International Journal of Mathematical Sciences and Computing (IJMSC)
IJMSC Vol. 12, No. 1, 8 Feb. 2026
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Ergodicity and the Emergence of Long-Term Balance in the Dynamical States of π
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Author(s)
Fethi Kadhi
1,*
Moncef Ghazel
2
Malek Ghazel
2
Index Terms
Markov Chain, Irrational Numbers, Uniform Distribution, Ergodic Theory, Statistical Programming Language, Computing
Abstract
This paper investigates the digits of π within a probabilistic framework based on Markov chains, proposing this model as a rigorous tool to support the conjecture of π’s uniformity. Unlike simple frequency analyses, the Markov approach captures the dynamic structure of transitions between digits, allowing us to compute empirical stationary distributions that reveal how local irregularities evolve toward global equilibrium. This ergodic behavior provides quantitative, model based evidence that the digits of π tend toward fairness in the long run. Beyond its mathematical significance, this convergence toward uniformity invites a broader conceptual interpretation.
Cite This Paper
Fethi Kadhi, Moncef Ghazel, Malek Ghazel, "Ergodicity and the Emergence of Long-Term Balance in the Dynamical States of π", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.12, No.1, pp. 42-54, 2026. DOI: 10.5815/ijmsc.2026.01.04
Reference
[1]David H. Bailey and Richard E. Crandall, “Random generators and normal numbers”, Experimental Mathematics, Vol.11, No.4, pp. 527-546, 2012.
[2]Emile Borel, “Sur les chiffres d´ecimaux de √2 et divers probl`emes de probabilit´es en chaˆıne”, Comptes Rendus de l’Acad´emie des Sciences, Vol.230, pp. 591-593, 1950.
[3]Yann Bugeaud, “Distribution Modulo One and Diophantine Approximation”, Cambridge University Press, 2012.
[4]Jonathan M. Borwein and David H. Bailey, “Mathematics by Experiment: Plausible Reasoning in the 21st Century”, A K Peters, 2004.
[5]Yasumasa Kanada, “Numerical computations of π and applications”, Journal of Computational and Applied Mathematics, Vol.159, No.1, pp. 1-8, 2003.
[6]Fethi Kadhi, “Bridging Category Theory and Functional Programming for Enhanced Learning”, International Journal of Mathematical Sciences and Computing (IJMSC), Vol.11, No.3, pp. 1-18, 2025. DOI: 10.5815/ijmsc.2025.03.01. James
[7]Franklin, “Logical Probability and the Strength of Mathematical Conjectures”, The Mathematical Intelligencer, Vol.38, No.3, pp. 14-19, 2016.
[8]Gottfried Wilhelm Leibniz, “Essais de Th´eodic´ee sur la bont´e de Dieu, la libert´e de l’homme et l’origine du mal”, Isaac Troyel, Amsterdam, 1710.
[9]James R. Norris, “Markov Chains”, Cambridge University Press, 1998.
[10]J. Liu and R. Wilson, “The Beauty of Mathematics”, The Mathematical Intelligencer, 2025. DOI: 10.1007/s00283025-10429-7.
[11]Ghazel M, Alraqad T, and Kadhi F (2020). Dynamical study of the class of difference equation Xn+1 =Xn-2q+1/A+BXn-2q+1 Xn-q+1 International Journal of Advanced and Applied Sciences, 7(5): 66–78.