In recent years there has been an upsurge of interest towards the study of
non equilibrium phenomena, relaxational problems and transport processes.
Different theoretical approaches have been developed in order to
understand such issues and many results have been obtained.
The behavior of a confined fluid can be different from that
of a bulk fluid in many important aspects. First of all the confinement
induces density inhomogeneities which may determine a variety of
phenomena having no counterpart in bulk systems.
The presence of surfaces, not only alters the average equilibrium
properties of fluids, but also affects their
time-dependent behavior such as diffusion, momentum and energy transport.
As a result, a great effort is currently devoted to the understanding
of fluid physics at the nanoscale.
In the last thirty years, a massive effort has been devoted to the
understanding of systems at thermodynamic equilibrium and new techniques
have been developed, among these Density Functional theory (DFT)
being perhaps the most versatile.
Recently, dynamical generalizations
of these equilibrium methods have been applied to non-equilibrium
problems such as Stokes drift, polymeric fluids confined to cavities
On the other hand, we do not have a similar control
over the behavior of non equilibrium systems but several
methods have been proposed.
A first community has considered an
approach which can be viewed as an extension of DFT to dynamical problems.
It assumes that the evolution of the microscopic density
is determined by some gradient of the local chemical potential, which
in turn depends on the free energy of the system.
The approach has been tested successfully in the case of colloidal
suspensions and of polymeric solutions,
where the presence of a solvent renders the dynamics dissipative,
and makes the density the only relevant field.
On the contrary, in the case of standard fluids one needs to fully account
for the momentum and energy transport.
The natural procedure seems therefore
to follow the evolution of both
positions and momenta and convey the necessary structural information
into an equation for the phase-space
distribution. Such a task can in principle be achieved
by using a modified Enskog-Boltzmann (EB) approach, as proposed by
The Enskog-Boltzmann approach
provides a satisfactory treatment of transport coefficients
and of many non equilibrium problems.
However, the treatment of inhomogeneous systems remains incredibly hard
within this approach. Non uniform solutions of the EB equation can be rarely
obtained. Nevertheless, in recent years a powerful numerical method
has been proposed, namely the Lattice Boltzmann equation (LBE), which
provides the possibility of solving minimal forms of the Boltzmannn
equation under external gradients by using an ingenious discretization
method. Several generalizations of LBE have been proposed to deal
with strongly interacting systems. Among the various versions of the
LBE, the Shan-Chen approach has rapidly gained an important place
because of its versatility, numerical stability and
limited computational effort required.
This success has been obtained at the price of drastic simplifications
concerning the treatment of the microscopic interactions.
Alternatively, some authors have put forward different
proposals but their methods are still the object of
Finally, a third approach has been also fruitfully employed
and consists in deriving the evolution of non uniform fluid systems
from a mesoscopic approach.
The need to use mesoscopic
methods to study the dynamics of complex fluids has lead to the
development of mixed strategies in which a kinetic method, such as LB,
is combined with simplified free energy functionals which allow a
versatile procedure to tune and control the thermodynamics of the
system of interest, incorporating the energy cost of sustaining
interfaces. The development and use of the hybrid schemes has been
based on simple, density gradient functionals. The use of such
approaches at smaller scales, where more detailed description of the
liquid structure may be needed, requires a critical scrutiny. From a
more fundamental point of view, it is also of interest to confront the
connection between such an approach and dynamical density functional
More recently, alternative hybrid schemes are being developed, in
which a mesoscopic solvent is coupled to an atomistic dynamics for
the solute particles whose trajectories need to be resolved individually. In
these approaches, as for the case of Stochastic Rotation Dynamics,
there is a more microscopic control of the interaction between
molecules coupled to a momentum conserving fluid. Although
usually particle-based mesoscopic approaches display very simple
thermodynamics, there is a on-going effort in providing them with
tunable equilibrium properties. For example, in Dissipative Particle
Dynamics phenomenological proposals close in spirit to density
functional theory have been proposed, and analogous efforts are being
carried out in Stochastic Rotation Dynamics.