Lifshitz tails and long-time decay in random systems with arbitrary disorder
Lifshitz tails and long-time decay in random systems with arbitrary disorder
复制标题
具有任意无序的随机系统中的 Lifshitz 尾部和长时间衰减
DOI：
10.1007/bf01016401
复制
发表时间：
1988
影响因子：
1.6
通讯作者：
T. Nieuwenhuizen
中科院分区：
文献类型：
--
作者：
J. Luck;T. Nieuwenhuizen
在随机系统中，各种线性问题的态密度，如声子、紧密结合电子或在具有陷阱的介质中的扩散，在带边缘表现出指数小的Liftshitz尾。当适当的随机变量（原子质量，位置能量，陷阱深度）的分布在其下界（上界）有一个δ函数时，Lifshitz奇点是纯指数。我们以定量的方式研究了从幂律开始的连续分布的普遍对数校正如何影响这些奇异点。在这个对数变量中，我们得到了Lifshitz尾对所有阶的渐近展开式。对于从本质奇点开始的分布，Lifshitz奇点本身的指数被修改。这些结果是在具有随机质量的谐波链的例子中得到的。有人认为，类似的结果在高维中也成立。本文还讨论了它们对其他模型的启示，如捕获问题中的长时间衰减。
In random systems, the density of states of various linear problems, such as phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits an exponentially small Liftshitz tail at band edges. When the distribution of the appropriate random variables (atomic masses, site energies, trap depths) has a delta function at its lower (upper) bound, the Lifshitz singularities are pure exponentials. We study in a quantitative way how these singularities are affected by a universal logarithmic correction for continuous distributions starting with a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders in this logarithmic variable. For distributions starting with an essential singularity, the exponent of the Lifshitz singularity itself is modified. These results are obtained in the example of harmonic chains with random masses. It is argued that analogous results hoid in higher dimensions. Their implications for other models, such as the long-time decay in trapping problems, are also discussed.