The Physical Basis for Parrondo’s Games

Several authors [1–4] have implied that the original inspiration for Parrondo’s games was a physical system called a “flashing Brownian ratchet [5, 6]” The relationship seems to be intuitively clear but, surprisingly, has not yet been established with rigor. The dynamics of a flashing Brownian ratchet can be described using a partial differential equation called the Fokker-Planck equation [7], that describes the probability density, of finding a particle at a certain place and time, under the influence of diffusion and externally applied fields. In this paper, we apply standard finite-difference methods of numerical analysis [8–10] to the Fokker-Planck equation. We derive a set of finite difference equations and show that they have the same form as Parrondo’s games. This justifies the claim that Parrondo’s games are a discrete-time, discrete-space version of a flashing Brownian ratchet. Parrondo’s games, are in effect, a particular way of sampling a Fokker-Planck equation. Our difference equations are a natural and physically motivated generalisation of Parrondo’s games. We refer to some well established theorems of numerical analysis to suggest conditions under which the solutions to the difference equations and partial differential equations would converge. The diffusion operator, implicitly assumed in Parrondo’s original games, reduces to the Schmidt formula for the integration of the diffusion equation. There is actually an infinite continuum of possible diffusion operators. The Schmidt formula is at one extreme of the feasible range. We suggest that an operator in the middle of the feasible range, with half-period binomial weightings, would be a better representation of the underlying physics. Physical Brownian ratchets have been constructed and have worked [11–15]. It is hoped that the finite element method presented here will be useful in the simulation and design of flashing Brownian ratchets.