Local ergodicity in coupled harmonic vibrators: classical and quantal treatments

Abstract

Ensemble-time ergodicity is proven under some restrictive assumptions for a classical system, comprising interacting harmonic oscillators. An atom in a monatomic chain or lattice is shown to behave ergodically, in the sense that the time average behavior of a lattice point is identical to the ensemble average of the behavior of a lattice point at any long time (in large excess of the inverse vibrational frequencies). This equivalence (for ‘local observables’) differs from the Fermi–Pasta–Ulam result for mode energies (which are non-local). Then, the analogous quantal result is derived, with extensions to wider instances. Relationships to canonical typicality and to the eigenstate thermalization hypothesis are discussed and possibilities of experimental verifications of the results are indicated.