Classical dynamics of the quantum harmonic chain

The origin of classical predictability is investigated for the one dimensional harmonic chain considered as a closed quantum mechanical system. By comparing the properties of a family of coarse-grained descriptions of the chain, we conclude that local coarse grainings in this family are more useful for prediction than nonlocal ones. A quantum mechanical system exhibits classical behavior when the probability is high for histories having the correlations in time implied by classical deterministic laws. But approximate classical determinism holds only for certain coarse grainings and then only if the initial state of the system is suitably restricted. Coarse grainings by the values of the hydrodynamic variables (integrals over suitable volumes of densities of approximately conserved quantities) define the histories usually used in classical physics. But what distinguishes this coarse graining from others? This paper approaches this question by analyzing a family of coarse grainings for the linear harmonic chain. At one extreme in the family the chain is divided into local groups of N atoms. At the other extreme the N atoms are distributed nonlocally over the whole chain. Each coarse graining follows the average (center of mass) positions of the groups and ignores the ``internal'' coordinates within each group, these constituting a different environment for each coarse graining. For an initial condition where long wavelength modes are excited and short wavelength modes are distributed thermally we find that the coarse-grained positions obey deterministic equations of motion accompanied by noise. The noise is greater the more nonlocal the coarse graining. Further, the deterministic equations require more time steps to evolve over a given time interval for the nonlocal coarse grainings than for the local ones. A continuum limit is possible only for the near local coarse grainings. For parameters of the model characteristic of realistic situations these features strongly favor the local coarse grainings over the nonlocal ones for prediction. Each of these differences can be traced to the approximate conservation of the local center of mass momentum. We then consider the chain quantum mechanically and show that, for realistic parameters, all the coarse grainings decohere rapidly compared to dynamical time scales. We conclude that noise, decoherence, and computational complexity favor locality over nonlocality for deterministic predictability.