Equilibrium solution for cold dynamical systems and self-similarity. (II)

TL;DR

This paper explores how cold initial conditions in dynamical systems evolve towards self-similarity near equilibrium due to the emergence of a new symmetry, with implications for various symmetric and energy-dependent systems.

Contribution

It demonstrates that the combination of initial cold symmetry and equilibrium symmetry leads to self-similarity, and shows how small perturbations can select specific self-similar solutions.

Findings

Cold initial conditions evolve towards self-similarity near equilibrium.

Emergence of a new symmetry at equilibrium facilitates self-similarity.

Small perturbations can break degeneracy and select specific solutions.

Abstract

Numerical simulations demonstrate a link between dynamically cold initial solutions and self-similarity. However the nature of this link is not fully understood. Cold initial conditions alone without further symmetry do not lead to self-similarity. Here we show that when the system approaches equilibrium a new symmetry appears. The combination of this equilibrium symmetry with the cold symmetry in the initial conditions leads to full self-similarity. As a consequence for any initially cold system even if the initial spatial distribution is not self-similar we will observe an evolution towards self-similarity near equilibrium. The case of one dimensional systems or spherically symmetric systems in 3D are discussed in detail. Systems depending on the energy and other integrals are also considered. The problem of the degeneracy of the self-similar solutions at equilibrium is tackled. It is shown that very small perturbations at the center of the system have the ability to break this degeneracy and lead to the convergence towards a specific auto-similar solution.