Dimensional crossover of hard parallel cylinders confined on cylindrical surfaces

We derive, from the dimensional-crossover criterion, a fundamental-measure density functional for parallel hard curved rectangles moving on a cylindrical surface. We derive it from the density functional of circular arcs of length σ with centers of mass located on an external circumference of radius R(0). The latter functional in turn is obtained from the corresponding two-dimensional functional for a fluid of hard disks of radius R on a flat surface with centers of mass confined onto a circumference of radius R(0). Thus the curved length of closest approach between the two centers of mass of hard disks on this circumference is σ=2R(0)sin(-1)(R/R(0)), the length of the circular arcs. From the density functional of circular arcs, and by applying a dimensional expansion procedure to the spatial dimension orthogonal to the plane of the circumference, we finally obtain the density functional of curved rectangles of edge lengths σ and L. Along with the derivation, we show that, when the centers of mass of the disks are confined to the exterior circumference of a circle of radius R(0),(i) for R(0)>R, the exact Percus one-dimensional (1D) density functional of circular arcs of length 2R(0)sin(-1)(R/R(0)) is obtained, and (ii) for R(0)R, the obtained functional is equivalent to that of parallel hard rectangles on a flat surface of the same lengths, except that now the density profile of curved rectangles is a periodic function of the azimuthal angle, ρ(φ,z)=ρ(φ+2π,z). The phase behavior of a fluid of aligned curved rectangles is obtained by calculating the free-energy branches of smectic, columnar, and crystalline phases for different values of the ratio R(0)/R in the range 1