Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

Abstract

How does a systematic time-dependence of the diffusion coefficient D(t) affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we answer this question via studying the characteristics of a set of standard statistical quantifiers relevant to single-particle-tracking (SPT) experiments. We examine, for instance, how the behavior of the ensemble- and time-averaged mean-squared displacements—denoted as the standard MSD 〈x²(Δ)〉 and TAMSD quantifiers—of FBM featuring (where H is the Hurst exponent and Δ is the [lag] time) changes in the presence of a power-law deterministically varying diffusivity D_α(t) ∝ t^α−1—germane to the process of scaled Brownian motion (SBM)—determining the strength of fractional Gaussian noise. The resulting compound “scaled-fractional” Brownian motion or FBM–SBM is found to be nonergodic, with 〈x²(Δ)〉 ∝ Δ^α+2H−1 and . We also detect a stalling behavior of the MSDs for very subdiffusive SBM and FBM, when α + 2H − 1 < 0. The distribution of particle displacements for FBM–SBM remains Gaussian, as that for the parent processes of FBM and SBM, in the entire region of scaling exponents (0 < α < 2 and 0 < H < 1). The FBM–SBM process is aging in a manner similar to SBM. The velocity autocorrelation function (ACF) of particle increments of FBM–SBM exhibits a dip when the parent FBM process is subdiffusive. Both for sub- and superdiffusive FBM contributions to the FBM–SBM process, the SBM exponent affects the long-time decay exponent of the ACF. Applications of the FBM–SBM-amalgamated process to the analysis of SPT data are discussed. A comparative tabulated overview of recent experimental (mainly SPT) and computational datasets amenable for interpretation in terms of FBM-, SBM-, and FBM–SBM-like models of diffusion culminates the presentation. The statistical aspects of the dynamics of a wide range of biological systems is compared in the table, from nanosized beads in living cells, to chromosomal loci, to water diffusion in the brain, and, finally, to patterns of animal movements.

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I Introduction

The flurry of new single-particle-tracking (SPT) datasets reporting on and novel theoretical-analysis tools assessing the properties of anomalous diffusion^1–27 has established an unprecedented need for novel theoretical models of diffusion and transport. Such models should desirably embody certain characteristic features of different “standard” anomalous-diffusion processes^28–33 such as, i.e., conventional Brownian motion (BM),^34–41 fractional BM (FBM)^42–48 governed by fractional Gaussian noise, scaled BM (SBM)^49–61 and ultraslow SBM^62,63 with a power-law time-dependent diffusivity D(t) ∝ t^α−1, continuous-time random walks (CTRWs),^31,64,65 Lévy walks and flights,⁶⁶ heterogeneous diffusion processes (HDPs)^{52,53,67–69} with a power-law position-dependent diffusivity, D(x) ∝ |x|^{[small alpha, Greek, macron]}, etc.

In recent years, certain combinations of models were proposed, including switching-diffusivity^{11,13,70–77} and annealed-transient-time models (ATTM),⁷⁸ BM with fluctuating or diffusing diffusivity (DD),^70,79–85 BM- and anomalous-diffusion-models with “superstatistically” distributed model parameters,^86–89 compound diffusion processes of SBM-DD,⁸⁴ SBM–HDPs,^90,91 FBM-DD,⁹² FBM–HDPs,^19,93 SBM with exponentially and logarithmically varying D(t),⁹⁴ CTRWs with random walks on fractal (RWFs),⁹⁵ CTRW–FBM,^16,96–98 as well as several other models,^74,99–113 including fractional-Langevin-equation (FLE) motion. Renewal processes involving alternation of different types of motions were proposed as well [with, e.g., switching between different diffusion processes,^{20,100,114,115} processes with intermittent mobility states,¹³ or with different anomalous scaling exponents^{12,103,116–119}]. For some recent examples, we refer the reader to, e.g., the hybrid models of alternating Lévy walks with BMs^120,121 and Lévy walks with CTRWs.¹²² Multifractional FBM-based processes with a Hurst exponent varying in time^49,123,124 were also considered recently.^125–127 Such processes were used, e.g., in mathematical-finance models with (multi-)fractional stochastic volatility.^128,129

This multifaceted picture of diffusion scenarios potentially realizable for SPT datasets is, however, far from being complete, and we develop here one more useful model of FBM–SBM. The main objective is to examine how the temporal correlations of fractional Gaussian noise—featuring persistent displacements for superdiffusive and antipersistent displacements of a particle for subdiffusive Hurst exponents H of FBM—“interfere” with a power-law deterministic D(t)-dependence of SBM.

The rest of the paper is organized as follows. In Section II we introduce the concepts of ensemble- and time-averaging and discuss ergodicity¹³⁰ of FBM and SBM. Some details of simulations are given in Section III A. The main results of the theoretical analysis and of computer simulations for the FBM–SBM model are presented in Section III. Specifically, the properties of the ensemble-averaged mean-squared displacement (MSD) and of the time-averaged MSD (TAMSD), the aging characteristics of the TAMSD, the probability-density function (PDF), and increment/velocity autocorrelation function (ACF) are considered in Section III B, III C, III D, and III E, respectively. The discussion and conclusions are presented in Section IV. The list of applications of FBM and SBM is overviewed in Section IV A. The main features of experimental SPT datasets and FBM-, SBM-, and FBM–SBM-related theoretical models are summarized in Table 1, while several additional figures are listed in Appendix A.

Table 1 Comparative overview of several recent (last decade) experimental SPT and simulations-based datasets, with their typical MSD and TAMSD scaling relations (at short and long time), the aging dependence of the TAMSD on the trajectory length, the observed scatter/dispersion of individual TAMSD realizations computed at short lag times, the degree of polydispersity of the tracer particles, the general form of the ACF, and the tentative mathematical model(s) of diffusion. These quantities are organized in the columns of the table. The entries are ordered in blocks [separated by delimiters] reporting the FBM-like experimental, SBM-like experimental, SBM-like simulational, and, finally, FBM–SBM-mixed studies as well as hybrid-model-based dynamics. A system with weak ergodicity breaking features MSD(Δ) ≠ TAMSD(Δ) at short lag times

Dataset	MSD, 〈x^2(Δ)〉	TAMSD,	Aging of	Scatter of	Polydispersity of tracers	Form of the ACF	Model of diffusion
a Dataset on nonergodic subdiffusion of chromosomal loci in live bacterial cells. b With distributed p(Kβ) and p(β) in the short-lag-time fit (4). c Diffusion of a fictitious zipping fork along a polymer chain undergoing a Rouse dynamics, as studied by computer simulations. d The decay of the negative part of the ACF for the subdiffusive dynamics, with the tail scaling ∝t^2H−2 = t^−1.12, observed in these simulations is consistent with the FBM-based exponents of the MSD growth as well as with the growth of the variance σ^2 of the detected PDF, namely σ^2(t) ∝ t^2H ∝ MSD(t). e The SPT-microscopy dataset on and the statistical analysis of diffusion of lipid granules in fission-yeast cells. f Telomere diffusion in mammalian U2OS cancer cells. g Ergodic model of polymer dynamics; with the observed irreproducibilities of the TAMSD diffusion coefficients (with ca. 20 times variation) attributable to inhomogeneities of the diffusion environment and to variabilities of local chromosome organization. h SPT dataset on the dynamics of telomeres in the nucleus of mammalian U2OS osteosarcoma cells. i A typical size of telomeres is 50 to 100 nm.^197 j See also ref. 198 for an independent confirmation of the FBM model for this dataset based on the statistics of tracer increments and the generalized p-variation test. k Statistical analysis and SPT in vivo measurements of diffusion of telomere in 3T3 mouse embryonic fibroblast cells. l After the correction terms due to the effects of measurement noise were taken into account. This has improved the initial assessment of the TAMSD growth with ∝Δ^0.24 at short and ∝Δ^0.65 at long lag times (in a heterogeneous population of tracers and noisy original data). m SPT dataset on subdiffusion of fluorescent beads (50 nm in diameter) in artificial crowded fluids, such as sucrose and dextran. n M. Weiss, personal communication, 2022. o Dataset on tracking of avian predators to study the statistics of their slow-dynamics area-restricted search patterns and of directed, nearly ballistic commuting paths. p M. Assaf, personal communication, 2022. q Note that fitting the MSD growth at short times (typical for the SPT-data analysis) would yield significantly smaller MSD and TAMSD scaling exponents of this mode. r Heterogeneities in the size distribution of tracked animals, in specific terrestrial Beschaffenheiten, as well as in individual behavioral traits can affect the degree of the TAMSD spread observed^25,200,201 in each mode of the spreading dynamics. s Note, however, the change in the detected MSD scaling behavior. t Molecular-dynamics-based large-scale computer simulations of diffusion of lipids and proteins in pure/noncrowded lipid membranes. u The MSD behavior was not reported in ref. 203 and 204, mainly employing sliding averaging (TAMSDs and mean TAMSDs). For the case of a homogeneous system in the liquid-disordered state the MSD-to-(mean TAMSD) equality in scaling and magnitude was confirmed by computer simulations at short-to-intermediate times (0.01…10 ns, where the results of ensemble averaging were still reliable) [J.-H. Jeon, personal communication, 2022]. For diffusion of cholesterol in lipid membranes as well as for the lipid dynamics in the gel-phase lipid systems—where the spread of the TAMSDs becomes significant, see Fig. S8 and S12 (ESI) in ref. 203—the MSD-to-TAMSD equivalence—being a prerequisite of the FBM- and FLE-based model descriptions—was not confirmed in simulations. v Although for diffusion in noncrowded membranes no aging was observed, under the crowded conditions some diffusing lipids did rarely display trapping events on their entire trajectories. This fact not only produced a significantly larger individual-TAMSD spread, but also resulted in aging of the TAMSDs and a slower than expected^31,44 EB ∼ 1/T decay of the ergodicity breaking parameter with the trace length T (see Fig. 8 in ref. 204). No aging was however observed after averaging over all diffusing particles [M. Javanainen, personal communication, 2022]. w Large-scale computer simulations of lateral diffusion of lipids in lipid membranes. x Purely ballistic initial diffusion with the exponent β = 2 was not observed at short times in simulations [M. Javanainen, personal communication, 2022]. y Large-scale computer simulations of lipid diffusion in membranes. z Two-dimensional SPT dataset of diffusion of insulin granules in live MIN6 cells. aa The granules are ca. 250…350 nm in diameter. ab FBM subordinated to CTRW [responsible for distributed waiting times of binding-unbinding events of the granules] is described in the supplement of ref. 96, see also ref. 95 for CTRW and RWF. ac SPT dataset of both ergodic and nonergodic anomalous-diffusion dynamics of Kv2.1 potassium channels in the plasma membrane. ad Examination of the SPT dataset on diffusion of chromosomal loci in the presence of particle-localization uncertainties. ae After both static and dynamical errors were taken into account, see Fig. S3 in ref. 142. af Examined both in terms of the minimum of and the overall form of the ACF, for H in close agreement with the inferred MSD exponents. ag Static SPT-localization errors^206 due to photon statistics and the dynamical errors due to finite exposure times [motion blur] were examined i.e. in ref. 142 in their effects onto the short-time MSD behavior and the ACF features for subdiffusive FBM-like motion. The minimum of the ACF becomes shallower at larger time shifts δ for static errors contributing to the data, corroborating a less subdiffusive MSD exponent at later times [and the up-shift of short-time MSD]. In the presence of dynamical localization errors, in contrast, the ACF minimum goes down for larger δ shifts, consistent with a more subdiffusive MSD exponent at later times [with the short-time MSD acquiring a downward shift] for FBM-like diffusion. For a confined diffusion, the same downward shift of the ACF negative peak at later time-lags is found, see ref. 207. Reexamination of the conclusions on the inferred anomalous exponents and the extent of the region of subdiffusion for a number already studied SPT datasets [i.e., some from this table] regarding the effects of localization errors—either unknown, considered not important, or neglected in the original studies—would thus be informative/compelling, especially for the datasets yielding low magnitudes of the short-time TAMSDs, time-varying TAMSD scaling exponent from a transiently subdiffusive to a more “normal” behavior, or featuring confined/adsorbed states with the internal dynamics of the tracers to be inferred. ah SPT experiments of QDs in the cytoplasm of living mammalian HeLa cells. ai With the precise form of the ACF, including the depth of its minimum due to subdiffusion, being in full correspondence with the theoretical FBM-based prediction for the exponent 2H as determined from the TAMSD-evolution data. aj Intermittent and heterogeneous FBM—with alternations between the two states characterized by different tracer mobilities—describes the data. Such dichotomous switching also gets reflected in the two states visible in the PDF (as, e.g., in ref. 163). ak SPT dataset on motor-driven highly superdiffusive active motion of endogenous intracellular particles in the amoeboid cells of Acanthamoeba castellanii. al SPT dataset of diffusion of transmembrane parts of the HIV-spike proteins in the plasma membrane of immune T-cells. am The mean over the TAMSD-aging exponents in each of 30 T-cells in the dataset is computed here, see ref. 27 for the detailed statistical analysis [complementary to that presented in ref. 1]. an With transient and permanent clusters of the traced proteins being possible. ao The computed ACF^1 is reminiscent of that of subdiffusive FBM. Note, however, that for slow- and medium-mobility subpopulations the depth of the initial dip [indicative of the degree of anticorrelations for FBM and RWF] does not change with the time shift, whereas for the fast subpopulation this minimum becomes deeper at later times (compare to the opposite trend observed in ref. 16). ap SPT dataset and the analysis of anomalous diffusion of network elements of the endoplasmic reticulum. aq With the depth of the ACF minimum increasing measurably as longer time-intervals δt are being used in the ACF analysis. ar With the distributed diffusivities p(Kβ) and skewed Rayleigh-like distribution of the TAMSD exponents, p(β). as SPT dataset and the statistical analysis of the nonergodic dynamics of membrane proteins on the somatic surface of hippocampal neurons. at Approximately, in magnitude and exponent [for the entire length of the recorded traces]. A nearly ergodic dynamics was observed after the removal of immobilized segments of the trajectories, whereas prior to that the strongly subdiffusive periods of motion yielded MSD ≫ TAMSD and also disparate anomalous scaling exponents.^103 au These mean scaling exponents were computed after removing strongly subdiffusive parts of the trajectories (bound tracers), based on the evaluated time-local TAMSD exponents. av With a transient formation of nanoclusters by the tracer molecules. aw FBM or RWF were proposed (but not confirmed) as the mathematical models of diffusion. Widely distributed and strongly positively correlated Kβ and β were experimentally detected [being computed for the full trajectories]. ax SPT dataset of nanosized-tracers diffusion in the cytoplasm of mammalian cells and the statistical analysis of particle-medium interactions and medium heterogeneities. ay See the detailed discussion in ref. 8 on implications of various preset conditions for a SPT dataset, specifically, for the effects of [distributed, minimal/maximal, etc.] trajectory lengths, the number of fitting points used for assessing the TAMSD exponents for individual SPT time series (see also ref. 142, 199 and 208–210), polydisperse tracer particles and heterogeneous environments (see also ref. 211), etc. az For NPs and QDs, thoroughly controlled; with a tracer-size dependent polydispersity. ba With a power-law distribution of the tracer-environment immobilization times, ψ(τ) ∼ 1/τ^γ+1, with large variabilities of the particle mobilities realized for 0 < γ < 1. bb Tracking of membraneless organelles (p-granules) in single-cell state of Caenorhabditis elegans embryos. bc For subdiffusive, nearly normal, and superdiffusive subpopulations of p-granules, respectively. bd With the characteristic ACF features expected for the respective values of FBM exponents of tracer subpopulations, see footnote bc. be Broadly distributed and strongly positively correlated trajectory-specific generalized TAMSD transport coefficients with the distribution p(Kβ) and TAMSD scaling exponents with the distribution p(β) were observed,^140 alike in ref. 8. bf SPT trajectories and statistical analysis of diffusion of endosomes in a heterogeneous media inside living eukaryotic cells, with the tracers displaying switching between persistent and anti-persistent modes of motion. bg In magnitude and exponent at intermediate times, whereas at short times a levelling off of the MSD and TAMSD curves is observed [see also footnote ag]. bh For slow subpopulation, all tracers, and fast tracers, respectively. bi But not studied in detail in ref. 19. The short-time values of the MSDs and TAMSDs for the trajectories with T > 2 s are larger than those for a subset of traces with T > 8 s. bj Dependent on the number of fitting points used to extract the generalized diffusivities and scaling exponents from the source data [N. Korabel, personal communication, 2022], see also the discussion in ref. 6 and 8. The transport coefficients and scaling exponents were found to be strongly positively correlated.^19 bk With a pronounced negative peak for a slow subpopulations of the tracers. bl The heterogeneity was observed in experiments and achieved in the accompanying computer simulations of a subdiffusive FBM in terms of particular distributions of trace lengths ϕ(T) ∝ T^−1.85 and diffusion coefficients D(T) ∝ T^−0.6 among the trajectories. These distributions can, e.g., yield the MSD function surprisingly decreasing with diffusion time, see Fig. 2a and 4 in ref. 19 [N. Korabel, personal communication, 2022]. How much bias such hard-to-control—but SPT-experimental-conditions-predisposed and thus inevitable (see also the discussion in ref. 8)—distributions of the trace lengths, particle mobilities, medium heterogeneities, etc. have on the results, conclusions, and interpretation of numerous existing/published SPT datasets regarding, first and foremost, the temporal duration and actual degree of anomalous diffusion remains to be understood via a thorough and comparative analysis of the effects of systematically varied “preparatory” conditions in and the data-analysis parameters of various SPT datasets. bm SPT dataset of diffusion of polystyrene colloids (0.25 and 0.75 micron in radius) in the polymerized solutions of G-actin polymers. bn SPT-data on and the statistical analysis of the two-dimensional spreading dynamics of shape-heterogeneous amoeboid cells of Dictyostelium discoideum. bo With distributed and negatively correlated Kβ and β values. bp Tracking and statistical analysis of self-propelled metastatic glioblastoma U87-MG brain-cancer cells invading a three-dimensional extracellular matrix. bq For gradient-containing and gradient-free migration assay, correspondingly; 2D projections of 3D migration patterns in Matrigel™ were recorded. br Heterogeneous distributions of the TAMSD trajectories were observed, with distributed trace-specific persistence times and propulsion speeds in terms of the underlying persistent random walk (PRW). With the cells moving radially and superdiffusively outward in the spheroid-induced gradient, the individual TAMSDs were fitted by the Fürth-Ornstein formula for PRWs,^212–214 to extract these individual diffusion parameters. bs As other statistical properties—such as TAMSD aging or ACF—were not studied, an interpretation of these data in terms of FBM-SBM model cannot be excluded too. bt The same system as for ref. 111. bu Variable MSD scaling exponents, depending on the concentrations of the extracellular matrix. bv A large spread of the TAMSDs for individual trajectories is detected, reflecting rather broad distributions of times of persistent motion and of migration speeds of the cells [L. Jauffred, personal communication, 2022]. bw A double-exponential decay of the observed ACF was proposed (rather than a single-exponent decay, as for standard PRWs^216), with the characteristic decay times varying with the penetrability of the diffusion matrix. Fitting the ACF with superdiffusive-FBM-like functional form might also be a plausible alternative. bx With different persistent times along the main and auxiliary direction of cell motion. Distinct subpopulations of the cancer cells—with the persistent time of about 15 minutes and a much more persistent subpopulation (superspreaders)—were detected, underlying a heterogeneity of the cells' proliferation properties. by SPT dataset reporting transient superdiffusion of polydisperse vacuoles in motile amoeboid cells of Acanthamoeba castellanii. bz The mean TAMSD is somewhat subdiffusive at short times (see footnotes ag and eb for possible reasons) and it turns superdiffusive at intermediate lag times, closely matching the MSD in these time domains. ca With broad distributions of p(Kβ) and p(β). The trajectory-specific diffusivities and scaling exponents are strongly positively correlated at the start of the TAMSDs of vacuoles and (surprisingly, see also footnote bl) reverting this behavior into pronounced anticorrelations of Kβ and β at the late stages of vacuole diffusion.^10 cb With a division of the time series into subpopulations corresponding to small, medium, and large vacuoles being performed in the analysis. cc With the FBM-commensurate behavior of the velocity-ACF in the region of short-time subdiffusion and intermediate-time superdiffusion of vacuoles. cd SPT dataset on diffusion of histone-like nucleoid-structuring H-NS proteins in living cells of Escherichia coli. ce With power-law distributed diffusivities of the trajectories, with p(Kβ) ∝ Kβ^−(1.94±0.07), see also footnote bl and ref. 89. The measured diffusivities and scaling exponents increase as the length of bacteria grows for older cells (being divided into 3 cell-length subpopulations in the analysis), being supported by the measurements of their more fluidized cytoplasm. cf With the measured distribution of the number of proteins per protein cluster being P(n) ∝ p^n, with p ≈ 0.95. cg With widely distributed tracer sizes and mobility parameters. ch SPT dataset and the statistical analysis of tracer diffusion in mucin hydrogels at varying pH. ci The physical age of samples was not precisely controlled and the trajectory-length-variations of the computed TAMSD magnitudes were not measured. cj Depending on the pH values, the ACFs are, respectively, BM-, subdiffusive-FBM-, and BM-like. ck FBM was shown to dominate the results of the Bayesian model-assessment analysis at all pH values used in the experiments,^8,163 with the distributed p(Kβ) and p(β) among the trajectories of the tracers. cl SPT dataset of nonergodic diffusion of receptor molecules on living-cell membranes. cm The diffusion model based on the overdamped Langevin equation with diffusion coefficients distributed along each trajectory according to a Gamma-distribution pD(D) ∼ D^σ−1e^−D/D0 and with the exponentially distributed transit times was used to rationalize the experimental data. cn All-atom supercomputer simulations and the statistical analysis of diffusion of doxorubicin drug molecules in silica nanoslits. co But with the dip of the ACF becoming shallower for longer time shifts δ, see Fig. 14 in ref. 16, as compared to the constant depth (10) for FBM. cp SPT dataset of diffusion of rhodamine molecules in water nanofilms on hydrated silica surfaces at varying humidity levels. cq Apart from a relatively small TAMSD decrease/aging present for very short trajectories, irrespective of the degree of subdiffusion. cr The heterogeneities of the diffusion environment in terms of the strength of the adsorption sites were however pronounced. cs The depth of the ACF negative peak decreases measurably at higher humidities (see Fig. S2 of ref. 218) corroborating the accompanying transition from subdiffusion to nearly normal diffusion observed for the MSD and TAMSD. ct Molecular-dynamics simulations of laterally diffusing M2-receptor proteins (at infinite dilution) in a hydrated mixed lipid membranes (with up to 50 percent of cholesterol). cu The velocity-ACF reflected the MSD behavior, with a region of negative correlations at ∼10…50 ps for diffusing proteins. The velocity-ACF- and the MSD-behaviors for lipids were also similar, with the respective regions of the dynamics shifted earlier by about one order of time. cv A model of this equation with specific memory kernels was used to described the crossovers in protein diffusion from the short-time ballistic, to the intermediate-time strongly subdiffusive, and, ultimately, to the long-time Brownian behavior of the MSD. cw Particle-collisions and events-based large-scale computer simulations of the dynamics in a free-cooling granular gas. cx Particles of the same size were used in simulations [A. Bodrova, personal communication, 2022]. cy With the MSD scaling exponent being α = 1/6 and time-dependence of the temperature being T(t) ∝ 1/t^2×5/6, as recently confirmed experimentally.^190 The MSD and TAMSD for the data^190 regarding the applicability of the SBM model were not analyzed, as the tracers were not tagged rendering a time-series-based analysis impossible [M. Sperl, personal communication, 2022]. cz Water diffusion and D(t) time-dependent diffusion coefficients in the brain tissues. da Possibly, SBM-like short-time diffusion is realized. In diffusion-magnetic-resonance imaging, the diffusion coefficient is measured and commonly reported as D(t) = 〈x^2(t)〉/(2t). If one defines the instantaneous diffusion coefficient as Dinst(t) = ∂〈x^2(t)〉/∂(2t), then at long times Dinst(t) = D∞ + Const × t^−ϑ, where ϑ = (p + d)/2 is the exponent of temporal correlations (here p is the structural exponent and d is the system dimensionality).^168 This SBM-like dependence of Dinst(t) − D∞ describes a correction to the asymptotically normal, Gaussian diffusion with the bulk diffusivity D∞ [D. Novikov, personal communication, 2022]. db Theory and computer simulations of the overdamped Langevin equation with the diffusivity D(t) ∼ 1/(1 + t/τ0). dc Expected to be SBM-like. dd Theory and simulations of the underdamped Langevin equation with the time-varying temperature T(t) ∼ (1 + t/τ0)^2α−2, the damping coefficient , and the diffusivity given by the time-dependent Einstein relation, D(t) ∼ T(t)/γ(t). de Expected to be small and SBM-like. df SBM for massive particles in the case α = 0. dg Stochastic computer simulations of both massless and massive particles with exponentially time-varying diffusivity, . dh For massless particles with , see eqn (A27) in ref. 94. di For massless particles with , see eqn (B22) in ref. 94. dj As in footnote dg, but for the diffusion coefficient varying logarithmically in time, Dlog(t) = D0 log[t/[small tau, Greek, tilde]0]. dk For massless particles, see eqn (C36) in ref. 94. dl Computer simulations and the theoretical analysis of SBM confined in a parabolic potential, U(x) = kx^2/2. dm Theory and computer simulations of renewal Poissonian-reset SBM with exponentially distributed waiting times between the reset events ψ(t) = re^−rt in the model of overdamped Langevin equation [here r is the reset rate]. dn Stochastic computer simulations of the FBM-DD system of equations in the overdamped limit. do ACF was not shown in ref. 92, but checked by us later via computer simulations [results not shown]. dp Simulations of the underdamped Langevin equation (particle mass m) driven by fractional Gaussian noise. dq The dispersion of individual TAMSDs for massive FBM decreases with T as for standard FBM, see footnote ds. In addition, for the particles of small-to-intermediate masses at short lag times^48 EB(m) ∝ m^1 for 0 < H < 3/4 and EB(m) ∝ m^4−4H for 1 > H > 3/4. dr Simulations of the overdamped Langevin equation driven by fractional Gaussian noise and confined to an interval [−L, L]. We refer the reader also to ref. 137, 138 and 221 where non-Gaussian PDFs of boundary-reflected FBM were observed, with depletion and accumulation of particles near the wall for reflected sub- and superdiffusive FBM, respectively. ds With the spread of the short-lag-time TAMSDs decreasing dramatically for smaller confining intervals. The ergodicity-breaking parameter EB^31 at short lag times decreases with the trace length T as EB(T) ∝ 1/T for 0 < H < 3/4 and as EB(T) ∝ 1/T^4−4H for 1 > H > 3/4. dt The same as in ref. 220, but for Poisson-reset FBM. du For strongly subdiffusive initial FBM the spread of TAMSDs of reset FBM is very small (FBM-like), while for strongly superdiffusive reset FBM it can reach 1–2 decades, see Fig. 2 and 13 of ref. 222. The TAMSD dispersion of frequently reset superdiffusive FBM, in addition, depends reciprocally on the reset rate: the respective ergodicity-breaking parameter^31 computed at short lag times decreases as EB(r) ∝ 1/r, see Fig. 4 and 5, and of ref. 222. dv The same as in ref. 222, but with Poisson-reset FBM considered in the stationary regime. dw The increment-MSD^222 computed in the nonequilibrium steady state equals the mean TAMSD at short times (both for sub- and superdiffusive FBM exponents) as well as at long times in the TAMSD-plateau region, restoring the ergodicity, as compared to the results of ref. 222. dx Provided the reset process is stationary and the trajectory length is much longer that the relaxation time. dy Theory and simulations of heavy-tailed subdiffusive CTRWs as subordinated walks on fractal structures (CTRW and RWF). The PDF of waiting times is ψ(τ) ∼ 1/τ^1+β, the MSD is 〈x^2(Δ)〉 ∝ Δ^2Hβ, and the short-lag-time TAMSD growth is . The TAMSD thus resembles that of FBM–SBM, being nonlinear in Δ and featuring a CTRW-like aging behavior for varying T, with the exponents of the source processes of CTRW and RWF entering the exponent of the MSD of the resulting process also multiplicatively, as in SBM–HDP^91 and FBM–HDP.^93 dz Diffusion of the massless particles with the diffusivity varying exponentially in space, . ea At long times, after the relaxation of the initial conditions, x0 = x(0). eb Splitting of the entire ensemble of trajectories into two subpopulations with distinctly different growth of the TAMSDs with the lag time Δ. Scalings are valid both at short and long lag times. ec Large scatter of the TAMSDs of 2…3 decades for large negative x0 values (in the region of exponentially high diffusivity) and vanishing scatter of the TAMSDs (with very low overall magnitudes) for large positive x0 values. ed The same as in footnote dz, but for logarithmically varying diffusivity, Dlog(x) ∝ log[(x/x̄)^2 + 1]. ee The scatter of 1…4 decades decreases for larger initial positions x0, where the space-variation of the diffusivity Dlog(x) is progressively smaller. ef Stochastic simulations of the SBM–HDP-related overdamped Langevin equation. eg A HDP-like ACF is expected, which itself is reminiscent of the ACF of FBM, see Fig. C1 in ref. 67, with the HDP subdiffusive exponent of the MSD 2/(2 − [small alpha, Greek, macron]) < 1 corresponding to the Hurst exponent 2H < 1. eh With the space-time dependence of the diffusivity of the form D(x, t) ∝ |x|^[small alpha, Greek, macron]t^α−1 and p([small alpha, Greek, macron]) = 2/(2 − [small alpha, Greek, macron]). Note that such power-law D(x, t) were proposed, e.g., to rationalize data on diffusion laws in turbulence,^90,224 that is one of the first examples of a non-Fickian and anomalously fast 〈x^2(t)〉 ∝ t^3 relative dispersion for a pair of molecules.^225,226 Power-law-like D(x) were also employed to describe animal dynamics.^227 ei Stochastic simulations of the overdamped FBM–HDP Langevin equation. ej Stochastic simulations of the FBM-SBM-related overdamped Langevin equation.
Ref. 156, exper.^a	∝Δ^0.4 (short times)	∝Δ^0.4 (short times)	Unknown	Large (≈1.5 decades)	Medium to large	FBM-like	FBM^b
Ref. 195, simul.^c	∝t^2/2.26 ∝ t^0.88	Not studied	Not studied	Not studied	None	FBM-like^d	FBM
Ref. 157, exper.^e	∝Δ^0.80…0.85 (short times)	∝Δ^1 (short times), ∝Δ^0.1…0.2 (long times)	None, ∝1/T^0	Medium (≈0.5 decades)	Present, not specified	Confined-FBM- or confined-CTRW-like	CTRW (short), FBM (long times)
Ref. 196, exper.^f	Not shown	∝Δ^0.32±0.12 (short times, 10^−2…10^0 s) ∝Δ^1.15±0.44 (long times, 3 × 10^2…10^4 s)	None, ∝1/T^0	Medium (≈1 decade)	Medium to large	Not studied	Polymer reptation^g
Ref. 197, exper.^h	Not shown	∝Δ^0.31±0.11 (short times, 10^−2…10^1 s)	∝1/T^0	Not presented	Medium to large^i	Not studied	FBM,^j FLE
Ref. 199, exper.^k	Not presented	∝Δ^0.4±0.04 (all times)^l	Not shown	Not shown	Medium to large	Not computed	FBM or obstructed diffusion
Ref. 161 and 162, exper.^m	∝Δ^0.82, ∝Δ^0.98 (short times)	∝Δ^0.82, ∝Δ^0.98 (short times)	None, ∝1/T^0 ^n	Not presented	Small, controlled	FBM-like	FBM
Ref. 200 and 201, exper.^o	∝Δ^0.6 (short times)	∝Δ^0.35…0.5 (short times)	∝1/T^0.26…0.28	Medium	Moderate^r	Not computed	Confined^143,202
	∝Δ^0.11 (long times)^p	∝Δ^0.35…0.5 (long times)	(Slow dynamics)	(≈1…1.5 decades)			subdiff. CTRWs^s
	Slow dynamics [nesting periods]	Slow dynamics	Unknown	(Slow dynamics)			(Slow dynamics)
	∝Δ^1.52…1.69 (all times)	∝Δ^1.6…1.8 (all times)^q	(Fast dynamics)	Not computed			Not examined
	Fast dynamics [commuting periods]	Fast dynamics		(Fast dynamics)			(Fast dynamics)
Ref. 203, simul.^t	Not reported^u	∝Δ^0.63…0.67 (short times, 0.01…1 ns) ∝Δ^0.94…0.97 (long times, 10…100 ns)	None, ∝1/T^0 ^v	Very small, BM-like	Small to none	FLE-like	FBM and FLE
Ref. 205, simul.^w	Not reported	∝Δ^≈1.5…1.7^x (short times, 1…100 fs), ∝Δ^≈0.3…0.7 (interm. times, 0.01…10 ns), ∝Δ^≈0.8…1.0 (long times, 0.1…10 µs)	Not presented	Small (≈0.5 decade)	Small to none	Not presented	FBM and FLE
Ref. 204, simul.^y	Not reported	∝Δ^0.8…0.9 (short/interm. times)	None, ∝1/T^0	Medium (≈0.5…1.5 decades)	Small	FLE-like	FBM, FLE
Ref. 96, exper.^z	Not presented	∝Δ^0.76…0.84 (short times)	∝1/T^0.2…0.28	Large (≈1…1.5 decades)	Moderate^aa	Not examined	FBM-CTRW^ab
Ref. 167, exper.^ac	∝Δ^0.9 (short times)	∝Δ^0.8 (short times)	∝1/T^0.1	Large (≈1.5 decades)	High, with clusters	Not presented	CTRW + fractal BM
Ref. 142, exper.^ad	Not studied	∝Δ^0.74±0.26 (0.1…1 s)^ae	Not studied	Medium (≈1 decade)	Moderate to large	FBM^af	FBM^ag
Ref. 13, exper.^ah	Not presented	∝Δ^0.59 (short times)	Not studied	Small-medium (≈0.5 decade)	Small	FBM-like^ai	FBM^aj
		∝Δ^0.46 (short times)
		(Nocodazole-treated cells)
Ref. 164, exper.^ak	Not specified	∝Δ^≈1.4…2	Not specified	Medium (≈1 decade)	Medium-large	Not presented	FBM with drift
Ref. 1, exper.^al	∝Δ^0.61, ∝Δ^0.63, ∝Δ^0.81	∝Δ^0.63, ∝Δ^0.50, ∝Δ^0.72	∝1/T^0.39±0…08 ^am	Very large (≈3…4 decades)	Medium to high^an	FBM-like^ao	FBM and RWF^1; FBM, BM, SBM, BM-DD, and HDPs^27
	Slow, medium, and fast diffusion “states” (short times)	3 diffusion “states” (short times)
Ref. 165, exper.^ap	Not presented	∝Δ^0.48±0.01 (0.1…10 s) (Polymerized microtubules) ∝Δ^0.3 (nocodazole-treated cells)	Not studied	Medium (≈1 decade)	Moderate to large	FBM-like^aq	FBM^ar
Ref. 103, exper.^as	MSD(Δ) ≈ TAMSD(Δ)^at	{∝Δ^0.90, ∝Δ^0.86, ∝Δ^0.92}^au	Not studied	Medium {≈1, ≈1.5, ≈1} decades for {Nav1.6, Kv1.4, CD4}	Medium to large^av	Not studied	FBM or RWF^aw
		for {Nav1.6, Kv1.4, CD4} (short times)
Ref. 4, exper.^ax	Not specified	∝Δ^0.92 (fast 25 nm Rho-NPs),	Not specified	Medium (≈1 decades)	Small^az	Not analyzed	CTRW^ba
	Not specified	∝Δ^0.18 (slow 25 nm Rho-NPs);	Not specified	Large (≈2 decades)
	Not specified	∝Δ^0.56 (75 nm Rho-NPs);	Not specified	Not specified
	Not specified	∝Δ^1 (20 nm QDs, short times [10 ms]),	∝1/T^0.3 undefined	Large (≈1.5 decades)
	Not specified	∝Δ^0.7 (20 nm QDs, long times [1 s])	∝1/T^0.3	Not well defined
	Not specified	∝Δ^0.82 (20 nm QDs, all times, T > T*)^ay		Large (≈2 decades)
Ref. 140, exper.^bb	Not shown	{∝Δ^0.1…0.9, ∝Δ^0.9…1.1, ∝Δ^1.1…1.6}^bc (short times)	Not studied	Medium (≈1.5 decades)	Moderate	FBM-like^bd	FBM^be
Ref. 19, exper.^bf	MSD(Δ) ≈ TAMSD(Δ)^bg	{∝Δ^1/2, ∝Δ^1, ∝Δ^1.26}^bh (interm. times)	Present^bi	Very large (≈1.5…2 decades)^bj	Large	FBM-like^bk	“Heterogeneous” FBM^bl
Ref. 97, exper.^bm	{∝Δ^1, ∝Δ^0.5, ∝Δ^0.6}	{∝Δ^1, ∝Δ^0.7, ∝Δ^0.6}	Unknown	Very small	Small, controlled size	Not studied	{BM, CTRW, FBM} 3 tracer sizes
	3 tracer sizes (short times)	3 tracer sizes (short times)
Ref. 6, exper.^bn	∝Δ^1.28 (short times)	∝Δ^1.37 (short times)	∝1/T^0.3	Large (≈1.5 decades)^bo	Medium-high	Not computed	Not examined
Ref. 111, exper.^bp	∝t^1.38…1.41 and ∝t^1.30…1.33 (short times, ≲100 min)^bq	Not presented	Not studied	Medium^br	Medium-to-large	Not shown	Biased random walk or PRW^bs
Ref. 215, exper.^bt	∝t^1.44…1.53 (short times, ≲100 min)^bu	Not shown	Not studied	Not shown	Medium-to-large^bv	Always positive, decaying^bw	PRW, anisotropic PRW^bx
Ref. 10, exper.^by	MSD(Δ) ≈ TAMSD(Δ)	∝Δ^1.2…1.4 (interm. times)^bz	Not examined	Medium (≈1 decade)^ca	Large^cb	FBM-like^cc	Parameter-distributed FBM
Ref. 217, exper.^cd	Not shown	∝Δ^0.57±0.02	Not studied	Very large^ce (≈2.5…3 decades)	Large^cf	Subdiffusive-FBM-like	FBM^cg
Ref. 8, exper.^ch	{∝Δ^0.46, ∝Δ^0.28, ∝Δ^0.36} 3 pH values (short times)	{∝Δ^1.09, ∝Δ^0.75, ∝Δ^0.94} 3 pH values (short times)	Unknown^ci	{Very large (≈3 decades), medium (≈1.5 decades), small (≈0.5 decades)} 3 pH values	Small, controlled	FBM-like^cj	FBM,^ck also BM and BM-DD
Ref. 70, exper.^cl	∝Δ^0.84±0.05 (all times)	∝Δ^0.95±0.05 (all times)	∝1/T^0.17±0.05	Medium-large (≈1.5 decades)	Small	Not presented	BM with pD(D)^cm
Ref. 16, simul.^cn	∝Δ^1 (short times), ∝Δ^0.2…0.4 (long times)	∝Δ^0.48 (short times), ∝Δ^0.9 (long times)	∝1/T^0.34	Medium (≈0.5 decades)	None or controlled	FBM-like^co	FBM-CTRW hybrid
Ref. 218, exper.^cp	{∝Δ^0.44, ∝Δ^0.62, ∝Δ^0.82, ∝Δ^0.92} 4 humidity values (short times)	{∝Δ^0.49, ∝Δ^0.66, ∝Δ^0.84, ∝Δ^0.95} 4 humidity values (short times)	∝1/T^0 ^cq	Not presented	Small^cr	FBM-like^cs	FBM, FBM subordinated to CTRW
Ref. 219, simul.^ct	∝Δ^2 (short times, 1 ps)	Not studied	Not studied	Not studied	Small	With a negative dip^cu	Generalized Langevin equation^cv
	∝Δ^0.3…0.6
	(interm. times, ∼10^1…10^4 ps)
	∝Δ^1 (long times, 400 ns)
Ref. 57, simul.^cw	∝Δ^1/6 (short times)	∝Δ^7/6, ∝Δ^1	∝1/T^5/6, ∝1/T^1	Very small, SBM-like	None^cx	Not analyzed	SBM^cy
Ref. 168 and 169, exper.^cz	∝Δ^1−ϑ (short times, 0 < ϑ < 1)	Not measured	Not measured	Not measured	Small or none	Not studied	Not examined^da
	∝Δ^1 (long times)
Ref. 62, simul.^db	∝Δ^1 (short times, Δ ≪ τ0)	∝Δ^1 (short times, τ0 ≪ Δ ≪ T)	∝1/T	Very small, SBM-like	None	Not shown^dc	Ultraslow SBM
	∝log[Δ] (long times, τ0 ≪ Δ ≪ T)	∝(Δ/T)log[T/Δ] (long times, Δ ≫ τ0)
Ref. 58 and 63, simul.^dd	∝Δ^2 (short times, Δ ≪ 1/γ0)	∝Δ^2 (short times, Δ ≪ 1/γ0) ∝ Δ^1+α (interm. times, Δ ≪ Δ* ≪ T) ∝ Δ^1 (interm. times, Δ ≫ Δ* ≫ τ0)	∝1/T^1−α (interm. times)	Not presented^de	None	Not analyzed	Underdamped SBM
	∝Δ^1 (interm. times, 1/γ0 ≪ Δ ≪ τ0)
	∝Δ^α (long times, Δ ≫ τ0, α > 0)
Ref. 58, simul.^df	∝Δ^2 (short times, Δ ≪ 1/γ0)	∝Δ^2 (short times, Δ ≪ 1/γ0)	∝1/T (short and interm. times)	Very small, SBM-like	None	Not shown	Ultraslow underdamped SBM
	∝Δ^1 (interm. times, 1/γ0 ≪ Δ ≪ τ0)	∝Δ^1 (interm. times, Δ ≪ Δ*)
	∝log[Δ] (long times, Δ ≫ τ0)	∝(Δ/T)log[T/Δ] (long times, Δ* ≪ Δ ≪ T)
Ref. 94, simul.^dg	See Table 1 in ref. 94	See Table 1 in ref. 94	∝1/T^di	Small-to-moderate, -dependent	None	Not studied	Exponential SBM
Ref. 94, simul.^dj	See Table 1 in ref. 94	See Table 1 in ref. 94	∝log[T/[small tau, Greek, tilde]0]^dk	Small	None	Not studied	Logarithmic SBM
Ref. 55, simul.^dl	∝Δ^α (short times, Δ ≪ 1/k)	∝Δ^1 (short times)	Not studied	Not presented	None	Not studied	Confined SBM
	∝Δ^α−1 (long times, Δ ≫ 1/k)	∝Δ^0 (long times)
Ref. 220, simul.^dm	∝Δ^α (short times, Δ ≪ 1/r)	Not studied	Not studied	Not studied	None	Not studied	Reset SBM
	∝1/r^α (long times, Δ ≫ 1/r)
Ref. 92, simul.^dn	∝Δ^2H (short times)	∝Δ^2H (short times)	None, ∝1/T^0	Small-medium (≈0.5 decades)	None	FBM-like^do	FBM-DD
	∝Δ^2H (long times, 1/2 < H)	∝Δ^2H (long times, 1/2 < H)
	∝Δ^1 (long times, H < 1/2)	∝Δ^1 (long times, H < 1/2)
Ref. 48, simul.^dp	∝Δ^2H+2 (short times)	∝Δ^2 (short times)	Not studied	FBM-like^dq	None	Not studied	Underdamped FBM
	∝Δ^2H (long times)	∝Δ^2H (long times)
Ref. 45 and 46, simul.^dr	MSD(Δ) = TAMSD(Δ)	∝Δ^2H (short times)	Not studied	FBM-like^ds	None	Not studied	Confined FBM
		∝Δ^0 × L^2/3 (long times, 2K2HΔ^2H ≫ L^2)
Ref. 222, simul.^dt	∝Δ^2H (short times, Δ ≪ 1/r)	∝Δ^2H (short times, 0 < H < 1/2)	Not studied	Small to medium^du	None	Not studied	Reset FBM
	∝1/r^2H (long times, Δ ≫ 1/r)	∝Δ^1 (short times, 1 > H > 1/2)
		∝2/r^2H (long times)
Ref. 154, simul.^dv	iMSD(Δ) = TAMSD(Δ)^dw	∝Δ^2H (short times, Δ ≪ 1/r)	None, ∝1/T^0 ^dx	Small to medium	None	Not studied	Stationary reset FBM
		for 0 < H < 1/2
		∝Δ^1 (short times, Δ ≪ 1/r)
		for 1/2 < H < 1
		∝1/r^2H (long times, Δ ≫ 1/r)
Ref. 95, simul.^dy	∝Δ^2Hβ	Δ ^1−β+2Hβ	1/T^1−β	Not reported	None	Not reported	CTRW with RWF
Ref. 223, simul.^dz	∝log^2(t)^ea	∝Δ^1, for large positive x0	Present, not studied	Dependent on x0 ^ec	None	Not studied	Exponential HDP
		∝Δ^1 and ∝Δ^1/2, for x (0) ≈ 5^eb
		∝Δ^1/2, for large negative x0
Ref. 223, simul.^ed	∝x0^2 + 2Dlog(x0) × t^1	∝2Dlog(x0) × Δ^1	Present, not studied	Medium-to-large^ee	None	Not studied	Logarithmic HDP
Ref. 91, simul.^ef	∝Δ^α×2/(2−[small alpha, Greek, macron])	∝Δ^1	∝1/T^1−α × 2/(2−[small alpha, Greek, macron])	Large, HDP-like, [small alpha, Greek, macron]-dependent	None	Not computed^eg	SBM–HDP^eh
Ref. 93, simul.^ei	∝Δ^2H×2/(2−[small alpha, Greek, macron])	∝Δ^2H	∝1/T^(1−2/(2−[small alpha, Greek, macron]))2H	Large, HDP-like, [small alpha, Greek, macron]-dependent	None	Not computed	FBM–HDP
This study, simul.^ej	∝Δ^2H+α−1	∝Δ^2H	∝1/T^1−α	Small, SBM- and FBM-like	None	FBM-like	FBM–SBM

---

II Statistical properties of FBM and SBM

FBM and SBM are formulated based on the Langevin equation,¹³¹driven by fractional Gaussian noise ξ_H(t) for a constant diffusion coefficient D for FBM and by white Gaussian noise ξ(t) for a time-dependent diffusion coefficient D_α(t) for SBM, see below.

A FBM

The parental process of FBM—as introduced in ref. 42 and 43 and developed in some of our recent studies,^45,46,48,92 see eqn (16) (below)—is stationary in increments and nearly as ergodic^31,44 as BM. For FBM, the exponent and magnitude of the power-law anomalous^{28,31,132–135} growth of the MSDand TAMSDare equal at short lag times, Δ ≪ T, whereHere, the exponent β = 2H is twice the Hurst exponent¹³⁶ and K_2H is the generalized diffusion coefficient. The exponent H = 1/2 demarcates the situations of persistent (positive) and antipersistent (negative) correlations of particle displacements realized for FBM at 1 > H > 1/2 and 0 < H < 1/2, correspondingly, see ref. 45, 46, 48 and 92 and Section III A below. These two regimes yield super- and subdiffusion respectively. The discrete version of expression (3), used to rationalize various SPT datasets, iswhere δt is the discretization step of a time series with N points.

For FBM, all individual TAMSD trajectories, being highly reproducible,^45,46,48,92 are equal to the MSD in the limit of long measurement times T,

The mean TAMSD for an ensemble of N independent [and statistically identical] trajectories for a given lag time Δ and measurement time T is computed as the arithmetic mean,The magnitude of the TAMSDs is insensitive to the trajectory length and thus FBM does not feature aging, see ref. 31 and 44–46.

The PDF of FBM is Gaussian and the distribution of particle displacement at time t has the form

provided the initial PDF is P(x, t = 0) = δ(x). We consider below the free-space spreading dynamics, but emphasize that implementing a physically self-consistent scheme for FBM in the presence of confinement^45,46 and reflecting boundaries^137–139 is a nontrivial task due to the nonlocality of FBM and possibly variable degree of memory loss upon a reflection from a planar boundary [depending, e.g., on the angle of incidence of the jump to be reflected].

The ACF computed from the increments of particle positions, normalized to the initial value at time t = 0, is

Here, δ is the period of increment measurement. This ACF for subdiffusive FBM starts with unity at the initial time τ = 0, exhibits a dip at short time lags with the depth^13,140

C^(δ)_v (τ = δ)/C^(δ)_v (0) = 2^2H−1 − 1, (10)

and relaxes to zero from below in a power-law manner at n ≫ 1 as,^67,141,142

C^(δ)_v (τ = nδt) ∝ n^2H−2. (11)

Here n is the number of elementary time-intervals δt used in the analysis. The depth of this dip reflects the degree of antipersistence of successive displacements for FBM with 0 < 2H < 1. For superdiffusive FBM, with 1 < 2H < 2, the velocity ACF drops from unity at the initial time τ = 0 to the same value (10) which is now positive, indicative of persistence in displacement correlations. At longer times the ACF approaches zero from above, again as a power law (11).

B SBM

The inherently nonstationary process of SBM—as introduced in ref. 51, 54 and 55 and developed recently, i.e., in ref. 56–58—features the diffusion coefficient of the form,

D_α(t) = αK_αt^α−1, (12)

with 0 < α < 2. SBM is a nonergodic process in the sense of MSD-to-TAMSD nonequivalence.^31,55,57 SBM, like FBM, is an athermal process that can additionally be considered as a “mean field” over CTRWs.⁵⁴ For subdiffusive SBM, for instance, we observe a power-law-like MSD growth,

MSD(Δ) ∝ Δ^α, (13)

while at short lag times Δ/T ≪ 1 the mean TAMSD is linear in lag time,

TAMSD(Δ, T) ∝ Δ¹/T^1−α. (14)

Note that the argument of the MSD is real physical time (even if Δ is used for it below), while the argument of the TAMSD [per definition] is the lag time. The decay of the TAMSD with the trajectory length T for α < 1—underlying a slower dynamics of subdiffusive SBM at later times—is a conspicuous feature of SBM aging. The functional dependence of this decay on T as a function of α is similar to that of subdiffusive CTRWs^65,143–147 and subdiffusive HDPs.^68,69,91

In contrast to CTRWs and HDPs, for SBM, the distribution of TAMSDs is, however, narrow and the respective EB parameter,^28,31

characterizing this small irreproducibility of the TAMSDs from one realization to the next behaves often similar to that for BM, see ref. 54, 56 and 91.

The process of SBM is Markovian and it features the same PDF as FBM, with a trivial substitution 2H → α.

The ACF of SBM for sub- and superdiffusive realizations is the same as that of BM for disjoint time intervals. Namely, the ACF—normalized to the ACF at zero time shift—starts at unity, drops within one step δ = δt to zero, and—due to the absence of temporal noise correlations on later stages—stays zero at longer lag times τ (see, e.g., Fig. 2 in ref. 60).

III Results for FBM–SBM

A Model and simulations

We examine the overdamped Langevin equation featuring the power-law SBM-like diffusion coefficient (12) and FBM-associated external fractional Gaussian noise ξ_H(t),Here, noise ξ_H(t) for time instances |t₁ − t₂| ≫ 0 features the correlation function

〈ξ_H(t₁)ξ_H(t₂)〉 ∝ K_2H2H(2H − 1)|t₁ − t₂|^2H−2. (17)

These long-range temporal correlations for non-Markovian FBM are contrasting memoryless white Gaussian noise giving rise to BM. The diffusion coefficient in eqn (16)—stemming from a subdiffusive source process of SBM—describes the particle dynamics slowing down with time.

We emphasize that the underdamped Langevin equation¹³¹ and FBM for massive particles were considered recently too⁴⁸ (see also ref. 148). The consideration of massive particles with both a power-law SBM-like time-varying diffusivity D_α(t) as well as with the exponential and logarithmic dependencies D(t) were recently presented, see ref. 57, 58, 62, 63 and 94, respectively. For exponential-SBM, for instance, the MSD grows with time as for a multiplicative process of geometric BM,^149,150 see also ref. 151.

For numerical simulation of eqn (16) we employ the same standard forward-running Ito-like^152,153 discrete iterative scheme with a variable time-step δt, see the detailed description in our previous studies for the simulations of HDPs,^68,69 SBM–HDPs,⁹¹ and, particularly, of FBM–HDPs⁹³ and FBM-DD⁹² processes.

B MSD and TAMSD

The compound FBM–SBM process is weakly nonergodic. Specifically, the ergodic FBM behavior with the equivalent MSD (2) and TAMSD (3) for FBM–SBM with 2 > α > 1 changes to

MSD(Δ) ∝ Δ^2H+α−1 (18)

and

TAMSD(Δ) ∝ Δ^2H. (19)

FBM–SBM with 0 < α < 1 is also nonergodic. For a growing-in-time SBM diffusivity realized at 2 > α > 1 and for superdiffusive FBM the MSD of FBM–SBM is considerably lower than its TAMSD. The situation is the opposite for 0 < α < 1 when the MSD at short times is much larger than the TAMSD of FBM–SBM, see Fig. 1a and b, respectively. Naturally, as the Hurst exponent approaches the BM limit H = 1/2, the MSD exponent of FBM–SBM is 2H and its MSD–TAMSD behavior turns into that of pure SBM.† The process of FBM–SBM, as its parental processes, is free of a typical time scale.

Fig. 1 MSD and TAMSD of FBM–SBM as function of time t and lag time Δ, respectively. Superdiffusive parental FBM is considered, with H = 0.8. The diffusion scenarios of super- and subdiffusive parental SBM correspond to panels (a) and (b), respectively. The SBM exponents are provided in the legend. Thick green and blue curves are the theoretical asymptotes (18) of the MSD and (19) of the TAMSD (plotted with the unit prefactors (see also eqn (21) and Fig. 3)), while the respective symbols are the results of simulations. The red curves are the individual TAMSDs from simulations. Other parameters are as follows: the initial position of the particles is chosen at x₀ = 0, the total length of the diffusion trajectory is T = 10², the time-step in the simulations is δt = 10⁻², and ensemble averaging is performed over N = 10³in silico-generated time series. For all plots, the noise strength is set at K_2H = 1/2 in eqn (17) and we use K_α = 1/2 in eqn (12). For this, and all other plots expect Fig. 9, the time-step is δt = 0.01.

Fig. 2 Variation of the MSD scaling exponent of FBM–SBM from expression (18). The region of the MSD plateau 2H + α − 1 < 0 is indicated in the blue color, while the green color designates the domain of growing-in-time MSDs.

The derivation of the MSD (18) and TAMSD (19) of FBM–SBM is as follows. Starting from (16), we get

At H = 1/2 the fractional noise reduces to the white Gaussian noise and one arrives at pure SBM with 〈x²(t)〉 ∼ t^α. At H > 1/2, using the approximate expression (17) for the noise correlations, the MSD readsThe prefactor of this MSD as a function of exponents α and H of the two parental processes is shown in Fig. 3a to demonstrate the simulations vs. theory agreement. We find that in a large region of exponents the MSD prefactor is of order unity, corroborating the unit asymptotics used in Fig. 1. In special cases of pure SBM (2H = 1 and K_2H = 1/2) or pure FBM (α = 1 and K_α = 1/2) the general MSD expression (21) for FBM–SBM yields, respectively, the expected dependencies 〈x²(t)〉 = 2αK_αt^α for SBM and 〈x²(t)〉 = 2K_2Ht^2H for superdiffusive FBM.

Fig. 3 Visualization of the MSD prefactors in eqn (21) for superdiffusive (panel a) and in eqn (23) for subdiffusive (panel b, after n = 25 steps) parental FBM process, both yielding a growing-in-time MSDs of FBM–SBM (2H + α − 1 > 0). The color scheme for the respective regions of exponents is as in Fig. 2.

At H < 1/2, using the exact correlator of the noise,

within the discretized scheme (see ref. 154), for the time step δt and with ε = δt the MSD after n steps and t = n × δt becomes

〈x²(t)〉 = 4αK_αK_2HC(α, H, n) × (n × δt)^α+2H−1, (23)

where the coefficientThe MSD prefactor in eqn (23) for the power law ∝t^α+2H−1 is shown in Fig. 3b. If α + 2H − 1 > 0, the C (α, H, n) ∼ 1 and thus 〈x²(t)〉 ∼ t^α+2H−1.

In contrast, for the region of exponents α + 2H − 1 < 0—shown as the blue domain in Fig. 3b—the MSD is almost a constant, see Fig. 7. The explicit dependence of the MSD (23) on the time step δt—reciprocal in this regime of the parental exponents—is a rationale for the time-step-dependent plateaus of the MSD shown in Fig. 8.

Regarding the TAMSD, based on the intuition we gained studying similarly constructed hybrid processes of SBM–HDP and FBM–HDP with the TAMSD-to-MSD ratios (28) and (29) provided below, respectively, the anticipated TAMSD-to-MSD ratio for FBM–SBM given by (30) is fully supported by the results of our computer simulations. We thus do not derive here the explicit TAMSD expression analytically.

As FBM and SBM reveal a narrow amplitude spread of their TAMSD trajectories, a similar behavior is also observed for their combination FBM–SBM, see Fig. 1.

For FBM–SBM this “independence” of the parental processes (characteristic, e.g., for FBM–HDP⁹³ and SBM–HDP⁹¹) is no longer present. The temporal correlations of fractional Gaussian noise do interfere with the temporal decay or increase of the SBM-related diffusion coefficient. The respective FBM and SBM exponents in the MSD of FBM–SBM are, thus, being summed instead of being multiplied, see the green domain in the diagram of Fig. 2. For the TAMSD of FBM–SBM, rather non-surprisingly, the same scaling exponent as for FBM is observed.

We observe, however, a less obvious behavior for the situations of subdiffusive SBM and strongly subdiffusive FBM, see the blue domain in Fig. 2. The TAMSD of the combined FBM–SBM process still exhibits FBM-like scaling ∝Δ^2H, whereas the MSD very quickly stalls or stagnates at a plateau, see Fig. 7 and 8. The smaller the simulation step, the higher is the observed MSD plateau, with the MSD-plateau level being reached in the simulations within first couple of time-steps, Fig. 8. The MSD quickly jumps to a plateau because the (effective) diffusivity of the particles decays very quickly at longer times.‡

As the exponents for subdiffusive SBM and subdiffusive FBM decrease, the crossover time—beyond which the stalling MSD behavior is detected in the simulations—increases. As the parental FBM process turns less subdiffusive, the MSD of FBM–SBM in the limit of long times starts approaching the asymptote (18), see Fig. 9. It is essential that the MSD exponent of FBM–SBM is positive in this regime, α − 1 + 2H > 0. In the limit of short times, however, the MSD still reveals a dependence on the step-size in simulations.

C Aging of the TAMSD

The aging behavior of FBM–SBM is consistent with our physical intuition. Namely, as FBM is a nonaging process, aging of FBM–SBM stems solely from that of SBM, yielding for the mean TAMSD at short lag times (Δ ≪ T) the relationTherefore, for subdiffusive SBM with 0 < α < 1 the magnitude of the observed TAMSD of FBM–SBM decreases with the trace length T, indicating a dynamics slowing down with time. The scaling behavior (25) is supported by our computer simulations for both sub- and superdiffusive parental SBM, see Fig. 4. The TAMSD amplitude at short lag times is considered as the most representative and statistically robust characteristics.^31,94

Fig. 4 Variation of the TAMSD magnitude at short lag times with the length of the trajectory, with the relation (25) shown as the asymptotes (here, the “normalization” trace-length is T₁ = 1). The values of FBM and SBM exponents are provided in the legend. Due to the TAMSD “normalization” employed here, the two different SBM exponents used define the two aging exponents of FBM–SBM in the computer-simulation results.

We also stress that—similarly to subdiffusive HDPs and subdiffusive CTRWs—pure SBM features a peculiar aging property. Namely, the ratio of the aged mean TAMSD³¹

to the nonaged mean TAMSD (3) is given for SBM by^63,91

Λ_α(t_a/T) = (1 + t_a/T)^α − (t_a/T)^α. (27)

Here, the aging time is denoted by t_a. The same law is observed for SBM–FBM, see Fig. 5, because FBM is a nonaging process. Note also that expression (27) is valid for the aging behavior of the TAMSD for subdiffusive HDPs and subdiffusive CTRWs too, see eqn (15) in ref. 68 and eqn (39) in ref. 145, respectively.

Fig. 5 Dependence of the normalized TAMSD magnitude on aging time , with the relation (27) shown as the solid curves. The scaling relations at long aging times are given by the dashed lines. Parental FBM and SBM exponents are provided in the legend.

There exist close similarities in scaling relations for the MSD-to-TAMSD ratios for the compound processes of SBM–HDP, FBM–HDP, and FBM–SBM. For SBM–HDP with D(x, t) ∝ |x|^{[small alpha, Greek, macron]}t^α−1 we found that⁹¹ the MSD grows as 〈x²(Δ)〉 ∝ Δ^αp, the TAMSD increases [at short lag times] as and their ratio is

where p = 2/(2 − [small alpha, Greek, macron]) is the MSD exponent of pure HDPs.^67,69 For FBM–HDP we demonstrated that⁹³ 〈x²(Δ)〉 ∝ Δ^2Hp, , andFor FBM–SBM [considered here] we observe that at short times 〈x²(Δ)〉 ∝ Δ^2H+α−1, andThese relations are valid for the MSD and TAMSD growing with (lag) time.§ The scaling relations of this paragraph embody the main characteristics of a family of SBM–HDP, FBM–HDP, and FBM–SBM hybrid processes, see Table 1.

D PDF

For FBM–SBM with a nonstalling MSD behavior we always observe PDFs of a Gaussian shape. This finding is intuitive (results not shown) because the parental processes of FBM and SBM are also Gaussian, eqn (8). The width of the PDFs characterizing the dispersion of the particles grows in time in full correspondence to the observed MSD-growth law, see Fig. 11. What is more surprising is that the Gaussian PDFs are also detected for situations with a stalling-MSD behavior of FBM–SBM. The integration of the fitted PDFs of particle displacements, eqn (2), in the stagnating/stalling regime of the MSD yields for the second moment the plateau values in full agreement with the simulations, as illustrated in Fig. 8.

E ACF

For FBM–SBM the ACF C^(δ)_v(τ)/C^(δ)_v(0) behaves similarly to that of FBM, see Fig. 12. Quantitatively, for subdiffusive FBM contributing to FBM–SBM the depth of the minimal value of the ACF shifts upwards for sub- and downwards for superdiffusive SBM contributions. For superdiffusive FBM in FBM–SBM the ACF curve shifts downwards and upwards for sub- and superdiffusive SBM in FBM–FBM, respectively. The decay law for the long-time tail of the ACF of FBM–SBM gets modified as compared to (11) for FBM. Namely, using the discretely defined ACF (9), we get

C^(δ)_v(τ = nδt)/C^(δ)_v(0) ∝ n^{2H−2+(α−1)/2}, (31)

as illustrated in Fig. 13.

In terms of the MSD exponent (18), for FBM–SBM with subdiffusive FBM and subdiffusive SBM the resulting process is more subdiffusive than original FBM. For superdiffusive FBM, the superdiffusive SBM contribution makes the resulting FBM–SBM more superdiffusive. The behavior of the ACF minimum, being computed naively as in Fig. 12 and described above, therefore, does not agree with these trends of the MSD exponents. This inconsistency gets “repaired/fixed” by a time-dependent normalization of the ACF, namely

Namely, as FBM–SBM is a nonstationary process, the time-dependence of the instantaneous mean-squared displacement-increments, scaling from eqn (16) as

〈(dx(t)/dt)²〉 ∝ D(t) ∝ t^α−1, (33)

should enter the ACF normalization (see, e.g., ref. 155 for the consideration of nonstationary ACFs). From eqn (16) for FBM–SBM it follows thatThe ACF of FBM–SBM normalized this way, that accounts for the nonstationarity of SBM, is (nearly) the same as that of its parental FBM, see the results of simulations shown in Fig. 6. This is expected because the displacement-ACF of SBM—being similar to that of BM, Section II B—is zero at long lag times. SBM, therefore, does not affect the long-time tails of the properly-normalized ACF of FBM–SBM (34): it renders the scaling behavior (31) equal to (11) for pure FBM, as confirmed by the results of simulations presented in Fig. 14.

Fig. 6 ACF of FBM–SBM after the time-dependent renormalization (34) due to the nonstationarity of SBM was taken into account (see the legend for the parameter values). The results of pure FBM for different times t are included as the FBM–SBM results with α = 1. The dashed and dot-dashed lines in panel (a) indicate, respectively, the minimum (9) for pure FBM and the ACF = 0 level.

IV Discussion and conclusions

A Applications of FBM and SBM

Some recently observed mainly biophysical phenomena describable (at least partially) by FBM and SBM are as follows.

The model of antipersistent subdiffusive FBM—including the situations with ensemble-distributed values of 2H and K_2H—has been successfully applied to the description of the spreading characteristics of tracers of various nature in living biological cells^{13,140,156–159} as well as in various prototypical in vitro crowded environments^8,160–163 (sucrose, dextran, mucin, etc.) mimicking the macromolecularly crowded cyto- and nucleoplasm of a cell. The list of endogenous and “introduced” tracers includes micron-sized beads (both inert and interacting with the medium), quantum dots (QDs), granules, polymer segments (mRNA, chromosomal loci and telomeres,¹⁵⁶etc.), vacuoles and particles,¹⁶⁴ p-granules and organelles.^12,140,165 Note that experimental studies of non-equilibrium cytoskeleton-induced forces impacting the subdiffusion of telomeres also exist.¹⁶⁶

Some concrete biophysical examples of FBM-type motions are as follows. The physical mechanisms potentially underlying the anomalous dynamics of (i) potassium channels in the plasma membrane,¹⁶⁷ (ii) intracellular transport of insulin granules,⁹⁶ (iii) envelope glycoprotein gp41 transmembrane proteins of human-immunodeficiency virus in the T-cell plasma membranes,¹ (iv) subdiffusion of chromosomal loci in bacterial cells,¹⁵⁶ (v) subdiffusion of endogenous lipid granules in the living cells of fission yeast,¹⁵⁷ (vi) lysosome and endosome intracellular movements [describable by FBM with a stochastic Hurst exponent],^12,19 to name a few, see Table 1.

As to SBM, one example concerns the in vivo observations and physical interpretations of time-dependent diffusion of water in white- and gray-matter tissues of the [human] brain. The respective diffusion process features an SBM-like variation of the time-dependent part of the diffusion coefficient on time, as thoroughly studied recently^168–175 [see also ref. 176 and our recent coverage;⁹⁴ we refer here to Table 1]. The experimentally measured finite long-time diffusion coefficients—in nonconfining compartments, such as in the extra-cellular space, with long-time diffusivity D_∞ > 0—are strongly supportive of the picture of normal macroscopic water diffusion in the brain tissues, with often Gaussian distribution of displacements.^{168,169,171,172} The detailed features of possible transient anomalous diffusion of water in neural tissues were discussed in ref. 177. For some earlier studies of time-dependent diffusion—with the diffusivities D(t) often decreasing with time—in barrier- or obstacle-containing, compartmentalized, and labyrinthine tissue environments we refer to ref. 168 and 178–185 for, D(t) in neural tissues, muscle tissues, and prostate cancer. These diffusion-MRI studies aim at unravelling the detailed microstructure of the respective tissues.

The second example of a physical system amenable for the SBM description is nonergodic diffusion of particles⁵⁷ in free-cooling granular gases,^186–190 see Table 1. Nonelastic collisions of particles/constituents in such gases lead to energy dissipation and, as a result, the effective temperature of the environments decreases in accord with (12), along with the particle diffusivity. Note, however, that for a cooling gas of rapidly aggregating particles the standard Haffs¹⁹¹ temperature law^57,190T(t) ∝ 1/t² can be inverted yielding an increasing—rather than decreasing—energy of an aggregate particle in time, as advocated recently.¹⁸⁹

Lastly, for the experiments on fluorescence recovery after photobleaching in the presence of passive and active particle motion the effective diffusion coefficient has, e.g., a time-dependent correction¹⁹²D(t) = D(1 + v²t/(4D)) to yield the MSD 〈x²(t)〉 = 4D(t) × t = 4Dt + v²t². Diffusion of BM-tracers in expanding or contracting media (like our Universe) produces a time-space-rescaled stochastic process similar to SBM (in terms of its long-time MSD).¹⁹³ Generally, the spreading kinetics slowing down in time—as, e.g., in the polymerization or aggregation reactions of polymers¹⁹⁴—where the diffusivity drops with the typical particle size [e.g., in a Stockesian ∝1/size fashion] and with diffusion time—can potentially be considered as SBM with a power-law (or more complicated) functional forms for D(t).

B FBM–SBM

The FBM–SBM process introduced here enriches the armamentarium of anomalous-diffusion processes applicable to the description of SPT datasets. Some “paradoxical”⁵⁵ features of nonstationary SBM coupled to a power-law decaying memory of FBM yields a so-called scaled-fractional process of FBM–SBM, certain characteristics of which are unveiled in the present study. The combination of SBM and FBM yields a nonlinear-in-lag-time FBM-like growth of the TAMSD and the presence of aging for FBM–SBM is reflected in the SBM-like dependence of the TAMSD magnitude [at short lag times] on the trajectory length, eqn (25). The MSD exponent for FBM–SBM is the “sum” of the SBM and FBM scaling exponents. The magnitude and scaling relation of the MSD are disparate from those of the TAMSD and, thus, FBM–SBM features weak ergodicity breaking.

These two essential attributes—nonequal scaling exponents of the MSD and TAMSD as well as a pronounced aging of the TAMSD—are particularly useful (indispensable) when selecting for or envisaging the most appropriate models of anomalous and nonergodic diffusion in a statistical analysis of SPT datasets. For instance, the TAMSD dependencies nonlinear in lag time and TAMSD aging with the trace length as in (25) were observed for diffusion of micron-sized beads in hydrogels of mucin polymers,⁸ propulsive chemotaxis-free dynamics of amoeboid cells,⁶ diffusion of doxorubicin drug molecules in silica nanoslits,¹⁶ as well as diffusion of gp41 transmembrane proteins in the plasma membranes of surface-adhered immune T-cells.^1,27 The process of FBM–SBM is thus applicable to nonstationary physical systems with power-law-like varying SBM-like diffusion coefficient and FBM-like memory-containing correlations of particle displacements in successive time steps.

To conclude, paraphrasing the classics, “knowing is not enough, we must apply” the mathematical models of anomalous diffusion to real experimental data to check if really “all models are wrong, but some are useful”. We hope that Table 1—targeting primarily the experimental SPT community—unveils the usefulness of some pure and hybrid mathematical models of anomalous diffusion as examination tools for rationalizing, understanding, and categorizing some relevant and measurable attributes of natural physical phenomena, as gained from the experimental SPT observations as well as from (toy-)model-based in silico studies.

Abbreviations

PDF	Probability-density function
MSD	Mean-squared displacement
TAMSD	Time-averaged MSD
SPT	Single-particle tracking
BM	Brownian motion
SBM	Scaled BM
FBM	Fractional Brownian motion
FLE motion	Fractional-Langevin-equation motion
HDPs	Heterogeneous diffusion processes
CTRW	Continuous-time random walk
ATTM	Annealed transient-time model
RWF	Random walk on a fractal
PRW	Persistent random walk
DD model	Diffusing-diffusivity model
QDs	Quantum dots
NPs	Nanoparticles

Conflicts of interest

There are no conflicts to declare.

Appendix A: Auxiliary figures

Here, we present some supplementary plots supporting the claims of the main text.

Fig. 7 The plateau behavior of the MSD for subdiffusive SBM and very subdiffusive FBM exponents such that α − 1 + 2H < 0 (see the legend for the values). The notations and colors for the curves and symbols are the same as in Fig. 1.

Fig. 8 The plateau behavior of the MSD for subdiffusive SBM and very subdiffusive FBM exponents such that α − 1 + 2H < 0, shown for the scaling exponents used in Fig. 7 for varying values of the time-step (see the legend).

Fig. 9 The same as in Fig. 8 but for somewhat larger Hurst exponent, plotted for different time-steps used in the simulations. The long-time MSD asymptote (18) is shown as the dashed line.

Fig. 10 Ratio of the TAMSD to MSD for FBM–SBM, with the asymptote (30) shown as the dashed line. Parameters are provided in the legend. At the last point, we find MSD(T) = TAMSD(T), as expected for the MSDs growing in time. For the model parameters in the blue region of Fig. 2 and for a stagnating MSD this statement regarding the last-point MSD-to-TAMSD equality is no longer valid, see also Fig. 7.³¹

Fig. 11 MSD growth in the simulations (solid symbols) as compared to the fitted MSDs, obtained via integrating the Gaussian PDFs extracted as a fit of the actual distributions of particle displacements (empty symbols connected by the lines). FBM and SBM parental exponents are given in the legend. The laws of the MSD growth are also consistent with (18).

Fig. 12 ACF of FBM–SBM for subdiffusive (panel a) and superdiffusive (panel b) parental FBM process, with the results being plotted for varying SBM scaling exponent and variable time shifts used in the computation. The legends report the values of the model parameters; t = 1 × δt.

Fig. 13 Scaling of the ACF tail of FBM–SBM for superdiffusive FBM (see the legend for actual parameters, t = 1 × δt), with expression (31) presented as the asymptote.

Fig. 14 The same as in Fig. 13 after the time-dependent ACF renormalization, shown for varying times t.

Acknowledgements

W. W. acknowledges the computational resources provided by the MPI-PKS in Dresden. A. G. C. thanks M. Assaf, M. Balcerek, A. Bodrova, N. Brilliantov, M. Javanainen, J.-H. Jeon, N. Korabel, L. Jauffred, D. Novikov, F. Spahn, M. Sperl, and M. Weiss for scientific correspondence. R. M. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG Grant ME 1535/12-1).

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Footnotes

† We remind the reader here that for FBM–HDP and SBM–HDP the scaling exponents of the MSD were found to be the products of the respective scaling exponents of the parental processes (see eqn (35) in ref. 93 and eqn (20) in ref. 91, correspondingly; see also Table 1). The position-dependent diffusion coefficient in these processes did not “interfere” with the properties of FBM-related noise and with the time-dependence of the SBM D_α(t). For FBM-SBM, the MSD exponent is the sum of the respective SBM and FBM exponents because both processes “mix” along the time axis introducing temporal changes of the diffusivity and temporal correlation of noise. We note also that for the combination FBM-HDP the TAMSDs scales always as [because the TAMSD of HDPs per se is always linear in Δ], while for SBM-HDP the TAMSD is always linear, [as the TAMSDs of SBM and HDPs are linear in Δ]. We refer the reader to eqn (36) in ref. 93 and eqn (22) in ref. 91 for these relations, respectively.

‡ The divergence of the respective MSD integral, as follows from eqn (16) for the displacement increments, is the reason for this. These MSD plateaus indicate a stalling dynamics of the particles: the subdiffusive SBM part progressively slows down the diffusivity and—in addition to that—the anticorrelated successive increments of the particle from parental subdiffusive FBM lead to nearly zero net displacement. The plateaus are realized in the region of FBM and SBM exponents where the cumulative MSD exponent of FBM–SBM is negative. Note also that for pure SBM, e.g., negative MSD exponents are outside of the region of allowed model parameters so that such stalling MSD situations were not observed.⁵⁵ Technically, however, the simulations of FBM–SBM in this parameter region presents no complications.

§ The prefactors in the MSDs and TAMSDs above are rather complicated, but the MSD-to-TAMSD ratio—characterizing the respective degree of ergodicity of a given process in this list—contains no prefactors. This ratio is thus a valuable tool to assess the nonergodicity of an unknown dataset of time-series, see Table 1. The absence of coefficients in these TAMSD-to-MSD ratios—proven in Fig. 10 for several values of the exponents of parental SBM and FBM processes—also renders the often tedious analytical calculations of the MSD and [especially] of the TAMSD as separate quantities less important.

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