Discrete and continuous ratchets: from coin toss to molecular motor

Pages: 153 - 167, Volume 2, Issue 2, May 2002

doi:10.3934/dcdsb.2002.2.153       Abstract        Full Text (483.7K)       Related Articles

David Heath - Department of Mathematical Sciences, Carnegie Mellon University, Pittsburg, PA 15213, United States (email)
David Kinderlehrer - Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, United States (email)
Michal Kowalczyk - Department of Mathematical Sciences, Kent State University, Kent, OH 44242, United States (email)

Abstract: 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.

Keywords:  Diffusion mediated transport, Brownian motor, Parrondo Paradox.
Mathematics Subject Classification:  35K15, 35K20, 35B40, 60J10, 60J20, 60J60, 91A60.

Revised: November 2001;      Published: February 2002.