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

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Abstract

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 Sp: t {mapping} Σn∈ℕsin(n-pt), 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 αp. As to chaos, the seemingly erratic fluctuations about this overall trend are discussed. Experts' commentaries are reproduced as to why the fluctuations of Sp(t) - αpsign(t) {pipe}t{pipe}1/p 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 ⌈t1/(p+1)⌉-th partial sum of Sp(t), 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 Sp(t) to properties of the Riemann ζ function is exhibited using the Mellin transform.

Original languageEnglish (US)
Pages (from-to)572-600
Number of pages29
JournalJournal of Statistical Physics
Volume150
Issue number3
DOIs
StatePublished - Feb 2013

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

  • Deterministic chaos
  • Fourier transform
  • Kac central limit theorem
  • Markov-Lévy method of characteristic functions
  • Mellin transform
  • Riemann ζ function
  • Sine series
  • Steinhaus notion of statistical independence of functions
  • Tempered distributions

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