An approximation by Parrondo games of the Brownian ratchet

The Brownian ratchet is a diffusion process that represents the dynamics of a Brownian particle moving toward a minimum of an asymmetric sawtooth potential. It motivated Parrondo’s paradox, in which two losing games can be combined in a certain manner to achieve a winning outcome. Recently it has been found that the Brownian ratchet can be approximated by discrete-time random walks with state-dependent transition probabilities derived from corresponding Parrondo games. We study the discretized Fokker–Planck equation of the Brownian ratchet so that we can determine whether the approximating Parrondo game is fair through tilting of the potential function. A fair Parrondo game corresponds to a periodic untilted potential function whereas a winning or losing Parrondo game induces a tilted potential function. As a result, we provide transition probabilities of a random walk that can be used to approximate a diffusion process with a periodic piecewise constant drift coefficient. •We provide winning probabilities of Parrondo games to approximate the Brownian ratchet.•Using this approximation we can explain the relation between the potential functions and the Parrondo paradox.•We can approximate the stationary distribution of the diffusion process with a periodic piecewise constant drift.•It generalizes the usual drift for the Brownian ratchet.