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

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 \textbf{112}, L042105 (2025)] is established via the Poisson summation formula, highlighting the versatility and conceptual advantages of the Laplace-pole framework.