Estudo analítico da condutividade termica em modelos microscopicos hamiltonianos fora do equilibrio - Humberto Cesar Fernandes Lemos (2008)
We study simple classical microscopic hamiltonian models out of thermal equilibrium. Thermal reservoirs are modelled mainly by stochastic noises acting on usual dynamical equations, e.g. for conservative models, we introduce noises in Hamiltons equations. Starting from the stochastic differential equations and using some wellknown tools from Itos calculus, we develop an integral formalism that allows us to evaluate the correlation functions for the models. First, we investigate the relaxation rate of the two-point correlation functionand its dependence on the temperature gradient for harmonic models with dissipative or conservative dynamics. Both systems have a similar behavior: if all stochastic noises have the same intensity, mimicking relaxation to the equilibrium state at a temperature T, the relaxation rate does not depend on T. On the other hand, in the case of relaxation to a non-equilibrium stationary state, relaxation rate depends on the temperature gradient. After this, we investigate heat conduction in stationary non-equilibrium states for conservative models, and consequently the validity of Fouriers law in these cases. We study a model similar to that of Frenkel-Kontorova, aiming to understand howa weak and limited anharmonic on-site potential changes the thermal conductivity. Using the integral formalism previously developed, a perturbative analysis up to first order in the anharmonicity coefficient shows us that, in any temperature regime, this model behaves as the harmonic one: if we have self-consistent thermal baths coupledvi to all chain sites, Fouriers law holds. Otherwise, if we take small coupling between inner sites of the chain and their respective reservoirs, in order to infer the behavior of the system with heat baths at the boundaries only, Fouriers law does not hold. Later on, we address another problem involving the heat flow in non-equilibrium stationary states: we momentarily abandon the derivation of Fouriers law and investigate properties of thermal conductivity of a model that has normal conductivity, the harmonic chain coupled to self-consistent thermal reservoirs. Changing the particle mass and/or on-site harmonic potential intensity, a new behavior arises: for alternating masses, thermal conductivity also depends on the square of masses difference. This new behavior can dramatically change the thermal conductivity. Finally, we investigate a harmonic quantum chain, looking to understand heat flow in non-equilibrium stationary states in this model, specially at low temperatures. Contrasting with classical models previously studied, the dynamics for this quantum chain is not stochastic: thermal reservoirs are modelled by hamiltonian systems. If the chain particles have alternate masses, we can easily see the importance of quantum behavior at low temperatures: thermal conductivity depends on the temperature and on the average mass, but not on the mass difference. On the other hand, in the high-temperature regime, we believe that the system behaves as its classical equivalent, as the thermal conductivity does not depend on temperature, but does depend on the square masses difference. vii