Self-organization without conservation: true or just apparent scale-invariance?

Juan A. Bonachela, Miguel A. Muñoz

Research output: Contribution to journalArticle

47 Citations (Scopus)

Abstract

The existence of true scale-invariance in slowly driven models of self-organized criticality without a conservation law, such as forest-fires or earthquake automata, is scrutinized in this paper. By using three different levels of description - (i) a simple mean field, (ii) a more detailed mean-field description in terms of a (self-organized) branching processes, and (iii) a full stochastic representation in terms of a Langevin equation - it is shown on general grounds that non-conserving dynamics does not lead to bona fide criticality. Contrary to the case for conserving systems, a parameter, which we term the 're-charging' rate (e.g. the tree-growth rate in forest-fire models), needs to be fine-tuned in non-conserving systems to obtain criticality. In the infinite-size limit, such a fine-tuning of the loading rate is easy to achieve, as it emerges by imposing a second separation of timescales but, for any finite size, a precise tuning is required to achieve criticality and a coherent finite-size scaling picture. Using the approaches above, we shed light on the common mechanisms by which 'apparent criticality' is observed in non-conserving systems, and explain in detail (both qualitatively and quantitatively) the difference with respect to true criticality obtained in conserving systems. We propose to call this self-organized quasi-criticality (SOqC). Some of the reported results are already known and some of them are new. We hope that the unified framework presented here will help to elucidate the confusing and contradictory literature in this field. In a forthcoming paper, we shall discuss the implications of the general results obtained here for models of neural avalanches in neuroscience for which self-organized scale-invariance in the absence of conservation has been claimed.

LanguageEnglish
Article numberP09009
Number of pages38
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2009
Issue number9
Early online date17 Sep 2009
DOIs
Publication statusPublished - 28 Dec 2009

Fingerprint

Scale Invariance
Criticality
Self-organization
forest fires
Conservation
conservation
invariance
tuning
neurology
Forest Fire
loading rate
conservation laws
Mean Field
avalanches
charging
Tuning
earthquakes
Stochastic Representation
scaling
Self-organized Criticality

Keywords

  • percolation problems
  • phase transitions into absorbing states
  • sandpile models
  • self-organized criticality

Cite this

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title = "Self-organization without conservation: true or just apparent scale-invariance?",
abstract = "The existence of true scale-invariance in slowly driven models of self-organized criticality without a conservation law, such as forest-fires or earthquake automata, is scrutinized in this paper. By using three different levels of description - (i) a simple mean field, (ii) a more detailed mean-field description in terms of a (self-organized) branching processes, and (iii) a full stochastic representation in terms of a Langevin equation - it is shown on general grounds that non-conserving dynamics does not lead to bona fide criticality. Contrary to the case for conserving systems, a parameter, which we term the 're-charging' rate (e.g. the tree-growth rate in forest-fire models), needs to be fine-tuned in non-conserving systems to obtain criticality. In the infinite-size limit, such a fine-tuning of the loading rate is easy to achieve, as it emerges by imposing a second separation of timescales but, for any finite size, a precise tuning is required to achieve criticality and a coherent finite-size scaling picture. Using the approaches above, we shed light on the common mechanisms by which 'apparent criticality' is observed in non-conserving systems, and explain in detail (both qualitatively and quantitatively) the difference with respect to true criticality obtained in conserving systems. We propose to call this self-organized quasi-criticality (SOqC). Some of the reported results are already known and some of them are new. We hope that the unified framework presented here will help to elucidate the confusing and contradictory literature in this field. In a forthcoming paper, we shall discuss the implications of the general results obtained here for models of neural avalanches in neuroscience for which self-organized scale-invariance in the absence of conservation has been claimed.",
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Self-organization without conservation : true or just apparent scale-invariance? . / Bonachela, Juan A.; Muñoz, Miguel A.

In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 2009, No. 9, P09009, 28.12.2009.

Research output: Contribution to journalArticle

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T2 - Journal of Statistical Mechanics: Theory and Experiment

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AU - Muñoz, Miguel A.

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