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Parrondo Games as Lattice Gas Automata

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Abstract

Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.

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REFERENCES

  1. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

    Google Scholar 

  2. D. A. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys. 85:55–574 (1996).

    Google Scholar 

  3. B. M. Boghosian and W. Taylor, IV, A quantum lattice-gas model for the many-particle Schrödinger equation in d dimensions, Phys. Rev. E 8:70–716 (1997).

    Google Scholar 

  4. D. A. Meyer, Quantum mechanics of lattice gas automata: One particle plane waves and potentials, Phys. Rev. E 55:526–5269 (1997).

    Google Scholar 

  5. D. A. Meyer, Quantum mechanics of lattice gas automata: Boundary conditions and other inhomogeneities, J. Phys. A: Math. Gen. 31:232–2340 (1998).

    Google Scholar 

  6. J. M. R. Parrondo, Parrondo's Paradoxical Games, http://seneca.fis.ucm.es/parr/GAMES.

  7. G. P. Harmer and D. Abbott, Parrondo's paradox, Stat. Science 14:20–213 (1999); G. P. Harmer and D. Abbott, Losing strategies can win by Parrondo's paradox, Nature 402:864 (1999).

    Google Scholar 

  8. D. A. Meyer, Quantum strategies, Phys. Rev. Lett. 82:105–1055 (1999).

    Google Scholar 

  9. D. Abbott, personal communication (February 2000).

  10. G. I. Taylor, Diffusion by continuous movements, Proc. Lond. Math. Soc. 20:19–212 (1920).

    Google Scholar 

  11. C. DeWitt-Morette and S. K. Foong, Path-integral solutions of wave equations with dissipation, Phys. Rev. Lett. 62:220–2204 (1989).

    Google Scholar 

  12. J. W. Strutt and Baron Rayleigh, The Theory of Sound, Vol. 1, 2nd edn. (Macmillan, London, 1894), Section 42; J. W. Strutt and Baron Rayleigh, On James Bernoulli's theorem in probabilities, Philos. Mag. (5) 47:24–251 (1899); M. von Smoluchowski, Ñber Brownschen Molekularbewegung unter Einwirkung äußerer Kräfte unde deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Ann. Physik 48:110–1112 (1915).

    Google Scholar 

  13. G. Zanetti, Hydrodynamics of lattice gas automata, Phys. Rev. A 40:153–1548 (1989); Z. Cheng, J. L. Lebowitz, and E. R. Speer, Microscopic shock structure in model particle systems: The Boghosian-Levermore cellular automaton revisited, Commun. Pure Appl. Math. XLIV:97–979 (1991); B. Hasslacher and D. A. Meyer, Lattice gases and exactly solvable models, J. Statist. Phys. 68:57–590 (1992).

    Google Scholar 

  14. C. R. Doering, Randomly rattled ratchets, Il Nuovo Cimento D 17:68–697 (1995).

    Google Scholar 

  15. M. von Smoluchowski, Experimentell nachweisbare, der üblichen Thermodynamik widersprechende Molekularphänomene, Phys. Z. XIII:106–1080 (1912).

    Google Scholar 

  16. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 1 (Addison-Wesley, Reading, MA, 1963), Sections 46.–46.9.

    Google Scholar 

  17. J. M. R. Parrondo and P. Español, Criticism of Feynman's analysis of the ratchet as an engine, Am. J. Phys. 64:112–1130 (1996).

    Google Scholar 

  18. D. Abbott, B. David, and J. M. R. Parrondo, The problem of detailed balance for the Feynman-Smoluchowski engine (FSE) and the multiple pawl paradox, Unsolved Problems of Noise and Fluctuations, UPoN'99, Proceedings of the Second International Conference, Adelaide, Australia, 1–15 July 1999, AIP Conference Proceedings 511 (2000), pp. 21–218.

    Google Scholar 

  19. J. Rousselet, L. Salome, A. Adjari, and J. Prost, Directional motion of Brownian particles induced by a periodic asymmetric potential, Nature 370:44–448 (1994).

    Google Scholar 

  20. L. P. Faucheux, L. S. Bourdieu, P. D. Kaplan, and A. J. Libchaber, Optical thermal ratchet, Phys. Rev. Lett. 74:150–1507 (1995).

    Google Scholar 

  21. P. Riemann, M. Grifoni, and P. Hänggi, Quantum ratchets, Phys. Rev. Lett. 79:1–13 (1997); S. Yukawa, M. Kikuchi, G. Tatara, and H. Matsukawa, Quantum ratchets, J. Phys. Soc. Jpn. 66:295–2956 (1997); G. Tatara, M. Kikuchi, S. Yukawa, and H. Matsukawa, Dissipation enhanced-asymmetric transport in quantum ratchets, J. Phys. Soc. Jpn. 67:109–1093 (1998); I. Goychuk, M. Grifoni, and P. Hänggi, Nonadiabatic quantum Brownian rectifiers, Phys. Rev. Lett. 81:64–652 (1998).

    Google Scholar 

  22. A. Lorke, S. Wimmer, B. Jager, J. P. Kotthaus, W. Wegscheider, and M. Bichler, Farinfrared and transport properties of antidot arrays with broken symmetry, Physica B 249-251:31–316 (1998); H. Linke, T. E. Humphrey, A. Löfgren, A. O, Sushkov, R. Newbury, R. P. Taylor, and P. Omling, Experimental tunneling ratchets, Science 286:231–2317 (1999).

    Google Scholar 

  23. D. A. Meyer, On the absence of homogeneous scalar unitary cellular automata, Phys. Lett. A 223:33–340 (1996).

    Google Scholar 

  24. J. Hardy, Y. Pomeau, and O. de Pazzis, Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions, J. Math. Phys. 14:174–1759 (1973); J. Hardy, O. de Pazzis, and Y. Pomeau, Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions, Phys. Rev. A 13:194–1961 (1976); U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice-gas automata for the Navier-Stokes equation, Phys. Rev. Lett. 56:150–1508 (1986).

    Google Scholar 

  25. S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math. 4:12–156 (1951).

    Google Scholar 

  26. M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain J. Math. 4:49–509 (1974); reprinted from Some Stochastic Problems in Physics and Mathematics, Colloquium Lectures in the Pure and Applied Sciences, No. 2 (Magnolia Petroleum and Socony Mobil Oil, Dallas, Texas, 1956).

    Google Scholar 

  27. J. M. R. Parrondo, G. P. Harmer, and D. Abbott, New paradoxical games based on Brownian ratchets, Phys. Rev. Lett. 85:522–5229 (2000).

    Google Scholar 

  28. D. A. Meyer, Quantum lattice gases and their invariants, Int. J. Mod. Phys. C 8:71–735 (1997).

    Google Scholar 

  29. F. Sauter, Ñber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs, Z. Physik 69:74–764 (1931); D. Îto, K. Mori, and E. Carriere, An example of dynamical systems with linear trajectory, Nuovo Cimento A 51:111–1121 (1967); P. A. Cook, Relativisitic harmonic oscillators with intrinsic spin structure, Lett. Nuovo Cimento 1:41–426 (1971); M. Moshinsky and A. Szczepaniak, The Dirac oscillator, J. Phys. A: Math. Gen. 22:L81–L819 (1989).

    Google Scholar 

  30. Y. Nogami and F. M. Toyama, Coherent states of the Dirac oscillator, Canadian J. Phys. 74:11–121 (1996); F. M. Toyama, Y. Nogami, and F. A. B. Continho, Behaviour of wavepackets of the ''Dirac oscillator:'' Dirac representation versus Foldy-Wouthuysen representation, J. Phys. A: Math. Gen. 30:258–2595 (1997).

    Google Scholar 

  31. S. C. Benjamin and P. M. Hayden, Multi-player quantum games, Phys. Rev. A 64:030301(R) (2001).

    Google Scholar 

  32. W. B. Arthur, Inductive reasoning and bounded rationality, Am. Econom. Rev. 84:40–411 (1994); D. Challet and Y.-C. Zhang, Emergence of cooperation and organization in an evolutionary game, Physica A 246:40–418 (1997).

    Google Scholar 

  33. D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, ''Quantum walks on graphs,'' Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC'01), Hersonissos, Crete, Greece, –8 July 2001 (New York, ACM Press, 2001), pp. 5–59.

    Google Scholar 

  34. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, ''One-dimensional quantum walks,'' Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC'01), Hersonissos, Crete, Greece, –8 July 2001 (New York, ACM Press, 2001), pp. 3–49.

    Google Scholar 

  35. A. M. Childs, E. Farhi, and S. Gutmann, An Example of the Difference Between Quantum and Classical Random Walks, quant-ph/0103020.

  36. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, New York, 2000).

    Google Scholar 

  37. D. A. Meyer, Physical Quantum Algorithms, UCSD preprint (2001).

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Authors and Affiliations

  1. Project in Geometry and Physics, Department of Mathematics, University of California/San Diego, La Jolla, California, 92093-0112

    David A. Meyer

  2. Institute for Physical Sciences, Los Alamos, New Mexico, 87544

    David A. Meyer & Heather Blumer

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  1. David A. Meyer
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  2. Heather Blumer
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Meyer, D.A., Blumer, H. Parrondo Games as Lattice Gas Automata. Journal of Statistical Physics 107, 225–239 (2002). https://doi.org/10.1023/A:1014566822448

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