Introduction

The theories of complex systems have provided an interesting perspective on the dynamics of growth. More specifically, the concept of self-organized criticality proposes that such systems evolve into a state where a slight perturbation can trigger changes of all magnitudes. This feature has been observed in systems that transition into a steady state far from equilibrium, such as the Diffusion Limited Aggregation (DLA) model.

The focus of our research is to explore the manner in which scale-invariant noise is produced as the complexity of the system changes. We will use a generalization of the DLA model known as the dielectric breakdown (DB) model with a scale-invariant growth law.

DB Model and Laplacian Growth

Within the DB model, like the DLA, the growth is determined by a scalar field, φ, which obeys the Laplace equation outside the aggregate. The growth velocity relies on the gradient of this field to an assigned power, v ∝ |∇φ|η. When η=1, we can replicate the DLA model, while η=0 simulates the Eden model. For η greater than 1, growth occurs only in surface areas of the aggregate where the scalar field, ∇φ, is at a maximum, thereby simplifying the system behavior.

What we found through extensive calculations was that the aggregates are indeed self-similar. The noise associated with the growth process can be reduced in a regulated manner, maintaining its inherent complexity. The resulting patterns are complex while maintaining regularity, mirroring the primary features of dendritic growth.

Role of Complexity and Noise in Growth

We investigated the changes in the noise spectrum as a function of η. This led us to redefine catastrophes linked to the growth events based on two factors: the spatial extent of the rearrangement which follows a growth event, and the duration to restore the diffusion field to stability after a growth event. The examination showed that the self-organized critical regime is largely independent of these defined thresholds.

An intriguing discovery was that the noise accompanying the growth of individual patterns could be registered in terms of the field distribution along the periphery of a static pattern. While the self-organized criticality hypothesis usually details the noise spectrum of a dynamic process, we found that better numerical precision can be achieved by studying static distributions instead.

Conclusions

We have identified conditions under which self-organized critical behavior may emerge. Analyzing models in which both noise and intrinsic complexities can be altered, we can reveal that self-organized criticality is linked directly to the complexity of the evolving system. Models resulting in straightforward patterns did not display scale-invariant noise potential despite the inclusion of noise in their evolution.

In contrast, growth laws fostering complex patterns displayed noise at all scales, independent of the stochasticity included in their simulations. We express our gratitude to V. Hakim and L. M. Sander for stimulating conversations about this subject. The funding for this research has been provided by the Comisión Interministerial de Ciencia y Tecnología.

This exploration of the dynamics of Laplacian growth models and the interplay between complexity and criticality has opened new insights into the behavior of complex systems, with implications for a wide range of physical and biological processes.