the flashing ratchet, the forced ratchet and the information ratchet. Historically, the flashing ratchet model, introduced by Ajdari and Prost (1992), has had a tremendous influence in the scientific community, motivating many experimental and theoretical works. Though a similar model had been previously introduced by Bug and Berne (1987) in a different context, it was the work by Ajdari and Prost that made the field of Brownian ratchets really take off. It was soon realized by Magnasco (1993) that other general schemes to generate directed motion are possible, such as that of the forced ratchet. The concept of the information ratchet was introduced by Astumian and Derenyi (1998) in order to describe in simple terms chemical powered motors such as those present in all living beings, the motor proteins.

The flashing ratchet

Consider the one-dimensional dynamics of a Brownian particle moving in the x-direction in the presence of a ratchet potential V(x), that is, a spatially asymmetric potential which is assumed to be spatially periodic, V(x + L) = V(x), as illustrated in the top panel of Fig. 2.1.