Discrete games of chance as models for continuous stochastic transport processes

Andrew G. Allison; Derek Abbott

doi:10.1117/12.501497

7 May 2003 Discrete games of chance as models for continuous stochastic transport processes

Andrew G. Allison, Derek Abbott

Proceedings Volume 5114, Noise in Complex Systems and Stochastic Dynamics; (2003) https://doi.org/10.1117/12.501497
Event: SPIE's First International Symposium on Fluctuations and Noise, 2003, Santa Fe, New Mexico, United States

Abstract

Discrete games of chance can be used to illustrate principles of stochastic processes. For example, most readers are familiar with the use of discrete random walks to model the microscopic phenomenon of Brownian motion. We show that discrete games of chance, such as those of Parrondo and Astumian, can be used to quantitatively model stochastic transport processes. Discrete games can be used as “toy” models for pedagogic purposes but they can be much more than “toys”. In principle we could perform accurate simulations and we could reduce the errors of approximation to any desired level, provided that we were prepared to pay the computational cost. We consider some different approaches to discrete games, in the literature, and we use partial differential equations to model the particle densities inside a Brownian Ratchet. We apply a finite difference approach and obtain finite difference equations, which are equivalent to the games of Parrondo. The new games generalize Parrondo's original games, in the context of stochastic transport problems. We provide a practical method for constructing sets of discrete games, which can be used to simulate stochastic transport processes. We also attempt to place discrete games, such as those of Parrondo and Astumian, on a more sound philosophical basis.

Citation Download Citation

Andrew G. Allison and Derek Abbott "Discrete games of chance as models for continuous stochastic transport processes", Proc. SPIE 5114, Noise in Complex Systems and Stochastic Dynamics, (7 May 2003); https://doi.org/10.1117/12.501497

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