Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity and aging

Aleksei Chechkin; Qing Wei; Ralf Metzler; Wei Wang; Yifa Tang

REVIEW 2 major objections 5 minor 106 references

Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity and aging

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For the fractional Langevin equation far from equilibrium, the time-averaged mean-squared displacement converges to the mean-squared increment rather than the mean-squared displacement in the regime $1/2<\alpha<3/2$—spurious nonergodicity…

desk verdict Solid fractional-α results, but the 'any α>1/2' claim misses the ordinary Brownian exception at α=1. read the letter →

arxiv 2412.11559 v1 pith:TGWE25YU submitted 2024-12-16 cond-mat.stat-mech

Qing Wei , Wei Wang , Yifa Tang , Ralf Metzler , Aleksei Chechkin This is my paper · ORCID

classification cond-mat.stat-mech MSC 60G2282C31 PACS 05.40.Fb05.40.-a

keywords fractionalLangevinequationRiemann-LiouvilleBrownianmotionFBMIIspuriousnonergodicitytime-averagedmeansquareddisplacementincrementaginganomalousdiffusion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Fractional Langevin equations are used to model diffusion with memory. This paper studies the version driven by white noise rather than by the equilibrium noise required by the fluctuation-dissipation theorem—the 'fractional Langevin equation far from equilibrium'—whose solution is Riemann-Liouville fractional Brownian motion. It establishes that for fractional-derivative order $1/2<\alpha<3/2$, the time-averaged mean-squared displacement (TAMSD) converges to the mean-squared increment (MSI, or structure function) rather than to the ensemble mean-squared displacement (MSD). The MSD and MSI differ by a constant prefactor, so a standard ergodicity test based on TAMSD-versus-MSD would incorrectly declare the process nonergodic even though its increments become stationary. For $\alpha\geq 3/2$ the first increments are genuinely nonergodic, but higher-order increments recover stationarity in successive $\alpha$-bands, and strong aging restores ergodicity in $1/2<\alpha<3/2$.

What carries the argument

The object carrying the argument is Riemann-Liouville fractional Brownian motion (RL-FBM, also called FBM II), defined as the fractional integral of white noise, $x(t)=\sqrt{2K_\alpha}\int_0^t \frac{(t-t')^{\alpha-1}}{\Gamma(\alpha)}\xi(t')dt'$, which is the zero-initial-condition solution of the FLEFE. The derivational machinery is the asymptotic expansion of the Fox H-function, reached through the hypergeometric form $_2F_1$, in Appendices A and B. The load-bearing identity is the long-time stationary limit of the MSI integral, $\int_0^\infty [(1+s)^{\alpha-1}-s^{\alpha-1}]^2\,ds+\frac{1}{2\alpha-1}=\frac{\Gamma(\alpha)^2}{\Gamma(2\alpha)|\cos(\pi\alpha)|}$ for $1/2<\alpha<3/2$, which produces the spurious-nonergodicity prefactor; the power-logarithmic variant of the H-function expansion handles the borderline case $\alpha=3/2$.

What would settle it

Simulate RL-FBM trajectories via the discretized integral in Appendix C for $\alpha=0.8$ and $\alpha=1.2$; fix a small lag $\Delta$ and measure the ratio $\langle\delta^2(\Delta)\rangle/\langle x^2(\Delta)\rangle$ as $T/\Delta$ grows. The paper's claim requires this ratio to tend to $R(\alpha)=\frac{(2\alpha-1)\Gamma(\alpha)^2}{\Gamma(2\alpha)|\cos(\pi\alpha)|}$ (about 1.12 for $\alpha=0.8$) rather than to 1. Equivalently, for $\alpha>3/2$ the mean TAMSD must scale as $T^{2\alpha-3}\Delta^2$ and not converge to the MSD; a direct evaluation of the exact H-function expression (41) that shows convergence to the MSD for large $T$ would falsify the central claim.

Watch

Extended reading notes

Core claim

The central discovery is that RL-FBM, the solution of the FLEFE, displays spurious nonergodicity. For all $\alpha>1/2$ the mean TAMSD and the MSD do not coincide even in the limit of long trajectories; specifically, for $1/2<\alpha<3/2$ the mean TAMSD converges to the stationary MSI, $\langle\delta^2(\Delta)\rangle\sim \langle x^2_\Delta(t)\rangle\sim \frac{2K_\alpha}{\Gamma(2\alpha)|\cos(\pi\alpha)|}\Delta^{2\alpha-1}$, which differs from the MSD $\langle x^2(t)\rangle = \frac{2K_\alpha}{(2\alpha-1)\Gamma(\alpha)^2} t^{2\alpha-1}$ by the factor $R(\alpha)=\frac{(2\alpha-1)\Gamma(\alpha)^2}{\Gamma(2\alpha)|\cos(\pi\alpha)|}$. Since the increments become asymptotically stationary in this regime, the TAMSD-to-MSI convergence is the physically appropriate ergodic criterion, and the TAMSD-to-MSD mismatch is therefore spurious. For $\alpha\geq 3/2$, the first-order increments are not ergodic, but the $(n+1)$th-order increments restore stationarity in the bands $(2n+1)/2<\alpha<(2n+3)/2$; under strong aging ($t_a\gg T$) the aged MSD and aged mean TAMSD coincide for $1/2<\alpha<3/2$, restoring ergodicity.

Load-bearing premise

The load-bearing premise is that the Fox H-function asymptotic expansions in Appendix B (Eqs. B9 and B14) remain valid uniformly in $\alpha$, including the special values $\alpha=1$, $\alpha=3/2$, and integer $\alpha$, with the stated cancellations of leading terms; the paper gives no error bounds or uniformity proof, so if these expansions fail in any regime the claimed prefactor in Eq. (43), and hence the spurious-nonergodicity ratio, would be affected.

Editorial extensions

If this is right

For single-particle-tracking data, a TAMSD that does not match the MSD is not by itself evidence of nonergodicity: for processes with asymptotically stationary increments, the structure function (MSI) is the correct reference, and RL-FBM is ergodic in that sense for 1/2 < α < 3/2.
In the regime α ≥ 3/2, ergodicity is recovered by moving to higher-order increments: the (n+1)th-order MSI becomes stationary in each band (2n+1)/2 < α < (2n+3)/2, so differentiating (or incrementing) trajectories more times restores standard ergodicity tests.
Strong aging (t_a ≫ T) makes the aged MSD and aged TAMSD coincide and the increment autocovariance function become lag-time-only dependent for 1/2 < α < 3/2; experiments that wait long enough after preparing a system will see apparent ergodicity even though the unaged process fails the usual test.
The FLEFE is a minimal Langevin-type model for systems that violate the fluctuation-dissipation theorem, remaining well-defined for all α > 1/2 and thus covering subdiffusive, superdiffusive, and ballistic-like regimes within one equation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the ratio R(α)=⟨δ²⟩/⟨x²⟩ = (2α−1)Γ(α)²/[Γ(2α)|cos(πα)|] is a closed-form function of α alone, an experimenter could use an observed TAMSD/MSD ratio to estimate α and to test whether a system is better described by RL-FBM than by FBM or FLE (which have R=1); the paper derives R(α) but does not propose this estimator.
The paper does not analyze confining potentials, but its closed-form MSI and TAMSD for the free process provide the ingredients to derive corresponding results for a harmonically trapped FLEFE, which would make contact with optical-tweezer experiments.
The strong-aging result implies that in systems where measurements start well after preparation (e.g., long climate or financial records), apparent ergodicity may reflect the aging time rather than equilibrium; distinguishing these requires varying the delay between preparation and measurement, an experimental protocol the paper does not discuss.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a useful, largely self-contained derivation of the TAMSD for Riemann-Liouville FBM (the FLEFE solution), covering all fractional orders and adding a clean aging analysis. The core result—that for 1/2<α<3/2 the mean TAMSD converges to the MSI rather than the MSD—is new and the simulations back it up. That said, the universal statements 'any α>1/2' and 'in the entire domain' are literally false at α=1, where the process is ordinary Brownian motion and Eq. (27) and Eq. (43) both give 2K₁Δ, so TAMSD=MSD and there is no spurious nonergodicity. The paper's own Appendix B treats α=1 separately and produces exactly the Brownian TAMSD, so this is an oversight in the wording, not a flaw in the fractional-order math. It should be fixed by excluding α=1 (or explicitly labeling it the ergodic Brownian limit) throughout the abstract and conclusions.

What's new: the TAMSD for all α>1/2 (Eqs. 43-46), the higher-order increment stationarity (Eqs. 52-53), and the strong-aging ergodicity restoration. The exact expressions for MSI and MSD were already in Lim and Eab-Lim, but the TAMSD is new, as is the distinction between MSD and MSI for nonstationary increments. The derivations are careful and the H-function machinery is used competently, even if the asymptotic expansions in App. B lack explicit uniformity bounds. That's a minor concern given the simulation agreement.

Soft spots: (i) the α=1 exception, which should be in the abstract; (ii) the higher-order increment argument relies on the approximation Δx(t;τ)/τ ≈ dx/dt without error control—this is heuristic but the result seems right and is verified numerically; (iii) no error bars or code are provided for the simulations, which is a reproducibility nit, not a correctness issue.

Bottom line: worth a serious referee. The fractional-order results are likely correct, and the paper fills a real gap in the RL-FBM literature. Send it to review with a request to fix the α=1 caveat and tighten the higher-order justification. I'd take it to reading group and would cite it for the TAMSD formulas, with the α=1 caveat noted.

Referee Report

2 major / 5 minor

Summary. This paper analyzes the fractional Langevin equation far from equilibrium (FLEFE), whose zero-initial-condition solution is Riemann-Liouville fractional Brownian motion (RL-FBM). The authors derive exact and asymptotic expressions for the MSD, the mean-squared increment (MSI), the time-averaged MSD (TAMSD), higher-order increments, and aging effects. The central claim is that for 1/2<α<3/2 the mean TAMSD does not converge to the MSD but instead to the stationary MSI, leading to ``spurious nonergodicity,'' while for α≥3/2 the first increments are nonergodic and higher-order increments restore stationarity in appropriate intervals of α. Strong aging is claimed to restore ergodicity in 1/2<α<3/2. The main results are derived via hypergeometric and H-function expansions and are compared with numerical simulations.

Significance. If the results are properly qualified, the paper is a valuable contribution: it provides closed-form expressions for central observables of RL-FBM with no fitted parameters, it draws attention to the distinction between MSD and MSI for processes with nonstationary increments, and it offers testable predictions for single-particle-tracking data. The higher-order increment stationarity and aging analysis extend the known properties of RL-FBM. The paper is self-contained, and the simulation results support the main asymptotic formulas. However, the headline universal claim fails at α=1, which is inside the stated spurious-nonergodicity interval, and the higher-order increment derivation relies on an unproved derivative approximation; both points need to be addressed before the claims can be accepted as stated.

major comments (2)

[Abstract; Sec. IV.C; Eq. (46)] The statement that the TAMSD and MSD differ for every α>1/2, and that spurious nonergodicity occurs throughout 1/2<α<3/2, is false at α=1. At α=1 the FLEFE reduces to ordinary Brownian motion: Eq. (27) gives ⟨x²(t)⟩=2K₁t, while Eq. (33) and Eq. (43) both give 2K₁Δ, so the TAMSD equals the MSD and the ratio is 1. This is confirmed by the special case in Appendix B: Eq. (B22) yields I₁=2K₁T(1−Δ/T), which together with Eq. (B1) gives ⟨δ²(Δ)⟩=2K₁Δ. The universal quantifiers ``any α>1/2'' and ``the entire domain'' in the abstract, Sec. IV.C, and the conclusions must therefore be qualified by excluding α=1, or α=1 must be explicitly identified as the ordinary ergodic Brownian limit.
[Sec. V; Eq. (50)] The derivation of higher-order increment stationarity rests on the approximation Δ^{(1)}x(t;τ₁) ≈ τ₁ dx(t)/dt for t≫τ₁, stated in Eq. (50), without an error estimate. Since Eq. (53), the general higher-order increment result, is a central claim, the authors should justify this replacement in the mean-square sense, for example by showing that the neglected terms are subdominant in the appropriate limit, or by deriving the second-order increment result directly from the stochastic integral representation. The numerical agreement in Fig. 3 is encouraging, but it does not replace a bound or a rigorous statement of the approximation's validity regime.

minor comments (5)

[Title] The title contains a typo: ``Rieman n-Liouville'' should be ``Riemann-Liouville''.
[Sec. V, text after Eq. (52)] The word ``revels'' should be ``reveals'' in the sentence describing the stationarity of the second-order increment.
[Appendix B] The text contains the typo ``TMASD'' where ``TAMSD'' is meant; this should be corrected.
[Fig. 2 caption] The caption says the mean TAMSD converges to the MSI ``in the strong aging limit,'' but the surrounding text concerns the long-time limit T/Δ≫1; please clarify whether aging is involved in this figure.
[Appendix D] The sentence ``the relation δ ≫ δ is not perfectly fulfilled'' appears to contain a typo; presumably it should read ``δ ≪ Δ'' or ``δ ≪ Δ is not perfectly fulfilled.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the TAMSD/MSI comparison is derived from the model's own definitions and covariance calculations, with a non-circular correctness caveat at alpha = 1.

full rationale

The derivation chain is self-contained. The FLEFE solution (23) is taken from Eab and Lim [39], but the paper's new quantities are computed directly: the MSD in Eq. (27) follows from the two-time covariance (24) and the hypergeometric formula (26); the MSI in Eqs. (28)-(30) is an explicit covariance calculation; and the mean TAMSD in Eq. (41) is the time average of that MSI. No empirical data, fitted coefficient, or assumed prefactor enters these formulas. The asymptotic reductions in Eqs. (33), (39), (43)-(46), and (B12)-(B17) are obtained from standard H-function and hypergeometric expansions, and the simulations in Figs. 1-4 are generated from the discretized integral representation (C3), so they are independent numerical checks rather than inputs. The claim that the TAMSD converges to the MSI rather than the MSD is not circular: it follows because the MSI is first shown to become t-independent (Eqs. (32)-(33)), after which the long-time average of the MSI necessarily approaches that stationary value; the MSD is a different, separately computed quantity. Self-citations such as [15], [42], and [65] supply definitions and context, but none of the central prefactors or scaling laws is justified solely by those citations. One non-circular caveat should be recorded: the universal wording in the abstract and Section IV.C that the TAMSD and MSD 'differ in the entire domain alpha > 1/2' is false at alpha = 1, where Eq. (27) and Eq. (43) both give 2*K1*Delta and the ratio equals one; this is a correctness overstatement, not a circularity. A further rigor limitation is that the Appendix B H-function expansions are used without explicit uniform-error bounds, but that affects proof completeness rather than the direction of the derivation.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The derivation is self-contained: no parameters are fitted to data, and the only inputs are the model parameters K_alpha and alpha. The central formulas rely on standard fractional calculus and H-function identities, plus one heuristic approximation for higher-order increments and the conventional ergodicity criterion. No new physical entities are postulated.

free parameters (2)

K_alpha (noise intensity) = not fitted
Overall scaling prefactor in the FLEFE; all observables are proportional to it. It is a model input, not determined by the paper.
alpha (fractional derivative order) = not fitted
Controls the memory exponent and the diffusion exponent 2α-1. The paper studies all α>1/2 and does not tune it to data.

assumptions (6)

domain assumption Fractional integral representation of the solution to the Caputo FDE (Eq. 22) is taken as the starting point; zero initial conditions are assumed.
This is standard for linear Caputo equations, but applying it to white noise defines the stochastic process; any other regularization could change results.
standard math Itô isometry and Gaussianity are used to compute variances and covariances.
These are standard tools for Gaussian stochastic integrals and are not proved in the paper.
domain assumption The asymptotic stationarity of RL-FBM increments for 1/2<α<3/2 is inherited from Lim [41], and the paper uses it to call the MSI stationary.
The paper's own Appendix A establishes the MSI asymptotics, but full distributional stationarity of increments is not proven in the non-aged limit.
standard math H-function and hypergeometric expansion formulas from Prudnikov and Kilbas-Saigo are used without proof in Appendices A and B.
These are standard reference identities, but no derivations or uniformity checks are provided.
ad hoc to paper For higher-order increments, the approximation x(t+τ)-x(t) ≈ τ dx(t)/dt is assumed for t≫τ (Eq. 50).
This is stated without error bounds and is used to derive stationarity of the (n+1)th order MSI, Eqs. (52)-(53).
domain assumption Ergodicity is assessed via the equality of mean TAMSD and MSD (Eq. 12), the conventional single-particle-tracking criterion.
The spurious-nonergodicity conclusion is framed within this criterion; if one instead compares TAMSD with MSI, the nonergodicity disappears for asymptotically stationary increments.

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Pith. "Pith review of Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity and aging." pith.science (2026). https://pith.science/paper/TGWE25YU

@misc{pith2026241211559,
  author       = {Pith},
  title        = {Pith review of: Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity and aging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TGWE25YU}},
  note         = {Machine review of arXiv:2412.11559}
}

abstract

We consider the fractional Langevin equation far from equilibrium (FLEFE) to describe stochastic dynamics which do not obey the fluctuation-dissipation theorem, unlike the conventional fractional Langevin equation (FLE). The solution of this equation is Riemann-Liouville fractional Brownian motion (RL-FBM), also known in the literature as FBM II. Spurious nonergodicity, stationarity, and aging properties of the solution are explored for all admissible values $\alpha>1/2$ of the order $\alpha$ of the time-fractional Caputo derivative in the FLEFE. The increments of the process are asymptotically stationary. However when $1/2<\alpha<3/2$, the time-averaged mean-squared displacement (TAMSD) does not converge to the mean-squared displacement (MSD). Instead, it converges to the mean-squared increment (MSI) or structure function, leading to the phenomenon of spurious nonergodicity. When $\alpha\ge 3/2$, the increments of FLEFE motion are nonergodic, however the higher order increments are asymptotically ergodic. We also discuss the aging effect in the FLEFE by investigating the influence of an aging time $t_a$ on the mean-squared displacement, time-averaged mean-squared displacement and autocovariance function of the increments. We find that under strong aging conditions the process becomes ergodic, and the increments become stationary in the domain $1/2<\alpha<3/2$.

Figures

Figures reproduced from arXiv: 2412.11559 by the authors.

**Figure 2.** FIG. 2. Ratio [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3. Simulations (symbols) for the second order MSI of RL- [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Simulations and analytical results for the aged MSD a [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

106 extracted references · 80 canonical work pages

[1]

Case 1/ 2 < α < 3/ 2 In the long time limit t ≫ ∆ , the integral in Eq. ( 28) converges as [ 67] Lα ( t ∆ ) + 1 2α − 1 ≈ ∫ ∞ 0 [ (1 + s)α −1 − sα −1] 2 ds + 1 2α − 1 = Γ( α )2 Γ(2 α )| cos(πα )| , (32) and thus we arrive at the the stationary MSI approximated as [ 41] ⟨x2 ∆ (t)⟩ ∼ 2Kα Γ(2 α )| cos(πα )| ∆ 2α −1, (33) which depends solely on the lag time ∆...
[2]

Case α > 3/ 2 For α > 3/ 2, the integral in Eq. ( 28) asymptotically reads Lα ( t ∆ ) = ∫ t/ ∆ 0 s2α −2[ (1 + s−1)α −1 − 1 ] 2 ds ∼ (α − 1)2 2α − 3 ( t ∆ )2α −3 , (34) and thus the MSI is given by ⟨x2 ∆ (t)⟩ ∼ 2(α − 1)2Kα (2α − 3)Γ( α )2 t2α −3∆ 2. (35) In this range of α in the FLEFE, the MSI depends ballistically on the lag time ∆ and keeps a dependence...
[3]

( 33) and ( 35) that the case α = 3/ 2 needs to be discussed separately

Case α = 3/ 2 Apparently, it follows from Eqs. ( 33) and ( 35) that the case α = 3/ 2 needs to be discussed separately. In this case the MSI ( 9) can be rewritten as ⟨x2 ∆ (t)⟩ = 8K3/ 2 π [ L3/ 2 ( t ∆ ) + 1 2 ] ∆ 2, (36) where the integral L3/ 2(z) can be evaluated explicitly as L3/ 2(z) = ∫ z 0 (√ 1 + s − √s )2 ds = z2 + z + 1 2 [ sinh−1(√z) − √z √ z + ...
[4]

( B12) in App

Case 1/ 2 < α < 3/ 2 From the derivation of Eq. ( B12) in App. B we obtain the mean TAMSD of RL-FBM in Eq. ( 41) in the long time limit T ≫ ∆ in the form ⟨ δ2(∆) ⟩ ∼ 2Kα Γ(2 α )| cos(πα )| ∆ 2α −1. (43) The mean TAMSD in this range of α deviates from the MSD ( 27) in the prefactor but has the same form as the MSI (33)
[5]

( B13) in App

Case α > 3/ 2 According to the derivation of Eq. ( B13) in App. B we obtain the mean TAMSD ( 41) of RL-FBM in the long time limit T ≫ ∆ as ⟨ δ2(∆) ⟩ ∼ (α − 1)Kα (2α − 3)Γ( α )2 T 2α −3∆ 2. (44) In this case the mean TAMSD depends ballistically on the lag t ime ∆ and keeps a dependence on the measurement time T with the scaling T 2α −3
[6]

( B14), one can obtain the approximation of the TAMSD (see the derivation of Eq

Case α = 3/ 2 Using the explicit power-logarithmic expansion of the H-function for α = 3/ 2 as shown in Eq. ( B14), one can obtain the approximation of the TAMSD (see the derivation of Eq. ( B17) in App. B for details) when T / ∆ ≫ 1, ⟨ δ2(∆) ⟩ ∼ 2K3/ 2 π ∆ 2 ln ( T ∆ ) . (45) 9 (a) 2 4 6 8 10 0 5 10 15 20 25 (b) 2 4 6 8 10 0 50 100 150 200 (c) 10-1 100 1...

2021
[7]

Case 1/ 2 < α < 3/ 2 For 1/ 2 < α < 3/ 2, for strong aging ( ta ≫ ∆ ) we use the following approximation for the ﬁrst integral in Eq. ( D5), ∫ ta ∆ 0 qα −1(1 + q)α −3dq ≈ ∫ ∞ 0 qα −1(1 + q)α −3dq = B(α, 3 − 2α ), (D9) 22 where the (complete) Beta function is deﬁned by [ 64] B(a, b ) = ∫ ∞ 0 sa−1(1 + s)−a−bds = Γ( a)Γ( b) Γ( a + b) . (D10) The second integ...
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( D5) diverges as t tends to inﬁnity

Case α > 3/ 2 For α > 3/ 2 and strong aging ( ta ≫ ∆ ), the ﬁrst integral in Eq. ( D5) diverges as t tends to inﬁnity. We rewrite the ﬁrst integral in two parts, ∫ ta ∆ 0 qα −1(1 + q)α −3dq = ∫ M 0 qα −1(1 + q)α −3dq + ∫ ta ∆ M qα −1(1 + q)α −3dq, for M ≫ 1. (D13) Then the integral in Eq. ( D13) is dominated by the second term and can be approximated as ∫...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Case 1/ 2 < α < 3/ 2 In the long time limit t ≫ ∆ , the integral in Eq. ( 28) converges as [ 67] Lα ( t ∆ ) + 1 2α − 1 ≈ ∫ ∞ 0 [ (1 + s)α −1 − sα −1] 2 ds + 1 2α − 1 = Γ( α )2 Γ(2 α )| cos(πα )| , (32) and thus we arrive at the the stationary MSI approximated as [ 41] ⟨x2 ∆ (t)⟩ ∼ 2Kα Γ(2 α )| cos(πα )| ∆ 2α −1, (33) which depends solely on the lag time ∆...

[2] [2]

Case α > 3/ 2 For α > 3/ 2, the integral in Eq. ( 28) asymptotically reads Lα ( t ∆ ) = ∫ t/ ∆ 0 s2α −2[ (1 + s−1)α −1 − 1 ] 2 ds ∼ (α − 1)2 2α − 3 ( t ∆ )2α −3 , (34) and thus the MSI is given by ⟨x2 ∆ (t)⟩ ∼ 2(α − 1)2Kα (2α − 3)Γ( α )2 t2α −3∆ 2. (35) In this range of α in the FLEFE, the MSI depends ballistically on the lag time ∆ and keeps a dependence...

[3] [3]

( 33) and ( 35) that the case α = 3/ 2 needs to be discussed separately

Case α = 3/ 2 Apparently, it follows from Eqs. ( 33) and ( 35) that the case α = 3/ 2 needs to be discussed separately. In this case the MSI ( 9) can be rewritten as ⟨x2 ∆ (t)⟩ = 8K3/ 2 π [ L3/ 2 ( t ∆ ) + 1 2 ] ∆ 2, (36) where the integral L3/ 2(z) can be evaluated explicitly as L3/ 2(z) = ∫ z 0 (√ 1 + s − √s )2 ds = z2 + z + 1 2 [ sinh−1(√z) − √z √ z + ...

[4] [4]

( B12) in App

Case 1/ 2 < α < 3/ 2 From the derivation of Eq. ( B12) in App. B we obtain the mean TAMSD of RL-FBM in Eq. ( 41) in the long time limit T ≫ ∆ in the form ⟨ δ2(∆) ⟩ ∼ 2Kα Γ(2 α )| cos(πα )| ∆ 2α −1. (43) The mean TAMSD in this range of α deviates from the MSD ( 27) in the prefactor but has the same form as the MSI (33)

[5] [5]

( B13) in App

Case α > 3/ 2 According to the derivation of Eq. ( B13) in App. B we obtain the mean TAMSD ( 41) of RL-FBM in the long time limit T ≫ ∆ as ⟨ δ2(∆) ⟩ ∼ (α − 1)Kα (2α − 3)Γ( α )2 T 2α −3∆ 2. (44) In this case the mean TAMSD depends ballistically on the lag t ime ∆ and keeps a dependence on the measurement time T with the scaling T 2α −3

[6] [6]

( B14), one can obtain the approximation of the TAMSD (see the derivation of Eq

Case α = 3/ 2 Using the explicit power-logarithmic expansion of the H-function for α = 3/ 2 as shown in Eq. ( B14), one can obtain the approximation of the TAMSD (see the derivation of Eq. ( B17) in App. B for details) when T / ∆ ≫ 1, ⟨ δ2(∆) ⟩ ∼ 2K3/ 2 π ∆ 2 ln ( T ∆ ) . (45) 9 (a) 2 4 6 8 10 0 5 10 15 20 25 (b) 2 4 6 8 10 0 50 100 150 200 (c) 10-1 100 1...

2021

[7] [7]

Case 1/ 2 < α < 3/ 2 For 1/ 2 < α < 3/ 2, for strong aging ( ta ≫ ∆ ) we use the following approximation for the ﬁrst integral in Eq. ( D5), ∫ ta ∆ 0 qα −1(1 + q)α −3dq ≈ ∫ ∞ 0 qα −1(1 + q)α −3dq = B(α, 3 − 2α ), (D9) 22 where the (complete) Beta function is deﬁned by [ 64] B(a, b ) = ∫ ∞ 0 sa−1(1 + s)−a−bds = Γ( a)Γ( b) Γ( a + b) . (D10) The second integ...

[8] [8]

( D5) diverges as t tends to inﬁnity

Case α > 3/ 2 For α > 3/ 2 and strong aging ( ta ≫ ∆ ), the ﬁrst integral in Eq. ( D5) diverges as t tends to inﬁnity. We rewrite the ﬁrst integral in two parts, ∫ ta ∆ 0 qα −1(1 + q)α −3dq = ∫ M 0 qα −1(1 + q)α −3dq + ∫ ta ∆ M qα −1(1 + q)α −3dq, for M ≫ 1. (D13) Then the integral in Eq. ( D13) is dominated by the second term and can be approximated as ∫...