Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

Abstract

We demonstrate the non-ergodicity of a simple Markovian stochastic process with space-dependent diffusion coefficient D(x). For power-law forms D(x) |x| α , this process yields anomalous diffusion of the form x 2 (t) t 2/(2−α) . Interestingly, in both the sub-and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time-averaged mean-squared displacement δ 2 ( ) remains linear in the lag time and thus differs from the corresponding ensemble average x 2 (t) . We analyse the non-ergodic behaviour of this process in terms of the time-averaged mean-squared displacement δ 2 and its random features, i.e. the statistical distribution of δ 2 and the ergodicity breaking parameters. The heterogeneous diffusion model represents an alternative approach to non-ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

“…1a,b. HDPs are weakly non-ergodic and ageing, that is, their dynamics depends explicitly on the time gap between original initiation of the system and start of the measurement [32][33][34][35][36]. We note that the ageing properties of HDPs [35] embodied in the ensemble and time averaged MSDs are in fact similar to those of subdiffusive continuous time random walks [37] and scaled Brownian motion [38].…”

“…For reference in what follows we also mention that the probability density function of HDPs obeys has the exponential form [33] …”

“…(2) is [D 0 ] = cm 2−α0 sec −1 . The exponent (3) designates subdiffusion for α 0 < 0 and superdiffusion for 0 < α 0 [32][33][34][35][36]. The profiles of the diffusivity for these cases are shown in Fig.…”

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.…”

Spatially dependent diffusion coefficient as a model for pH sensitive microgel particles in microchannels

“…The position dependence of diffusivity can also significantly influence properties of the dispersion, such as viscosity and the mixing facility of multicomponent systems. In fact, inhomogeneity can lead to fundamentally new behavior such as anomalous diffusion (super-or subdiffusion), nonergodic effects, 13,14 and the occurrence of multiplicative noise. 15 Therefore, when describing the system dynamics and explaining related experimental results, it is important to consider any diffusion inhomogeneity effects and their potential interplay with any imposed flow profiles, as this will affect the spatial distribution of the particles along the channel with time and hence the flux of particles.…”