arXiv:cond-mat/9902074v1 [cond-mat.stat-mech] 4 Feb 1999 1/f Noise and Long Configuration Memory in Bak-Tang-Wiesenfeld Models on Narrow Stripes Sergei Maslov 1 , Chao Tang 2 , and Yi-Cheng Zhang 3 1 Department of Physics, Brookhaven National Laboratory, Upton, New York 11973 2 NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540 3 Institut de Physique Th´ eorique, Universit´ e de Fribourg, Fribourg CH-1700, Switzerland (February 5, 1999) We report our findings of an 1/f power spectrum for the total amount of sand in directed and undi- rected Bak-Tang-Wiesenfeld models confined on narrow stripes and driven locally. The underlying mechanism for the 1/f noise in these systems is an exponentially long configuration memory giving rise to a very broad distribution of time scales. Both models are solved analytically with the help of an operator algebra to explicitly show the appearance of the long configuration memory. PACS number(s): 05.65.+b, 05.45.-a, 05.40.-a I. INTRODUCTION The ubiquitous 1/f noise fascinated physicists for gen- erations [1,2]. There are many examples from wildly dif- ferent systems in which the power spectra S(f ) ∼ 1/f α with α close to one. This phenomenon usually indicates the presense of a broad distribution of time scales in the system. The common case where α = 1 is particularly intriguing, in that it implies a kind of “equal partition” of power among every decade of frequency range, i.e. the integral  10f f S(f )df ∼  10f f 1 f df =  10f f d ln f = ln 10 is independent of f . One mechanism to generate such a distribution of time scales in systems at thermal equi- librium is through thermal activation events over a suffi- ciently broad and flat distribution of energy barriers [2]. The “local” power spectrum generated by any single bar- rier has a characteristic frequency which decreases expo- nentially with increasing barrier height. But the super- position of the power spectra from all the barriers gives rise to an 1/f spectrum. This mechanism is often em- ployed to explain, for example, the 1/f spectrum of low frequency voltage fluctuations in semi-conductors [2]. In search for a more general answer, applicable to nonequi- librium and dynamic systems, Bak, Tang, and Wiesen- feld (BTW) introduced the notion of Self-Organized Crit- icality (SOC) [3]. In particular, they proposed a simple “sandpile” (BTW) model which shows the emergent scale free behavior in both space and time. However, the origi- nal BTW model did not exhibit the 1/f noise [4]. In this paper we report the observation of 1/f noise for directed and standard (undirected) BTW models confined on nar- row stripes (quasi-one-dimensional geometries). In these models, sand flows in the long direction, with periodic or closed boundary conditions in the other direction. The system is driven locally by randomly adding sand to a unique set of sites that have the same coordinate along the long axis. The total amount of sand in the sandpile as a function of time, measured by the number of added grains, exhibits a clean 1/f power spectrum with an ex- ponentially small lower cutoff. Surprisingly, the mecha- nism for the 1/f noise in this athermal nonequilibrium model is rather similar to the above mentioned thermal mechanism. In our model the local characteristic fre- quency also falls off exponentially as a function of some parameter, which in this case is the distance from the driving point. II. DIRECTED MODELS Let us first consider the simpler directed model, defined as follows. An integer variable z (x, y) is assigned on every site (x, y) of a two-dimensional lattice of size L x × L y (1 ≤ x ≤ L x ,1 ≤ y ≤ L y ). Throughout the paper, we refer to z (x, y) as the number of grains of sand (or height) at the site (x, y). The dynamics consists of the following steps: (i) Add a grain of sand to a randomly selected site in the first column, (1,y): z (1,y) → z (1,y) + 1. (ii) If as a result of the process the height z (x, y) ex- ceeds a critical value z c = 2, the site topples and three grains of sand are redistributed from this site to three of its nearest neighbors up, down, and to the right, that is z (x, y + 1) → z (x, y + 1) + 1, z (x, y − 1) → z (x, y − 1) + 1, z (x +1,y) → z (x +1,y)+1, z (x, y) → z (x, y) − 3. (iii) Repeat step (ii) until all sites are stable, i.e. z (x, y) ≤ 2 everywhere. This chain reaction of up- dates is referred to as an avalanche. (iv) When the avalanche is over, measure the total amount of sand in the system Z (t)= ∑ z (x, y). Then go to step (i). 1