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.

布朗棘轮是一个扩散过程，代表布朗粒子向不对称锯齿电位的最小值移动的动力学。这激发了帕隆多的悖论，在这场悖论中，可以以某种方式将两场失败的比赛组合在一起，以取得胜利。最近已经发现，布朗棘轮可以通过离散时间随机游走来近似，该离散游走具有从相应的帕隆多博弈导出的状态相关的转移概率。我们研究了布朗棘轮的离散化Fokker-Planck方程，以便我们可以通过势函数的倾斜来确定近似Parrondo博弈是否公平。公平的Parrondo游戏对应于周期性的直到位势函数，而获胜或失败的Parrondo游戏则诱发倾斜的势函数。结果是，