Exact density functional for hard-rod mixtures derived from Markov chain approach

Abstract

Using a Markov chain approach we rederive the exact density functional for hard rod mixtures on a one-dimensional lattice, which forms the basis of the lattice fundamental measure theory. The transition probability in the Markov chain depends on a set of occupation numbers, which reflects the property of a zero-dimensional cavity to hold at most one particle. For given mean occupation numbers (density profile), an exact expression for the equilibrium distribution of microstates is obtained, that means an expression for the unique external potential that generates the density profile in equilibrium. By considering the rod ends to fall onto lattice sites, the mixture is always additive. In 2002 Lafuente and Cuesta extended Rosenfeld's fundamental measure theory to lattice models based on a derivation of an exact density functional for hard rod mixtures in one dimension [16,17]. This derivation was carried out following a procedure developed by Vanderlick et al.[18] for continuum fluids. Since the excess free energy part of the functional could be expressed in terms of differences between parts that agree in their functional form with the excess free energy functional of a zerodimensional cavity, approximate functionals in higher dimensions were obtained by dimensional expansion of the corresponding difference operator. By construction these fundamental measure functionals have the property to become exact under dimensional reduction and their impressive power was first shown by determining phase diagrams of hard squares [17,19] and hard cube mixtures [16,17,20] with good quality. The fundamental measure functionals moreover allow one to apply the method of dimensional crossover and the merit of this was demonstrated by deriving functionals for lattice gases with nearest neighbor exclusion for different lattice types (square, triangular, face-and body-centered cubic) from the functional for cubes in (d + 1) dimensions [21]. The structure of the corresponding results led to a suggestion how to construct fundamental measure functionals for hard core lattice gases for any type of lattice, shape of the particles, and arbitrary dimension [22].In this report we rederive the exact density functional for hard rod mixtures in one dimension, that means the * philipp.maass@uni-osnabrueck.de

Thermodynamics of interacting hard rods on a lattice

“…System of pure hard rods has been investigated in the literature both in continuum 23,[39][40][41][42][43] and on a lattice [14][15][16][17][18] for mono and multicomponent cases. However, to the best of our knowledge, there is no thermodynamic analysis of this system.…”

“…One dimensional hard rods played a central role in the development of classical density functional theory [14][15][16][17][18][19] and it is paradigm model for the statistical mechanics of an extended molecules or polymers. It has been extensively studied and many exact results have been derived.…”

Interacting hard-sphere fluids in an external field

“…Inspired by work of Percus on hard rods [26] and by the probabilistic modeling of lattice hard rods with additional interactions, we derive a recurrence relation for the pair distribution function (PDF) of interacting hardsphere fluids which is valid in any dimension [27][28][29][30][31][32]. Using this recurrence relation, we infer explicit expressions for entropy and free energy as functionals of density and PDF.…”

Monodisperse hard rods in external potentials

“…To tackle this question analytically, exact results for inhomogeneous systems are required. For hard rods with first-neighbor interactions in one dimension, exact treatments are possible via recursion relations for partition functions [9,12] or density functional theory (DFT) [13][14][15]. These methods allow for the exact derivation of density profiles as well as many-body distribution functions.…”