Energy transport in a chain of highly anharmonic classical oscillators

Abstract In this work we examine heat transport in one-dimensional chains of anharmonic oscillators coupled to stochastic heat baths at each site. The main objective is to calculate the thermal conductivity of the system. The probability measure of the anharmonic system is calculated using an integral formalism based on Girsanov’s Theorem. To overcome the mathematical difficulties of the continuous-time problem, a time discretization is implemented. This discretization is justified by the agreement with the results obtained for the harmonic case and allows the use of a convergent polymer expansion. The polymer expansion is an essential tool for calculating the correlation functions of the system and, consequently, obtaining the thermal conductivity. Precisely, the convergent polymer expansion rigorously supports the perturbative computation carried out here. In the specific case of a ϕ 4 anharmonic potential, the obtained thermal conductivity ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> ) is consistent with previous analytical and numerical results. The study also investigates the presence of thermal rectification in the system, showing that inhomogeneity in the anharmonic couplings, but not in the harmonic interaction constants, induces thermal rectification, at least to a first approximation. The work concludes by highlighting the effectiveness of time discretization and polymer expansion as tools for investigating heat transport in anharmonic systems and points out potential avenues for future research in this area.