Equilibrium and dynamical properties of two dimensional self-gravitating systems

TL;DR

This paper investigates the phase transition, dynamical behavior, and chaos in a 2D self-gravitating particle system, revealing a first-order transition, anomalous diffusion, and size-dependent chaos properties.

Contribution

It provides a comprehensive analysis of the equilibrium phases, dynamical regimes, and chaos characteristics of 2D self-gravitating systems, including finite-size effects and their impact on diffusion and Lyapunov exponents.

Findings

Identifies a first-order phase transition at critical energy U_c.

Reveals anomalous diffusion below U_c and ballistic motion above U_c.

Shows Lyapunov exponent decreases as N^{-1/3} for U > U_c.

Abstract

A system of N classical particles in a 2D periodic cell interacting via long-range attractive potential is studied. For low energy density $U$ a collapsed phase is identified, while in the high energy limit the particles are homogeneously distributed. A phase transition from the collapsed to the homogeneous state occurs at critical energy U_c. A theoretical analysis within the canonical ensemble identifies such a transition as first order. But microcanonical simulations reveal a negative specific heat regime near $U_c$. The dynamical behaviour of the system is affected by this transition : below U_c anomalous diffusion is observed, while for U > U_c the motion of the particles is almost ballistic. In the collapsed phase, finite $N$-effects act like a noise source of variance O(1/N), that restores normal diffusion on a time scale diverging with N. As a consequence, the asymptotic diffusion coefficient will also diverge algebraically with N and superdiffusion will be observable at any time in the limit N \to \infty. A Lyapunov analysis reveals that for U > U_c the maximal exponent \lambda decreases proportionally to N^{-1/3} and vanishes in the mean-field limit. For sufficiently small energy, in spite of a clear non ergodicity of the system, a common scaling law \lambda \propto U^{1/2} is observed for any initial conditions.