Algebraic to exponential decay of spatial correlations in one-dimensional and confined hard-core fluids: A Laplace-pole analysis

TL;DR

This work addresses the spatial decay of correlations in 1D hard-core fluids and quasi-1D confinements at high packing. It develops a Laplace-transform and pole-analysis framework that yields a compact Jacobi theta function, $\theta_3$, representation of the RDF, linking oscillation frequencies and decay rates to Laplace-space poles. The approach reproduces the known Tonks-gas behavior and extends to symmetric binary mixtures and confined hard disks, revealing a universal scaling $\zeta_n \sim n^2\bar{\phi}^2$ and oscillation frequencies $\omega_n \sim n\pi$, while providing accurate envelopes and extending algebraic-to-exponential crossover insights. By unifying the RDF representations and connecting them to pole structures, the method offers a versatile, generalizable tool for analyzing 1D and confined fluids, with potential implications for dynamical properties and time-dependent responses.

Abstract

We derive the asymptotic behavior of the radial distribution function $g(x)$ for one-dimensional (1D) hard-rod systems and related quasi-one-dimensional geometries at high packing fractions using Laplace transform techniques and pole analysis. By identifying the poles and residues of the Laplace transform in the limit of small void fraction, we obtain compact representations of $g(x)$ in terms of the Jacobi elliptic theta function $θ_3$. This formulation naturally captures the two regimes governing the oscillatory decay toward unity: an intermediate algebraic decay and a long-distance exponential decay, consistent with previous results for the Tonks gas. Our approach provides a unified framework that (i) expresses $g(x)$ in a single well-tabulated special function, (ii) links spatial correlations directly to the pole structure in complex Laplace space, offering clear physical insight into decay rates and oscillation frequencies, and (iii) generalizes straightforwardly to 1D binary mixtures and confined hard-disk systems, where direct Gaussian decompositions are cumbersome. The equivalence between the theta-function representation and the Gaussian superposition of Bouzar and Messina [Phys. Rev. E 112, L042105 (2025)] is established via the Poisson summation formula, highlighting the versatility and conceptual advantages of the Laplace-pole framework.

Figures (5)

• Figure 1: Monocomponent system. (a) $\zeta_n/2n^2\pi^2\bar{\phi}^2$ and (b) $\omega_n/2n\pi$ as functions of $\bar{\phi}$ for $n=1,2,\ldots,10$ [from top to bottom in panel (a) and from bottom to top in panel (b)].
• Figure 2: Monocomponent system. Plot of $\widetilde{g}(\widetilde{x})$ for $\phi=0.9$. The solid, dotted, and dashed lines correspond to the exact function, the Gaussian approximation [Eq. \ref{['gauss']}], and our approximation [Eq. \ref{['Myg']}], respectively. The envelope [Eq. \ref{['envelope']}] is also shown (dashed-dotted line). The dashed and dotted lines are nearly indistinguishable.
• Figure 3: Symmetric binary mixture with $a=\frac{1}{2}$. (a) $\zeta_n/(n^2\pi^2\bar{\phi}^2/2)$ and (b) $\omega_n/n\pi$ as functions of $\bar{\phi}$ for $n=1,2,\ldots,10$ [from top to bottom in panel (a) and from bottom to top in panel (b)].
• Figure 4: Symmetric binary mixture with $a=\frac{1}{2}$. Plot of $\sqrt{\widetilde{x}}\widetilde{g}_{11}(\widetilde{x})$ (maxima at $\widetilde{x}\approx 1,3,5,\ldots$), $\sqrt{\widetilde{x}}\widetilde{g}_{12}(\widetilde{x})$ (maxima at $\widetilde{x}\approx 0,2,4,\ldots$), and $\sqrt{\widetilde{x}}\widetilde{g}(\widetilde{x})$ (maxima at $\widetilde{x}\approx 0,1,2,\ldots$) for $\phi=0.9$. Solid lines correspond to the exact functions, and dashed lines to our approximations [Eqs. \ref{['25']}]. The horizontal dashed lines indicate $2/\sqrt{2\pi\bar{\phi}^2}\simeq 7.98$ and $1/\sqrt{2\pi\bar{\phi}^2}\simeq 3.99$. Panel (a) shows $3\leq\widetilde{x}\leq 8$, and panel (b) shows $0\leq\widetilde{x}\leq 40$.
• Figure 5: Confined hard disks with $\epsilon=\frac{\sqrt{3}}{2}$. Plot of $\sqrt{\widetilde{x}}\widetilde{g}_{++}(\widetilde{x})$ (maxima at $\widetilde{x}\approx 1,3,5,\ldots$), $\sqrt{\widetilde{x}}\widetilde{g}_{+-}(\widetilde{x})$ (maxima at $\widetilde{x}\approx 0,2,4,\ldots$), and $\sqrt{\widetilde{x}}\widetilde{g}(\widetilde{x})$ (maxima at $\widetilde{x}\approx 0,1,2,\ldots$) for $\phi=0.9$. Solid lines correspond to the exact functions, and dashed lines to our approximations [Eqs. \ref{['29']}]. The horizontal dashed lines indicate $2/\sqrt{\pi\bar{\phi}^2}\simeq 11.28$ and $1/\sqrt{\pi\bar{\phi}^2}\simeq 5.64$. Panel (a) shows $3\leq\widetilde{x}\leq 8$, and panel (b) shows $0\leq\widetilde{x}\leq 40$.