Directed transport has been recognized as a major issue in a number of physical and biological contexts. The crucial phenomenon is the so-called ratchet behavior, in which currents may flow in a counter-intuitive direction. An interesting model of directed transport is the Parrondo game, in which random combination of two loosing games (motion to the negative direction) may result in a winning strategy (motion to the positive direction). This paradoxical dynamics has been successfully explained with Markovian chain analysis in the papers by Hammer, Abbott, Taylor and Parrondo. However, the importance of this subtle transport phenomena naturally calls for further studies from different perspectives.
In this paper, the authors supply just such alternative analysis using a deterministic analogue model of Parrondo games. In the model, the capital $x$ gets transformed by random combination of two piecewise linear maps $F_A(x)$ and $F_B(x)$, which consist of two and six segments respectively. The authors then argue that the random combination of $F_A(x)$ and $F_B(x)$ can be represented by a single six segments piecewise linear map $F_{A*B}(x)$.
The transport properties of this chaotic map can be fully analyzed with standard periodic orbit expansions. Specifically, the transport velocity $v=\left< x_n - x_0\right>_0 / n$ and the diffusion constant $D=(\left< (x_n - x_0)^2 \right>_0 - v^2 n^2)/ (2n)$, in which $x_n$ denotes the value of $x$ after $n$-th iterated mapping and $\left< \right>_0$ is the ensemble average over initial conditions, are shown to be fully expressed in terms of integer values and slopes of six segments of the map $F_{A*B}$ through the dynamical zeta function and its symbolic coding. Based on the obtained formulae for $v$ and $D$, the authors give a definite and quantitative proof of the existence of counter intuitive directed transport in the model trough several numerical examples.
The authors foresee further detailed analyses of deterministic Parrondo games both in its current piecewise linear version and also in nonlinear variant which is expected to possess intriguing sticking effects.
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