The physical basis for Parrondo's games

TL;DR

This paper rigorously links Parrondo's games to physical Brownian ratchets by deriving finite difference equations from the Fokker-Planck equation, showing they are mathematically equivalent and useful for simulating ratchets.

Contribution

It establishes a rigorous mathematical connection between Parrondo's games and physical Brownian ratchets using finite difference methods.

Findings

Finite difference equations match the form of Parrondo's games.

Parrondo's games sample the Fokker-Planck equation.

Finite element method can aid in ratchet simulation and design.

Abstract

Several authors have implied that the original inspiration for Parrondo's games was a physical system called a ``flashing Brownian ratchet''. The relationship seems to be intuitively clear but, surprisingly, has not yet been established with rigor. 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. Parrondo's games, are in effect, a particular way of sampling a Fokker-Planck equation. Physical Brownian ratchets have been constructed and have worked. It is hoped that the finite element method presented here will be useful in the simulation and design of flashing Brownian ratchets.