Invariant Measure: Power Laws

Invariant Measure: Power Laws

Copyright: © 2013 |Pages: 20
DOI: 10.4018/978-1-4666-2202-9.ch005
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Bounded randomness of mass/energy exchange rates neither presuppose nor selects any specific time scale, thresholds of stability included. Nonetheless, the boundedness of the rates sets certain non-physical correlations among successive increments and thus justifies formation of “sub-walks” on the finest scale. Further, the “U-turns” at the thresholds of stability set certain correlations on the biggest possible scale of a relevant variable. The major question now is how the balance between the universal correlations, set by the “U-turns,” and those of the specific “sub-walks,” set by the bounded randomness, shapes the structure of a BIS that represents the evolutionary pattern of a relevant variable. It is proven that this issue is inherently related to another universal property of complex systems behavior that is power law distributions. It is demonstrated that power law distributions acquire novel understanding in the setting of boundedness: they appear as universal criterion for hierarchical structuring implemented under boundedness.
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The major aim of the previous chapter has been to demonstrate that the self-organization of a complex system in a multi-level hierarchical manner provides strengthening of the response by means of its diversification: different levels respond to different stimuli. The connectivity of a system implies that the hierarchy of levels is “linked” by inter-level feedbacks each of which operates in both directions, i.e. bottom up and top down. In Chapter 3 we considered the basic inter-level feedback, namely the one that operates at quantum level and makes atoms and molecules to self-organize in a” flow” with certain concentration. Moreover, we established that at the quantum level the concentration controls the local accumulation of energy/matter so that to intensify the dispersion of any local exceeding of matter/energy accumulation; in turn, on the next hierarchical level the “controlling mechanism” appears as boundedness of the local fluctuations of the concentration. Thus the inter-level feedback operates in both directions. The high non-triviality of the matter is 3-fold:

  • 1.

    It starts with imposing boundedness at the quantum level by means of assuming boundedness over local accumulation of matter/energy as leading property. It has been proven that this could not be achieved by means of imposing boundedness over the velocities alone. To the most surprise it turns out that this boundedness not only does not prevent arbitrary accumulation of matter/energy but, on the contrary, it produces amplification of the local fluctuations. The successful approach turns out to be the consideration of non-unitary interactions in the setting of the concept of boundedness so that their generic property, dissipativeness, to be used for dispersing any extra-accumulated matter/energy.

  • 2.

    Thus at the next hierarchical level the inter-level feedback operates as a source of inevitable fluctuations of the concentrations whose generic property is their boundedness. What sets their appearance as an inter-level feedback but not as a bounded environment is their participation in the choice of units. It is worth reminding that the choice of units is of primary importance for non-ambiguous setting of the structure of the state space when the evolution is described by non-linear equations of any type. Then, as considered in section Metrics in the State Space of Chapter 4 the inter-level feedback participates with units determined at the lower level while a bounded environment sets units external for the system. Thus, in the latter case a system does not retain robustness to an ever-changing environment and thus it would not be time-translational invariant.

  • 3.

    The above properties of the basic inter-level feedback along with the property of bringing about emergent phenomena make self-organization the appropriate generic implement for diversification of the response through further hierarchical super-structuring. Therefore the diversity of properties is reached by diversification of the hierarchical structuring. It should be stressed that this setting makes our approach fundamentally different from the traditional reductionist philosophy of physics. Indeed, the latter reduces the complexity to separate levels of organizations (elementary particles, atoms and molecules, large systems, biological systems) each of which is considered at specific pre-determined external constraints so that the hierarchical order goes only bottom up i.e., from elementary particles to living organisms. These assumptions constitute its fundamental difference from the concept of boundedness which, on the contrary, sets hierarchical self-organization at non-specified external constraints so that its maintaining goes both bottom up and top down by means of loop-like inter-level feedbacks that serve as constraints imposed by one level to the nearest (both upper and lower) ones. Further, contrary to the reductionist approach which adopts the idea of dimension expansion as a tool for reaching diversity of properties, the diversification under boundednes happens through hierarchical super structuring in constraint dimensionality.

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