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## The flashing ratchet

by Greg Harmer and Derek Abbott

### The mechanism of the flashing Brownian ratchet

Using the ratchet and pawl machine as a basis for a Brownian ratchet, several ideas have been used to get directed movement from Brownian particles. A common mechanism,  referred to as a flashing ratchet, which may prove fruitful when comparing it to Parrondo's games, is described below. It was first published by Ajdari and Prost in 1992.

Figure 2. This shows the how the mechanism of the ratchet potential works. The diagrams on the left, (a)-(c) show when there is no macroscopic gradient present and the net movement of particles is in the forward direction (defined by arrow). The diagrams on the right, (d)-(f) have a slight gradient present, and this causes the particles to drift backwards while Uoff is acting. Hence the nett flow of particles in the forward direction is reduced.

Consider a system where there exists two one-dimensional potentials, Uon and Uoff, as shown in Figure 2. The asymmetry of the potential Uon is determined by a, where 0 < alpha < 1. Having alpha = 1/2 creates a triangular symmetric potential otherwise the potential is asymmetrical like Uon in Figure 2 where alpha < 1/2. Let there be Brownian particles existing in the potential diffusing to a position of least energy.

When the Uon is applied, the particles are trapped in the minima of the potential so the concentration of the particles is peaked. Switching the potential off allows the particles to diffuse freely so the concentration is a set of Gaussian curves centered around the minima. When Uon is switched on again there is a probability Pfwd that is proportional to the green area of the curve that some particles are to the right of alpha*L. These particles move forwards to the minima located at L. Similarly there is a probability Pbck for particles that are to the left of (1-alpha)L. Since  a < 1/2 then Pfwd > Pbck the nett motion of the particles is to the right. Thus we have directed motion!!

Here is a good Java applet that shows a flashing Brownian ratchet (BR) working and allows you to change the parameters.