Abstract
In this paper we consider the dynamics of extended nonequilibrium systems. An example is the dynamics of interfaces when these are pushed or dragged in a noisy medium1–6. Here we discuss the limit where this motion is sufficiently slow to allow the development of long range correlations in the dynamics 7–9. This then imposes novel long range correlations also in the static snapshots of the interface10,11. The scaling of the interface profile appears to be closer to the experimentally observed ones1,2 than that of the KPZ equation3. Furthermore, we will see that the dynamics leading to long range correlations also imply intermittency. The emerging dynamics with bursts on all scales is called self organized criticality7, and is suggested to characterize phenomena ranging from plate tectonics with earthquakes to the punctuated dynamics of evolving life12,13.
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References
M.A. Rubio, C. Edwards, A. Dougherty and J.P. Gollup, Phys. Rev. Lett. 63, 1685 (1989).
M.A. Rubio, C. Edwards, A. Dougherty and J.P. Gollup, Phys. Rev. Lett. 65, 1389 (1990).
V.K. Horváth, F. Family and T. Vicsek, Phys. Rev. Lett. 65, 1388 (1990).
V.K. Horváth, F. Family and T. Vicsek, J. Phys A 24, L–25 (1991).
M. Kardar, G. Parisi and Y.-C. Zhang, Phys.Rev.Lett. 56, 889 (1986).
G. Parisi, Europhys.Lett. 17 673 (1992).
N. Martys, M. Cieplak and M.O. Robbins, Phys. Rev. Lett. 66 (1991) 1058.
L.-H. Tang and H. Leschhorn, Phys. Rev. A 45, R8309 (1992).
S.V. Buldyrev et. al, Phys. Rev. A 45, R8313 (1992).
P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).
P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988).
S. I. Zeitsev, Physica A 189 411 (1992).
K. Sneppen, Phys.Rev.Lett. 69, 3539 (1992).
K. Sneppen and M. H. Jensen, Phys. Rev. Lett. 70, 3833 (1993).
K. Sneppen and M. H. Jensen, Phys. Rev. Lett. 71, 101 (1993).
J. Falk, M.H. Jensen and K. Sneppen, Phys. Rev. E 49, 2804 (1994).
P. Bak and K. Sneppen, Phys. Rev. Lett. 71, 4083 (1993).
H. Flyvbjerg, K. Sneppen and P. Bak, Phys. Rev. Lett. 71, 4087 (1993).
D. Wilkinson and J.F. Willemsen, J.Phys.A 16, 3365 (1983).
L. Furuberg et al., Phys. Rev. Lett. 61 2117 (1988).
S. Roux and E. Guyon, J. Phys. A: Math. Gen. 22, 3693 (1989).
J.F. Gouyet, Physica A 168 581 (1990).
N. Martys, M.O. Robbins and M. Cieplak, Phys. Rev. B44 12294 (1991).
S. Havlin et ai, in Growth Patterns in Physical Sciences and Biology, eds. J. M. Garcia-Ruiz et. all. (Plenum, New York, 1993).
L.H. Tang and H. Leschhorn, Phys. Rev. Lett. 70, 3833 (1993).
Percolation Structures and Process edited by G. Deutscher, R. Zallen, and J. Adler, Annals of the Israel Physical Society Vol. 5 (1983).
J.W. Essam et al., Phys. Rev. B 33, 1982 (1986).
Z. Olami, I. Procaccia, and R. Zeitak, Phys. Rev. E 49, 1232 (1994).
A.-L. Barabási et al, in Surface Disordering: Growth, Roughening, and Phase Transitions, eds. R. Jullien et. all. (Nova Science, New York, 1992).
H. Leschhorn and L.-H. Tang, Phys. Rev. E 49, 1238 (1994).
K. Sneppen and M.H. Jensen, Scalings for Self Interactions in Branching Processes. In preparation. Investigates scaling of areas enclosed between neighbour branches of a 1 + 1 d directed percolating cluster. Finds that the length l of such “holes” along “time” direction scales as P(l) ∝ l −2. 6±00.1. This distribution includes all holes, also holes within dangling bonds. It is at present not clear how the distribution of hole length in the total network of directed percolation is related to the distribution of hole length in the backbone network considered by23: P backb (l) ∝ l −2+1/v ∥ DP = l −1.42.
S. Maslov and M. Paczuski, Phys. Rev. E50, R1 (1994).
Alternative to OPZ19 and25 for the derivation of the avalanche size distribution associated to ηn = ηc − Δ: P(s) = s −τ f(s/Δ−v ). An avalanche is identified as the area enclosed between two branches of directed percolation (in dimension d = 1). The avalanche cut off may therefore scale as the product of horizontal and perpendicular correlation length of these branches: v = dv ∥ +v ⊥. The scaling of average avalanche size ‹s› ∝ Δ−γ identify γ = v(2−r) γ is deduced from time to reach saturation. Transient ends when the last part of the interface starts moving, giving a time s ∝ L d+x. Alternatively the time for developing a correlation length equal the system size L equals the time it takes to develop a gap G ∝ L −1/v∥. ¿From integrating the PMB gap equation24 s ∝ L d G 1−γ. Equating transient time expressions: γ = 1 + v ⊥. ¿From γ and v one finds: τ = 1 + (dv ∥ − 1)/(dv ∥ + v ¿).
M. Paczuski, S. Maslov, and P. Bak, Brookhaven National Laboratory report BNL-49916 (1993); Europhys. lett. 27, 97 (1994).
O. Narayan and D. S. Fisher, Phys. Rev. B 48, 7030 (1993).
T. Sams, M.H. Jensen and K. Sneppen “On the connection between Self Organized Depinning and Directed Percolation”. In preparation. For system size L = 2 17 we observe π = 2.19 ±0.05, τ = 1.26 ± 0.03 and τ∥ = 1.44 ± 0.05, all close to the MP23 prediction. Other observation is that the scaling of width with segment size l ≪ L depends whether we measure arbitrarely (w ∝ l 0.645±0.005) or only when all η > ηc within segment l (w η l 0.625±0.005) Thus the critically evolving interface is perturbed slightly away from the critical network of directed percolation.
Z. Olami, I. Procaccia and R. Zeitak. “Interface Roughening in Systems with Quenched Disorder” Submitted to Phys. Rev. Lett.
T. Halpin-Healy and Y.-C. Zhang, submitted to Phys. Rep. (1994).
R. Bruinsma and G. Aeppli, Phys. Rev. Lett. 52, 1547 (1984).
T. Natterman et al., J. Phys. II, France 2, 1483 (1992).
H. Leschorn, Physica A195, 324 (1993).
J. de Boer, B. Derrida, H. Flyvbjerg, A.D.Jackson and T. Wettig (1994), To appear in Phys.Rev.Lett.
de Boer, A.D. Jackson and T. Wettig, Criticality in Simple Models of Evolution, Preprint (1994).
K. Sneppen and M. H. Jensen, Phys. Rev. E49 919 (1994).
T.S. Ray and N. Jan, Phys. Rev. Lett. 72 4045 (1994).
M.H.Jensen and K.Sneppen, “Multiscaling in associated processes of extremum dynamics.” For the BS model the associated processes scales as P(s) = s −τ(Δ) f(s/Δ−v) were also τ = τ(Δ) depends on distance Δ = B c − B t to B c (= 2/3 in 1− d with two neighbours).
J. Kertesz, V.K. Horvath and F. Weber, Fractals, Vol. 1, no. 1 (1993) 67.
K. Christensen, L. Furuberg, K.Sneppen, “Directed Invasion Percolation”. Numerical simulations of associated processes for an ever expanding directed invasion shows avalanche scaling that is very different from that of directed percolation. This is because the subsequent avalanches limit each other’s geometrical extension.
L.-H. Tang, Private communication; Suggested as a model for a flux line dragged slowly trough a superconductor with pinning impurities.
K. Sneppen, P.Bak, H.Flyvbjerg, M.H.Jensen A Self-Organized Critical model for evolution, BNL Preprint, submitted to PNAS.
D.M. Raub, Science 231, 1528 (1986).
J.J. Jr. Sepkoski, Paleobiology 19 43 (1993).
S.J. Gould and N. Eldredge, Nature 366 223 (1993).
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Sneppen, K. (1995). Minimal SOC: Intermittency in growth and evolution. In: McKane, A., Droz, M., Vannimenus, J., Wolf, D. (eds) Scale Invariance, Interfaces, and Non-Equilibrium Dynamics. NATO ASI Series, vol 344. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1421-7_12
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DOI: https://doi.org/10.1007/978-1-4899-1421-7_12
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