Transporte de energia em uma cadeia de osciladores clássicos altamente anarmônicos

Abstract

This master’s dissertation presents an analytical study of heat transport along a one-dimensional chain of classical anharmonic oscillators. The focus is on the effects of a local anharmonic potential of the form U_{an} = λP = λq^{2n}/2n (λ large and n ∈ N). The work begins by presenting a general model of oscillators with quadratic interaction and on-site harmonic and anharmonic potentials, which implements heat sources in the system through a model of stochastic reservoirs at all sites. Next, a version of the continuity equation for local energy is derived using the Itô calculus formalism, and the probability distribution of the stochastic variables is calculated for the case of non-interacting harmonic oscillators. A methodology to study the heat flow along the chain is then developed by introducing an integral formalism for the system correlations. This formalism is based on a version of Girsanov’s theorem and allows the study of interacting and anharmonic chains perturbatively from the non-interacting harmonic model. The harmonic problem with interaction is then approached as an example of the application of integral formalism and as a justification for its application to the anharmonic model. A temporal discretization of the integral perturbative formalism is then introduced to solve, again as an example, the interacting harmonic chain and compare the heat flow obtained in this case with the heat flow obtained without temporal discretization. This comparison is then used to justify a temporal discretization for the anharmonic case U_{an} = λP = λq^{2n}/2n (λ large and n ∈ N). The technique used to calculate the system correlations is based on a rigorous proof of the convergence of expansion in polymers for the perturbative analysis of the problem with n=2. Once the heat flow is obtained in first approximation, we are able to study conduction properties such as the validity of Fourier’s Law and thermal rectification. Finally, the results obtained for the heat flow throughout the proposed anharmonic system are investigated for consistency. The results of this dissertation were published in [1].