Temperature response of heat conducting models

In the realm of statistical physics, a system can be studied for its response to an external stimulus; the equilibrium fluctation-dissipation theorem relates this response to the correlation at equilibrium between the observable we want to study and the entropy flux created by the perturbation. This is, of course, a meaningful approach for driven system with a non-equilibrium stationary state, such as an open systems in contact with different reservoirs at different temperatures. While the linear response theory for a system in an equilibrium state has a long history that dates back to the works of Einstein, Nyquist and Onsager, and it is generally systemized, for non-equilibrium states the theory has not progressed as fast and most of the results are usually restricted to the study of the response to mechanical perturbation. Mathematical problems arise when studying the response to a thermal perturbation: the temperature specifies the noise amplitude, so there is no absolute continuity between the two processes. Different solutions have been proposed to this problem, but the problem remains open for inertial systems, which is why we are interested in this issue in this thesis. We will present a different solution to this problem based on an algorithmic definition of the noise via Andersen thermostats hence, for the first time we obtain a fluctuation-response relation for thermal perturbations of a system with full Hamiltonian dynamics and operating out of equilibrium, thus extending previous results available only for systems evolving via overdamped stochastic equations or Markov jump processes.