Trajectory analysis of anomalous dynamics in optical lattice

We apply the trajectory formulation to analyze the anomalous dynamics of cold atoms in an optical lattice. The phase space probability density function of cold atoms, their dynamics, and the mechanism of dynamic evolution from an initial Gaussian distribution to a power-law distribution are analyzed. The results of the trajectory formulation are in good agreement with the previously reported experimental results for the exponent of position variance for a long time and the position-momentum correlation. The self-similar natures of trajectories in phase space are found for Lévy distributions. Our results unify the raw moments that can be expressed as the summation of a number of independent, identically distributed variables and the anomalous dynamics, which holds promise for an intuitive interpretation anomalous behavior and their kinetic mechanisms from initial Gaussian to anomalous distributions for a long time.