| Abstract: | We present, on a simple model of aone-dimensional crystal lattice, the consequences of theassumption that the phases in the action-anglerepresentation are random. We prove that this assumptionamounts to the introduction of a stochastic measurewhich can be interpreted as a Gaussian noise. Thepresence of noise gives rise to a new spectralrepresentation of states of the lattice. It is shownthat this new spectral representation of states can alsobe extended on an infinite lattice through a rigorouslydefined transition to the thermodynamic limit. Thetraditional spectral representation, as a superposition of independent modes, of such states as atomicdisplacements leads to meaningless expressions in thethermodynamic limit. One of the main results is thatunder the random phase assumption the interactions lead to the appearance of equilibrium states.We obtain an explicit spectral representation of suchstates. This specific model illustrates howprobabilistic behavior of an infinite system can bederived from classical laws of dynamics. |
