Ewald sums for one dimension

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 is 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 to provide clarification over differences these issues have induced.

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

“…The algorithms employ analytic expressions for the time dependencies of the relative separations Z j (t) and relative velocities W j (t) between two consecutive particles in the primitive cell, where Z j = (x j+1 − x j ) and W j = (v j+1 − v j ), with x j and v j representing, respectively, the position and velocity of the j-th particle, whereas x j+1 and v j+1 representing those of the (j + 1)-th particle. Combining the results of [12,21], we find that…”

“…Simulations are performed by rescaling the system parameters using dimensionless units such that the number density N/2L = 1 and the characteristic frequencies ω, Λ equal unity [12,21]. Energies H c and H g (respectively for Coulombic and gravitational systems) are measured with respect to the minimum values of the corresponding potential energies allowed for each N.…”

“…where κ = 2πkq 2 for the case of the Coulombic system [12] (with each sheet, henceforth referred to as a particle or a body, having a surface charge density q in addition to the surface mass density m) and κ = −2πGm 2 for the gravitational system [21].…”

“…Positions and velocities of the particles are obtained using event-driven algorithms based on the approaches proposed in [12] for the Coulombic system and in [21] for the gravitational system. The algorithms employ analytic expressions for the time dependencies of the relative separations Z j (t) and relative velocities W j (t) between two consecutive particles in the primitive cell, where Z j = (x j+1 − x j ) and W j = (v j+1 − v j ), with x j and v j representing, respectively, the position and velocity of the j-th particle, whereas x j+1 and v j+1 representing those of the (j + 1)-th particle.…”

“…In the analysis of large systems considered in plasma and gravitational physics, periodic boundary conditions are preferred [15][16][17][18] and have been utilized in the study of one-dimensional Coulombic and gravitational systems [12,[19][20][21]. Such studies often rely on numerical simulations to validate the predictions made by theory.…”