By combining methods of kinetic and density functional theory, we present a description of molecular fluids which accounts for their microscopic structure and thermodynamic properties as well as for the hydrodynamic behavior. We focus on the evolution of the one particle phase space distribution, $f(r,v,t)$ rather than on the evolution of the average particle density, $\rho(r,t)$, which features in dynamic density functional theory often employed to describe colloidal systems.
In order to describe with sufficient accuracy the fluid structure at length scales comparable with the size of the particles we shall resort to methods similar to those of density functional theory (DFT) employed in the study of equilibrium and non equilibrium properties. In the case of hard-core fluids, DFT and its dynamical extension give excellent results and can be extended to more realistic fluids by using the van der Waals picture of decomposing the total inter-particle potential into a short-range repulsive potential and a long-range attractive potential tail. The first is treated by means of a reference hard-sphere system whilst the second is considered within the random phase approximation (RPA).
A simple analysis of the equations is used to derive explicit expressions both for equilibrium thermodynamic quantities, such as pressure, compressibility etc., and for non equilibrium transport coefficients.
In the second part of our presentation we shall introduce a a multicomponent extension of our theory and describe miscible and immiscible liquid mixtures under inhomogeneous, non steady conditions typical of confined fluid flows. We first derive from a microscopic level the evolution equations of the phase space distribution function of each component in terms of a set of self consistent fields, representing both body forces and viscous forces. Secondly, we solve numerically the resulting governing equations by means of the Lattice Boltzmann method whose implementation contains novel features with respect to existing approaches. Our model incorporates hydrodynamic flow, diffusion, surface tension,and the possibility for global and local viscosity variations. We validate our model by studying the bulk viscosity dependence of the mixture on concentration, packing fraction and size ratio. Finally we consider inhomogeneous systems and study the dynamics of mixtures in slits of molecular thickness and relate structural and flow properties.
The resulting equation for $f(r,v,t)$ is studied in two different physical limits: diffusive dynamics, typical of colloidal fluids without hydrodynamic interaction, where particles are subject to overdamped motion resulting from the coupling with a solvent at rest, and inertial dynamics, typical of molecular fluids .

The role of interplay between noise sources and nonlinearity is investigated in three different classical and quantum systems. (i) The role of a non-Gaussian Lévy noise on the nonlinear transient dynamics of a short overdamped Josephson junction is analyzed. The mean escape time of the junction is investigated considering Gaussian, Cauchy-Lorentz and Lévy-Smirnov probability distributions of the noise signals. In these conditions we find resonant activation and the first evidence of noise enhanced stability in a metastable system in the presence of Lévy noise. For Cauchy-Lorentz noise source, trapping phenomena and power law dependence on the noise intensity are observed. (ii) The phenomena of dissonance and consonance in a simple auditory sensory model composed of three neurons are considered. Two of them, here so-called sensory neurons, are driven by noise and subthreshold periodic signals with different ratio of frequencies, and its outputs plus noise are applied synaptically to a third neuron, so-called interneuron. We propose a theoretical analysis with a probabilistic approach to investigate the interspike intervals (ISI) statistics of the spike train generated by the interneuron. We find that at the output of the interneuron, inharmonious signals give rise to blurry spike trains, while the harmonious signals produce more regular, less noisy, spike trains. Theoretical results are compared with numerical simulations. (iii) Finally the dynamics of a quantum particle subject to an asymmetric bistable potential and interacting with a thermal reservoir is investigated. We obtain the time evolution of the population distributions in the position eigenstates of the particle, for different values of the coupling strength with the thermal bath. The calculation is carried out by using the Feynman-Vernon functional under the discrete variable representation.