Order and Chaos in Some Trigonometric Series: Curious Adventures of a Statistical Mechanic

• Published in: Journal of statistical physics Vol. 150; no. 3; pp. 572 - 600
• Main Author: Kiessling, Michael K.-H.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.02.2013; Springer
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Tempered distributions; Markov-Lévy method of characteristic functions; function; Sine series; Kac central limit theorem; Fourier transform; Steinhaus notion of statistical independence of functions; Riemann; Deterministic chaos; Mellin transform
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-012-0578-7

This paper tells the story how a MAPLE-assisted quest for an interesting undergraduate problem in trigonometric series led some “amateurs” to the discovery that the one-parameter family of deterministic trigonometric series , p >1, exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. As to order, an elementary (undergraduate) proof is given that ∀ t ∈ℝ, with explicitly computed constant α p . As to chaos, the seemingly erratic fluctuations about this overall trend are discussed. Experts’ commentaries are reproduced as to why the fluctuations of are presumably not Gaussian. Inspired by a central limit type theorem of Marc Kac, a well-motivated conjecture is formulated to the effect that the fluctuations of the ⌈ t 1/( p +1) ⌉-th partial sum of , when properly scaled, do converge in distribution to a standard Gaussian when t →∞, though—provided that p is chosen so that the frequencies { n − p } n ∈ℕ are rationally linear independent; no conjecture has been forthcoming for rationally dependent { n − p } n ∈ℕ . Moreover, following other experts’ tip-offs, the interesting relationship of the asymptotics of to properties of the Riemann ζ function is exhibited using the Mellin transform.