CECAM - Common trends between Kinetic theory, Dynamical Density Functional methods and mesoscopic methods based on effective free energy models

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 etc. 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 different authors. 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 intense debate. 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 theories. 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.

Common trends between Kinetic theory, Dynamical Density Functional methods and mesoscopic methods based on effective free energy models

CECAM-HQ-EPFL, Lausanne, Switzerland

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