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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]

The effect of particle overtaking on transport in a narrow channel is studied using a 1d model of a driven tracer in a quiescent bath. In contrast with the well-studied non-driven case, where the tracer's long-time dynamics is diffusive whenever overtaking is allowed, the driven tracer is shown to retain the hallmark single-file, sub-diffusive behavior for finite overtaking rates. Beyond a critical rate, the model exhibits a non-equilibrium phase transition from single-file to ballistic transport. The tracer velocity and the bath density profile are studied in both the single file and ballistic phases, demonstrating the distinct features of these phases.

A recently developed non-linear fluctuating hydrodynamics theory has been quite successful in describing various features of anomalous energy transport. However the diffusion and the noise terms present in this theory are not derived from microscopic descriptions but rather added phenomenologically. We here derive these hydrodynamic equations with explicit calculation of the diffusion and noise terms in a one-dimensional model. We show that in this model the energy current scales anomalously with system size L as ∼ L −2/3 in the leading order with a diffusive correction of order ∼ L −1 . The crossover length c from diffusive to anomalous transport is expressed in terms of microscopic parameters. Our theoretical predictions are verified numerically.

Anomalous dynamics in which local perturbations spread faster than diffusion are ubiquitously observed in the long-time behavior of a wide variety of systems. Here, the manner by which such systems evolve towards their asymptotic superdiffusive behavior is explored using the 1d Levy walk of order 1 < β < 2. The approach towards superdiffusion, as captured by the leading correction to the asymptotic behavior, is shown to remarkably undergo a transition as β crosses the critical value βc = 3/2. Above βc, this correction scales as |x| ∼ t 1/2 , describing simple diffusion. However, below βc it is instead found to remain superdiffusive, scaling as |x| ∼ t 1/(2β−1) . This transition is shown to be independent of the precise model details and is thus argued to be universal.