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In the framework of statistical mechanics the properties of macroscopicsystems are deduced starting from the laws of their microscopic dynamics. Oneof the key assumptions in this procedure is the ergodic property, namely theequivalence between time averages and ensemble averages. This property can beproved only for a limited number of systems; however, as proved by Khinchin[1], weak forms of it hold even in systems that are not ergodic at themicroscopic scale, provided that extensive observables are considered. Here weshow in a pedagogical way the validity of the ergodic hypothesis, at apractical level, in the paradigmatic case of a chain of harmonic oscillators.By using analytical results and numerical computations, we provide evidencethat this non-chaotic integrable system shows ergodic behavior in the limit ofmany degrees of freedom. In particular, the Maxwell-Boltzmann distributionturns out to fairly describe the statistics of the single particle velocity. Astudy of the typical time-scales for relaxation is also provided.

Ergodic observables in non-ergodic systems: the example of the harmonic chain