Ergodic characterization of nonergodic anomalous diffusion processes

Abstract

Anomalous diffusion in various complex systems abounds in nature and spans multiple space and time scales. Canonical characterization techniques that rely upon mean squared displacement break down for nonergodic processes, making it challenging to characterize anomalous diffusion from an individual time-series measurement. Nonergodicity reigns when the time-averaged mean square displacement differs from the ensemble-averaged mean squared displacement even in the limit of long measurement series. In these cases, the typical theoretical results for ensemble averages cannot be used to understand and interpret data acquired from time averages. The difficulty then lies in obtaining statistical descriptors of the measured diffusion process that are not nonergodic. We show that linear descriptors such as the standard deviation, coefficient of variation, and root mean square break ergodicity in proportion to nonergodicity in the diffusion process. In contrast, time series of descriptors addressing sequential structure and its potential nonlinearity: multifractality change in a time-independent way and fulfill the ergodic assumption, largely independent of the time series' nonergodicity. We show that these findings follow the multiplicative cascades underlying these diffusion processes. Adding fractal and multifractal descriptors to typical linear descriptors would improve the characterization of anomalous diffusion processes. Two particular points bear emphasis here. First, as an appropriate formalism for encoding the nonlinearity that might generate nonergodicity, multifractal modeling offers descriptors that can behave ergodically enough to meet the needs of linear modeling. Second, this capacity to describe nonergodic processes in ergodic terms offers the possibility that multifractal modeling could unify several disparate nonergodic diffusion processes into a common framework.