Hard particles in narrow pores. Transfer-matrix solution and the periodic narrow box

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Abstract
We derive an exact transfer‐matrix solution for an infinite system of hard particles confined in a manner that precludes non‐nearest‐neighbor interactions. The solution takes the form of a functional eigenvalue equation which may be solved numerically for the thermodynamic and structural properties of the confined fluid. Barker [Aust. J. Phys. 15, 127 (1962)] originally derived this solution by a different route, and we apply it in a number of new ways. We present the first calculations based on this solution for systems of hard disks between parallel lines, and for hard spheres in a cylindrical pore. Through comparison with Monte Carlo simulations, we examine the range of validity of the solution when applied to systems in which non‐nearest‐neighbor interactions may occur. We find that the transfer‐matrix approach provides acceptable results for pore widths up to two particle diameters, and that the approximation becomes quite poor as the pore is widened further, particularly at high density. This solution may be used to test the narrow‐pore limit of more versatile theories of confined fluids. We further apply the solution to the so‐called ‘‘periodic narrow box,’’ which has previously been solved in only two dimensions. We reproduce the two‐dimensional result, and present the solution (which may be formulated explicitly rather than as an eigenvalue equation) for the three‐dimensional, hard sphere version of the model. This simple model provides a remarkably accurate description of the thermodynamic behavior of a bulk hard sphere crystal.

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