  In the recent years the lattice Boltzmann (LB) methodology has been
fruitfully extended to include the effects of thermal fluctuations. So far, all
studied cases pertain equilibrium fluctuations, i.e. fluctuations with respect
to an equilibrium background state. In this paper we take a step further and
present results of fluctuating LB simulations of a binary mixture confined
between two parallel walls in presence of a constant concentration gradient in
the wall-to-wall direction. This is a paradigmatic set-up for the study of
non-equilibrium (NE) fluctuations, i.e. fluctuations with respect to a non-
equilibrium state. We analyze the dependence of the structure factors for the
hydrodynamical fields on the wave vector $\boldsymbol{q}$ in both the
directions parallel and perpendicular to the walls, as well as the finite-size
effects induced by confinement, highlighting the long-range ($\sim
|\boldsymbol{q}|^{-4}$) nature of correlations in the NE framework. Results
quantitatively agree with the predictions of fluctuating hydrodynamics.
Moreover, in presence of a non-ideal (NI) equation of state of the mixture, we
also observe that the (spatially homogeneous) average pressure changes, due to
a genuinely new contribution triggered by the long-range nature of NE
fluctuations. These NE pressure effects are studied at changing the system size
and the concentration gradient. Taken all together, we argue that these
findings are instrumental to boost the applicability of the fluctuating LB
methodology in the framework of NE fluctuations, possibly in conjunction with
experiments.
