On the relationship between the density functional formalism and the potential distribution theory for nonuniform fluids

Density Functional Theories of Hard Particle Systems

“…The free energy DF of hard core systems is purely entropic, and it may be split in the ideal gas contribution plus an excess free energy, F ex [ρ], to account for the entropy reduction due to the non-overlap of the molecular cores. Hence, it is usually written in k B T units as 11) with the ideal free energy density as a function, Φ id (ρ) = ρ(ln ρ−1), evaluated at the local density, while the excess contribution is expressed as the volume integral of an excess free energy density Φ [ρ]; r , which is a function of r and a functional of ρ(r). Notice that there could be multiple choices of Φ [ρ]; r leading to the same F ex [ρ], since there are many ways to separate the total free energy excess of the system in terms of local contributions, and we will see that different approaches to F ex [ρ] may give similar results with very different Φ [ρ]; r .…”

Untitled

“…An alternative way to treat the hard-rod lattice gas has been followed in [16]. In this approach, which relates to density functional theory of classical fluids, one considers the grand-canonical ensemble and defines the occupation numbers x i , where x i = 1 if site i is occupied by a rod center, and x i = 0 else.…”

“…This was subsequently generalized to finite-range forces between neighboring rods [15]. A generalized discrete version of the continuum hard-rod fluid on a linear chain was studied by Robledo and Varea [16]. They derived an exact functional for the mean occupation numbers of the rod centers on the chain, which, by taking the continuum limit, allowed them to recover the continuum density functional of Percus (for a review on classical density functionals, see [17]).…”

“…We first show that the canonical and grand canonical partition functions of this "Takahashi lattice gas" (TLG) obey simple recursion relations and that density profiles and density correlations can be conveniently calculated from the partition functions due to the one-dimensional nature of the model. Since the TLG contains as a special case the hard-rod model studied by Robledo and Varea [16] (for which the interaction potential would be infinite for interparticle distances smaller than the rod length and zero else), we can give a simple solution of the nonlinear difference equations derived in [16]. In this context we found it worthwhile to rederive the central formulae given in [16].…”

“…Since the TLG contains as a special case the hard-rod model studied by Robledo and Varea [16] (for which the interaction potential would be infinite for interparticle distances smaller than the rod length and zero else), we can give a simple solution of the nonlinear difference equations derived in [16]. In this context we found it worthwhile to rederive the central formulae given in [16]. We will apply our formalism to a system of hard-rods confined both by hard and soft walls, and also to a system of particles, which in addition to the athermal hard-rod repulsion experience a finite interaction potential over a limited range.…”