相变处的标度不变性、分形动力学与临界指数
Scale invariance, fractal dynamics, and critical exponents at the phase transition
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中文总结 AI 辅助
提出临界动力学在分形子空间上演化，用分形维数几何解释反常临界指数，并推导关联函数与标度关系。
中文摘要 AI 辅助
我们提出，在临界点，平衡动力学实际上是在一个分形子空间上演化，而非遍布整个欧几里得空间。从Fisher的有序参数关联函数表述出发，我们通过与该有效子空间关联的关联分形维数，对反常临界指数$\eta$进行了几何解释。利用分数阶微积分工具，我们推导出关联函数的一种形式，该形式恢复了低于上临界维度时的临界行为，并得到了$\eta$与关联到Riesz分数阶导数的分形维数$d_R$之间的显式关系。我们还考察了Rushbrooke标度关系，并研究了非整数维和 disordered 系统中的临界行为。在由参数$\sigma$控制的 disordered 情形中，我们检验了所提出的几何解释的有效范围。这些结果将标度律、临界指数、关联和分形几何联系起来，表明在临界点观察到的反常行为可以理解为受限于有效分形子空间的动力学的结果。
英文摘要
We propose that, at criticality, equilibrium dynamics effectively evolves on a fractal subspace rather than throughout the full Euclidean space. Starting from Fisher's formulation of the order-parameter correlation function, we interpret the anomalous critical exponent $η$ geometrically through a correlation fractal dimension associated with this effective subspace. Using fractional-calculus tools, we derive a form of the correlation function that recovers critical behavior below the upper critical dimension and obtain an explicit relation between $η$ and a fractal dimension $d_R$ linked to the Riesz fractional derivative. We also examine the Rushbrooke scaling relation and investigate critical behavior in non-integer-dimensional and disordered systems. In the disordered case, controlled by a parameter $σ$, we test the range over which the proposed geometric interpretation remains valid. The results connect scaling laws, critical exponents, correlations, and fractal geometry, suggesting that the anomalous behavior observed at criticality can be understood as a consequence of dynamics constrained to an effective fractal subspace.
发表机构
- University of Brasília（巴西利亚大学）
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