A Brownian ratchet is described here at Wikipedia.
The "why it fails" section reads:
Feynman demonstrated that if the entire device is at the same temperature, the ratchet will not rotate continuously in one direction but will move randomly back and forth, and therefore will not produce any useful work.
The reason is that the pawl, since it is at the same temperature as the paddle, will also undergo Brownian motion, "bouncing" up and down. It therefore will intermittently fail by allowing a ratchet tooth to slip backward under the pawl while it is up.
Another issue is that when the pawl rests on the sloping face of the tooth, the spring which returns the pawl exerts a sideways force on the tooth which tends to rotate the ratchet in a backwards direction.
These reasons sounds strange as physical argument to me:
-The pawl also undergos Brownian motion is a trivial fact, but I don't see any reason why it is relevant. This sound more of a practical problem rather than a theoretical one. Moreover, the forces exerted on the pawl and the paddle are applied at different direction - one radially (with a spring counteracting against it), one tangentially. How can they be compared directly?
-The sideway force would restores the energy back to the gas in the paddle, leaving the total energy unchanged. and it doesn't matter in a statistical sense anyway even if this force is comparable to, or bigger than the Brownion motion of the gas - most likely the expectation just becomes a shifted Gaussian distribution.
So, why doesn't this setup violate the second law of thermodynamics?