---
slug: paper-bak-p-tang-c-wiesenfeld-k-1987-self-organized-criticality-an-explanation-of-the
title: Bak, Tang, and Wiesenfeld on Self-Organized Criticality (1987)
canonical: https://miscsubjects.com/a/paper-bak-p-tang-c-wiesenfeld-k-1987-self-organized-criticality-an-explanation-of-the
---

## What the work establishes

The 1987 paper by Per Bak, Chao Tang, and Kurt Wiesenfeld introduces self-organized criticality as a mechanism in extended dissipative dynamical systems. These systems evolve spontaneously to a critical state under slow driving. The critical state produces power-law distributions in space and time without external parameter tuning.

The core result identifies 1/f noise with the dynamics at this self-organized critical point. It also connects the process to the formation of fractal structures.

## Core results and passages

The abstract states: "We show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or 1/f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects." (Bak, Tang, Wiesenfeld, Phys. Rev. Lett. 59, 381, 1987).

The authors demonstrate the claim through a cellular automaton sandpile model. Grains of sand are added slowly. Avalanches occur when local slopes exceed a threshold. The system reaches a minimally stable state where small perturbations trigger events of all sizes.

Cluster size distribution follows D(s) ~ s^−τ with τ near 1 in two dimensions. Lifetime distributions yield a 1/f power spectrum in the frequency response. The paper reports numerical results on arrays up to 100×100 sites showing straight-line behavior on log-log plots over two decades.

A later related passage in the 1988 expansion notes: "the emergence of the self-organized critical state provides a connection between nonlinear dynamics, the appearance of spatial self-similarity, and 1/f noise in a natural and robust way."

## Convergence patterns evidenced

The model exhibits branching avalanches. Local interactions propagate through domino-like relaxations across the lattice. This produces scale-invariant power laws. Energy input drives dissipation through events spanning all scales. The resulting structures show fractal geometry and memory in the form of long-range correlations built by successive additions.

These match observed patterns in river networks, earthquakes, and turbulence where slow driving meets threshold instabilities.

## Distance from the full synthesis

The work supplies a mechanistic account of how energy flows in dissipative systems generate branching and scale-invariant structures. It stops at physical and mathematical description. It does not address the transition to memory storage, replication, or minded observation. The sandpile remains a toy model of local rules producing global statistics.

## Honest limits

The original sandpile yields avalanche statistics consistent with power laws, yet later analysis debates whether the spectrum is precisely 1/f or closer to 1/f² in some regimes. Finite-size effects cut off the largest events. The model requires specific local thresholds and slow driving; not every driven system reaches the same attractor. Universality classes remain under investigation across variants.

The paper provides no direct empirical data from natural systems. Its strength lies in the numerical demonstration that criticality emerges without fine-tuning.

The synthesis treats the result as one concrete realization of energy-driven pattern formation at the level of physical flow and structure. Readers inside the system can invoke the same local rules to test further instances through simulation or measurement.
