Exactly solvable sandpile with fractal avalanching

Power spectrum of mass and activity fluctuations in a sandpile

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 timesteps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros 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.

Assessment of long-range correlation in time series: how to avoid pitfalls

Due to the ubiquity of time series with long-range correlation in many areas of science and engineering, analysis and modeling of such data is an important problem. While the field seems to be mature, three major issues have not been satisfactorily resolved. (i) Many methods have been proposed to assess long-range correlation in time series. Under what circumstances do they yield consistent results? (ii) The mathematical theory of long-range correlation concerns the behavior of the correlation of the time series for very large times. A measured time series is finite, however. How can we relate the fractal scaling break at a specific time scale to important parameters of the data? (iii) An important technique in assessing long-range correlation in a time series is to construct a random walk process from the data, under the assumption that the data are like a stationary noise process. Due to the difficulty in determining whether a time series is stationary or not, however, one cannot be 100% sure whether the data should be treated as a noise or a random walk process. Is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process, and vice versa? In this paper, we seek to gain important insights into these issues by examining three model systems, the autoregressive process of order 1, on-off intermittency, and Lévy motions, and considering an important engineering problem, target detection within sea-clutter radar returns. We also provide a few rules of thumb to safeguard against misinterpretations of long-range correlation in a time series, and discuss relevance of this study to pattern recognition.

Inverse cascade avalanche model with limit cycle exhibiting period doubling, intermittency, and self-similarity

A one-dimensional avalanche "sandpile" algorithm is presented for transport in a driven dissipative confinement system. Sand is added at the closed inflow boundary and redistributed when the local gradient exceeds a threshold. The redistribution rule is conservative, nonlocal, and linear and is chosen to mimic fluid flow. Potential energy is dissipated by avalanches that also expel matter at the open outflow boundary. The system then evolves through an inverse cascade. A "fluidization" parameter L(f) specifies the length scale over which the algorithm operates. The limiting case of L(f)=1 cell and L(f)=N, the system size, are analytically soluble. For other values of L(f) the emergent, large-scale dynamics of the system shows a variety of behavior including a limit cycle that has a period-doubling sequence, intermittency, and a random walk.