Temperature profile and boundary conditions in an anomalous heat transport model

Abstract

Abstract. A framework for studying the effect of the coupling to the heat bath in models exhibiting anomalous heat conduction is described. The framework is applied to the harmonic chain with momentum exchange model where the non-trivial temperature profile is calculated. In this approach one first uses the hydrodynamic (HD) equations to calculate the equilibrium current-current correlation function in large but finite chains, explicitly taking into account the BCs resulting from the coupling to the heat reservoirs. Making use of a linear response relation, the anomalous conductivity exponent α and an integral equation for the temperature profile are obtained. The temperature profile is found to be singular at the boundaries with an exponent which varies continuously with the coupling to the heat reservoirs expressed by the BCs. In addition, the relation between the harmonic chain and a system of noninteracting Lévy walkers is made explicit, where different BCs of the chain correspond to different reflection coefficients of the Lévy particles. Keywords: Heat conduction, Fluctuating hydrodynamics, Current fluctuationsIf one imposes two temperatures T + and T − at the left and right ends of a large 1D system of length N , a naive application of Fourier's law predicts a heat current J = κ, where κ is an N -independent quantity called the conductivity. Anomalous heat conduction, typically characterized by κ = aN α with 0 < α ≤ 1, has however been shown to be a generic feature of one-dimensional momentum-conserving systems [31,17,29]. Violations of Fourier's law have been reported for a large variety of systems over the last decades both in theoretical models [30,22,36,11,23,48,15,7,38,42,40,2,24,26,3] and in nanotube experiments [8]. This phenomenon is usually accompanied by other puzzling features such as divergence of the time integral of the equilibrium currentcurrent correlations, superdiffusive propagation of local energy perturbations [35] and boundary singularities in the nonequilibrium stationary temperature profile [29].arXiv:1609.06614v2 [cond-mat.stat-mech]

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

“…left) boundary, it gets reflected as a φ − (resp. φ + ) Gaussian peak with area under the peak reduced by a factor w. This feature has been observed in numerical simulations and the validity of (108) has been confirmed [36]. There are two interesting cases w = 0 and w → 1.…”

“…It is important to note that non-linear temperature profiles can also be obtained in case of diffusive transport if the thermal conductivity κ is temperature-dependent and ∆T is large. On the other hand, for many systems with AHT, one finds a strongly non-linear temperature profile even when ∆T is made arbitrary small [5,10,11,26,34,36,65]. Quite often the profiles are characterized by divergent slopes at the boundaries.…”

“…(74,75,76) is a difficult problem [28] and has so far not been possible. A different approach, based on linear response theory and NFH was proposed in [36] and we present some details here.…”

“…which is based on a linear response calculation as done in [67] but around a local equilibrium state characterized by a temperature profile. According to this calculation the Kernel is related to the equilibrium current-current correlation [36] K N (x, y)…”

Thermal conductance of one-dimensional disordered harmonic chains

“…We show that a system with strong disorder, characterized by a 'heavy-tailed' probability distribution, and with large impedance mismatch between the bath and the system satisfies Fourier's law. We identify a dimensionless scaling parameter, related to the temperature scale and the localization length of the phonons, through which the thermal conductance for different models of disorder and different temperatures follows a universal behavior.Introduction-The study of heat transport via phonons in low dimensional (spatial dimension d < 3) classical and quantum mechanical systems has attracted considerable theoretical and experimental attention in recent years [1][2][3][4][5][6][7][8][9][10][11]. One of the main objectives of these studies is to understand the scaling of heat flux J which, according to Fourier's law [4], should scale with the system size L as J ∝ L −1 (L is measured along the direction of heat propagation).…”