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

摘要

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. Copyright (C) EPLA, 2008.