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As I understand, Brownian ratchets do not work because the paddle, which is supposed to keep the gear spinning in a single direction, also oscillates with thermal noise. A the paddle can dissipate the movement in one direction into thermal noise, but then thermal noise is also going to be spontaneously converted into movement in the opposite-direction, as stated in the fluctuation-dissipation theorem.

However I still do not understand why the following Brownian motor does not work, other than the fact that it would convert thermal noise into rotational energy if it worked, violating the second law of thermodynamics.

Brownian motor

The machine is quite simple. It is a gear with asymmetric, sigsaw tooth in vacuum. Other than the shaft (not shown), the gear only touches a needle. The needle is connected to a hook and allowed to move up and down, only hitting the gear at right angles. Because of the asymmetric tooth of the gear, whenever it hits it is more likely to hit the "right side" of a teeth and to cause the gear to gain counter-clockwise angular momentum.

I understand the needle will also vibrate in the horizontal direction and that this vibration, while with higher frequency, has the same energy as the vertical movement, but this does not seem to transfer clockwise angular momentum to the gear on average.

Can someone help me understand what am I getting wrong? I have had this question for several years already and until today have not had enough courage to ask for an explanation.

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You could just as well make an argument for spontaneous clockwise rotation: thermal vibration of the gear bumps the needle and causes it to glide up the gradual ratchet slope much easier than up the sharp ratchet slope. Dropping of the needle behind the sharp ratchet slope prevents reverse (counterclockwise) motion.

In reality, both effects are relevant (the needle driving the gear and the gear driving the needle), resulting in no consistent rotation in one direction or the other.

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