Horowitz Claudio M, Romá Federico, Albano Ezequiel V
Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA), UNLP, CCT La Plata-CONICET, Sucursal 4, Casilla de Correo 16, (1900) La Plata, Argentina.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 1):061118. doi: 10.1103/PhysRevE.78.061118. Epub 2008 Dec 16.
The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees, i.e., structures inhibited for further growth, lying below the growing interface. It is shown that the rms height (h_{s}) and width (w_{s}) of the trees of size s obey power laws of the form h_{s} proportional, variants;{nu_{ parallel}} and w_{s} proportional, variants;{nu_{ perpendicular}} , respectively. Also, the tree-size distribution (n_{s}) behaves according to n_{s} approximately s;{-tau} . Here, nu_{ parallel} and nu_{ perpendicular} are the correlation length exponents in the directions parallel and perpendicular to the interface, respectively. Also, tau is a critical exponent. However, due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process, all these power laws are modulated by logarithmic periodic oscillations. The fundamental scaling ratios, characteristic of these oscillations, can be linked to the (spatial) fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents. We argue that the interplay between the spatial discrete scale invariance of the fractal substrate and the dynamics of the physical process occurring in those media is a quite general phenomenon that leads to the observation of logarithmic-periodic modulations of physical observables.
通过数值模拟研究了确定性分形基底上弹道聚集体的生长。首先，我们尝试运用成熟的法米利-维谢克动态标度方法来描述聚集体不断演化的界面。观察到与该标准标度律的系统性偏差，这表明必须引入显著的标度修正，以便更准确地理解界面的行为。随后，我们研究了正在生长的聚集体的内部结构，这种结构可以根据冻结树的标度行为来合理化，即位于生长界面下方、被抑制进一步生长的结构。结果表明，尺寸为s的树的均方根高度（h_s）和宽度（w_s）分别服从形式为h_s∝s^{ν_∥}和w_s∝s^{ν_⊥}的幂律。此外，树的尺寸分布（n_s）的行为符合n_s∝s^{-τ}。这里，ν_∥和ν_⊥分别是平行于和垂直于界面方向的关联长度指数。同样，τ是一个临界指数。然而，由于底层分形基底的离散标度不变性与生长过程动力学之间的相互作用，所有这些幂律都受到对数周期振荡的调制。这些振荡的特征性基本标度比，可以通过涉及临界指数的关系与底层分形的（空间）基本标度比联系起来。我们认为，分形基底的空间离散标度不变性与那些介质中发生的物理过程动力学之间的相互作用是一种相当普遍的现象，它导致了对物理可观测量的对数周期调制的观测。
