  One of the main features of statistical systems out of equilibrium is the
currents they exhibit in their stationary state: microscopic currents of
probability between configurations, which translate into macroscopic currents
of mass, charge, etc. Understanding the general behaviour of these currents is
an important step towards building a universal framework for non-equilibrium
steady states akin to the Gibbs-Boltzmann distribution for equilibrium systems.
In this review, we consider one-dimensional bulk-driven particle gases, and in
particular the asymmetric simple exclusion process (ASEP) with open boundaries,
which is one of the most popular models of one-dimensional transport. We focus,
in particular, on the current of particles flowing through the system in its
steady state, and on its fluctuations. We show how one can obtain the complete
statistics of that current, through its large deviation function, by combining
results from various methods: exact calculation of the cumulants of the
current, using the integrability of the model ; direct diagonalisation of a
biased process in the limits of very high or low current ; hydrodynamic
description of the model in the continuous limit using the macroscopic
fluctuation theory (MFT). We give a pedagogical account of these techniques,
starting with a quick introduction to the necessary mathematical tools, as well
as a short overview of the existing works relating to the ASEP. We conclude by
drawing the complete dynamical phase diagram of the current. We also remark on
a few possible generalisations of these results.
