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 .

We consider the response of a memoryless nonlinear device that converts an input signal ξ(t) into an output η(t) that only depends on the value of the input at the same time, t. For input Gaussian noise with power spectrum 1/f α , the nonlinearity modifies the spectral index of the output togive a spectrum that varies as 1/f α with α = α. We show that the value of α depends on the nonlinear transformation and can be tuned continuously. This provides a general mechanism for the ubiquitous '1/f ' noise found in nature.PACS numbers: 05.40. Ca, 05.10.Gg, 05.45.Tp In a very wide variety of natural systems, temporal fluctuations in observables are found to be long-ranged, with an approximately 1/f α divergence in the powerspectrum at small frequencies f . When the exponent α lies between 1 and 2, this is generally termed '1/f ' noise, and understanding the origin of such fluctuations as well as its ubiquity has been a long-standing problem in physics The difficulty of generating non-integer values of α for the noise exponent within linear response theory has contributed to keeping this problem somewhat enigmatic and unresolved. In the past nine decades that the problem has been extant, several different explanations have been proposed (see below), but none of these have been fully satisfactory. 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 . Thus, if the lower cutoff on the frequencies is ∼ 1/T , the net power in the signal increases as T a , where a = α − 1. Our main observation in this paper is easily described. We consider a discrete-time Gaussian stochastic process ξ(t), which has the power-law spectral densitywith a lower-cutoff 1/T . The variance of the signal (which is also equal to the total power P ξ in the signal) increases as T α−1 when the cutoff T is increased. We generate the output signal η(t) by applying a instantaneous nonlinear transform to the input signal,where the function R is typically sigmoidal. A representative example of such a transform is given byThen the total power in the output process is P η = |η(t)| ...

We study a neural network model of interacting stochastic discrete two-state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint and marginal probability distributions are computed. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean-field theory.

Stochastic processes wherein the size of the state space is changing as a function of time offer models for the emergence of scale-invariant features observed in complex systems. I consider such a sample-space reducing (SSR) stochastic process that results in a random sequence of strictly decreasing integers {x(t)},0≤t≤τ, with boundary conditions x(0)=N and x(τ) = 1. This model is shown to be exactly solvable: P_{N}(τ), the probability that the process survives for time τ is analytically evaluated. In the limit of large N, the asymptotic form of this probability distribution is Gaussian, with mean and variance both varying logarithmically with system size: 〈τ〉∼lnN and σ_{τ}^{2}∼lnN. Correspondence can be made between survival-time statistics in the SSR process and record statistics of independent and identically distributed random variables.

We propose a variant model of the Pólya urn process, where the dynamics consist of two competing elements: suppression of growth and enhancement of dormant character. Here the level of such features is controlled by an internal parameter in a random manner, so that the system self-organizes and characteristic observables exhibit scale invariance, suggesting the existence of criticality. Varying the internal control parameter, one can explain interesting universality classes for avalanche-type events. We also discuss different versions of the model. It is pointed out that such an underlying mechanism has earlier been noted to operate in a neural network.

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The Himalayan region is one of the most crucial mountain systems across the globe, which has significant importance in terms of the largest depository of snow and glaciers for fresh water supply, river runoff, hydropower, rich biodiversity, climate, and many more socioeconomic developments. This region directly or indirectly affects millions of lives and their livelihoods but has been considered one of the most climatically sensitive parts of the world. This study investigates the spatiotemporal variation in maximum extent of snow cover area (SCA) and its response to temperature, precipitation, and elevation over the northwest Himalaya (NWH) during 2000–2019. The analysis uses Moderate Resolution Imaging Spectroradiometer (MODIS)/Terra 8-day composite snow Cover product (MOD10A2), MODIS/Terra/V6 daily land surface temperature product (MOD11A1), Climate Hazards Infrared Precipitation with Station data (CHIRPS) precipitation product, and Shuttle Radar Topography Mission (SRTM) DEM product for the investigation. Modified Mann-Kendall (mMK) test and Spearman’s correlation methods were employed to examine the trends and the interrelationships between SCA and climatic parameters. Results indicate a significant increasing trend in annual mean SCA (663.88 km2/year) between 2000 and 2019. The seasonal and monthly analyses were also carried out for the study region. The Zone-wise analysis showed that the lower Himalaya (184.5 km2/year) and the middle Himalaya (232.1 km2/year) revealed significant increasing mean annual SCA trends. In contrast, the upper Himalaya showed no trend during the study period over the NWH region. Statistically significant negative correlation (−0.81) was observed between annual SCA and temperature, whereas a nonsignificant positive correlation (0.47) existed between annual SCA and precipitation in the past 20 years. It was also noticed that the SCA variability over the past 20 years has mainly been driven by temperature, whereas the influence of precipitation has been limited. A decline in average annual temperature (−0.039 °C/year) and a rise in precipitation (24.56 mm/year) was detected over the region. The results indicate that climate plays a vital role in controlling the SCA over the NWH region. The maximum and minimum snow cover frequency (SCF) was observed during the winter (74.42%) and monsoon (46.01%) season, respectively, while the average SCF was recorded to be 59.11% during the study period. Of the SCA, 54.81% had a SCF above 60% and could be considered as the perennial snow. The elevation-based analysis showed that 84% of the upper Himalaya (UH) experienced perennial snow, while the seasonal snow mostly dominated over the lower Himalaya (LH) and the middle Himalaya (MH).

We show that for fluctuations in a sandpile with $1/f^{\alpha}$ spectrum, the power may also exhibit a scaling with its cutoff time. Despite significant progress, this generic behaviour remains so far overlooked and poorly understood. An intriguing example is the power spectrum of a simple random walk (normal diffusive process) on a ring with $L$ sites, showing $L/f^{3/2}$ behaviour instead of $1/f^2$ (Brownian noise) in the frequency regime $1/L^2 \ll f \ll 1/2$. We address the intriguing problem by scaling method and discuss its relevance in the context of the sandpile system and a class of stochastic processes.