Abstract
Classically, a phase transition occurs when the properties of a system change upon tuning an external parameter, like the temperature. Is it possible, that a complex system regulates an internal parameter on its own, self-organized, such that it approaches a critical point all by itself? This is the central question discussed in this chapter.
Starting with an introduction to the Landau theory of phase transitions, particular attention will be devoted to cellular automata, an important and popular class of standardized dynamical systems. Cellular automata allow for an intuitive construction of models, such as the forest fire mode, the game of life, and the sandpile model, which exhibits “self-organized criticality”. Mathematically, a further understanding will be attained with the help of random branching theory. The chapter concludes with a discussion of whether self-organized criticality occurs in the most adaptive dynamical system of all, namely in the context of long-term evolution.
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Notes
- 1.
Spontaneous symmetry breaking is not present for first-order transitions, at which properties change discontinuous, see Table 6.1.
- 2.
- 3.
The stability of the three solutions are treated in exercise (6.1).
- 4.
- 5.
A particularly important class of dynamical networks are studied in Chap. 7.
- 6.
The concept of life at the edge of chaos was first develop in the context of information-processing networks, as discussed Chap. 7.
- 7.
- 8.
- 9.
- 10.
- 11.
The evaluation of avalanche durations is the topic of exercise (6.8).
- 12.
See Chap. 8.
- 13.
The term “genotype” denotes the ensemble of genes. The actual form of an organism, the “phenotype”, is determined by the genotype plus environmental factors, like food supply during growth.
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- 15.
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Gros, C. (2024). Self-Organized Criticality. In: Complex and Adaptive Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-55076-8_6
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