Lévy动力学对湮灭过程中的涨落效应
Fluctuation Effects in the Pair-Annihilation Process with Lévy Dynamics
摘要：
本文研究了粒子在立方晶格上进行反常扩散时，对湮灭过程A+A->0中的密度衰减。反常扩散是通过Lévy飞行实现的，其特征在于长程跳跃并导致超扩散行为。因此，临界尺寸连续依赖于Lévy飞行分布的控制参数。本文利用非微扰重整化群理论研究了接近临界维数的系统。接近临界尺寸时，粒子间的相互作用关系违反了反应物充分搅拌的假设，质量作用定律也就失效了。众所周知，质量作用定律的崩溃是由长程涨落引起的。我们确定了这些波动的三个相互关联的后果。首先，尽管是一个非普适量，因此取决于微观细节，重整化反应速率可以近似为一个接近临界尺寸的普适定律。普遍性的出现依赖于这样一个事实，即长程涨落抑制了底层微观细节的影响。第二，当接近临界时，宏观反应速率降低，使得质量作用定律失去其意义。第三，附加的非解析幂律修正补充了质量作用项的解析定律。随着临界尺寸的接近，这些修正的数量越来越多，并积累起来，作出重要贡献。我们测试我们的研究结果的两个实现的Lévy航班，不同的方式，他们跨越正常的扩散。
英文摘要：
We investigate the density decay in the pair-annihilation process A+A->0 in the case when the particles perform anomalous diffusion on a cubic lattice. The anomalous diffusion is realized via Lévy flights, which are characterized by long-range jumps and lead to superdiffusive behavior. As a consequence, the critical dimension depends continuously on the control parameter of the Lévy flight distribution. This instance is used to study the system close to the critical dimension by means of the nonperturbative renormalization group theory. Close to the critical dimension, the assumption of well-stirred reactants is violated by anticorrelations between the particles, and the law of mass action breaks down. The breakdown of the law of mass action is known to be caused by long-range fluctuations. We identify three interrelated consequences of these fluctuations. First, despite being a nonuniversal quantity and thus depending on the microscopic details, the renormalized reaction rate can be approximated by a universal law close to the critical dimension. The emergence of universality relies on the fact that long-range fluctuations suppress the influence of the underlying microscopic details. Second, as criticality is approached, the macroscopic reaction rate decreases such that the law of mass action loses its significance. And third, additional nonanalytic power law corrections complement the analytic law of mass action term. An increasing number of those corrections accumulate and give an essential contribution as the critical dimension is approached. We test our findings for two implementations of Lévy flights that differ in the way they cross over to the normal diffusion.
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