  The main subject of the thesis is the study of stationary nonequilibrium
states trough the use of microscopic stochastic models that encode the physical
interaction in the rules of Markovian dynamics for particles configurations.
These models are known as interacting particles systems and are simple enough
to be treated analytically but also complex enough to capture essential
physical behaviours. The thesis is organized in two parts. The part 1 is
devoted to the microscopic theory of the stationary states. We characterize
these states developing some general structures that have an interest in
themselves. In this part there is an interlude dedicated to discrete calculus
on discrete manifolds with an exposition a little bit different to the one
available in literature and some original definitions. The part 2 studies the
problem macroscopically. In particular we consider the large deviations
asymptotic behavior for a class of solvable one dimensional models of heat
conduction. Both part 1 and 2 begin with an introduction of motivational
character followed by an overview of the relevant results and a summary
explaining the organization. Even tough the two parts are strictly connected
they can be read independently after chapter 1. The material is presented in
such a way to be self-consistent as much as possible.
