Reversible ratchets as Brownian particles in an adiabatically changing periodic potential

The existence of transport of Brownian particles in a one-dimensional periodic potential which changes adiabatically is proven. The net fraction of particles crossing a given point toward a given direction during an adiabatic process can be expressed as a contour integral of a nonexact differential in the space of parameters of the potential. Since the work done to change the potential is an exact differential in the space of parameters, cycles can be designed where transport of particles is induced without any energy consumption. These cycles can be called reversible ratchets, and a concrete example is described. The repercussions of these results on equilibrium thermodynamics are discussed.