Relaxation to a Perpetually Pulsating Equilibrium

TL;DR

This paper studies a unique N-body system with inverse cubic repulsion and harmonic attraction that pulsates indefinitely, yet its degrees of freedom relax to a pulsating Maxwellian distribution, confirmed by numerical simulations.

Contribution

It introduces a new pulsating equilibrium concept for a specific N-body system and demonstrates its relaxation properties through analytical and numerical methods.

Findings

System exhibits perpetual pulsation with a constant reduced Hamiltonian.

Degrees of freedom relax to a pulsating Maxwellian distribution.

Numerical simulations confirm rapid relaxation to this pulsating equilibrium.

Abstract

Paper in honour of Freeman Dyson on the occasion of his 80th birthday. Normal N-body systems relax to equilibrium distributions in which classical kinetic energy components are 1/2 kT, but, when inter-particle forces are an inverse cubic repulsion together with a linear (simple harmonic) attraction, the system pulsates for ever. In spite of this pulsation in scale, r(t), other degrees of freedom relax to an ever-changing Maxwellian distribution. With a new time, tau, defined so that r^2d/dt =d/d tau it is shown that the remaining degrees of freedom evolve with an unchanging reduced Hamiltonian. The distribution predicted by equilibrium statistical mechanics applied to the reduced Hamiltonian is an ever-pulsating Maxwellian in which the temperature pulsates like r^-2. Numerical simulation with 1000 particles demonstrate a rapid relaxation to this pulsating equilibrium.