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Modular Transformations, Order-Chaos Transitions and Pseudo-Random Number Generation

Author

Listed:
  • Antonio Bonelli

    (Dipartimento di Chimica, Università della Basilicata, via Nazario Sauro 85, I-85100, Potenza, Italy)

  • Stefano Ruffo

    (Dipartimento di Energetica, Università di Firenze, via s. Marta 3, I-50139 Firenze, Italy;
    INFN, sez. Firenze, Italy)

Abstract

Successive pairs of pseudo-random numbers generated by standard linear congruential transformations display ordered patterns of parallel lines. We study the "ordered" and "chaotic" distribution of such pairs by solving the eigenvalue problem for two-dimensional modular transformations over integers. We conjecture that the optimal uniformity for pair distribution is obtained when the slope of linear modular eigenspaces takes the value$n_{\rm opt} ={\rm maxint} (p/\sqrt{p-1})$, wherepis a prime number. We then propose a new generator of pairs of independent pseudo-random numbers, which realizes an optimal uniform distribution (in the "statistical" sense) of points on the unit square(0, 1] × (0, 1]. The method can be easily generalized to the generation ofk-tuples of random numbers (withk>2).

Suggested Citation

  • Antonio Bonelli & Stefano Ruffo, 1998. "Modular Transformations, Order-Chaos Transitions and Pseudo-Random Number Generation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 9(07), pages 987-997.
    • Handle: RePEc:wsi:ijmpcx:v:09:y:1998:i:07:n:s0129183198000923
    DOI: 10.1142/S0129183198000923
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