Transport coefficients for a confined Brownian ratchet operating between   two heat reservoirs
  We discuss two-dimensional diffusion of a Brownian particle confined to a periodic asymmetric channel with soft walls modeled by a parabolic potential. In the channel, the particle experiences different noise intensities, or temperatures, in the transversal and longitudinal directions. The model is inspired by the famous Feynman's ratchet and pawl. Using ideas of the Fick-Jacobs approximation we derive the mean velocity of the particle, the effective diffusion coefficient, and the stationary probability density in the potential unit cell. The derived results are exact for small channel width. Yet, we check by exact numerical calculation that they qualitatively describe the ratchet effect observed for an arbitrary width of the channel. 
Artem Ryabov, Viktor Holubec, Mohammad Hassan Yaghoubi, Martin Varga,   M. Ebrahim Foulaadvand and Petr Chvosta
cond-mat.stat-mech