Thermodynamic properties of quasi-one-dimensional fluids

We calculate  thermodynamic  and structural quantities of  a fluid of  hard spheres of diameter  $\sigma$ in a quasi-one-dimensional pore with accessible pore width $W $ smaller than $\sigma$ by applying  a perturbative method worked out earlier for a confined fluid  in a slit pore [Phys. Rev. Lett. \textbf{109}, 240601 (2012)]. In a first step, we prove that the  thermodynamic and a certain class of structural quantities of the hard-sphere fluid in the pore can be obtained from  a  purely one-dimensional fluid of rods of length $ \sigma$ with a central hard core of size  $\sigma_W =\sqrt{\sigma^2 - W^2}$ and a soft part  at both ends of length $(\sigma-\sigma_W)/2$. These rods  interact via effective $k$-body potentials $v^{(k)}_\text{eff}$ ($k \geq 2$) . The two- and the three-body potential will be calculated explicitly.  In a second step, the free energy of this effective one-dimensional  fluid  is calculated up to leading order in $ (W/\sigma)^2$. Explicit results for, e.g. the perpendicular pressure,   surface tension, and the density profile as a function of density, temperature,  and pore width are presented presented and partly compared with results from Monte-Carlo simulations and standard virial expansions.  Despite the perturbative character of our approach it encompasses the singularity of the thermodynamic quantities at the jamming transition point.