Gluzman S, Sornette D
Institute of Geophysics and Planetary Physics, University of California Los Angeles, Los Angeles, California 90095-1567, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Mar;65(3 Pt 2A):036142. doi: 10.1103/PhysRevE.65.036142. Epub 2002 Mar 7.
Log-periodic oscillations have been found to decorate the usual power-law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance symmetry. For Ising or Potts spins with ferromagnetic interactions on hierarchical systems, the relative magnitude of the log-periodic corrections are usually very small, of order 10(-5). In growth processes [diffusion limited aggregation (DLA)], rupture, earthquake, and financial crashes, log-periodic oscillations with amplitudes of the order of 10% have been reported. We suggest a "technical" explanation for this 4 order-of-magnitude difference based on the property of the "regular function" g(x) embodying the effect of the microscopic degrees of freedom summed over in a renormalization group (RG) approach F(x)=g(x)+mu(-1)F(gamma x) of an observable F as a function of a control parameter x. For systems for which the RG equation has not been derived, the previous equation can be understood as a Jackson q integral, which is the natural tool for describing discrete-scale invariance. We classify the "Weierstrass-type" solutions of the RG into two classes characterized by the amplitudes A(n) of the power-law series expansion. These two classes are separated by a novel "critical" point. Growth processes (DLA), rupture, earthquake, and financial crashes thus seem to be characterized by oscillatory or bounded regular microscopic functions that lead to a slow power-law decay of A(n), giving strong log-periodic amplitudes. If in addition, the phases of A(n) are ergodic and mixing, the observable presents self-affine nondifferentiable properties. In contrast, the regular function of statistical physics models with "ferromagnetic"-type interactions at equilibrium involves unbound logarithms of polynomials of the control variable that lead to a fast exponential decay of A(n) giving weak log-periodic amplitudes and smoothed observables.
当连续尺度不变性对称性部分破缺为离散尺度不变性对称性时，人们发现对数周期振荡修饰了通常用于描述临界点趋近过程的幂律行为。对于具有铁磁相互作用的分层系统上的伊辛或Potts自旋，对数周期修正的相对大小通常非常小，约为10^(-5)。在生长过程[扩散限制聚集（DLA）]、破裂、地震和金融崩溃中，已报道存在幅度约为10%的对数周期振荡。我们基于“正则函数”g(x)的性质，对这种4个数量级的差异提出一种“技术”解释，该正则函数体现了在重正化群（RG）方法F(x)=g(x)+μ^(-1)F(γx)中对微观自由度求和的效应，其中可观测量F是控制参数x的函数。对于尚未推导出RG方程的系统，前一个方程可理解为杰克逊q积分，它是描述离散尺度不变性的自然工具。我们将RG的“Weierstrass型”解分为两类，其特征在于幂律级数展开的幅度A(n)。这两类由一个新的“临界点”分隔。因此，生长过程（DLA）、破裂、地震和金融崩溃似乎具有振荡或有界的正则微观函数的特征，这些函数导致A(n)缓慢的幂律衰减，产生强烈的对数周期幅度。此外，如果A(n)的相位是遍历且混合的，则可观测量呈现自仿射不可微性质。相比之下，平衡态下具有“铁磁”型相互作用的统计物理模型的正则函数涉及控制变量多项式的无界对数，这导致A(n)快速指数衰减，产生微弱的对数周期幅度和平滑的可观测量。
