Departamento de Ingeniería, Universidad Popular Autónoma del Estado de Puebla, Puebla, Pue., 72410, México.
Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil and Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado postal J-48, Puebla 72570, México.
Phys Rev E. 2019 Nov;100(5-1):052109. doi: 10.1103/PhysRevE.100.052109.
We consider heat transport in a one-dimensional harmonic chain attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation d between any two successive impurities is randomly distributed according to a power-law distribution P(d)∼1/d^{α+1}, being α>0. In the regime where the first moment of the distribution is well defined (1<α<2) the thermal conductivity κ scales with the system size N as κ∼N^{(α-3)/α} for fixed boundary conditions, whereas for free boundary conditions κ∼N^{(α-1)/α} if N≫1. When α=2, the inverse localization length λ scales with the frequency ω as λ∼ω^{2}lnω in the low-frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a nonclosed form. When α>2, the thermal conductivity scales as in the uncorrelated disorder case. The situation α<1 is only analyzed numerically, where λ(ω)∼ω^{2-α}, which leads to the following asymptotic thermal conductivity: κ∼N^{-(α+1)/(2-α)} for fixed boundary conditions and κ∼N^{(1-α)/(2-α)} for free boundary conditions.
我们考虑一维调和链的热输运，其两端连接到朗之万热浴上。该调和链具有质量杂质，其中任意两个相邻杂质之间的间距 d 根据幂律分布 P(d)∼1/d^{α+1}随机分布，其中 α>0。在分布的第一矩有明确定义的区域（1<α<2）中，对于固定边界条件，热导率 κ 与系统尺寸 N 呈 κ∼N^{(α-3)/α}的比例关系，而对于自由边界条件，κ∼N^{(α-1)/α}，如果 N≫1。当 α=2 时，由于对数修正，在低频区，反定位长度 λ 与频率 ω 呈 λ∼ω^{2}lnω 的关系，热导率的大小标度定律获得非封闭形式。当 α>2 时，热导率的标度与无关联无序情况相同。α<1 的情况仅通过数值分析进行了分析，其中 λ(ω)∼ω^{2-α}，这导致以下渐近热导率：对于固定边界条件，κ∼N^{-(α+1)/(2-α)}，对于自由边界条件，κ∼N^{(1-α)/(2-α)}。
