没有守恒的自组织：真的还是只是表面的标度不变性？
Self-organization without conservation: true or just apparent scale-invariance?
摘要：
本文仔细研究了在没有守恒定律的缓慢驱动的自组织临界性模型（例如森林火灾或地震自动机）中是否存在真正的标度不变性。通过使用三个不同层次的描述-（i）一个简单的平均场，（ii）一个更详细的平均场描述（自组织）分支过程，（iii）一个完整的随机表示的朗之万方程-，它是在一般的理由，非保守的动力学不会导致真正的临界性。与保护系统类似，我们称之为“充电”率的参数（例如森林火灾模型中的树木生长率）需要在非保护系统中进行微调以获得临界性。在无限尺寸的限制，这样的加载速率的微调是很容易实现的，因为它出现通过施加第二个分离的时间尺度，但对于任何有限的大小，需要精确的调整，以实现临界性和连贯的有限尺寸标度图片。使用上述方法，我们揭示了在非守恒系统中观察到的“表观临界性”的常见机制，并详细解释了（定性和定量）守恒系统中获得的真实临界性的差异。我们建议称之为自组织准临界（SOqC）。报告的一些结果是已知的，其中一些是新的。我们希望这里提出的统一框架有助于阐明这一领域令人困惑和矛盾的文献。在第二篇随附的论文中，我们将讨论这里获得的一般结果对神经科学中的神经雪崩模型的影响，其中在没有守恒的情况下，自组织尺度不变性已经被要求。
英文摘要：
The existence of true scale-invariance in slowly driven models of self-organized criticality without a conservation law, 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. Contrarily to conserving systems, a parameter, which we term "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 time-scales 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 the unified framework presented here helps to elucidate the confusing and contradictory literature in this field. In a second accompanying 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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