Recovery of normal heat conduction in harmonic chains with correlated disorder

Abstract

We address the general problem of heat conduction in one dimensional harmonic chain, with correlated isotopic disorder, attached at its ends to white noise or oscillator heat baths. When the low wavelength µ behavior of the power spectrum W (of the fluctuations of the random masses around their common mean value) scales as W (µ) ∼ µ β , the asymptotic thermal conductivity κ scales with the system size N as κ ∼ N (1+β)/(2+β) for free boundary conditions, whereas for fixed boundary conditions κ ∼ N (β−1)/(2+β) ; where β > −1, which is the usual power law scaling for one dimensional systems. Nevertheless, if W does not scale as a power law in the low wavelength limit, the thermal conductivity may not scale in its usual form κ ∼ N α , where the value of α depends on the particular one dimensional model. As an example of the latter statement, if W (µ) ∼ exp(−1/µ)/µ 2 , κ ∼ N/(log N ) 3 for fixed boundary conditions and κ ∼ N/ log(N ) for free boundary conditions, which represent non-standard scalings of the thermal conductivity.

Fast and slow thermal processes in harmonic scalar lattices

“…where J 0 is the Bessel function of the first kind. Formula (48) shows that T F asymptotically tends to zero inversely proportional to the square root of time 19 . This result has originally been obtained in paper [39].…”

Thermal transport in disordered harmonic chains revisited: Formulation of thermal conductivity and local temperatures

“…In this paper, we restrict our attention to uncorrelated disorders. It is shown in the literature [16] that a longrange spatial correlation of random masses leads to a localization length with a power-law exponent α = 1 + δ and hence the finite-size conductivity scales as κ N ∝ N δ/(1+δ) . The factor δ is positive, but arbitrarily small, so that normal scaling is recoverable for the correlated disorder in the limit of δ → 0.…”

“…The previous studies, which generally assume spatially uncorrelated mass disorder, support anomalous transport in momentum-conserving systems. Nonetheless, κ N scaling normally with N has been found in recent years for particular classes of disorders such as correlated mass disorder [16] and uncorrelated bond disorder [17,18]. The recovery of normal conductivity despite total momentum conservation seems to disprove the prevailing conjecture, if the normality is also verified for local temperatures in the interior of the system.…”

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].…”