Orientational ordering and correlations in a quasi-one-dimensional hard-dumbbell fluid

TL;DR

This paper provides an exact analytical study of a quasi-one-dimensional hard-dumbbell fluid, revealing detailed thermodynamic, orientational, and correlation properties, including a continuous crossover in ordering and correlation decay behaviors.

Contribution

It introduces an exact transfer-matrix approach for analyzing orientational ordering and correlations in a hard-dumbbell fluid, offering new analytical expressions and insights.

Findings

Exact expressions for thermodynamic properties and correlation functions.

Identification of a continuous crossover from weak to strong orientational order.

Universal pressure ratio at high density, twice the Tonks pressure.

Abstract

We study a quasi-one-dimensional fluid of hard dumbbells with continuous orientational degrees of freedom using an exact transfer-matrix formulation. The model allows for a complete analytical characterization of thermodynamic properties, orientational ordering, and correlation functions in terms of the spectral properties of an integral operator. We derive exact expressions for the equation of state, the orientational distribution function, and both partial and total radial distribution functions. Their asymptotic behavior is governed by the complex poles of the Laplace-transformed correlation functions, which determine the positional and orientational correlation lengths. As density increases, the system exhibits a continuous crossover from a weakly ordered regime with a unimodal orientational distribution to a strongly constrained regime characterized by bimodal orientational ordering. This crossover is accompanied by a nonmonotonic behavior of the pressure relative to the Tonks gas and by a qualitative change in the decay of correlation functions from oscillatory to monotonic. In the high-pressure limit, we show that orientational and positional fluctuations contribute equally to the pressure, leading to a universal ratio of twice the Tonks pressure. The theoretical predictions are supported by numerical solutions of the discretized transfer operator and by scaling arguments that elucidate the high-pressure behavior of ordering and correlation lengths.