Discrete scale invariance, complex fractal dimensions, and log‐periodic fluctuations in seismicity
Discrete scale invariance, complex fractal dimensions, and log‐periodic fluctuations in seismicity
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地震活动的离散尺度不变性、复杂分形维数和对数周期波动
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
D. Sornette
中科院分区：
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作者：
H. Saleur;C. Sammis;D. Sornette
我们详细讨论了离散尺度不变性的概念，并展示了它如何导致复杂的临界指数，从而导致对接近大地震奇点的地震活动的各种测量所表现出的尺度的对数周期校正。离散尺度不变性首先在几何分形（谢尔宾斯基垫片）上进行说明，它被证明可以通过复杂分形维数来完全描述，其虚部是离散比例因子的简单函数（对数的倒数）。然后，分析了分层格上的一组简单物理系统（自旋和渗滤），以举例说明为解决该问题而引入的离散重整化群形式主义中不同项的起源。作为与地震相关的破裂的更具体的例子，我们提出了 Newman 等人的分层时间相关纤维束的解决方案。 [1994]它明确地展示了一个离散重正化群，从中进行对数周期校正。最后我们指出，离散尺度不变性不一定需要底层的几何层次结构。层次结构可以从欧几里得（非层次）异构系统中的物理和/或动力学中“自发地”出现。我们根据随机异质系统中地震能量的随机游走（或扩散）来简要讨论这种机制的简单动力学模型。
We discuss in detail the concept of discrete scale invariance and show how it leads to complex critical exponents and hence to the log-periodic corrections to scaling exhibited by various measures of seismic activity close to a large earthquake singularity. Discrete scale invariance is first illustrated on a geometrical fractal, the Sierpinsky gasket, which is shown to be fully described by a complex fractal dimension whose imaginary part is a simple function (inverse of the logarithm) of the discrete scaling factor. Then, a set of simple physical systems (spins and percolation) on hierarchical lattices is analyzed to exemplify the origin of the different terms in the discrete renormalization group formalism introduced to tackle this problem. As a more specific example of rupture relevant for earthquakes, we propose a solution of the hierarchical time-dependent fiber bundle of Newman et al. [1994] which exhibits explicitly a discrete renormalization group from which log-periodic corrections follow. We end by pointing out that discrete scale invariance does not necessarily require an underlying geometrical hierarchical structure. A hierarchy may appear “spontaneously” from the physics and/or the dynamics in a Euclidean (nonhierarchical) heterogeneous system. We briefly discuss a simple dynamical model of such mechanism, in terms of a random walk (or diffusion) of the seismic energy in a random heterogeneous system.