Dynamics and statistics of simple models with infinite-range attractive interaction

TL;DR

This paper reviews 1D and 2D simple N-body models with infinite-range attractive interactions, analyzing phase transitions, negative specific heat, anomalous diffusion, and Lyapunov instability, revealing different dynamical mechanisms in each dimension.

Contribution

It provides an exact analytical and numerical study of phase transitions and dynamical properties in simple models with infinite-range attraction, highlighting differences between 1D and 2D cases.

Findings

Both models exhibit phase transitions: second order in 1D, first order in 2D.

Negative specific heat appears near the phase transition in both models.

Superdiffusion arises from particle trapping and channelling mechanisms in the models.

Abstract

In this paper we review a series of results obtained for 1D and 2D simple N-body dynamical models with infinite-range attractive interactions and without short distance singularities. The free energy of both models can be exactly obtained in the canonical ensemble, while microcanonical results can be derived from numerical simulations. Both models show a phase transition from a low energy clustered phase to a high energy gaseous state, in analogy with the models introduced in the early 70's by Thirring and Hertel. The phase transition is second order for the 1D model, first order for the 2D model. Negative specific heat appears in both models near the phase transition point. For both models, in the presence of a negative specific heat, a cluster of collapsed particles coexists with a halo of higher energy particles which perform long correlated flights, which lead to anomalous diffusion. The dynamical origin of the "superdiffusion" is different in the two models, being related to particle trapping and untrapping in the cluster in 1D, while in 2D the channelling of particles in an egg-crate effective potential is responsible of the effect. Both models are Lyapunov unstable and the maximal Lyapunov exponent $\lambda$ has a peak just in the region preceeding the phase transition. Moreover, in the low energy limit $\lambda$ increases proportionally to the square root of the internal energy, while in the high energy region it vanishes as $N^{-1/3}$.