Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

Abstract

Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

“…Here ∆ is the lag time and T is the measurement time (length) of the trajectory x(t) [7,27,28]. Often, also the additional average…”

“…In such studies one is mainly interested in the quantitative behavior of the particle MSD (1) as well as the ergodic properties of the system: is the information from time averages of physical observables typically garnered as time series by modern particle tracking assays equivalent to those of the corresponding ensemble averages known from the theoretical models? It turns out that a large variety of anomalous diffusion processes involve weak ergodicity breaking [7,[25][26][27][28][29], the disparity between (long) time averages and ensemble averages of physical observables such as the MSD, and that in those cases the Khintchine theorem needs to be substituted by generalized versions [30,31]. Here, we study the dynamics and the ergodic properties of heterogeneous diffusion processes (HDPs) with position dependent diffusivity D(x), in the presence of piece-wise deterministic quenched and annealed disorder.…”

Fluctuation analysis of time-averaged mean-square displacement for the Langevin equation with time-dependent and fluctuating diffusivity

“…In some literature, the squared RSD is utilized as the ergodicity breaking parameter [14,[18][19][20].) The RF and RSD analyses for the TAMSD are useful if the systems are non-ergodic.…”

Langevin formulation of a subdiffusive continuous-time random walk in physical time

“…The here-defined noise enables us to define a class of CTRW-like processes with forces acting for all times, which are different from the corresponding standard CTRWs. Furthermore, we revisit the behavior of the SBM under confinement and show that its MSD correctly converges to a plateau as it is typical of confined motion [37], provided that we use more general time changes with truncated power-law tails. This suggest that the anomaly observed in [30] is mainly due to the localizing effect of the external linear force, which is able to trap the particle in the zero position if we allow for infinitely long waiting times between the jumps to eventually occur in the long time limit.…”