Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions

Details Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions

2008-05-12

arxiv.org/abs/0805.1568v1

Physics

Condensed Matter

Statistical Mechanics

13 pages,6 figures

Scientific paper

10.1209/0295-5075/83/30011

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 life-time 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.

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