Atomic vibrations in the thermodynamic limit

We present, on a simple model of a one-dimensional crystal lattice, the con sequences of the assumption that the phases in the action-angle representat ion are random. We prove that this assumption amounts to the introduction o f a stochastic measure which can be interpreted as a Gaussian noise. The pr esence of noise gives rise to a new spectral representation of states of th e lattice. It is shown that this new spectral representation of states can also be extended on an infinite lattice through a rigorously defined transi tion to the thermodynamic limit. The traditional spectral representation, a s a superposition of independent modes, of such states as atomic displaceme nts leads to meaningless expressions in the thermodynamic limit. One of the main results is that under the random phase assumption the interactions le ad to the appearance of equilibrium states. We obtain an explicit spectral representation of such states. This specific model illustrates how probabil istic behavior of an infinite system can be derived from classical laws of dynamics.