Ergodicity and central-limit theorem in systems with long-range interactionsA. 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.
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