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

Antonio Bonelli () and Stefano Ruffo ()
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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;

International Journal of Modern Physics C (IJMPC), 1998, vol. 09, issue 07, 987-997

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).

Keywords: Modular Transformations; Order and Chaos; Pseudo-random Number Generators (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1142/S0129183198000923

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