Monodisperse hard rods in external potentials

Abstract

We consider linear arrays of cells of volume V c populated by monodisperse rods of size σ V c , σ = 1,2, . . ., subject to hardcore exclusion interaction. Each rod experiences a position-dependent external potential. In one application we also examine effects of contact forces between rods. We employ two distinct methods of exact analysis with complementary strengths and different limits of spatial resolution to calculate profiles of pressure and density on mesoscopic and microscopic length scales at thermal equilibrium. One method uses density functionals and the other statistically interacting vacancy particles. The applications worked out include gravity, power-law traps, and hard walls. We identify oscillations in the profiles on a microscopic length scale and show how they are systematically averaged out on a well-defined mesoscopic length scale to establish full consistency between the two approaches. The continuum limit, realized as V c → 0, σ → ∞ at nonzero and finite σ V c , connects our highest-resolution results with known exact results for monodisperse rods in a continuum. We also compare the pressure profiles obtained from density functionals with the average microscopic pressure profiles derived from the pair distribution function.

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

“…When we now combine Eqs. ( 28), (32), and (33), the free energy functional acquires the form, βΩ[ρ, ρ (2) ] = dr ρ(r) ln ρ(r) − dr ρ (2) (r, r ) + ρ(r) ln ρ(r) − dr ρ (2) (r , r)…”

“…For 3D sticky-core fluids, the interaction range is reduced to the hard sphere diameter (ξ → σ). In consequence, the volume integrals V in Eq (20) or (32) have to be replaced by surface integrals S over spheres of di-ameter σ. The regions S and V represent the surface and the interior space of the sphere with diameter σ centered at position r, respectively.…”

“…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.…”