Monodisperse hard rods in external potentials

We consider linear arrays of cells of volume $V_{\mathrm{c}}$ populated by monodisperse rods of size $\sigma V_{\mathrm{c}},\sigma =1,2,...$, 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 $V_{\mathrm{c}}\to 0,\sigma \to \infty$ at nonzero and finite $\sigma V_{\mathrm{c}}$, 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.

Figure 1

Four rods of size $\sigma =3$ at positions $i=2,7,14,17$ on a lattice of size $L=20$. The relevant external potentials ${\mathcal{U}}_i$ and interaction potentials ${\mathcal{V}}_{i,j}$ are stated. One is a contact potential and the other represents an interaction of maximum range $(\sigma -1)V_{\mathrm{c}}$.

Figure 2

Profiles of (a) pressure and (b) density of rods in a uniform gravitational field at different temperatures. The solid lines represent rods of size $\sigma =1$ on a lattice and the dashed lines represent rods (of any size) in a continuum.

Figure 3

Density and pressure profiles for rods of size $\sigma =2$ in a uniform gravitational field at different temperatures. The solid curves represent solutions of Eqs. (28) and (29) as predicted by SIVP for a system with $N\gg 1$. The ${\overline{n}}_i$ data on the left (circles) originate from Eq. (37) with $\zeta$ from Eq. (36) and ${\lambda }_i$ from Eq. (32) with $\langle N\rangle =5$. These data are transcribed into the circles on the right via Eq. (10) and into the circles of panel (g) via Eq. (39).

Figure 4

Profiles of kinematic, interaction, and total pressure ${\overline{p}}_{\mathrm{mic}}$ for rods of size $\sigma =10$ at low $\widehat{T}$ as inferred from Eq. (24) and the pair distribution function as derived in Appendix pp3.

Figure 5

Profiles of the local pressure calculated from Eq. (24) in conjunction with Eq. (C7) (circles connected by dashed lines) and calculated from Eq. (11) (solid lines) for high $\widehat{T}$ (left) and low $\widehat{T}$ (right).

Figure 6

Profiles of (a) pressure and (b) density of rods (of size $\sigma =1)$ in a uniform gravitational field at temperatures $\widehat{T}=0.05$ and for (repulsive) contact interaction $\widehat{v}=0,-0.5,-1,-5$. Panel (c) shows the $\widehat{p}$ vs $\widehat{z}$ data near $\widehat{z}=0$. The dashed line has slope $-2/3$.

Figure 7

Profiles of pressure (left) and density (right) in a power-law trap with $\alpha =0.5$ (top) and $\alpha =2$ (bottom) at temperatures $\widehat{T}=0.02,0.5,1$ for rods of sizes $\sigma =1$ (solid curves) and $\sigma =\infty$ (dashed curves).

Figure 8

Density profile of rods ${\overline{n}}_i$ and mass ${\rho }_i$ for rods of size $\sigma =2$ in a harmonic trap $(\alpha =2)$ and in a power-law trap with firmer walls $(\alpha =10)$. In both traps we use $x_0=10V_{\mathrm{c}}$. All data are for $\overline{T}=0.1$. The chemical potential $\overline{\mu }$ has the values (a) 1.0, (b) 1.55, (c) 0.3, and (d) 0.5.

Figure 9

Density profile ${\overline{n}}_i$ for rods of size $\sigma =3$ in a power-law trap with $\alpha =10$. We use $x_0=20V_{\mathrm{c}}$. All data are for $\overline{T}=0.1$. The scaled chemical potentials are (a) $\overline{\mu }=1.1,1.15,1.2$ and (b) $\overline{\mu }=2.1,2.14,2.2$. In each plot the second and third data sets are vertically displaced by 0.2 and 0.4, respectively.

Figure 10

Density profile for rods of size $\sigma =2$ with center at coordinate $i=0,\pm 1,...,I$ in a box of width $I=10$. Full circles connected by lines: solution of Eq. (B10). Open circles: result of Eq. (62) for a box of infinite width with one wall at $i=10$.

Figure 11

Scaled density of rods of size $\sigma =5$ (big circles) and $\sigma =200$ (small circles) with first cell at position $x_i=(i-1)/\sigma$ near a rigid wall at $i=0$.

Figure 12

Scaled density of rods of size $\sigma =20$ on a lattice (big circles) and rods of (scaled) unit size in a continuum (solid lines).

Figure 13

(a) Scaled density of rods in a continuum with ${\rho }^{\left(\mathrm{mes}\right)}=0.9$. (b) Spectrum (80) of the attenuated oscillation (77) at three values of ${\rho }^{\left(\mathrm{mes}\right)}$.

Figure 14

Profiles of mass density of rods in a continuum at three different average values ${\rho }^{\left(\mathrm{mes}\right)}$.