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

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. At high densities, an extended intermediate regime with an approximate algebraic decay of the radial distribution function precedes the pole-dominated asymptotic exponential behavior. 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.