# Abelian Sandpile - 阿贝尔沙堆模型

Interactive visualization of the Abelian Sandpile Model — Self-Organized Criticality, cascade avalanches, and power-law distributions

> Canonical page: https://elysiatools.com/en/visualizations/abelian-sandpile

- **Category:** Math

## Overview

Interactive visualization of the Abelian Sandpile Model (Bak-Tang-Wiesenfeld 1987) — Self-Organized Criticality (SOC), cascade avalanches, and power-law distributions. Drop sand grains onto a 2D grid; when any cell reaches the critical threshold (default 4), it topples by distributing one grain to each of its four neighbors, potentially triggering chain-reaction avalanches. Features adjustable grid size (21-301), critical threshold (3-8), drop rate (1-100 grains/frame), drop mode (center/random), simulation speed (1-20x), three view modes (height map with color-coded sand heights, activity heatmap showing cumulative topple intensity, recurrence plot highlighting active cells), click-to-drop grains at any grid position, burst mode (1000 grains at once), single-grain stepping, real-time statistics (total grains, avalanche count, last/max avalanche size, active cells, boundary losses), log-log avalanche size distribution plot with power-law exponent τ estimation via least-squares regression, avalanche time series with color-coded bar chart (cyan/amber/red by size relative to maximum), and comprehensive educational content covering the Abelian Sandpile Model rules (drop, topple, cascade, open boundary, commutativity), Self-Organized Criticality theory (scale-free distributions, 1/f noise, marginal stability, universality, robustness), natural SOC phenomena (earthquakes/Gutenberg-Richter law, forest fires, stock market crashes, solar flares, river networks, punctuated equilibrium), the Abelian property and sandpile group (finite Abelian group on recurrent configurations, reduced Laplacian, connection to tropical geometry and chip-firing games), fractal identity patterns (self-similar structures at grid sizes 2^n-1), and applications in geophysics (earthquake/landslide/wildfire modeling), network cascades (power grid blackouts, bank failures, viral spreading, neural avalanches), and computer science (parallel computing, image processing, tropical geometry, cryptography, load balancing). Perfect for complex systems education, statistical physics exploration, fractal pattern discovery, and understanding how simple local rules produce emergent global behavior.

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