The phase behavior of a quasi-one-dimensional fluid of hard spheres is studied in hard cylindrical and rectangular straight channels using the transfer operator method. The radius of the channel ($R$) is chosen such that only first-neighbor interactions are present, which allows only fluidlike and zigzag solidlike structures to form. It is managed to show that the fluid-solid structural change occurs approximately at $\rho =1/\langle {\sigma }_{\parallel }\rangle$, where $\rho$ is the one-dimensional density and $\langle {\sigma }_{\left|\right|}\rangle$ is the average of the longitudinal contact distance in the case of uniform particle distribution. At this density, the pressure of positionally disordered structure diverges, but the pressure ratio ($P/P_T$), where $P$ is the pressure of the system and $P_T$ is the pressure of an effective one-dimensional hard rod fluid, exhibits a peak. At the close packing, while the positional fluctuation $\left(\langle \mathrm{\Delta }{r}^2\rangle \right)$ vanishes with $P^{\tilde{\beta }}$, where $\tilde{\beta }=-2$ for any $R$ and cross section, $P/P_T$ (we call it $\tilde{\alpha }$) goes to 2 in the rectangular and 2.5 in the cylindrical channel. The most striking difference can be observed in the transversal position correlation length (ξ). It exhibits $exp\left(\mathrm{\Delta }P\right)$ dependence in the rectangular channel with $\mathrm{\Delta }$ being the extra length necessary to create a defect. However, ξ goes with $P^{\tilde{\gamma }}$ in the cylindrical channel, where $\tilde{\gamma }=1$. Interestingly, the cylindrically confined hard spheres obey the sum rules $\tilde{\alpha }+\tilde{\beta }=1/2$ and $\tilde{\beta }+\tilde{\gamma }=-1$ for any $R$.

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• Classical statistical mechanics
• Confinement
• Equations of state
• Colloids
• Exact solutions for many-body systems