Leibovich N, Dechant A, Lutz E, Barkai E
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel.
Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany.
Phys Rev E. 2016 Nov;94(5-1):052130. doi: 10.1103/PhysRevE.94.052130. Epub 2016 Nov 17.
The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum 〈S_{t_{m}}(ω)〉 where t_{m} is the measurement time. For processes with an aging autocorrelation function of the form 〈I(t)I(t+τ)〉=t^{Υ}ϕ_{EA}(τ/t), where ϕ_{EA}(x) is a nonanalytic function when x is small, we find aging 1/f^{β} noise. Aging 1/f^{β} noise is characterized by five critical exponents. We derive the relations between the scaled autocorrelation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of 1/f^{β} noise. We illustrate our results for blinking-quantum-dot models, single-file diffusion, and Brownian motion in a logarithmic potential.
平稳过程的功率谱可以使用维纳 - 辛钦定理根据自相关函数来计算。我们在此将维纳 - 辛钦定理推广到非平稳过程，并引入一个随时间变化的功率谱〈S_{t_{m}}(ω)〉，其中t_{m}是测量时间。对于具有形如〈I(t)I(t + τ)〉 = t^{Υ}ϕ_{EA}(τ/t)的老化自相关函数的过程，其中当x很小时ϕ_{EA}(x)是一个非解析函数，我们发现了老化1/f^{β}噪声。老化1/f^{β}噪声由五个临界指数表征。我们推导了缩放后的自相关函数与这些指数之间的关系。我们表明，我们对随时间变化谱的定义保留了其作为傅里叶模式密度的解释，并讨论了与1/f^{β}噪声明显的红外发散的关系。我们针对闪烁量子点模型、单文件扩散以及对数势中的布朗运动说明了我们的结果。
