Why is it so hard to explain that the Brownian Ratchet doesn't work? The Brownian Ratchet stood up to a lot of scrutiny before it was finally shown why it would not work as a perpetual motion machine, but it seems weird to me that all of that was necessary. If the particles hitting the paddle caused it to move they would lose energy and slow down. Because of this there is a finite amount of work that the system can do which would make it not a perpetual motion machine. Why is it necessary to show that the pawl undergoes Brownian motion to prove that it won't work?
 A: Most proposed perpetual-motion machines are perpetual motion machines of the first kind (that is, they violate the First Law of Thermodynamics, conservation of energy).  It's easy to understand why these shouldn't work, and the missing energy source (or the fact that the machine isn't really perpetual) can usually be found quickly.
The Brownian Ratchet is an example of the less-common perpetual motion machine of the second kind.  It apparently violates the Second Law of Thermodynamics (entropy in an isolated system cannot decrease), in this case, by generating motion directly from a heat source.  Entropy is a much harder concept to understand than energy (it can be thought of as a measure of the "sameness" of the universe), and working out why the Brownian Ratchet doesn't work requires complicated statistical-mechanics calculations.
The key thing to remember when analyzing the Brownian Ratchet is that the Second Law applies to isolated systems as a whole.  The paddles aren't isolated from the pawl; a correct analysis needs to take into account the whole paddle/pawl/working fluid system.  If you do this, you'll find that, over time, the entropy decrease from molecules hitting the paddles is exactly offset by the entropy increase from molecules hitting the pawl or the pawl's own thermal motion, letting the paddles slip backwards.
A: Here is another reason why a ratchet should not work: it would define a directional arrow of time even in thermal equilibrium. 
To see this look at the modified ratchet in the graph (the triangular piece is attached to a spring). the brownian particle would move easier to the right as it can push the triangular piece down, but if it is at the right and tries to move to the left it will be stopped by the vertical surface (no way to push it down). Our intuition tells us that the particle then will circulate clockwise, even in thermal equilibrium. This defines an arrow of time: you can tell if a movie is shown in reverse. Which cannot be true.

