Discrete-time ratchets, the Fokker-Planck equation and Parrondo's games

Abstract

We present a detailed analysis of the so called Parrondo's games. They were originally devised as a pedagogical example of the ashing ratchet, but the relation was merely qualitative. Basically it is based on the combination of two games, both of them losing games when played alone, whereas if we combine them either periodically or randomly we obtain a winning game. In this work we establish a quantitative relation between the parameters describing Parrondo's games and the physical model of the Brownian ratchet. In the following sections of the present Chapter we will introduce some basic concepts on stochastic processes Sec. 1.2 , Markov processes Sec. 1.3 , and we will also explain some basic concepts about information theory Sec. 1.5 that will be used in a next Chapter. We will also present brie y the theory of Brownian motion and explain in more detail the ashing ratchet as well as the ratchet effect in Chapter 2; in this Chapter we also explain in detail the Parrondo's games, as well as other versions that exist in the literature. In Chapter 3 a new variation of the games is introduced: Parrondo's games with self transition. Chapter 4 is dedicated to the relation between Parrondo's games and the ashing ratchet model, showing that they can be related in a rigorous way; in Chapter 5 we will explain the relation between Parrondo's games and information theory. Finally in Chapter 6 we will present the main conclusions of the work.