Nonergodicity of $d$-dimensional generalized L\'evy walks and their relation to other space-time coupled models

TL;DR

This paper analyzes the nonergodic behavior of generalized Lévy walks, revealing divergence in mean-squared displacements, contrasting ensemble and individual trajectory diffusion, and introducing a broader model to connect various space-time coupled processes.

Contribution

The study provides detailed analytical insights into the nonergodicity of generalized Lévy walks and introduces a variable speed model linking multiple space-time coupled anomalous diffusion models.

Findings

Mean-squared displacements can diverge for certain parameters.

Individual trajectories can show superdiffusion while ensemble spreads subdiffusively.

Time-averaged squared displacement fluctuations can diverge, indicating infinitely strong ergodicity breaking.

Abstract

We investigate the nonergodicity of the generalized L\'evy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent Letter [Phys. Rev. Lett. 120, 104501 (2018)], give profound interpretations, and especially emphasize three surprising results: First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. The latter phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our new findings into existing results in the literature, we define a more general model which we call variable speed generalized L\'evy walk and which includes well known models from the literature as special cases such as the space-time coupled L\'evy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized L\'evy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.