# American Institute of Mathematical Sciences

May  2002, 2(2): 153-167. doi: 10.3934/dcdsb.2002.2.153

## Discrete and continuous ratchets: from coin toss to molecular motor

 1 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburg, PA 15213, United States 2 Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, United States 3 Department of Mathematical Sciences, Kent State University, Kent, OH 44242

Revised  November 2001 Published  February 2002

Directed motion or ratchet-like behavior in many molecular scale systems is a consequence of diffusion mediated transport. The Brownian motor serves as a paradigm. The Parrondo Paradox is a pair of coin toss games, each of which is fair, or even losing, but become winning with a schedule of playing them in alternation. It has been proposed as a discrete analog of the Brownian motor. We examine the relationship between these two systems. We discover a class of Parrondo games with unusual ratchet-like behavior and for which diffusion plays a fundamentally different role than it does in the Brownian motor. Detailed balance is an important feature in these considerations.
The Brownian motor depends on details of the potential landscape in the system but the Parrondo game is decided on the potential difference alone. There are winning Parrondo games whose Brownian motor analogs move in the opposite direction. A general framework is discussed in section 7. The original Parrondo game, here in section 7.2, is completely determined by detailed balance.
Citation: David Heath, David Kinderlehrer, Michal Kowalczyk. Discrete and continuous ratchets: from coin toss to molecular motor. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 153-167. doi: 10.3934/dcdsb.2002.2.153
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