Abstract
We consider linear arrays of cells of volume V(c) populated by monodisperse rods of size σV(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 σV(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.
Highlights
In classical statistical mechanics, particles with shapes are ubiquitous
The work reported in the following is motivated by this chain of reasoning. It deals with hard rods in one dimension at thermal equilibrium in external potentials
For hard rods with firstneighbor interactions in one dimension, exact treatments are possible via recursion relations for partition functions [9,12] or density functional theory (DFT) [13,14,15]
Summary
Particles with shapes are ubiquitous. Their prominence in granular matter [1,2], soft condensed matter [3,4], and, biological matter [5,6], is well established. For hard rods with firstneighbor 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. The DFT for lattice fluids [12,14,18,19,20,21] is used to determine exact density profiles in external fields and the SIVP approach [10,22,23,24,25,26,27] is used as a realization of the EOS method. Appendix B presents a highly practical method of calculating exact density profiles within the DFT framework for arbitrary external potentials
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