Statistical Dynamical Behaviors of a One-Dimensional Lattice with an Isotopic Impurity

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Abstract
Equations of motion of a harmonically coupled one-dimensional lattice are exactly solved in the case that a lattice contains an isotopic impurity particle of an arbitrary mass. Statistical dynamical properties of the system are also investigated by introducing an initially canonical ensemble. The time series of displacement and velocity of a particle in the lattice constitute the stationary Gaussian processes, whose statistical properties, i. e. the ergodicity, non-Markovian properties and the approach to the equilibrium distribution are discussed. It is shown that the approach to the equilibrium (Maxwellian distribution) fails when the system has a localized vibrational mode.

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