Disentangling Sources of Anomalous Diffusion

Thiel, Felix; Flegel, Franziska; Sokolov, Igor M.

doi:10.1103/physrevlett.111.010601

Cited by 40 publications

(32 citation statements)

References 18 publications

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“…Thus, in HDPs with 0 < p < 1 the distribution φ(ξ) of the relative amplitude δ 2 of individual realizations decays to zero at ξ = 0, in contrast to the CTRW's finite fraction of immobile particles reflected in the finite value φ(0) > 0. Indeed the scatter distribution φ(ξ) was previously advocated as a good diagnosis tool for different stochastic processes [29,63], complementing other stochastic analysis methods [67,69,70].…”

Section: Discussionmentioning

confidence: 99%

“…HDPs are physically meaningful alternatives to the standard stochastic models for anomalous diffusion processes, namely, the CTRW process with long-tailed waiting time distribution, FBM and fractional Langevin equation motion based on Gaussian yet long-ranged in time correlations, diffusion in fractal environments, or their combinations [14,15,20,69]. HPDs are weakly nonergodic, sharing some yet not all properties of CTRW processes.…”

Section: Discussionmentioning

confidence: 99%

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Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

2014

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We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) ∼ |x| α of the generalized diffusion coefficient encompassing sub-and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time averaged mean squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function P (x, t) we analyze the weakly non-ergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realization. As we show, the fluctuations between individual realizations increase with growing modulus |α| of the scaling exponent. The fluctuations appear to diverge when the critical value α = 2 is approached, while for even larger α the fluctuations decrease, again. At criticality, the power-law behavior of the mean squared displacement changes to an exponentially fast growth, and the fluctuations of the time averaged mean squared displacement do not seem to converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

2014

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“…Thus, infinite densities might become important tools in unraveling the origin of anomalous diffusion, a topic which attracted considerable theoretical attention, e.g., [67][68][69] and ref. therein.…”

Section: Graphical Examplesmentioning

confidence: 99%

Infinite densities for Lévy walks

et al. 2014

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Motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion. In many cases, such systems exhibit strong anomalous diffusion, where low order moments |x(t)| q with q below a critical value qc exhibit diffusive scaling while for q > qc a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent α and the diffusion coefficient Kα. We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

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“…it arises from the combination of different mechanisms. This entanglement of different mechanism poses a new challenge to the theorist and raises the question, how the process may be decomposed and understood [16]. …”

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confidence: 99%

Weak ergodicity breaking in an anomalous diffusion process of mixed origins

Thiel

Sokolov

2014

Phys. Rev. E

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The ergodicity breaking parameter is a measure for the heterogeneity among different trajectories of one ensemble. In this report this parameter is calculated for fractional Brownian motion with a random change of time scale, often called "subordination". We proceed to show that this quantity is the same as the known CTRW case.PACS numbers: 05.40.Fb,05.40.Fb "Weak ergodicity breaking" is a term which occurred in the spotlight a few years ago, see for instance [1][2][3][4][5][6] with respect to observables showing aging behavior. The term "aging" applies to the specific behavior pertinent to the values of an observable measured at two distinct instants of time, like the mean squared displacement (MSD) in the time interval between t 1 and t 2 , ∆X 2 (t 1 , t 2 ) as depending on the age t 1 of the system (assumed to be prepared at t = 0) at the beginning of observation. Aging here essentially refers to a non-stationarity of the observed process, in which ∆X 2 (t 1 , t 2 ) cannot be expressed via the time lag τ = t 2 − t 1 only and keeps a considerable dependence on t 1 as a stand-alone variable.Let us concentrate now on the ergodicity properties the squared displacement during the time interval τ in a measurement starting at t. In the case when the data acquisition in each single run of the process takes time from t = 0 to t = T , the time average of ∆X 2 (t, t + τ ) can be performed:The process is considered ergodic provided ∆X 2 (τ ) = ∆X 2 (t, t + τ ) . The discussion of the ergodicity implies that the corresponding limit does exist in some probabilistic sense (say, in probability) and is equal to the ensemble mean. Note that the time-average over the data acquisition interval removes the explicit t-dependence. For non-stationary processes the ensemble mean at any t depends on t, while the time-integration over the data acquisition interval removes this dependence. Therefore, even provided the integral above converges in a whatever sense, it cannot converge to all ∆X 2 (t, t + τ ) simultaneously, and our process is trivially non-ergodic.Whether the random process is stationary or not, one can ask how diverse are its different realizations, i.e. how different are two trajectories of the process with respect to some time-averaged observable O(τ ; t) (in our case the MSD, O(τ ; t) = ∆X 2 (t, t + τ )). To this purpose one considers a fixed-T approximation to eq.(1)At any finite T the value of O(τ ) T is a random variable. For both, stationary and non-stationary processes O(τ, t), one can consider a parameter describing the strength of fluctuations of O(τ ) T . As a measure of homogeneity or heterogeneity of different trajectories one can take the relative dispersion of the O(τ ) T in different realizations of the process:The parameter J (often called "ergodicity breaking parameter") shows how different are different trajectories of the process with respect to the observable O. For stationary processes, vanishing of J in the long time limit indeed implies ergodicity [7], and its non-vanishing witnesses against erg...

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Disentangling Sources of Anomalous Diffusion

Cited by 40 publications

References 18 publications

Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

Infinite densities for Lévy walks

Weak ergodicity breaking in an anomalous diffusion process of mixed origins

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