Power spectrum of mass and activity fluctuations in a sandpile

Abstract

We consider a directed abelian sandpile on a strip of size 2 × n, driven by adding a grain randomly at the left boundary after every T time-steps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeroes in the ternary-base representation of the position of a random walker on a ring of size 3 n . We find that while the fluctuations of mass have a power spectrum that varies as 1/f for frequencies in the range 3 −2n ≪ f ≪ 1/T , the activity fluctuations in the same frequency range have a power spectrum that is linear in f .

Aging Wiener-Khinchin theorem and critical exponents of 1/fβ noise

“…1/f noise, with a range of different exponents, occurs in many systems in a variety of disciplines. A partial list includes electronic, solid and condensed matter devices [34][35][36][37], sand-pile models [38], blinking quantum dots [39,40], nanoscale electrodes [41], experimental data of voltage-dependent anion channel in rats brains [42] and processes modeled by nonlinearstochastic-differential equations [43].…”

General mechanism for the 1/f noise

“…The search for a general mechanism of the 1/f noise was the main motivation behind the proposal of self-organized criticality (SOC) by Bak, Tang and Wiesenfeld in 1987 [13] that yields long-ranged correlations in time and thus can generate 1/f α dependence in the power spectrum of fluctuations. Several SOC models have been studied from this viewpoint [14][15][16][17][18][19], but since 1/f α noise may be found in models with only a small number of degrees of freedom, clearly self-organized criticality is not a necessary condition for obtaining 1/f noise.The 1/f α spectrum, with α ≥ 1 is not integrable near f = 0, and the variance of the signal would be infinite, if there were no cutoffs. In practice, one finds that if the signal is studied for a duration T , then if one doubles * Present address: Department of Physics, Indian Institute of Science, Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411 008, India the duration, the observed mean of the signal appears to drift, and the variance of the signal increases with T .…”

“…The search for a general mechanism of the 1/f noise was the main motivation behind the proposal of self-organized criticality (SOC) by Bak, Tang and Wiesenfeld in 1987 [13] that yields long-ranged correlations in time and thus can generate 1/f α dependence in the power spectrum of fluctuations. Several SOC models have been studied from this viewpoint [14][15][16][17][18][19], but since 1/f α noise may be found in models with only a small number of degrees of freedom, clearly self-organized criticality is not a necessary condition for obtaining 1/f noise.…”

Memoryless nonlinear response: A simple mechanism for the 1/f noise

“…5 is a response that is a sum of square waves of periods 2, 4, 8, 16 and 32. Such a functional form arises in the study of toppling activity in an Abelian sandpile on a strip, and measures the number of zeros in a binary representation of a random walk on a ring [17]. For this R, it can be shown that the power spectrum varies exactly as 1/f for 1/B 2 ≪ f ≪ 1.…”

“…The power spectrum of activity in such models can be determined in terms of the distribution of inter-pulse intervals and the detailed characteristics of the pulse shape. Mass fluctuations in a one-dimensional sandpile have been shown to give a 1/f power spectrum over a very wide range of f [16,17].…”