自相似谐相互作用的线性色散关系
Relations de dispersion pour chaînes linéaires comportant des interactions harmoniques auto-similaires
摘要：
自然界中的许多系统都具有树状和分叉结构，如树木、蕨类植物、蜗牛、肺、血管系统等，并且在很大范围内看起来都是自相似的。进化通过选择自相似性优化了哪些机械和动力学特性？如何在静态和动态框架下描述自相似结构的力学行为？具有自相似对称性的物理系统需要引入非局域粒子-粒子相互作用和质量的（准）连续分布。我们构造自相似函数和线性算子，如拉普拉斯算子和达朗伯波算子的自相似变体。所得到的自相似线性波动方程描述了具有空间自相似分布的非局部粒子间弹簧的无限长准连续线性链的动力学。非局域谐振子-粒子相互作用的自相似性导致了具有自相似性和分形特征的Weierstrass-Mandelbrot函数形式的色散关系。我们推导出一个连续近似，链接的自相似拉普拉斯分数积分，并在低频制度的振荡器密度的幂律频率依赖性的收益率。对于本模型的细节，我们参考我们最近的论文（Michelitsch等人，E80，011135（2009））。
英文摘要：
Many systems in nature have arborescent and bifurcated structures such as trees, fern, snails, lungs, the blood vessel system, etc. and look self-similar over a wide range of scales. Which are the mechanical and dynamic properties that evolution has optimized by choosing self-similarity? How can we describe the mechanics of self-similar structures in the static and dynamic framework? Physical systems with self-similarity as a symmetry property require the introduction of non-local particle-particle interactions and a (quasi-) continuous distribution of mass. We construct self-similar functions and linear operators such as a self-similar variant of the Laplacian and of the D'Alembertian wave operator. The obtained self-similar linear wave equation describes the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We deduce a continuum approximation that links the self-similar Laplacian to fractional integrals and which yields in the low-frequency regime a power law frequency dependence for the oscillator density. For details of the present model we refer to our recent paper (Michelitsch et al., Phys. Rev. E 80, 011135 (2009)).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
