Power spectra for both interrupted and perennial aging processes

Abstract

We study the power spectrum of a random telegraphic noise with the distribution density of waiting times tau given by psi(tau) proportional to 1tau(mu), with mu approximately 2. The condition mu<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)=Kf(eta), with eta=3-mu, namely, the 1f noise lives at border between the ergodic mu>2 and nonergodic mu<2 condition. We study sequences with the finite length L. In the case mu<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 psi(tau) is truncated at tau approximately T(max) and L>>T(max) ensures the condition of interrupted aging. In this case, we find that K is a number independent of L. The latter case, L<<T(max), is more challenging. It was already solved by Margolin and Barkai, who used time asymptotic arguments based on the ergodicity breakdown and obtained K proportional to 1L(2-mu), proving that the out-of-equilibrium nature of the condition mu<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 mu>2. We do not limit our treatment to the time asymptotic case, thereby producing a prediction that accounts for the transition from the 1f(eta) to the 1f(2) 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<<T(max) or by the L>>T(max) condition.