In the present work I make an analysis about statistical quantities for linear and nonlinear chains in the stationary state. We start the discussion from a general linear model, with arbitrary couplings and connected to Gaussian reservoirs. A detailed analysis for quantities like heat ow and site temperatures is obtained, where all analytical expressions respective to those quantities are derived and a compared with numerical results. Then I study the quantitative and qualitative changes presented by the aforementioned quantities when the pinnings related to the reservoirs are modified. The changes in temperature profiles are related with the extrema of heat flux cumulants, motivating the investigation of whether phase transitions in the chain might occur. In order to investigate possible critical behaviors, I define velocity correlation functions between pair particles and squared velocities correlation functions. From where, one is able to estabilish a correlation length respective to these quantities. This study leads to one of the most remarkable achievements of this work, which is the connection made between the changes presented by some important statistical quantities of heat flux, the system s temperature, vibrational modes and the reservoirs pinnings. By treating the phenomenon of heat conduction in a more realistic and rigorous way, I develop a study to describe the transport properties in an anharmonic chain. Pondering that an exact solution for this sort of system is unfeasible, I use perturbation theory and other mathematical tools to discuss the main features of heat flux in a nonlinear chain. The technique developed throughout this thesis allows one to compute the heat current for a chain of arbitrary size, and is valid for systems under in fluence of reservoirs of any nature. We apply the method for chains governed by Gaussian and Poissonian reservoirs, verifying the impact of the nonlinearities over those systems, and comparing the obtained results to the linear case. In the case where there is a Poissonian bath injecting energy into the system, I shed some light on the effects of higher order cumulants related to the discontinous noise in the heat flux and I show how these new elements can lead to some results that at first glance seem physically incoherents.