Heat conduction in a weakly anharmonic chain: an analytical approach

Abstract

The analytical study of heat conduction in an anharmonic chain is considered here. We investigate an one-dimensional system (directly related to the Frenkel-Kontorova model) with anharmonic cosine on-site potential and harmonic interparticle interaction. We start with a stochastic thermal reservoir connected to each site of the system, and analyse the behaviour of the conductivity in the steady state with all the heat baths as we turn off the interior reservoirs, i.e., as we keep the heat baths at the boundaries only. For a weak interparticle potential and small anharmonicity, in a perturbative computation, we derive an analytic expression for the heat conductivity which indicates that the Fourier's law holds only when each site is connected to a heat bath. To show the trustworthiness of our perturbative computation, we also derive an expression for the conductivity by starting from the exact solution of the linear part of the dynamics and compare with the result which comes from the previous perturbative analysis.