Ergodic properties of heterogeneous diffusion processes in a potential well

Abstract

Heterogeneous diffusion processes can be well described by an overdamped Langevin equation with space-dependent diffusivity D(x). We investigate the ergodic and non-ergodic behavior of these processes in an arbitrary potential well U(x) in terms of the observable-occupation time. Since our main concern is the large-x behavior for long times, the diffusivity and potential are, respectively, assumed as the power-law forms D(x) = D 0 |x| α and U(x) = U 0 |x| β for simplicity. Based on the competition roles played by D(x) and U(x), three different cases, β > α, β = α, and β < α, are discussed. The system is ergodic for the first case β > α, where the time average agrees with the ensemble average, being both determined by the steady solution for long times. In contrast, the system is non-ergodic for β < α, where the relation between time average and ensemble average is uncovered by infinite-ergodic theory. For the middle case β = α, the ergodic property, depending on the prefactors D 0 and U 0 , becomes more delicate. The probability density distribution of the time averaged occupation time for three different cases are also evaluated from Monte Carlo simulations.

Time-averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

“…with > 0, was solved by Pattle 174 (see also some recent "reincarnations" 175,176 ). Contemporary models of diffusion with space-dependent diffusion coefficients 154,[177][178][179][180][181][182][183][184] -with HDPs being a specific example that assumes the functional diffusivity form (17)can be used to describe (•) the non-Brownian diffusion in crowded, porous, and heterogeneous media [185][186][187][188][189][190][191][192][193][194][195][196][197][198][199][200][201][202] (such as densely macromolecularly crowded cell cytoplasm), (•) the reduction of a critical "patch size" required for survival of a population in the case of heterogeneous diffusion of its individuals 181 , (•) diffusion in heterogeneous comb-like and fractal structures 182 , (•) escalated polymerization of RNA nucleotides by a spatially confined thermal (and diffusivity) gradient in thermophoresis setups 203 , (•) motion of active particles with space-dependent friction in potentials [both of power-law forms] 204 , and (•) transient subdiffusion in disordered space-inhomogeneous quantum walks 205,206 . We mention also a class of diffusion models with (•) particle-spreading scenarios with concentration-dependent power-law-like diffusivity (20) 175,207 , (•) concentration-dependent dispersion in the population dynamics, with a nonlinear dependence of mobility on particle density, D(ρ) ∼ ρ κ (yielding a migration from more-to less-populated areas) [208]…”

Extreme value statistics and arcsine laws for heterogeneous diffusion processes

“…Depending on the exponent α, this model displays super-diffusive (−1 < α < 0), diffusive (α = 0) or sub-diffusive (0 < α < ∞) scaling of the mean-squared displacement (MSD). Curiously, this simple Markov process also exhibits weak ergodicity breaking in the sense that the time-averaged and ensemble averaged MSDs are not equal even at large times [10][11][12][13][14][15][16].…”

“…Descriptions with space-dependent diffusion coefficient have also been used in modelling the diffusion in turbulent media [8] and on fractal objects [9]. Several studies on heterogeneous diffusive processes (HDPs) have revealed anomalous scaling of the mean squared displacement (MSD) and weak ergodicity breaking between time averaged and ensemble averaged MSDs [10][11][12][13][14][15][16]. Persistent properties of HDPs have also been investigated in [17,18].…”

Ergodic property of random diffusivity system with trapping events

“…For ergodic Brownian motion, its TAMSD converges to a constant for long time. However, the TAMSDs of many anomalous diffusion processes are random variables and present pronounced trajectory-to-trajectory variations, such as Lévy walk [64][65][66], Lévy flight [7,65,67], quenched models [68,69], heterogeneous diffusion processes [19,21,22,70] and so on. The stochasticity of TAMSD can be measured by the scatter of the dimensionless random variable η, which in our model is…”

“…The simple models presenting superdiffusion with β > 1 include Lévy flight with divergent second moment of jump length [6,7] and Lévy walk with heavy-tailed duration time of each running event [8][9][10][11]. There are still many anomalous diffusion processes, which present subdiffusion or superdiffusion depending on the specific value of system parameters, such as fractional Brownian motion [12][13][14][15], scaled Brownian motion [16][17][18], and heterogenous diffusion process [19][20][21][22]. In addition, a large number of exotic and hybrid processes have been invented in recent years, such as the diffusivity can be exponentially increasing and decreasing in time or logarithmically increasing [23], or depending on both the position and time [24], or a combined model of heterogenous diffusion process and fractional Brownian motion to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise [25].…”