Heat conduction in harmonic chains with Lévy-type disorder

Abstract

We consider heat transport in one-dimensional harmonic chains 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 ln ω in the low frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a non-closed 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.

Localization in a one-dimensional alloy with an arbitrary distribution of spacing between impurities: Application to Lévy glass

“…[6] it is argued that the Lyapunov exponent should be zero in this range of α but the numerical simulations of Ref. [8] show a nonzero Lyapunov exponent. Our result shows the explicit dependence on s in each term of the expansion.…”

“…In Ref. [8] the discrete model is reconsidered and an analytical formula for the power spectrum of the mass distribution of this model is obtained.…”

“…The studies on the discrete model [6,8] rely on the second-order perturbative expression for the Lyapunov exponent which is obtained [9] in terms of the correlation functions of the disorder. For the Lévy-type distribution of impurities, this method does not lead to a conclusive result in the entire range of the power-law exponent α.…”

Aspects of the disordered harmonic chain

“…Regarding ongoing studies of the dependence of J(N ) as function of boundary conditions and the spectral properties of the heat baths, see e.g. [32][33][34][35][36][37][38][39][40][41].…”

Aspects of the disordered harmonic chain

“…Regarding ongoing studies of the dependence of J(N ) as function of boundary conditions and the spectral properties of the heat baths, see e.g. [32][33][34][35][36][37][38][39][40][41].…”