A Fokker--Planck description for Parrondo's games

TL;DR

This paper explores the connection between Parrondo's games and the Fokker-Planck equation, revealing how game probabilities relate to particle potential landscapes and conditions for fair games.

Contribution

It provides an exact relation between game probabilities and potential landscapes, clarifying when potentials correspond to fair or unfair games.

Findings

Probabilities relate exactly to potential landscapes

Fair games correspond to current-less potentials

Conditions for fair games identified

Abstract

We discuss in detail two recently proposed relations between the Parrondo's games and the Fokker--Planck equation describing the flashing ratchet as the overdamped motion of a particle in a potential landscape. In both cases it is possible to relate exactly the probabilities of the games to the potential in which the overdamped particle moves. We will discuss under which conditions current-less potentials correspond to fair games and vice versa.