Ewald Sums for One Dimension

TL;DR

This paper derives analytic Ewald sums for one-dimensional periodic systems, introduces tools for system evolution analysis, and compares boundary condition approaches, with implications for cosmological simulations.

Contribution

It provides the first analytic solutions for 1D Ewald sums and develops efficient algorithms for simulating such systems with periodic boundaries.

Findings

Two boundary condition approaches yield similar clustering until few clusters remain.

The influence of boundary size becomes significant as clusters diminish.

Modern physics formulations require well-defined potentials, which are clarified by this work.

Abstract

We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words Ewald sums for one dimension. We also provide a set of tools for exploring the system evolution and show that it's possible to construct an efficient algorithm for carrying out simulations. In the cosmological setting we show that two approaches for satisfying periodic boundary conditions, one overly specified and the other completely general, provide a nearly identical clustering evolution until the number of clusters becomes small, at which time the influence of any size-dependent boundary cannot be ignored. Finally we compare the results with other recent work with the hope of providing clarification over differences these issues have induced. We explain that modern formulations of physics require a well defined potential which is not available if the forces are screened directly.