Dynamic Boltzmann free-energy functional theory

We present a generalization of the Density Functional Theory to distributions in μ-space rather than in configuration space. This equilibrium theory is the basic ingredient for constructing a dynamic theory with projection operators. The reversible part of the dynamics is computed exactly while the irreversible part is approximated with a fast momentum relaxation assumption. As a result the irreversible operator is given in terms of a viscosity tensor. We show that the kinetic equation has an H-theorem. Copyright c © EPLA, 2007 Introduction. – Fluids display structure at short length scales and behave hydrodynamically at large scales. An unexpected outcome of early computer simulations was that the scales where hydrodynamics is already valid are in practice rather small [1]. In fact, a fluid starts to behave hydrodynamically at a few atomic distances. Given the recent interest in micro and nanofluidic devices, the description of the dynamics of structured fluids flowing in confined geometries deserves some attention, in particular because a simple application of Navier-Stokes equations for structureless fluids is doomed to fail. A powerful approach for the study of structural static equilibrium properties of classical fluids is through Density Functional Theory (DFT) [2,3]. The use of sophisticated non-local models [4,5] has allowed to investigate a number of interesting phenomena, ranging from wetting phase transitions, to solid-liquid transitions and highly structured fluids in micropores (see [6] for a recent survey). Unfortunately, only equilibrium situations can be investigated with DFT and there is clearly a need to extend the theory to non-equilibrium situations [7]. Phenomenological theories of the Ginzburg-Landau or Cahn-Hilliard type have been proposed to study non-equilibrium situations, but these theories are too coarse-grained to capture the ordered structures that occur at nanoscales in fluids. Attempts to generalize DFT to non-equilibrium situations have been taken in refs. [8–12], and most notoriously in ref. [13] where the theory has been validated through Brownian dynamics simulations [13–15]. However, the theory is valid only if the underlying dynamics is stochastic and the range of application is limited to colloidal and other mesoscopic systems. Of course, one would like to have a similar dynamic equation that would apply to simple fluids and not only to colloidal systems. For simple fluids one expects to have a richer hydrodynamic phenomenology as compared to the purely diffusive dynamics of colloids. Very recently, DFT has been generalized to dynamic situations and hydrodynamics has been described through a Kohn-Sham approximation [16]. We propose in this paper an alternative approach. On intuitive grounds, one expects the momentum of the particles to play a role in a dynamic equation, and this suggests that one may focus on the one-particle distribution function in μ-space f(r,p) rather than on configuration space (i.e. rather than on the density n(r). While it is in principle possible to construct a dynamic equation based on n(r) as the relevant variable alone, it does not produce a hydrodynamic description where issues like momentum transport can be discussed. Clearly, the selection of relevant variables is a crucial and by no means trivial step in going from equilibrium to non-equilibrium situations. In dynamical situations, completely new variables may be required which are absent in equilibrium. Whether the selected relevant variables are the proper ones to describe the dynamics of a system or not is an issue that must be resolved a posteriori, from the predictions of the resulting dynamic equation and comparison with experiments or simulations. The selection of f(r,p) as relevant variable is not new. The classic Boltzmann’s equation is just a kinetic equation governing the dynamics of f(r,p) which, unfortunately, only applies to dilute gases. Extensions to dense situations have been also considered in the Revised Enskog