Systems with many degrees of freedom are notoriously hard to grasp intuitively

as well as capture mathematically. Thermodynamics and statistical mechanics

ideas allowed for a leap in the understanding of such complex systems at equilibrium

over a century ago. Despite this success, little is known when our system

is taken out of equilibrium. Moreover, most of the knowledge concentrates on

system driven slightly out of equilibrium.

In the last 15 years, the macroscopic fluctuation theory was shown to be a

successful description of out of equilibrium diffusive systems. It was shown to

successfully capture the behavior of the few solvable models in the field. In this

Thesis, I will focus on the study of two properties of diffusive boundary driven

systems within the scope of the macroscopic fluctuation theory.

The first, current fluctuations, allows intuitive understanding of the physics

governing the system through the noise statistics of the steady state. Generally,

calculating the current fluctuations is hard. However, a clever guess, known

as the additivity principle, allows to obtain analytically an expression for the

current fluctuations. This Thesis presents a sufficient and necessary condition

for the validity of the additivity principle guess. Moreover, assuming the validity

of the additivity principle, the universality of current fluctuations is shown for

systems of arbitrary geometry.

The second property discussed is the density correlations. While in equilibrium

- away from a phase transition, correlation functions are known to decay

exponentially, for systems driven out of equilibrium, correlation functions are

generically long ranged. Using known results for diffusive classical systems, it is

shown that transport in disordered quantum systems can also be studied using

the macroscopic fluctuation theory. Moreover, an exact correspondence between

classical processes and some quantum processes is found.