Non-Ergodic behaviour of a particle in long range interaction with all the particles of a heat bath

In this work, we investigate the ergodic properties of a test particle placed in long range interactions with all the particles of a given heat bath. The heat bath is constituted by N punctual Brownian particles that do not interact among themselves and their trajectories may be determined by Langevin equations with gaussian random noise. The non-ergodic behaviour and the issue about the thermalization of the test particle is analysed in the framework of the Weakly Non-Ergodic Statistical Physics (A. Rebenshtok\& E. Barkay, J. Stat. 133, 565 (2008) Phys. DOI 10.1007/s10955-008-9610-3).

Many of the methods developed in this present work are the natural development of two works: 1) Ergodicity and central-limit theorem in systems
with long-range interactions - EPL, 83 (2008) 30011 and 2) Truncated L'evy flights and weak ergodicity breaking in the Hamiltonian mean-field model - PHYSICAL REVIEW E 89, 022106 (2014).
We stress that our focus is to understand the transient regime of convergence toward a non-ergodic equilibrium or the sluggish convergence process toward the ergodic equilibrium.

Finaly we try to set up a model of CTRW (Continuous Time Random Walk), where the ocupation time is described by a truncated levy distribution,
in order to explain the ergodic properties associated to the test particle.