Volume 83, Number 3, August 2008
Article Number 30011
Number of page(s) 6
Section General
Published online 25 July 2008
EPL, 83 (2008) 30011
DOI: 10.1209/0295-5075/83/30011

Ergodicity and central-limit theorem in systems with long-range interactions

A. Figueiredo, T. M. Rocha Filho and M. A. Amato

Instituto de Física, Universidade de Brasília - CP 04455, 70919-970, Brasília, Brazil

received 18 March 2008; accepted in final form 20 June 2008; published August 2008
published online 25 July 2008

In this letter we discuss the validity of the ergodicity hypothesis in theories of violent relaxation in long-range interacting systems. We base our reasoning on the Hamiltonian mean-field model and show that the lifetime of quasi-stationary states resulting from the violent relaxation does not allow the system to reach a complete mixed state. We also discuss the applicability of a generalization of the central-limit theorem. In this context, we show that no attractor exists in distribution space for the sum of velocities of a particle other than the Gaussian distribution. The long-range nature of the interaction leads in fact to a new instance of sluggish convergence to a Gaussian distribution.

02.50.-r - Probability theory, stochastic processes, and statistics.
05.20.Dd - Kinetic theory.
05.90.+m - Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems.

© EPLA 2008