Equilibrium solution for cold dynamical systems and self-similarity

TL;DR

This paper analytically investigates how cold initial conditions in dynamical systems evolve towards self-similarity, revealing the mechanisms and conditions that facilitate this convergence near equilibrium.

Contribution

It establishes a theoretical link between cold initial states and self-similarity, extending the analysis with perturbative methods and practical induction mechanisms.

Findings

Cold solutions can evolve into self-similar states after several dynamical times.

Perturbative analysis shows the power-law potential tends to strengthen and propagate.

A broad range of initial conditions are compatible with self-similar solutions.

Abstract

Numerical simulations demonstrate a link between dynamically cold initial solutions and an evolution towards self-similarity. However the nature of this link is not fully understood. In this work the link between cold initial conditions and self-similarity near equilibrium is established. The evolution towards self-similarity is analyzed using an analytical solution in a power-law potential. The analytical solution indicates a convergence towards self-similarity after a number of dynamical times even if the inital conditions are far from self-similarity. The power-law model is extended by using perturbative analysis. The perturbative analysis shows that once the power-law potential is initiated it tends to become stronger and propagate. This behavior demonstrates the mechanism behind the convergence towards auto-similarity. The cold solutions are compatible with a broad range of self-similar solutions. As a consequence some seed of a specific self similarity class must appear to induce a convergence mechanism. In practice some local induction of a power-law potential is necessary and some examples of such inductive mechanisms are given.