  We give through pseudodifferential operator calculus a proof that the quantum
dynamics of a class of infinite harmonic crystals becomes ergodic and mixing
with respect to the quantum Gibbs measure if the classical infinite dynamics is
respectively ergodic and mixing with respect to the classical infinite Gibbs
measure. The classical ergodicity and mixing properties are recovered as
$\hbar\to 0$, and the infinitely many particles limits of the quantum Gibbs
averages are proved to be the averages over a classical infinite Gibbs measure
of the symbols generating the quantum observables under Weyl quantization.
