Search Results (7)

The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes. Full article

This paper is concerned with intriguing possibilities for non-conventional critical behavior that arise when a nearly critical strongly non-equilibrium system is subjected to chaotic or turbulent motion of the environment. We briefly explain the connection between the critical behavior theory and the quantum field theory that allows the application of the powerful methods of the latter to the study of stochastic systems. Then, we use the results of our recent research to illustrate several interesting effects of turbulent environment on the non-equilibrium critical behavior. Specifically, we couple the Kazantsev–Kraichnan “rapid-change” velocity ensemble that describes the environment to the three different stochastic models: the Kardar–Parisi–Zhang equation with time-independent random noise for randomly growing surface, the Hwa–Kardar model of a “running sandpile” and the generalized Pavlik model of non-linear diffusion with infinite number of coupling constants. Using field-theoretic renormalization group analysis, we show that the effect can be quite significant leading to the emergence of induced non-linearity or making the original anisotropic scaling appear only through certain “dimensional transmutation”. Full article

Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph G, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph, the asymptotic shape is known to be a ball in the graph metric. In this paper, we improve the sublinear bounds for the fluctuations known from its known asymptotic shape result by establishing bounds for the odometer function for a divisible sandpile model. Full article

Inspired by recent observations on active self-organized critical (SOC) systems, we designed an active pile (or ant pile) model with two ingredients: beyond-threshold toppling and under-threshold active motions. By including the latter component, we were able to replace the typical power-law distribution for geometric observables with a stretched exponential fat-tailed distribution, where the exponent and decay rate are dependent on the activity’s strength ($\zeta$). This observation helped us to uncover a hidden connection between active SOC systems and $\alpha$-stable Levy systems. We demonstrate that one can partially sweep $\alpha$-stable Levy distributions by changing $\zeta$. The system undergoes a crossover towards Bak–Tang–Weisenfeld (BTW) sandpiles with a power-law behavior (SOC fixed point) below a crossover point $\zeta <{\zeta }^{*}\approx 0.1$. Full article

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini (g) index and the Kolkata (k) index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as k, indicates the proportion of the ‘wealth’ owned by $(1-k)$ fraction of the ‘people’. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around $g=k\approx 0.87$, starting from the point of perfect equality, where $g=0$ and $k=0.5$) as competition increases in different social institutions, such as markets, movies, elections, universities, prize winning, battle fields, sports (Olympics), etc., under conditions of unrestricted competition (no social welfare or support mechanism). In this review, we present the concept of a generalized form of Pareto’s 80/20 law ($k=0.80$), where the coincidence of inequality indices is observed. The observation of this coincidence is consistent with the precursor values of the g and k indices for the self-organized critical (SOC) state in self-tuned physical systems such as sand piles. These results provide quantitative support for the view that interacting socioeconomic systems can be understood within the framework of SOC, which has been hypothesized for many years. These findings suggest that the SOC model can be extended to capture the dynamics of complex socioeconomic systems and help us better understand their behavior. Full article

Based on daily precipitation data from 1960 to 2017 in the rainy season in east China, to a given percentile threshold of one observation station, the time that precipitation spends below threshold is defined as quiet time $\tau$. The probability density functions $\tau$ in different thresholds follow power-law distributions with exponent $\mathsf{\beta }$ of approximately 1.2 in the day, pentad and ten-day period time scales, respectively. The probability density functions $\tau$ in different regions follow the same rules, too. Compared with sandpile model, Γ function describing the collapse behavior can effectively scale the quiet time distribution of precipitation events. These results confirm the assumption that for observation station data and low-resolution precipitation data, even in China, affected by complex weather and climate systems, precipitation is still a real world example of self-organized criticality in synoptic. Moreover, exponent $\beta$ of the probability density function $\tau$, mean quiet time ${\overline{\tau }}_q$ and hazard function $H_q$ of quiet times can give sensitive regions of precipitation events in China. Usual intensity precipitation events (UPEs) easily occur and cluster mainly in the middle Yangtze River basin, east of the Sichuan Province and north of the Gansu Province. Extreme intensity precipitation events (EPEs) more easily occur in northern China in the rainy season. UPEs in the Hubei Province and the Hunan Province are more likely to occur in the future. EPEs in the eastern Sichuan Province, the Guizhou Province, the Guangxi Province and Northeast China are more likely to occur. Full article

The present paper provides a qualitative discussion of the evolution of contact traction fields beneath rigid shallow foundations resting on granular materials. A phenomenological similarity is recognized in the measured contact traction fields of rigid footings and at the bases of sandpiles. This observation leads to the hypothesis that the stress distributions are brought about by the same physical phenomena, namely the development of arching effects through force chains and mobilized intergranular friction. A set of semi-empirical equations are suggested for the normal and tangential components of this contact traction based on past experimental measurements and phenomenological assumptions of frictional behaviors at the foundation system scale. These equations are then applied to the prescribed boundary conditions for the analysis of the settlement, resistance, and stress fields in supporting granular materials beneath the footing. A parametric sensitivity study is performed on the proposed modelling method, highlighting solutions to the boundary-value problems in an isotropic, homogeneous elastic half-space. Full article