# Triangle-Well and Ramp Interactions in One-Dimensional Fluids: A Fully Analytic Exact Solution

## TL;DR

This paper provides a fully analytic solution for the equilibrium properties of one-dimensional fluids with triangle-well and ramp interactions, enabling detailed analysis of thermodynamics and structure without numerical inversion.

## Contribution

It offers the first fully analytic expression for the radial distribution function in these models, surpassing previous numerical approaches and enabling comprehensive structural and thermodynamic analysis.

## Key findings

- Analytic expressions for $g(r)$, equation of state, and structure factors.
- Validation of closure relations through bridge and correlation functions.
- Identification of Fisher--Widom and Widom lines for the models.

## Abstract

The exact statistical-mechanical solution for the equilibrium properties, both thermodynamic and structural, of one-dimensional fluids of particles interacting via the triangle-well and the ramp potentials is worked out. In contrast to previous studies, where the radial distribution function $g(r)$ was obtained numerically from the structure factor by Fourier inversion, we provide a fully analytic representation of $g(r)$ up to any desired distance. The solution is employed to perform an extensive study of the equation of state, the excess internal energy per particle, the residual multiparticle entropy, the structure factor, the radial distribution function, and the direct correlation function. In addition, scatter plots of the bridge function versus the indirect correlation function are used to gauge the reliability of the hypernetted-chain, Percus--Yevick, and Martynov--Sarkisov closures. Finally, the Fisher--Widom and Widom lines are obtained in the case of the triangle-well model.