arXiv:cond-mat/0002270v2 [cond-mat.stat-mech] 23 May 2000 rbto n rni-iedsrbtoswr discussed dis- were avalanche distributions authors; was of transit-time ricepile group and of same tribution many model the has 1D be by thus A could proposed ricepile grains states. the heterogeneity metastable rice [8], different ways the elongated different Because is in here packed system. ingredient the key be- of SOC The found exhibit was grains real rice It haviors. if elongated investigate with behaviors. to ricepiles done SOC that were display grains systems rice granular of Experiments piles behavior. on sandpile SOC BTW [8] exhibit 1D even the not Especially does its in model [7]. noise versions flick 2D exhibit However, and not 1D does model. proposed they sandpile model cellular BTW the a the with SOC noise. model, of flick automaton idea underlying their mechanism demonstrated also the They be criticality may self-organized [6] and that Tang, (SOC) Bak, proposed electrical lacking. ex- have still and universal is Wiesenfeld noise a [3], flick ubiquity, the speech of its planation and water Despite to music [4,5]. [1] [2], quasars measurement variety rivers of a light in in the flows appears from noise ranging flick systems example, of For noise. flick ore rnfr ftesignal the of transform Fourier codn oWee-hnhnTerm Here Theorem. Wiener-Khinchin to according ftesga sra-aud t oe pcrmi just is spectrum power its real-valued, is signal the If a eatraieycluae as calculated alternatively be can ne td ssainr n t uocreainfunc- auto-correlation its on depends and only stationary tion is study under ···i · h· tlwfeuny ic h exponent the Since frequency. low at rmo inli enda h ore rnfr of transform Fourier the as defined function is auto-correlation signal its a of trum ,flc os sas ald1 called also is noise flick 1, signal nntr n aoaoy aypyia ytm show systems physical many laboratory, and nature In h term The eoe nebeaeae sal h signal the Usually average. ensemble denotes s ( t utae ihapwrspectrum power a with fluctuates ) ASnmes 54.a 56.b 57.n 05.70.Ln 45.70.-n, 05.65.+b, 05.40.-a, numbers: PACS h rfieflcutoso h ieudrdffrn rvn me driving different under pile the of fluctuations profile The h oe spectrum power the ftepl.I risaernol de otepl,tepowe the pile, the to added randomly are grains If pile. the of iknoise flick C s b ( h eprlflcuto fteaeaesoeo ieiemo ricepile a of slope average the of fluctuation temporal The ( nttt fLwEeg ula hsc,BiigNra Uni Normal Beijing Physics, Nuclear Energy Low of Institute τ f = ) = ) S nttt fTertclPyis ejn omlUniversi Normal Beijing Physics, Theoretical of Institute ( t Z f Z − −∞ −∞ = ) ∞ eest h hnmnnta a that phenomenon the to refers t ∞ 0 h uocreainfunction auto-correlation The . s s | ( s b ( t C t ( /f ) ) f s ( e ( ) t S i t | 0 2 os.Tepwrspec- power The noise. 2 ( + πft t , f 1 clsa 1 as scales ) = ) τ /f ejn aito etr ejn 008 China 100088, Beijing Center, Radiation Beijing dt. ) dt. α α h s sotncoeto close often is utain narcpl model ricepile a in fluctuations ( t 0 S ) s ( ( /f f s b t ( ) ) f α i ∝ where , sthe is ) with h-ogZhang Shu-dong 1 /f s (3) (2) (1) ( t α α ) ≈ 1 1 . n-iesoa ra fsts1 2, 1, sites of array one-dimensional behavior. flick-like or want noise We flick temporal has system. with it the if concerned of investigate slope are to average we the paper, of to this fluctuation the paid In of were other statistics model. Attentions transit-time several and by dynamics avalanche [10–12]. investigated the us later was including model authors The [9]. h oa egto h ieiea site at ricepile the of height local the ntaie ysetting by initialized an nitgrnumber integer an tains ieiei ob ul pfo cac.Tesse is system grain site The one at If dropped scratch. pile. is the from rice onto grains up of rice built dropping by be driven to is ricepile ilices y1, by increase will ris ieiei ul p h oa lp ftepile the of slope local The up. as built defined is is ricepile a grains, xed eti threshold certain exceeds eimdsre.Telre the larger parameter The The disorder. 1. medium than less no ger eoe h DBWsnpl oe.When model. If sandpile BTW system. 1D the the of becomes disorder medium of nvraiyo h vlnh iedsrbto n transit and distribution. In the distribution time on size [9]. avalanche disorder of the reference effect of in the universality investigated studied have we model [10], the ref. to reduces model oe.A nRf 1] vr iesite time every [10], Ref. in As model. ieo naaacei enda h ubro topplings of event. number avalanche the the as in defined involved is avalanche an of size rm1to 1 from h etedo h iei lsd otebudr con- boundary the So closed. is as while pile pile, kept the the is of of dition end end right the left from the only pile the leave is, to That right. the on site neighbor its to hehl slope threshold site ieol hntepl ssal.I nusal tt,all state, unstable an In stable. is pile sites the when only pile n rmr ie scle an called is sites more or one h adt esal fn oa lp xed hehl value, threshold exceeds slope local is, no that if stable be to said hngan frc r de nyt n end one to only added are rice of grains when 3 i +1 h ieiemdli enda olw:Cnie a Consider follows: as defined is model ricepile The h ausof values The i → i iltpl n n ri frc ilb transferred be will rice of grain one and topple will with z h hnssaeas discussed. also are chanisms i i +1 ≤ r pcrmehbt 1 exhibits spectrum r e sivsiae.I sfudthat found is It investigated. is del z est,Biig107,China 100875, Beijing versity, y ejn 085 China 100875, Beijing ty, z iheulpoaiiy Here probability. equal with z .I hsmdl iegan r allowed are grains rice model, this In 1. + i i i c z > = z o all for z i c h i c h h iltk e au admychosen randomly value new a take will saeesnilt h ento fthe of definition the to essential are ’s 0 i c i i = − opei aall h opig of topplings The parallel. in topple → h i h i h iegan r rpe othe to dropped are grains Rice . h h o l ie.Ti en the means This sites. all for 0 = i 1 +1 i i i and z 1 ihtedopn frice of dropping the With +1. frc ris Here grains. rice of hntehih fcolumn of height the then , hnvrtelclslope local the Whenever . i c seie ntefollowing), the in (specified h avalanche /f L +1 2 r · · · h ihrtelevel the higher the , behavior. r .Tercpl is ricepile The 0. = eet h ee of level the reflects , i L r h ytmis system The . ahst con- site Each . h h model the 1 = vn,adthe and event, i i → r ope,the topples, sa inte- an is h r h i i ,the 2, = scalled is − and 1 z i i In the present work, we performed extensive numerical made statistics on deviation of the ricepile slope from simulations on the system evolution. Let us first study its average value in the stationary state. We found that the case where rice grains are added only to the left end the greater the r, the greater the deviation. Note that of the pile, i.e., only to the site i = 1, which is in accor- the value of r > 1 does not affect the exponent α, as it dance with the experiment setup [8]. We shall refer to does not alter the universal avalanche exponent τ for the this driving mechanism as the fixed driving. When the avalanche distribution [10]. stationary state is reached, we record the average slope It is also interesting to study the effect of driving mech- z(t) = h1(t)/L of the pile after every avalanche. Here anism on the temporal behavior of the system. Now, in- time t is measured in the number of grained added to stead of dropping rice grains to the left end of the pile, we the system. Thus we got a time series z(t). Typical re- drop rice grains to randomly chosen sites of the system. sults about the fluctuation of the slope z(t) can be seen This way of dropping rice grains represents a different in Fig.1.a. We calculate the power spectrum of this time driving mechanism, which we shall refer to as the ran- series according to Eq.(2). We find that for not too small dom driving. Numerical simulations with this random systems the slope fluctuation displays 1/f-like behavior. driving were made and the time series z(t) was recorded. In fact, we got a power spectrum S(f) scaling as 1/f α at For this way of driving, typical result about the fluctua- low frequency, with the exponent α =1.3 ± 0.1. tion of z(t) can be seen in Fig. 1.b. It seems that z(t) We note the following points: now fluctuates more regularly than it does for the fixed (a) the temporal behavior of the ricepile model is dra- driving. We found that under random driving the tem- matically different from the 1D BTW sandpile model. In poral fluctuation has a trivial 1/f 2 behavior as in the the 1D BTW model, the system has a stable state [6], case of various previously studied sandpiles [7]. In Fig. which, once reached, cannot be altered by subsequent 3, we compare the power spectra of the two cases with droppings of sand grains. Thus the 1D BTW model does different driving mechanisms. Two groups of curves are not display SOC behaviors. The slope of the 1D sandpile shown in this figure, each with a different exponent α. model assumes constant values. The power spectrum is For the fixed driving, we have α ≈ 1.25, while for the thus of the form S(f) = δ(f). In words, there is only random driving we have α ≈ 2.0. The distinction of the dc component in the power spectrum for the 1D BTW two behaviors is quite clear. This shows that the driving model. mechanism, as an integrated part of the model, has an c (b) With the introduction of varying threshold zi into important role in the temporal behaviors of the system. the ricepile model the behavior of the system becomes Recent work on the continuous version of BTW sandpile much more rich. The avalanche distribution follows model [13], and that on quasi-1D BTW model [14] also power law [10], which is an important signature of SOC. showed that certain driving mechanisms are necessary Besides that, the temporal fluctuation of z(t) has a power conditions for these models to display 1/f fluctuation for spectrum of flick type at low frequency. the total amount of sand (or say, energy). We notice that the power spectrum is flattened at low Here we present an heuristic discussion on why the ex- frequency for small system. This is a size effect. It is be- ponent α is smaller for fixed driving than for random lieved that flick noise fluctuation is closely related to the driving. For fixed driving, the height h1 at site i = 1 long range spatial correlation in the system. For small changes almost every dropping, and so is the average system, the spatial correlation that can be built up is slope z(t). Thus the high frequency component of the limited by the system size, thus the long range temporal z(t) fluctuation has a heavier weight in its power spec- correlation required by flick noise is truncated off at low trum, and this will make the exponent α smaller. While frequency. Our numerical results verified the above dis- for the random driving, the rice grains are dropped to the cussion. In Fig. 2, we show the power spectrum of z(t) pile at random sites, the spatial correlations previously for different system sizes. It is clear that when the sys- built up can be easily destroyed, making the z(t) behave tem size increases the 1/f α behaviors extends to lower more or less as an random walk. So the exponent α for and lower frequency. this case shall be very close to 2. For random driving, We also investigated the effect of the level of disorder each site has the same probability in receiving a grain in on the power spectrum. This was done by simulating the every drop. For the fixed driving, however, only the left system with different values of r. In Fig. 3 we show the end site receive grains, so there is in some sense breaking power spectra for the system with different r. It can be of symmetry, which would lead to different scalings in the seen from this figure, that the power-law (straight in the slope fluctuations. log-log plot) part extends to lower frequency for higher To see more about the effect of driving mechanism, it value of r. The effect can be understood by the following is helpful to investigate the profile fluctuations of the pile discussion. When the level of disorder is increased, the under different drivings. Because avalanches change the amplitude of the slope fluctuation also increases. Larger surface of the ricepile, the pile fluctuates around an av- amplitude fluctuation of z(t) requires more grains to be erage profile, and the size of the fluctuations characterize added to the system. Hence the period of this fluctu- the active zone of the ricepile surface. Let the standard 2 2 ation increase, which gives rise to the increase of low deviation of height at site i be σh(i,L)= hhi i − hhii . frequency components in the power spectrum. We have Here h· · ·i represents average over time. Thenp we calcu-

2 1 272 late the profile width of the ricepile, w = L i σh(i,L), Zhang, Physica A , 1 (1999). which is a function of the system size. In Fig.P 4, we show [12] L. A. N. Amaral, and K. B. Lauritsen, Phys. Rev. E 56, the profile width of the ricepile for different r and differ- 231 (1997); Physica A 231, 608(1996). ent driving mechanisms. It can be seen that w scales with [13] P. De Los Rios and Y. -C. Zhang, Phys. Rev. Lett. 82, L as w ∝ Lχ, and that χ =0.25 ± 0.01 for fixed driving, 472, (1999). χ = 0.09 ± 0.01 for random driving. For given driving [14] S. Maslov, C. Tang, and Y. -C. Zhang, Phys. Rev. Lett. mechanism, w increases with increasing r. As we stated 83, 2449, (1999). in Ref. [10], the parameter r reflects the level of medium [15] A. Malthe-Sorenssen, J. Feder, K. Christensen, V. Frette, 83 disorder in the rice pile. Although the level of disorder T. Jossang, Phys. Rev. Lett. , 764 (1999). does not change the scaling exponent χ, it does affect the amplitude of fluctuations. For greater r, the profile width is larger. It can be seen in the figure, that the data points for r = 4 are above that for r = 2, for given driv- ing mechanism. A recent experiment showed that piles of polished rice grains has a smaller profile width than unpolished ones [15], which have higher level of medium disorder. In conclusion, we have studied the temporal fluctua- tion of the slope of ricepile model in its critical stationary state. The power spectrum of this fluctuation is closely related to the driving mechanism. When the rice grains are dropped to the left end of the pile, the slope fluctu- ates with a flick-type power spectrum, with the exponent α = 1.3 ± 0.1. When the driving mechanism is changed to random driving, the model displays 1/f 2 behaviors. Greater system size and higher level of disorder will ex- tend the frequency range where the power spectrum has the form 1/f α. The author thanks Prof. Vespignani for helpful discus- sions. This work was supported by the National Natural Science Foundation of China under Grant No. 19705002, and the Research Fund for the Doctoral Program of Higher Education (RFDP). FIG. 1. The time evolution of the average slope z(t) of the ricepile after the stationary state has been reached. The parameters used for this figure are L = 20, r = 4. (a) rice grains are added to the pile at site i = 1. (b) rice grains are added randomly to the pile.

[1] W. H. Press, Comments Astrophys. 7, 103(1978). [2] B. B. Mandelbrot and J. R. Wallis, Water Resour. Res. 4, 909 (1968); 5, 321 (1969). [3] R. F. Voss and J. Clarke, Nature 258, 317 (1975); J. Acoust. Soc. Am. 63, 258 (1978). [4] P. Dutta and P. M. Horn, Rev. Mod. Phys. 53, 497 (1981). [5] M. B. Weissman, Rev. Mod. Phys. 60, 537 (1988). [6] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988). [7] H. J. Jensen, K. Christensen, H. C. Fogedby, Phys. Rev. B 40, 7425 (1989). [8] V. Frette, K. Christensen, A. Malthe-Sorenssen, J. Feder, T. Jossang, and P. Meakin, Nature(London)379, 49 (1996). [9] K. Christensen,A. Corral, V. Frette, J. Feder and T. Jos- sang, Phys. Rev. Lett. 77, 107 (1996). [10] Shu-dong Zhang, Phys. Lett. A 233, 317 (1997). [11] M. Bengrine, A. Benyoussef, F. Mhirech, and S. D.

3 FIG. 2. The power spectrum of z(t) for different system FIG. 3. The power spectrum of the slope fluctuation for sizes, with fixed driving and r = 2. The curves (from top to a ricepile with L = 100. Two groups of curves are shown bottom) are for L = 40, 60, and 100 respectively. The dashed in this figure, with different exponents α. For reference, the − . −1.25 line is a curve for y ∝ x 1 25 for reference. The number of data upper dashed line is a curve for y ∝ x , while the lower −2 points for Fourier transformation is N = 2048, and the results dotted line shows a curve y ∝ x . In each group of curves, are obtained by averaging over 500 samples. fc ≡ 1/(2∆) is the upper curve is for the case r = 4, and the lower one is for the Nyquist critical frequency, where ∆ is the time interval r = 2. between two successive points in the Fourier transformation.

FIG. 4. The profile width of the ricepile scales with the system size. Data from numerical simulations are shown in open (for r = 2) or filled ( for r = 4) symbols. Solid lines are power law fit to the data for fixed driving, with power law exponent χ = 0.25 ± 0.01, and the dashed lines are for random driving, with χ = 0.09±0.01. For every run of numer- ical simulation, statistics were made over at least 105 grain dropping.

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