Power spectra for both interrupted and perennial aging processes

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Article on power spectra for both interrupted and perennial aging processes.

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11 p.

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Lukovic, Mirko & Grigolini, Paolo November 10, 2008.


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  • Lukovic, Mirko Università di Pisa
  • Grigolini, Paolo University of North Texas; Università di Pisa; Istituto dei Processi Chimico Fisici del Consiglio Nazionale delle Ricerche


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Article on power spectra for both interrupted and perennial aging processes.

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11 p.


Abstract: We study the power spectrum of a random telegraphic noise with the distribution density of waiting times τ given by ψ(τ)∝1/τμ, with μ≈2. The condition μ<2 violates the ergodic hypothesis, and in this case the adoption of Wiener–Khintchine (WK) theorem for the spectrum evaluation requires some caution. We study this problem theoretically and numerically and we prove that the power spectrum obeys the prescription S(f)=K/fη, with η=3−μ, namely, the 1/fnoise lives at border between the ergodic μ>2 and nonergodic μ<2 condition. We study sequences with the finite length L. In the case μ<2 the adoption of WK theorem is made legitimate by two different kinds of truncation effects: the physical and observation-induced effect. In the former case ψ(τ) is truncated at τ≈Tmax and L⪢Tmax ensures the condition of interrupted aging. In this case, we find that K is a number independent of L. The latter case, L⪡Tmax, is more challenging. It was already solved by Margolin and Barkai, who used time asymptotic arguments based on the ergodicity breakdown and obtained K∝1/L2−μ, proving that the out-of-equilibrium nature of the condition μ<2 is signaled by the decrease of K with the increase of L. We use a generalized version of the Onsager principle that leads us to the same conclusion from a somewhat more extended view valid also for the transient out-of-equilibrium case of μ>2. We do not limit our treatment to the time asymptotic case, thereby producing a prediction that accounts for the transition from the 1/fη to the 1/f2 regime, recently observed in an experiment on blinking quantum dots. Our theoretical approach allows us to discuss some other recent experiments on molecular intermittent fluorescence and affords indications that should help to assess whether the spectrum is determined by the L⪡Tmax or by the L⪢Tmax condition.

Copyright 2008 American Institute of Physics. Journal of Chemical Physics, 129:18, DOI: 10.1063/1.3006051. http://dx.doi.org/10.1063/1.3006051


  • Journal of Chemical Physics, 2008, College Park: American Institute of Physics


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  • Publication Title: Journal of Chemical Physics
  • Volume: 129
  • Issue: 18
  • Peer Reviewed: Yes


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  • November 10, 2008

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  • Nov. 8, 2013, 8:29 a.m.

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Lukovic, Mirko & Grigolini, Paolo. Power spectra for both interrupted and perennial aging processes, article, November 10, 2008; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc228286/: accessed November 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.