The physical basis for Parrondo's games

Abstract

It has been reported that the original inspiration for Parrondo's games was a physical system called a "flashing Brownian ratchet." The relationship seems to be intuitively plausible but has not previously established with rigor. This is the problem that we address in this paper. The dynamics of a Brownian particle in a flashing Brownian ratchet are the result of diffusion and of externally applied forces. The probability density, of finding the particle at a certain place and time, can be mathematically modelled using a Partial Differential Equation (PDE) namely the Fokker-Planck Equation. In this paper, we apply standard finite-difference methods of numerical analysis 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 which justifies the claim that Parrondo's games are a discrete-time, discrete-space version of a flashing Brownian ratchet. We claim that Parrondo's games are effectively a particular way of sampling a Fokker-Planck Equation. Our difference equations are a natural and physically motivated generalization 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 to the same solution. The diffusion operator, implicitly assumed in Parrondo's original games, reduces to a pre-existing numerical method called "The Schmidt formula." There is actually an infinite continuum of possible diffusion operators and the Schmidt formula is at one extreme of the feasible range. We point out that an operator in the middle of the feasible range, with half-period binomial weightings, would be a better representation of the underlying physics. We present a modified form of Parrondo's games based on the central diffusion operator. We suggest that the finite difference method presented here will be useful in the simulation and design of real physical flashing Brownian ratchets.