On Exact Summations in Long-Range Interactions

In this work, simple exact results are presented for summations in two-particle potential with long-range interactions. Polygamma function is used to evaluate summations. Results are found when a periodic media is consider. Periodic boundary conditions are applied by symmetric repetitions of a central cell. Contributions over all space are used in the problem. The potential depends strongly on the size of the system; however, forces are convergents anywhere. Introduction The thermodynamical extensivity imposes short-range interactions in classical systems. Standard theoretical and experimental behaviors have been discussed in details since several decades [1, 2]. However, anomalous behavior is obtained when the potential attractive tail behaves very slow. In recent years, much attention has been paid to physical systems with microscopic long-range interactions (i.e., see [3, 4] and references therein). The most frequently applied way to discuss this type of systems is the Ewald method[5], where neutralizing counter charges are introduced to ensure convergence of the energy between one particle and the infinite replications of other particle. A more recent approach is the Lekner method [6, 7], 1 where the symmetry of the lattice of identical computational cells is used to guarantee the convergence of the relevant interaction force. Here, no neutralizing charges are considered and therefore, we adopt a Lekner-like procedure to evaluate the resulting interactions. We get in all cases (short and longrange interactions) convergent thermodynamical quantities. We discuss the most appropriate size of the system to evaluate such quantity. We have a central computational cell in one dimension with normalized size L = 1, this is −1/2 < z < 1/2 where z is the variable of the position. This work is an analytical extension of a computational one introduced in [8]. Systems in one dimension are very important due to effects about ferromagnetism in one-dimensional monoatomic metal chains which has been reported recently[9]. Periodic boundary conditions are computed by repetition of the central cell to infinity. Particles interact with a potential given by v(z) = −J0 ∞