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

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 460, Issue 2048, 2004, Pages 2269-2284

  Amengual, P ^a,b   Allison, A ^b   Toral, R ^a   Abbott, D ^b  

^a INSTITUTO MEDITERRÁNEO DE ESTUDIOS AVANZADOS   (Spain)

^b UNIVERSITY OF ADELAIDE   (Australia)

Author keywords

Brownian ratchet; Fokker Planck equation; Parrondo's paradox

Indexed keywords

GAME; MATHEMATICAL ANALYSIS; MATHEMATICAL PARAMETERS; METHODOLOGY; PHYSICAL MODEL; PROBABILITY; REVIEW; TIME;

EID: 20944432257     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2004.1283     Document Type: Review

References (51)

1. Abbott, D. 2001 Overview: unsolved problems of noise and fluctuations. Chaos 11, 526-538.
2. Abbott, D., Davies, P. C. W. & Shalizi, C. R. 2002 Order from disorder: the role of noise in creative processes. Fluct. Noise Lett. 2, C1-C12.
3. Ait-Haddou, R. & Herzog, W. 2002 Force and motion generation of myosin motors: muscle contraction. J. Electromyogr. Kinesiol. 12, 435-445.
4. Ajdari, A. & Prost, J. 1992 Drift induced by a periodic potential of low symmetry: pulsed dielectrophoresis. C. R. Acad. Sci. Paris Sér. II 315, 1635-1639.
5. Allison, A. & Abbott, D. 2001a Control systems with stochastic feedback. Chaos 11, 715-724.
6. Allison, A. & Abbott, D. 2001b Stochastic resonance on a Brownian ratchet. Fluct. Noise Lett. 1, L239-L244.
7. Allison, A. & Abbott, D. 2002 The physical basis for Parrondo's games. Fluct. Noise Lett. 2, L327-L342.
8. Arena, P., Fazzino, S., Fortuna, L. & Maniscalco, P. 2003 Game theory and non-linear dynamics: the Parrondo paradox case study. Chaos Solitons Fractals 17, 545-555.
9. Boman, M., Johansson, S. J. & Lyback, D. 2001 Parrondo strategies for artificial traders. In Intelligent agent technology (ed. N. Zhong, J. Liu, S. Ohsuga & J. Bradshaw), pp. 150-159. World Scientific.
10. Buceta, J. & Lindenberg, K. 2002 Switching-induced Turing instability. Phys. Rev. E 66, 046202.
11. Buceta, J., Lindenberg, K. & Parrondo, J. M. R. 2002a Spatial patterns induced by random switching. Fluct. Noise Lett. 2, L21-L29.
12. Buceta, J., Lindenberg, K. & Parrondo, J. M. R. 2002b Pattern formation induced by nonequilibrium global alternation of dynamics. Phys. Rev. E 66, 036216.
13. Cleuren, B. & van den Broeck, C. 2002 Random walks with absolute negative mobility. Phys. Rev. E 65, 030101.
14. Davies, P. C. W. 2001 Physics and life: the Abdus Salam memorial lecture. In 6th Trieste Conf. Chemical Evolution, Trieste, Italy (ed. J. Chela-Flores, T. Tobias & F. Raulin), pp. 13-20. Dordrecht: Kluwer Academic.
15. Dinis, L. & Parrondo, J. M. R. 2003 Optimal strategies in collective Parondo games. Europhys. Lett. 63, 319-325.
16. Flitney, A. P., Ng, J. & Abbott, D. 2002 Quantum Parrondo's games. Physica A 314, 35-42.
17. Harmer, G. P. & Abbott, D. 1999a Parrondo's paradox. Statist. Sci. 14, 206-213.
18. Harmer, G. P. & Abbott, D. 1999b Losing strategies can win by Parrondo's paradox. Nature 402, 864.
19. Harmer, G. P. & Abbott, D. 2002 A review of Parrondo's paradox. Fluct. Noise Lett. 2, R71-R107.
20. Harmer, G. P., Abbott, D. & Taylor, P. G. 2000 The paradox of Parrondo's games. Proc. R. Soc. Lond. A 456, 247-259.
21. Harmer, G. P., Abbott, D., Taylor, P. G. & Parrondo, J. M. R. 2001 Brownian ratchets and Parrondo's games. Chaos 11, 705-714.
22. Heath, D., Kinderlehrer, D. & Kowalczyk, M. 2002 Discrete and continuous ratchets: from coin toss to molecular motor. Discrete Contin. Dynam. Syst. B 2, 153-167.
23. Horsthemke, W. & Lefever, R. 1984 Noise-induced transitions: theory and applications in physics, chemistry and biology. Springer.
24. Karlin, S. 1973 A first course in stochastic processes. Academic.
25. Kay, R. J. & Johnson, N. F. 2002 Winning combinations of history-dependent games. Phys. Rev. E 67, 056128.
26. Kestenbaum, D. 1997 Sand castles and cocktail nuts. New Scient. 154, 25-28.
27. Key, E. S. 1987 Computable examples of the maximal Lyapunov exponent. Probab. Theory Relat. Fields 75, 97-107.
28. Key, E. S., Klosek, M. & Abbott, D. 2002 On Parrondo's paradox: how to construct unfair games by composing fair games. (Preprint math/0206151.)
29. Kinderlehrer, D. & Kowalczyk, M. 2002 Diffusion-mediated transport and the flashing ratchet. Arch. Ration. Mech. Analysis 161, 149-179.
30. Kocarev, L. & Tasev, Z. 2002 Lyapunov exponents, noise-induced synchronization, and Parrondo's paradox. Phys. Rev. E 65, 046215.
31. Latouche, G. & Taylor, P. G. 2003 Drift conditions for matrix-analytic models. Math. Operat. Res. 28, 346-360.
32. Lee, C. F., Johnson, N. F., Rodriguez, F. & Quiroga, L. 2002a Quantum coherence, correlated noise and Parrondo games. Fluct. Noise Lett. 2, L293-L298.
33. Lee, Y., Allison, A., Abbott, D. & Stanley, H. E. 2002b Minimal Brownian ratchet: an exactly solvable model. Phys. Rev. Lett. 91, 220601.
34. Maslov, S. & Zhang, Y. 1998 Optimal investment strategy for risky assests. Int. J. Theor. Appl. Finance 1, 377-387.
35. Meyer, D. A. & Blumer, H. 2002a Quantum Parrondo games: biased and unbiased. Fluct. Noise Lett. 2, L257-L262.
36. Meyer, D. A. & Blumer, H. 2002b Parrondo games as a lattice gas automata. J. Stat. Phys. 107, 225-239.
37. Mihailović, Z. & Rajković, M. 2003a One dimensional asynchronous cooperative Parrondo's games. Fluct. Noise Lett. 3, L389-L398.
38. Mihailović, Z. & Rajković, M. 2003b Synchronous cooperative Parrondo's games. (Preprint condmat/0307212.) (Submitted.)
39. Moraal, H. 2000 Counterintuitive behaviour in games based on spin models. J. Phys. A 33, L203-L206.
40. Parrondo, J. M. R. 1996 How to cheat a bad mathematician. EEC HC&M Network Workshop on Complexity and Chaos (contract no. ERBCHRX-CT 940546), Turin, Italy.
41. Parrondo, J. M. R., Harmer, G. P. & Abbott, D. 2000 New paradoxical games based on Brownian ratchets. Phys. Rev. Lett. 85, 5226-5229.
42. Percus, O. E. & Percus, J. K. 2002 Can two wrongs make a right? Coin-tossing games and Parrondo's paradox. Math. Intelligencer 24, 68-72.
43. Pinsky, R. & Scheutzow, M. 1992 Some remarks and examples concerning the transient and recurrence of random diffusions. Ann. Inst. H. Poincaré B 28, 519-536.
44. Pyke, R. 2002 On random walks related to Parrondo's games. (Preprint math/0206150.)
45. Reimann, P. 2002 Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57-265.
46. Rosato, A., Strandburg, K. J., Prinz, F. & Swendsen, R. H. 1987 Why the Brazil nuts are on top: size segregation of particulate matter by shaking. Phys. Rev. Lett. 58, 1038-1040.
47. Toral, R. 2001 Cooperative Parrondo's games. Fluct. Noise Lett. 1, L7-L12.
48. Toral, R. 2002 Capital redistribution brings wealth by Parrondo's paradox. Fluct. Noise Lett. 2, L305-L311.
49. Toral, R., Amengual, P. & Mangioni, S. 2003a Parrondo's games as a discrete ratchet. Physica A 327, 105-110.
50. Toral, R., Amengual, P. & Mangioni, S. 2003b A Fokker Planck description for Parrondo's games. In Proc. SPIE Conf. on Noise in Complex Systems and Stochastic Dynamics, Santa Fe, NM, USA (ed. L. Schimansky-Geier, D. Abbott, A. Neiman & C. van den Broeck), vol. 5114, pp. 309-317. Bellingham, WA: SPIE.
51. Westerhoff, H. V., Tsong, T. Y., Chock, P. B., Chen, Y. & Astumian, R. D. 1986 How enzymes can capture and transmit free energy contained in an oscillating electric field. Proc. Natl Acad. Sci. 83, 4734-4738.