Universality in Anomalous Diffusion: Power Laws and Faster-Decaying Tails

Introduction

Great theoretical efforts have been devoted to unveiling the physical mechanisms that induce anomalous diffusion (see, e.g., Refs. [1, 2, 3] and references therein) in systems where the stochastic motion cannot be described by standard Brownian motion. They have been accompanied by a rapid increase of highly resolved experimental observations, particularly of video microscopy and particle tracking, as well as by ever more refined numerical simulations of ensembles of trajectories [4]. Processes exhibiting anomalous diffusion are observed in a broad range of phenomena, including molecules moving in the living cell [5, 6], dynamics on cell membranes [7], crowded environments [3], soil transport [8], heat transport in low-dimensional systems [9], and certain classes of billiards [10, 11], to mention but a few.

In these processes the probability distribution function (PDF) of the displacement is commonly non-Gaussian, meaning that the signatures of anomalous diffusion appear not only in the mean square displacement (MSD), but essentially in all the moments of the distribution. Deviations from Gaussian behavior are observed in the tails and, in certain dynamics, the large fluctuations of the tails scale differently from the small fluctuations described by the bulk of the distribution. In general, denoting by $r(t):=|x(t)-x(0)|$ the displacement accumulated over a time $t$—a random variable whose generic value we denote by $r$—the time scaling of its $q$th moment can be written as

where $\nu_q=1$ indicates normal diffusion, as derived from Fick’s law, and $\nu_q\ne1$ anomalous diffusion. A scale-invariant transport with a constant $\nu_q\ne1$ is called (weakly) anomalous. While, if $\nu_q$ varies with the moment order $q$, scale invariance breaks down, and transport is said to be strongly anomalous [12, 13]. As already noticed in [12] there are processes where $\nu_{q=2}=1$ but $\nu_q$ depends on the moment $q$; therefore a strong anomalous behavior can be observed even if $\langle r^2\rangle\sim t$ and the MSD seems to suggests a regular diffusion.

The most commonly investigated strong anomalous situation is characterized by two regimes, in which the power of $t$ is given by two different expressions that are linear in the moment order:

where $\theta$, $\zeta$, and $q_c$ are given parameters. Indeed, strong anomalous diffusion repeatedly arises as the combination of anomalous diffusion determining a local scale invariance for the small fluctuations of the displacement (with $q\le q_c$), and rare but long-lasting ballistic excursions determining a faster scaling of the large fluctuations, for which $\zeta=1$. The piecewise-linear behavior of Equation (1b) has been observed in the transport in polygonal billiards [14, 15, 16], billiards with infinite horizon [10, 17, 18], diffusion in laser-cooled atoms [19], one-dimensional maps [20, 21], intermittent maps [12, 22], experiments on the mobility of particles inside living cancer cells [23], the motion of particles passively advected by dynamical membranes [24], running sand piles [25], and stochastic models of inhomogeneous media [26, 27], among a host of other phenomena. A great deal of insight into strong anomalous diffusion has been gained by means of generalizations of the central limit theorem in terms of non-normalizable densities [28, 29, 30, 31] and the big jump principle [32, 33, 34]. The above scenario is not unique—for instance, in the bulk-mediated diffusion on lipid bilayers of Ref. [35] the spectrum of moments is not piecewise linear—but the double linear scaling of Equation (1b) is the most commonly reported situation in experimental observations and theoretical studies, and it is the one investigated here, for a set of different deterministic and stochastic systems.

We revisit the theory of strong anomalous diffusion from the point of view of scaling theory [36] and asymptotic matching [37] of the PDF $P(r,t)$ of the displacement, developing the approach of Ref. [38], which is summarized below. In that paper, the process $R(t)$, which takes values $r$, was assumed to evolve on two different asymptotic scales, described by two rescaled versions of the displacement, $z := r/t^{\theta}$ and $y := r/t^{\zeta}$, which pertain to two different regions of space: for sufficiently large times $t$, the PDF is subdivided into two branches, corresponding to two distinct asymptotic regimes, which coincide over an interval that gets larger and larger as $t$ grows. This was formalized through the following ansatz:

where $\theta$, $\iota$, $\zeta$, and $c$ are four parameters obeying $0 < \theta< \iota< \zeta$ and $c > 0$, and the scaled variables $z$ and $y$ concern the “bulk” and the “tail” of $\mathcal{P}$, respectively defined by $r \in[0,ct^{\iota})$ and $r \in(ct^{\iota},\infty)$. The functions $\mathcal{F}$ and $g\mathcal{J}$ are non-negative and integrable on their respective domains, so that $\mathcal{P}$ is non-negative and normalized over the half real line, for all $t$. The domain of $\mathcal{F}(z)$ grows without bounds with $t$, since the value of $z$ at its right border, $ct^{\iota-\theta}$, approaches infinity in the large-time limit. Analogously, the domain of $\mathcal{J}(y)$ extends to arbitrarily small values, since the smallest admissible value $ct^{\iota-\zeta}$ of $y$ approaches 0 for $t \to\infty$. Throughout the paper, only these three variables are needed: $r$ is the physical displacement, while $z$ and $y$ measure it in units of the bulk scale $t^{\theta}$ and of the tail scale $t^{\zeta}$, respectively.

Here, we illustrate first the validity of this hypothesis for the Spatial Diffusion of Cold Atoms (SDCA) in shallow optical lattices, an experimentally realized system of anomalous transport that is described in § 3.1. Figure 1 shows how, in this case, the scaling function $\mathcal{F}(z)$ takes the form of a power law for $z > z_c \simeq2$, and $\mathcal{J}(y)$ follows a power law for $y < y_c \simeq0.25$. Reference [38] then used the fact that the two branches of $\mathcal{P}(r,t)$ overlap as a power law in the interval $z_c t^{\theta} \lesssim r \lesssim y_c t^{\zeta}$, and that this interval expands for increasing $t$. The PDF thus enjoys asymptotic matching: as long as $\theta< \iota< \zeta$ and $t$ is large, the matching position of the two branches, $r = ct^{\ell}$, resides well inside this interval, irrespective of the particular choice of $c$ and $\ell$.

The present paper builds on these results, as well as on those summarized in the next sections and in the appendices: the hyper-scaling relation (3d) with the ensuing piecewise-linear spectrum; the corrections to scaling of Appendix § 7; and the fitting scheme of Appendix § 8. These were tested in Ref. [38] on quenched disordered systems of the Lévy–Lorentz type, on Lorentz gases, and on polygonal billiards. Here we report new numerical results on the SDCA—an experimentally realized dynamics whose non-ballistic cutoff exponent puts the full three-exponent fitting scheme to work—together with the finite-sampling criterion of § 3.1; and the entire converse theory of § 4, which establishes that power-law tails, carried by a weight $g(t)$ that itself decays as a power of time, are not only sufficient but also necessary for strong anomalous diffusion. In particular for exponentially rare ballistic flights our analysis characterizes a phenomenon that we call transient imitation.

In § 2 we show how the piecewise-linear scaling of $\nu_q$ emerges as a consequence of Equation (3). Moreover, an asymptotic matching of the bulk and the tail of $P(r,t)$ can be imposed in the presence of power-law decay of $\mathcal{F}(z)$, $\mathcal{J}(y)$ and $g(t)$. This matching establishes universal hyper-scaling relations between the exponents characterizing the system. In particular the crossover moment $q_c$ equals the exponent $\alpha$ that describes the decay of the scaling functions. In § 3 we explore these findings for three models that feature anomalous transport: besides elaborating on the SDCA, we revisit the exact solution of the Slicer Map (SM) [13, 21], and we verify the theory on numerical data for Lévy walks (LW) [1, 31]. The distributions and the spectra of moments of all these systems are described by the theory exemplified in § 2. In § 4 we then present the converse theory: we prove that asymptotic matching itself forces the intermediate asymptotics—the large-$z$ tail of $\mathcal{F}(z)$— to be a power law. In particular if the bulk $\mathcal{F}(z)$ is Gaussian and the excursions are exponentially rare, the apparent crossover of moments drifts as the observation time grows and the moment spectrum is single-linear. For a Gaussian bulk coexisting with a power-law tail, instead, asymptotic matching cannot be realized: strong anomalous diffusion persists, with a spectrum of the form of Equation (1b), but the hyper-scaling relation between the crossover moment and the tail exponent is violated. § 5 is devoted to exactly solvable deterministic models, within the Slicer Map class and its variants, that realize all these scenarios. In § 6 we summarize our main findings and provide an outlook on future perspectives. Appendix § 9 contains the proof that no strong anomalous diffusion can arise when the tails of the distribution decay faster than any power law.

Scaling and Asymptotic Matching

In this section we summarize the results of Ref. [38], whose novel application to the SDCA is presented in § 3.1. We begin by assuming that at large times $t$ the time evolution of the bulk of the probability density obeys scaling with the scaling variable $z = r/t^\theta$ (Equation (3)), and that it assumes a power law for large displacements,

Asymptotic Matching and Hyper-Scaling

Given the definition (2), the power law (3a) may be written for the two branches of the PDF. The bulk is described by the scaling function

where the parameter $z_c$ is deduced from the data. For the tail, we rely on the function $g$ introduced in Equation (3), noting that Equation (3a) implies

Then, taking

one obtains

where $y_c$ is deduced from the data, like $z_c$ is. At late times the bulk and the tail of the distribution overlap as power laws in a wide interval whose width diverges asymptotically in time.

There are two important things to note here. Firstly, in terms of the scaled variables and distributions, the deviation from the power law can be pushed below any desired bound by selecting larger values of $z_c$ and smaller values of $y_c$. Even when $y_c \ll z_c$, it suffices to wait for large values of $t$ until the interval $z_c t^\theta< r < y_c t^\zeta$ is non-empty. Eventually, in the large-$t$ limit, the length of the interval diverges because $\theta< \zeta$. Secondly, in Equations (3b) and (3e) not only the exponent $-(\alpha+1)$ of the power laws, but also the prefactors, $\alpha k$, agree. This holds for every set of exponents $\{\alpha,\beta,\theta,\zeta\}$ characterizing the PDF. The constraint (3d) is a hyper-scaling relation that expresses the exponent $\beta$, governing the temporal decay of the tail amplitude, in terms of $\alpha$, $\zeta$, and $\theta$; the agreement of the prefactors will become the cornerstone of the data analysis presented in Appendix § 8.

One may wonder whether some non-algebraic shape—for instance, a Gaussian—could play the role of the intermediate asymptotics joining the bulk and the tail branches, with the power law (3a) suitably replaced. The answer is negative: the matching structure by itself selects power laws.

Proof. Taking the logarithm of (9) and differentiating with respect to $\ln r$ at fixed $t$ gives $\Lambda_\varphi(z)=\Lambda_\psi(y)$, where $\Lambda_f(u):=u f'(u)/f(u)$ is the logarithmic slope, $z=r t^{-\theta}$, and $y=r t^{-\zeta}$ are the scaled variable in the "bulk" and the "tail" of $P$. Fix $\bar{z}>z_c$ and set $r=\bar{z}t^\theta$: this point remains in the overlap for all sufficiently large $t$, while $y=\bar{z}t^{\theta-\zeta}$ sweeps, continuously in $t$, an interval accumulating at 0. Therefore, since every point in that interval can be written as $y=\bar{z}t^{\theta-\zeta}$, we have $\Lambda_\psi(y)=\Lambda_\varphi(\bar{z})$. Hence $\Lambda_\psi(y)$ is constant, on that interval; and since $\bar{z}$ is arbitrary, $\Lambda_\varphi(z)$ is constant on $(z_c,\infty)$ as well. Call the common value of the logarithmic slopes $-(\alpha+1)$. Integrating $\Lambda_f(u)=-(\alpha+1)$, we obtain $\varphi(z)=k_1 z^{-\alpha-1}$ and $\psi(y)=k_2 y^{-\alpha-1}$; integrability of $\mathcal{F}$ at large $z$, required by the normalization of $P$, imposes $\alpha>0$. Substituting back into (9) yields $k_1t^{\alpha\theta}=g(t)k_2t^{\alpha\zeta}$, whence $g(t)=(k_1/k_2)t^{-\alpha(\zeta-\theta)}$. Normalizing $g$ as in (7) absorbs the constant $k_1/k_2$, and gives $k_1=k_2=\alpha k$ together with the hyper-scaling relation $\beta=\alpha(\zeta-\theta)$. ◂

Proposition 1 shows that the power law (3a) is not an assumption of convenience: it is the only intermediate asymptotics compatible with the matching of two different self-similar branches, in the spirit of intermediate-asymptotics theory [36]. In this sense, power-law tails are necessary and, as it will be shown in the next subsection, sufficient — for a time-persistent, bi-linear spectrum of moments. That the absence of the tail branch, considered in the first alternative of the previous Corollary, leads to the single linear spectrum follows from the analysis carried out in Appendix § 9. An example with a Gaussian bulk is provided in § 4.1.

Moments

The $q^{\mathrm{th}}$ moment of the displacement is defined by

For large $t$, where $z_c t^\theta< c t^\iota< y_c t^\zeta$, the integral can be broken into three pieces: the bulk contribution up to $z_c t^\theta$, the intermediate power-law region, and the tail contribution beyond $y_c t^\zeta$. In view of Equation (3), in which the Jacobians of the coordinate transformations are already accounted for, and applying Equation (3a) in the intermediate region, the $q^{\mathrm{th}}$ moment can be approximated asymptotically in time as

The first two integrals take fixed, finite values that do not depend on time. Moreover, for $q\ne\alpha$ the last integral in Equation (10) can be expressed as the difference of two power laws with the same exponents as those appearing in front of the other two integrals. Collecting terms with the same time dependence provides

with

For $q=\alpha$, on the other hand, the third integral in Equation (10) takes the form of a logarithm, and one finds

where

In either case the coefficients $N_q$ and $I_q$ take finite values as long as $\langle r^q\rangle_t$ exists, which is always the case at finite $t$ when the speed of the moving objects is bounded, so that the distribution has a cutoff. Moreover, in the scaling regime these values do not depend on the particular choice of $z_c$ and $y_c$, as one verifies by direct computation using Equations (3b) and (3e).

It is useful to note the geometric interpretation of the coefficients. Normalization of the PDF implies that $\mathcal{F}$, and hence $z^q\mathcal{F}$ for all $q>0$, is integrable at small $z$. For $q<\alpha$ the power-law tail is integrable at infinity, and $N_q$ amounts to the integral of $z^q\mathcal{F}(z)$ over $\mathbb{R}_+$; it takes a positive value. For $q>\alpha$, instead, $N_q$ amounts to the integral of the difference $z^q\mathcal{F}(z)-\alpha k z^{q-\alpha-1}$, and it can take positive or negative values, depending on whether $\mathcal{F}$ lies above or below the power law. Similarly, $I_q$ is positive for $q>\alpha$, where it amounts to the integral of $y^q\mathcal{J}(y)$ over $\mathbb{R}_+$, while for $q<\alpha$ it is the integral of $y^q\mathcal{J}(y)-\alpha k y^{q-\alpha-1}$, whose sign signifies which portion of $\mathcal{J}$ lies below or above the power law.

There are three scenarios for the asymptotic scaling of the moments. For $q<\alpha$ one has $q\zeta-\beta=q\zeta-\alpha(\zeta-\theta)<q\theta$, so that at large $t$ the right-hand side of Equation (6a) is dominated by the term containing $N_q$. Conversely, for $q>\alpha$ the exponent $q\zeta-\beta$ is the larger one, and the term containing $I_q$ dominates. For $q=\alpha$ the logarithmic term dominates in Equation (7). Asymptotically in time we may thus write

In particular, we recover the piecewise-linear spectrum (1b); moreover, a direct consequence of the asymptotic matching (i.e., of the hyper-scaling relation (3d)) is that the crossover moment $q_c$ equals $\alpha$. The result is purely based on the constraints imposed by matching the power laws of the bulk and of the tail of the distribution. In § 3 we present numerical and analytical results confirming that this theoretical framework applies to a vast variety of cases.

Comparison with Numerical Data

In applications one is often interested in the scaling of the MSD, $q = 2$. Figure 2a shows this data for the SDCA of § 3.1, for which $\zeta= 3/2$ and $q = 2$ lies above the crossover moment throughout the strong anomalous phase, so that $\nu_2 = 2\zeta- \beta$ takes the superballistic values of Equation (21), up to $\nu_2 \to3$ as $p_6 \to2/3$. It is then tempting to rewrite Equation (1b) in the form of a universal scaling curve that admits a data collapse for all moments and all parameter values of any model showing strong anomalous diffusion with two linear regimes of the spectrum:

Figure 2b shows this representation for a collection of moments obtained by fitting SDCA data at five values of $p_6$ (as discussed in § 3.1). The agreement is only approximate, in spite of clean numerical data: the fitted exponents deviate systematically from the universal curve, most strongly for the large moments, which are dominated by rare, extreme excursions, and for the moments closest to the crossover. These systematic errors are quantitatively explained by the corrections to scaling derived in Ref. [38] and recalled in Appendix § 7: every moment is the superposition of two power laws, Equation (6a), and the subleading one biases power-law fits up to the crossover time (A2), which grows without bounds as $q \to\alpha$, and becomes very large whenever $\zeta- \theta$ is small. Precisely where strong anomalous diffusion is most delicate, fitting individual moments is therefore unreliable, and the exponents are better determined from the PDF as a whole: the scheme of Ref. [38] that does so, recalled in Appendix § 8, collapses the bulk and the tails of the distribution simultaneously, and it is put to work on the SDCA in § 3.1.

Specific Models

We test the theory on three model systems featuring strong anomalous transport: the SDCA in shallow optical lattices [19, 39, 40], the Slicer Map (SM) [21], and Lévy walks (LW) [1, 31]. They feature different values of the exponents $\theta$ and $\alpha$, and they cover complementary situations: an experimentally realized Langevin dynamics with unbounded velocities and a non-ballistic cutoff exponent, $\zeta= 3/2$ (SDCA), an exactly solvable deterministic map (SM), and the classical annealed random walk with heavy-tailed flights, both with $\zeta= 1$ (LW). For reference, Table 1 also summarizes the exponents reported in Ref. [38]: the Lévy–Lorentz gas (LLg) [21, 26], the Lorentz gas with infinite (LGi) [10] and finite (LGf) [41] horizon, and polygonal billiard channels (PBC) [14, 15, 16, 17]. In the LGf, the tails of the distribution decay faster than any power law ($\alpha= \infty$), such that it cannot feature strong anomalous transport (cf. Appendix § 9 and § 4).

Spatial Diffusion of Cold Atoms

The SDCA describes the motion of an atom in a shallow optical lattice under Sisyphus laser cooling. In the semiclassical approximation [19, 39], the trajectory of an atom starting at the origin with $v(0)=0$ is determined by the Langevin equations

written in the dimensionless units of Ref. [19], where $\Gamma$ is Gaussian white noise and the momentum diffusion constant $D$, proportional to the ratio of the recoil energy and the lattice depth, is the control parameter. The cooling force is harmonic for $|v|\ll1$ but decays as $-1/v$ for $|v|\gg1$: fast atoms are only weakly damped, and perform long, nearly free flights. For $D<1$ the marginal velocity distribution reaches the power-law equilibrium $P_{\mathrm{eq}}(v)\sim|v|^{-1/D}$ [19, 42], and in the range $1/5<D<1$ the displacement features strong anomalous diffusion [19, 40, 43], with

The bulk of the packet obeys the Lévy scaling $r\sim t^{1/p_6}$ [40], while the tails are populated by atoms whose velocities have themselves diffused up to $v\sim t^{1/2}$, giving displacements $r\sim t^{3/2}$. A Richardson-like scaling that makes the SDCA the only system of Table 1 whose cutoff exponent is neither diffusive nor ballistic—so that $\zeta$ must be fitted. The non-normalizable limit function $\lim_{t\to\infty}t^{\zeta+\beta}P(r,t)$ is the infinite covariant density of Refs. [19, 42], whose exponent $1+1/(2D)$ coincides with our $\zeta+\beta=3p_6/2+1/2$, by the hyper-scaling relation Equation (3d). For the SDCA, Equation (1b) entails, in the strong anomalous phase,

Model	Parameter	$\theta$	$\zeta$	$\alpha$
LLg	$0 < p_1 < 1$	$(1+\alpha)^{-1}$	$1$	$p_1$
LLg	$1 < p_1$	$1/2$	$1$	$2p_1-1$
SM	$0 < p_2$	$0$	$1$	$p_2$
LW	$1 < p_3 < 2$	$p_3^{-1}$	$1$	$p_3$
LGi	$0 < p_4 < 1$	$1/2$	$1$	$2$
LGf	$1 < p_4$	$1/2$	$1$	$\infty$
PBC	$p_5 \approx1.0649$	$\approx0.5$	$1$	$\approx2$
PBC	$p_5 \approx0.935$	$\approx0.5$	$1$	$\approx3$
SDCA	$1 < p_6 < 2$	$p_6^{-1}$	$3/2$	$p_6$

and $\nu_2=1$ for $p_6\geq2$ ($D\leq1/5$, the Gaussian phase [19]). Remarkably, $\nu_2$ exceeds the ballistic value 2 for $p_6<4/3$: the MSD is superballistic, although the bulk of the packet spreads subballistically, with $\theta=1/p_6<1$—an extreme form of bi-scaling (Figure 2a; cf. § 2.3).

We integrated Equation (19) with the Euler–Maruyama scheme and time step 0.01, as in Ref. [19], for an ensemble of $10^5$ trajectories up to time $10^5$ at $D=0.4$, i.e. $p_6=7/6$, the value of Figure 1 of Ref. [19], and for ensembles of $3\times10^4$ to $10^6$ trajectories, with time horizons between $10^3$ and $10^5$, at ten further values of $D\in[0.15,0.8]$, spanning the strong anomalous phase and reaching into the Gaussian phase. Horizons and ensemble sizes are tied to each other by the requirement that the rare flights remain statistically resolved: since the fraction of ensemble members populating the tail branch decays as $g(t)=t^{-\beta}$ (Equation (3d)), an ensemble of $N$ trajectories tracks the flight branch of the moments only as long as $Nt^{-\beta}\gg1$, beyond which the sampled moments sag towards the bulk scaling—a finite-sample counterpart of the crossover time ((A2)). All moment exponents are therefore fitted over the latest window of one and a half decades satisfying $Nt^{-\beta}\geq30$, and the smallest values of $D$ in the strong anomalous phase, which carry the largest $\beta=3p_6/2-1$, are assigned the largest ensembles over the shortest horizons.

The joint collapse of $\mathcal{F}$ and $\mathcal{J}$ for the main run is shown in Figure 1 and Figure 3: the two branches match on the intermediate power law with exponent $-(1+p_6)=-13/6$, and the reduced plots single out $\alpha=p_6$ against the deliberately wrong values of Figure 3c,Figure 3d. The bulk-plateau fit of Appendix § 8 yields $\theta=0.845\pm0.006$ against the expected $1/p_6=6/7\simeq0.857$; a linear fit of the measured high-moment exponents, $q\nu_q/2=\zeta_q-\beta$ for $q\in[2.5,5]$, returns $\zeta=1.43\pm0.04$ and $\beta=0.76\pm0.05$ against the predicted $3/2$ and $3p_6/2-1=3/4$, the uncertainties reflecting the dependence on the admissible fit window. The hyper-scaling estimate $\alpha=\beta/(\zeta-\theta)=1.3\pm0.2$ is compatible with the crossover moment $p_6=7/6\simeq1.17$, with an accuracy limited by the slowly decaying corrections of Equation (A1): as anticipated in Appendix § 7, the PDF-based reduced plots of Figure 3 constrain $\alpha$ far more sharply than the moment fits do.

The scatter of the data in Figure 2b is due to the corrections to the power laws reported in Equation (A1). For the SDCA the relations $\zeta= 3/2$ and $\theta= 1/p_6$ imply

so that the correction terms decay slowly when $q \approx p_6$, and, at every $q$, when $p_6$ approaches $2/3$, where the bulk and tail scalings merge. Consistently, in Figure 2b the largest deviations from the universal curve occur for the moments closest to the crossover.

The Slicer Map

The Slicer Map was introduced in [21] as an analytically solvable dynamical system that features all regimes of anomalous transport, from localization to ballistic motion, and it was shown to exhibit strong anomalous transport. Its displacement autocorrelation function was obtained analytically [44] and, more recently, a universal scaling of the moments of displacement and of their correlation functions was established in Ref. [13]. From a coarse-grained perspective that focuses on the scaling of the distribution of the displacement, the SM is characterized as follows: the initial distribution is an approximation of a delta function at the origin, and transport proceeds in a one-dimensional space where the displacement from the origin is denoted by $r$. Each member $x$ of the ensemble is characterized by a life time $t_c(x)$. For $t < t_c(x)$ it propagates with unit velocity; then it stops and remains at its terminal position $r = t_c(x)$, up to negligible fluctuations, for all times $t > t_c(x)$. The distribution of life times obeys a power law, $\operatorname{Prob}(t_c > t) = (t/t_0)^{-p_2}$, where $t_0$ is the smallest admissible life time.

For $t > t_0$ the dynamics provides a PDF of displacements that can be written in the form of Equation (3) with

such that $\theta=0$, $\zeta=1$, and $\alpha=\beta=p_2$, in agreement with the hyper-scaling relation (7). In Ref. [44] the MSD was used to identify the relation between the parameters of the SM and of the Lévy–Lorentz gas, and it was pointed out that this entails a matching of all other moments. From the perspective of our present results, this is an immediate consequence of Equation (17). We return to the SM with more general, non-algebraic slicer sequences in Section 5.1.

Lévy Walks

Lévy walks are arguably the most elaborately studied model system featuring strong anomalous transport [28, 31, 45, 46]. We consider one-dimensional LWs where particles move with unit velocity along a line: all particles start at the origin and move in a given direction for a time sampled from a Lévy distribution $f(t) \propto p_3 t^{-p_3-1}$ with $t \in[1,\infty)$, reversing direction at the end of each step with probability $1/2$. For $1 < p_3 < 2$, the different scaling of the bulk ($\theta=1/p_3$) and of the tails ($\zeta=1$, $\beta=p_3-1$ and $\alpha=p_3$) of the PDF was first observed in this context, where the distribution $\mathcal{J}(y)$ of Equation (3c) has been denoted a non-normalizable density [28, 29, 31, 32, 33]. The moments of displacement, reviewed in Refs. [1, 31, 32, 33], follow the behavior anticipated in Equation (1b) with $\alpha=p_3$, $\zeta=1$, and $\theta=1/p_3$. The bulk $\mathcal{F}(r/t^\theta)$ perfectly matches the tail $g(t)\mathcal{J}(r/t^\zeta)$; in particular, $\beta=\alpha(\zeta-\theta)$. We simulated an ensemble of $2 \times10^{5}$ trajectories up to time $10^{5}$ for $p_{3} = \alpha= 1.5$ (data not shown). The joint collapse of $\mathcal{F}$ and $\mathcal{J}$ robustly recovers the exponents of Table 1 together with the analytical prefactors—in particular, the bulk-plateau fit yields $\theta= 0.665 \pm0.003$, against the expected $1/p_{3} = 2/3$—while reduced plots of the moments reveal that, even for such high-quality data, it is challenging to determine the asymptotic scaling of the moments by fitting power laws: the corrections to scaling of Equation (A1) remain visible up to the largest simulated times. The matching of the intermediate power laws of the bulk and of the tail, on the other hand, provides robust results in the super-diffusive regime $1 < p_{3} < 2$.

Gaussian Distributions

The theory of Section 2 rests on the algebraic intermediate asymptotics (3a). It is natural to ask what remains of the two-branch structure (2) when the distribution features no power-law regime—most importantly, when the scaling function is Gaussian. In this section we give four complementary answers: a Gaussian scaling function produces a single-linear spectrum of moments (Section 4.1); no non-algebraic shape can play the role of the intermediate asymptotics, because asymptotic matching by itself forces a power law (Section 4.1); ballistic excursions that are exponentially rare, rather than power-law distributed, produce only a transient imitation of strong anomalous diffusion, which can be recognized as such (Section 4.2); and power-law flights over a Gaussian bulk yield bona fide strong anomalous diffusion, with a crossover moment decoupled from the tail exponent (Section 4.3). Exactly solvable deterministic realizations of all these scenarios are constructed in Section 5.

Gaussian Scaling Functions Imply Single-Linear Spectrum

Consider the ansatz (2) with a Gaussian scaling function for the bulk, normalized on the half line,

The Gaussian decays faster than any power law, so that the assumptions of Appendix C are satisfied, and all moments are governed by the bulk branch:

Hence $q\nu_{q}/2 = \theta q$ for all moments: the spectrum is single-linear, transport is scale invariant, and, in the language of Table 1, $\alpha= \infty$, as for the Lorentz gas with finite horizon. For $\theta= 1/2$ one recovers normal diffusion; for $\theta\ne1/2$ one obtains weak anomalous diffusion with Gaussian statistics, as realized, for instance, by fractional Brownian motion with Hurst exponent $\theta= H \in(0,1)$ [47]. In this case no crossover moment exists, no logarithmic correction of the type of Equation (7) arises at any $q$, and the MSD alone determines the complete spectrum of moments.

Rare Ballistic Flights and Transient Bi-Scaling

Consider a diffusive Gaussian background coexisting with ballistic flights that survive scattering for exponentially distributed times. With unit speed ($\zeta= 1$) and $\theta= 1/2$, a minimal model is

(25) prescribes only the one-time distribution of the displacement: it selects no dynamics, and infinitely many evolutions share these marginals.¹ This structure—$\delta$ peaks at the ballistic front $r=t$, the largest displacement reachable at the finite speed of the dynamics, with exponentially decaying weight, plus a smooth part converging to a Gaussian—arises, for instance, in the telegraph (Goldstein–Kac) process, in which a particle moves at constant speed and reverses its direction at exponentially distributed times [48, 49]. Truncating the Gaussian at $r \le t$ does not affect the asymptotic statements below. The moments read:

with $N_q$ as in Equation (24): the same two-power-law structure as Equation (6a), except that the amplitude of the flight term decays faster than any power $t^{-\beta}$, consistently with Proposition 1. This difference has sharp consequences.

Firstly, for every fixed $q$ the flight term is asymptotically subdominant: as $t \to\infty$ the spectrum is single-linear, in agreement with § 4.1. Equating the logarithms of the two terms in Equation (26), and using $\ln N_q \simeq(q/2)[\ln(\sigma^2q)-1]$ for large $q$, one finds that the crossover structure is governed by a single constant $a$: the flight term dominates the value of the $q$th moment up to the time

where $a$ solves

Such a dominance window exists only if $\tau> 2\sigma^{2}$ (in units of unit speed), as one verifies by maximizing over $q$ at fixed $t$: rare flights leave a visible trace in the moments only when their persistence time exceeds the diffusive scale of the background. For the telegraph process itself, $\sigma^{2} = \tau$ at unit speed, so the criterion fails, and the flight term never dominates at late times; pronounced transients require flights that live much longer than the bulk scattering time. For $\tau= 10\sigma^{2}$ one finds $a \simeq20\sigma^{2}$, and the numerically determined dominance times $t_q = \{36, 76, 155, 315\}$ for $q = \{2, 4, 8, 16\}$ closely follow $t_q \simeq aq$; see Figure 4c.

Secondly, and most importantly for data analysis, consider fitting power laws over an observation window centered at a time $\bar{t}$. The local logarithmic slope of the flight term is

so that the moments dominated by it display an apparently linear second branch, $\zeta_q - \beta_{\mathrm{eff}}$, with effective offset $\beta_{\mathrm{eff}} = \bar{t}/\tau$. An observer interpreting such data through Equation (1b) would extract an apparent crossover moment $\alpha_{\mathrm{eff}} = \beta_{\mathrm{eff}}/(\zeta- \theta) = \bar{t}/[\tau(\zeta- \theta)]$, and the fitted pair $(\alpha_{\mathrm{eff}}, \beta_{\mathrm{eff}})$ would even satisfy the hyper-scaling relation (7). At any single observation time, therefore, the transient scenario is indistinguishable from genuine strong anomalous diffusion—even by the hyperscaling check. The give-away is the drift: both $\alpha_{\mathrm{eff}}$ and $\beta_{\mathrm{eff}}$ grow linearly with the observation time, whereas for power-law tails they are constants. Figure 4d displays the apparent spectrum of the minimal model at $\bar{t} = 50$, $100$, and $200$: the knee between the bulk line $\theta_q$ and the drifting branches $\zeta_q - \bar{t}/\tau$ moves to the right proportionally to $\bar{t}$. Between $\tilde{q}(\bar{t})$ and the knee, the local exponents even sag below the bulk line: these are the descending flanks of the humps of Figure 4c, where the flight term still dominates the value of the moment while already decaying. For $\bar{t}/\tau$ large enough, the sag drives the local exponents below zero—the $\bar{t} = 200$ curve of Figure 4d is negative precisely for $\tilde{q}(\bar{t}) \lesssim q < \bar{t}/\tau$: in the minimal model the mass leaving the flight term is redistributed instantaneously over the whole bulk, so that these moments transiently decrease. This is an artifact of that idealization. In the particle realizations of § 5.1 and § 5.3 the moments never decrease: for the slicer class, $\mathrm{d}\langle r^q\rangle_t/\mathrm{d}t = qt^{q-1}W(t) > 0$ exactly, and in the fly-and-diffuse model the stopped mass populates the region just behind the front before spreading, so that the measured exponents sag toward, but not below, zero.

The Gaussian bulk is, in fact, the marginal member of a whole family. For an exponential scaling function, $\mathcal{F}(z) = \sigma^{-1}e^{-z/\sigma}$, the bulk amplitudes $N_q = \sigma^q\Gamma(q+1)$ grow as $e^{q\ln q}$, twice as fast, in the exponent, as in the Gaussian case (24), and the flight term loses its niche within a finite time: maximizing the flight-to-bulk ratio over $q$ shows that the dominance window closes, up to subleading corrections, at

and that moments with $q \gtrsim\tau/\sigma^2$ are never dominated by the flights at all. The humps of the reduced moments are then bounded, and they all terminate by $t_{\mathrm{end}}$ (Figure 4a). The apparent second branch—which retains the local slope $\zeta q-\bar{t}/\tau$ of (29), since the latter is a property of the flight term alone—becomes a bounded bubble that rejoins the bulk line at large $q$, and it disappears altogether for $\bar{t}>t_{\mathrm{end}}$ (Figure 4b): with an exponential bulk, the imitation of strong anomalous diffusion does not drift, it dies.

In practice, three tests distinguish genuine strong anomalous diffusion from its finite-time imitation. (i) Stability: upon doubling the observation time, the fitted crossover moment of genuine strong anomalous diffusion is stable, while in the transient scenario it doubles (Gaussian bulk, cf. Equation (29)) or the second branch disappears altogether (heavier bulk tails, Remark 1). (ii) Collapse: the data analysis of Appendix § 8 collapses the tail branch with a single, time-independent exponent $\beta$ only for power-law tails; for an exponentially damped flight term, $t^{\zeta+\beta}P$ at fixed $y$ is proportional to $t^\beta e^{-t/\tau}$, so the curves at successive times sag for every choice of $\beta$, and no collapse is possible. (iii) Crossover times: reduced moments that reach their asymptotic plateaus at times growing linearly with the moment order, Equation (18), are the fingerprint of an exponentially damped ballistic component, whereas in genuine strong anomalous diffusion the crossover time (A2) diverges only for $q$ near $\alpha$. In the Lorentz gas with finite horizon the flight lengths are strictly bounded, so that there even the transient branch is cut off beyond a fixed time [41].

This analysis reinforces the practical message of Appendix § 7 and Appendix § 8: an apparent bi-linear spectrum at a single observation time does not, by itself, establish strong anomalous diffusion; the scaling analysis of the PDF, and the stability of the fitted exponents under an increase of the observation time, do.

A Gaussian Bulk with Power-Law Flights: the Crossover Rule

The third possibility is a Gaussian bulk coexisting with flights whose survival is itself a power law: take $W(t)=(1+t/t_0)^{-\beta}$, with $\beta>1$, so that the mean flight time is finite and the bulk spreads diffusively, $\theta=1/2$, $\zeta=1$, and let each member, when its flight ends, diffuse from the position it has reached. The marginals then carry three components, not the two of (25): between the Gaussian bulk and the ballistic front one finds the record of the terminated flights. By this we mean that, at unit speed, a flight ending at time $s$ also ends at position $r=s$, so the death times are recorded as positions. The three regions are then characterized by a probability density of the following form, where the middle component is new as compared to (25):

where $A(t)$ is an intermediate scale, $t^{1/2}\ll A(t)\ll t$, and $K(t)$ is a suitable normalization factor: since the mass beyond the bulk is exactly $\int_{A(t)}^\infty\mathcal{P}(r,t)\,\mathrm{d}r=W(A(t))$, which vanishes as $t\to\infty$, one has $K(t)\to1$. By recasting $\mathcal{P}(r,t)$ in the form of the ansatz (3), as for the Slicer Map of § 3.2, one can write:

with $z=r/t^{1/2}$ and $y=r/t$. Integrating $r^q$ against the three pieces, the bulk contributes $N_qt^{q/2}$, the record $\beta t_0^\beta t^{q-\beta}/(q-\beta)$, and the front $t_0^\beta t^{q-\beta}$, so that, to leading order,

where the record and the front combine into the first form, which holds for any survival function with regularly varying tail of index $-\beta$—by the rescaling of Appendix § 10—and the second form evaluates it to leading order in $t_0/t$. Below $\beta$, the record only adds a constant. Had the dead members instead been reinjected into the bulk, as in (25), the record would be absent and the flight term would carry unit amplitude, $t^qW(t)$: same exponents, same knee, but a different amplitude—the moments remember the bookkeeping. Each moment is dominated by the faster-growing branch, and the knee sits where the two growth rates cross:

In the matched scenario of § 2 the overlap of the two branches forces, in addition, $\alpha=q_c$: the crossover moment coincides with the exponent of the shared power law, and the hyper-scaling relation (7) expresses this coincidence. Here no such overlap exists—the Gaussian bulk has no power-law tail to share, realizing the second alternative of Corollary 1, in full consistency with Proposition 1; and Appendix § 9 is not contradicted, because its hypothesis—faster-than-power decay beyond the bulk scale—fails. The tail exponent is therefore fixed by the flight kine matics alone, and the crossover decouples from it: $q_c = 2\beta$, twice the tail exponent for the diffusive bulk. Two signatures separate the scenarios. At $q = q_c$ the matched case carries the logarithmic enhancement of Equation (7), while here the two amplitudes of (21) simply add, and no logarithm appears. And since the pair $(q_c, \beta)$ satisfies $\beta= q_c(\zeta- \theta)$ by construction, the hyper-scaling check alone cannot tell the two scenarios apart; the scaling analysis of the PDF can. Finally, we notice that $\nu_2 = 1$: the MSD grows linearly in time, and yet the diffusion is strongly anomalous. An exact realization, amplitudes included, is constructed in § 5.4.

Exactly Solvable Deterministic Models

The scenarios of § 4 are realized exactly, amplitudes included, within the Slicer Map class of § 3.2 and its fly-and-die and fly-and-diffuse variants, and the modified maps can be written explicitly. The coarse-grained dynamics of the whole family is specified by a single function, the survival probability $W(t)$ of the ballistic flights, and each behavior corresponds to a choice of $W$: power laws yield strong anomalous diffusion, faster-decaying survival functions its transient imitation, and a diffusing variant with power-law $W$ a Gaussian bulk coexisting with power-law flights.

Slicer Maps with Generic Survival Functions

The SM acts on the chain of unit cells $\hat{M} = [0,1] \times\mathbb{Z}$, with the ensemble initially uniform in cell $m = 0$, as [21]

where the “slicers” $\ell_{\lvert j\rvert}$ and $1-\ell_{\lvert j\rvert}$ cut cell $j$, and the original choice $\ell_{\lvert j\rvert} = ( \lvert j\rvert+ 2^{1/p_2})^{-p_2}$ yields the power-law survival of § 3.2. Points travel ballistically until their coordinate first falls on the inner side of a slicer, after which they bounce between two neighboring cells—the coarse-grained “death”. The mechanism only requires $\ell_0 = 1/2$, as in the original sequence, and monotonic decay. The coarse-grained dynamics is thus specified entirely by the survival function $W(t)$—the fraction of the ensemble still flying at time $t$, which is the survival probability $\operatorname{Prob}(t_c > t)$ of the life time $t_c$. For the Slicer Map, we have

whatever law describes $\ell_{\lvert j\rvert}$; $W$ may then be read as a step function of the continuous time $t$. Equivalently, in the fly-and-die formulation [13]—where the member of initial coordinate $x_0$, uniform on $(0,1]$, has position $x(x_0,t) = x_0 + t$ for $t \le t_c(x_0)$, frozen at $x_0 + t_c(x_0)$ afterwards—the original life-time function is $t_c(x_0) = t_0 x_0^{-1/p_2}$. The probability that the life ends in $[t,t+\mathrm{d}t)$ is $-\mathrm{d}W(t) = -W'(t)\,\mathrm{d}t$ when $W$ is differentiable; if it is not, like the step function of the Slicer Map, $-\mathrm{d}W(t)$ is defined just as easily, as illustrated in Appendix § 10. In a few words: (23) and the slicer sequences completely specify the map. Everything from (24) to (28) involves only $W$, and holds verbatim for the continuous-time fly-and-die process, of which the map is the integer-time sampling. This characterization of the ensemble is summarized by the cumulative distribution of $R(t)$, the displacement at time $t$. For a displacement value $r < t$, $R(t)$ does not exceed $r$ precisely when the life ended by time $r$, so that

which is the fraction $1-W$ behind the front and jumps by the still-flying fraction $W(t)$ at $r=t$. For the sake of illustration take $W$ differentiable; then differentiation with respect to $r$ yields the displacement PDF:

where $\Theta$ is the Heaviside step function and the $\delta$ term—the derivative of the jump—carries the still-flying fraction, cf. Appendix § 10. Of course, $\delta$ is not a function: Equation (35) is a formal but useful rewriting of Equation (34), and multiplication by the volume element $dr$ restores a bona fide probability element—the life-time element, which is $-dW(r)$ on $(0,t)$, plus an atom of mass $W(t)$ at the front—while every use made of $\mathcal{P}$ below amounts to an integral against observables.

Observe that, at unit speed, position and death time coincide for every dead member: an integral over death times is simultaneously an integral over the frozen positions, which fill exactly $(0,t)$, with unit Jacobian—the special gift of the slicer class. A detailed derivation of Equations (35), (36), (37) and (38) is provided in Appendix § 10. Since only the flying members advance, all at unit speed, the moments obey

exactly, whence

and

Introducing the running exponent of the survival function,

the exact spectrum (27) takes the asymptotic form

with the exponents identified as in § 4.2. Here, $\zeta= 1$, because the surviving members fly ballistically, so that the front $r = t$ is the moving cutoff, and $\theta= 0$—in place of the diffusive $1/2$—because the bulk is now the frozen record $-W'(r)$, which does not spread at all: below the knee the moments saturate to constants, and the lower branch of the spectrum is flat. The knee at $q \approx\varepsilon(t)$ is smeared over a range of moments of order $\sqrt{\varepsilon}$—the width over which the endpoint $s = t$ of the integral in Equation (36) competes with the interior maximum of its integrand—and it is therefore relatively sharp, since $\sqrt{\varepsilon}/\varepsilon\to0$ as $\varepsilon$ grows.

Stretched-Exponential Slicers: the Drifting Knee

For the present purpose, the original law for the slicers can be simply modified with the single replacement

which leaves (23), the phase space, and the non-chaotic, measure-preserving character of the map untouched, and gives the flying fraction $W(m) = 2\ell_m = e^{-(m/\tau)^\gamma}$ after $m$ steps. Analogously, the fly-and-die map requires

for which $\operatorname{Prob}(t_c > t) = e^{-(t/\tau)^\gamma}$ holds exactly at all $t \ge0$.

Equation (38) condenses the dichotomy of the present analysis into a single statement: the apparent offset of the second branch is the running exponent of the survival function. It is time independent—and the spectrum bi-linear at all times—if and only if $W$ is a power law: for $W = (t/t_0)^{-p_2}$ one has $\varepsilon\equiv p_2$, and (28) reproduces the SM spectrum of Section 3.2, with $\alpha= \beta= p_2$ (Figure 5a). For the stretched-exponential sequence (29), instead, $\varepsilon(t) = \gamma(t/\tau)^\gamma$: the knee drifts linearly in time for exponential slicers ($\gamma= 1$, Figure 5b), and quadratically for Gaussian ones ($\gamma= 2$), realizing at $\theta= 0$, and in closed form, the two columns of Figure 4. For $0 < \gamma< 1$, the properly stretched case, the knee drifts sublinearly, as $t^\gamma$ ($\gamma= 1/2$, Figure 5c): the smaller $\gamma$, the longer-lived the imitation. All members of the family (29) with fixed $\gamma> 0$ are thus equivalent in kind, however different in appearance: exponential, Gaussian and stretched slicers differ in the law of the knee’s drift, but share the same ultimate fate, the saturation of every moment—for each of them, the imitation is transient.

The equivalence breaks only at the boundary of the family. In fact, Equation (39) interpolates continuously between these two scenarios: as long as $\gamma\ln(t/\tau) \ll1$, one has $e^{-(t/\tau)^\gamma} \simeq e^{-1}(t/\tau)^{-\gamma}$, a power law with exponent $\gamma$, so that $\varepsilon(t) \simeq\gamma$ re mains constant over the window $t \ll\tau e^{1/\gamma}$, which exceeds any accessible time scale for small $\gamma$. In that limit the stretched-exponential slicer reduces to the power-law slicer with $p_2 = \gamma$: the knee does not drift, and genuine strong anomalous diffusion is recovered. At every fixed $t$ the apparent pair $\alpha_{\mathrm{eff}} = \beta_{\mathrm{eff}} = \varepsilon(t)$ again satisfies the hyper-scaling relation (7), since $\zeta- \theta= 1$. Furthermore, since the frozen record does not grow at all while the front contributes $\sim t^{q-\varepsilon(t)}$, the two branches compete at the knee at zero growth rate—the branch-value crossing of § 5.4 is here exactly marginal, the moment at the knee advancing only logarithmically—so that the crossing is decided by logarithms rather than by powers of $t$. The knee is nevertheless sharp, its width, of order $\sqrt{\varepsilon}$, remaining small compared with its drifting position $\varepsilon(t)$.

The Fly-and-Diffuse Model: the Escaping Knee

The slicer class realizes the scenarios of § 4 with a frozen bulk, $\theta= 0$: asymptotically, transport stops altogether. To realize the Gaussian minimal model (16) itself—exponentially rare flights over a bulk that keeps transporting—it suffices to let the members survive the end of their flight. In the fly-and-diffuse model, a member flies ballistically, at unit speed, for a life time drawn from the survival function $W(t) = e^{-t/\tau}$, and then continues with diffusive motion of diffusivity $D$. The diffusive stage can itself be generated by a deterministic chaotic map [50, 51, 52], keeping the dynamics fully deterministic. For the sake of illustration, here we take the Baker Map, which is two-dimensional—the minimum necessary to be time-reversal invariant—and preserves the Lebesgue measure. Like its numerous variations invented to investigate the foundations of statistical mechanics, it is analyzable in full detail, see, e.g., [53, 54], and can be adapted to associate diffusion of points with further nonequilibrium thermodynamic properties and their fluctuation theorems [55].

Here, we engraft the Baker Map into the Slicer Map, giving each member a second coordinate $y$, uniform on $[0,1]$ and untouched during the flight. More precisely, we enlarge the phase space of the Slicer Map by introducing a new phase space $[0,2) \times[0,1] \times\mathbb{Z}$ whose coordinates are $x$, $y$, and $m$. The dynamics proceeds as follows. For the points belonging to the initial ensemble, namely those with $x \in[0,1)$ and $m=0$, the Slicer Map acts until a slicing event occurs. At that point, the $x$-coordinate of the moving element is transformed into $x+1$: the member is thereby marked as dead, once and for all, and its $x$ remains in $[1,2)$, evolving under the contracting half of the Baker Map (Appendix E) with no effect on the displacement. During this first stage of the dynamics, the coordinate $y$, which is initially uniformly distributed over $[0,1]$, does not change. In the second stage, that is, after the slicing event has occurred, the coordinates $y$ and, correspondingly, $m$, evolve according to the following rule:

one cell per step, to the right or to the left according to the leading binary digit of $y_n$. Since $y$ is independent of $x$ and its binary digits can be seen as the result of independent tosses of a fair coin, the post-death motion turns out to be a simple random walk with $D=1/2$, in units of cells per step, with binomial propagators differing from the Gaussian ones at relative order $1/n$ only; the dynamics is completed by the contracting half of the Baker Map, Appendix E. The flight phase has zero Lyapunov exponent, the diffusive phase $\ln2$: the two extremes of chaos share the same chain of cells. The MSD obeys the exact identity

the frozen record of the Slicer Map plus the fresh diffusion accumulated since the end of each member’s flight; the higher moments follow from a single quadrature over the life-time distribution.

Three regimes result (Figure 6). For $t \lesssim\tau$ the motion is ballistic. For $\tau\lesssim t \lesssim t_{\mathrm{wake}} := 2\tau^2/\sigma^2$, with $\sigma^2=2D$, the frozen flight record dominates the bulk, and the model displays the slicer phenomenology of § 5.1: an apparently bi-linear spectrum with branches close to $0$ and $\zeta_q-t/\tau$, and a knee drifting as $\epsilon(t)=t/\tau$. For $t \gg t_{\mathrm{wake}}$, diffusion has washed the record out, the PDF converges to the minimal model (16), and the MSD resumes growing, $\langle r^2\rangle_t \simeq2\tau^2+\sigma^2(t-\tau)$. With the parameters of Figure 6, the flight branch has by then migrated to unobservably high moments, and the measured spectrum is single-linear, $\zeta_q=1$: the imitation of strong anomalous diffusion has died out, and what remains is normal transport, carried by the bulk. A single particle model thus traverses the entire scenario of § 4.2, from the transient bi-scaling of Figure 4 to its resolution.

Power-Law Flights over a Diffusive Bulk: the Fixed Knee

The rule of Section 4.3 is realized exactly within the family. It suffices to revert the survival function of the fly-and-diffuse model to the power law of Section 3.2: the slicer sequence

exactly at integer times, and the life-time function $t_c(x_0) = t_0\left(x_0^{-1/p_2}-1\right)$ realizes the same $W$ at all continuous times in the fly-and-die form. With $p_2 > 1$ the mean flight time is finite and the late-time bulk is diffusive: $\theta= 1/2$, with the Gaussian scaling function (23), $\sigma^2 = 2D$. To leading order the moments are the sum of the Gaussian bulk, Equation (24), and of the frozen flight record of the Slicer Map, Equation (36):

The record integral is evaluated by the rescaling $s = ut$ of Appendix § 10: for $q > p_2$ it is dominated by $u = O(1)$, where the shift in $W$ matters only to relative order $t_0/t$, and the $u$-integral converges at $u = 0$ by the usual criterion; for $q < p_2$ the region $u \approx0$ dominates instead, and the record only adds a constant. Both branches are now time independent: this is time-persistent strong anomalous diffusion, not the transient imitation of § 4.2, and the spectrum reproduces (32) with $\beta= p_2$:

with $\zeta= 1$ and $\beta= p_2$: the MSD is normal for every $p_2 > 1$, while all moments beyond $q_c = 2p_2$ are flight dominated.

The distribution cannot have the matched structure of Equation (3). Near the front the tail branch is a clean power law, $\mathcal{J}(y) \simeq p_2 t_0^{p_2} y^{-1-p_2}$ with $g(t) = t^{-\beta}$ and $\beta= p_2$, extending down to $y \sim t^{\theta-\zeta}$; but under bulk rescaling the corresponding large-$z$ tail of $t^\theta\mathcal{P}$ decays as $t^{-p_2/2}$. Clearly, the Gaussian scaling function does not develop a persistent power-law tail, and the two branches do not overlap on a common intermediate asymptotics.

In the data analysis of Appendix § 8, the collapse of $\mathcal{J}$ with the fixed exponent $\beta$ is perfect, but the joint bulk–tail plot of Figure 3a cannot be built, because the bulk has no power-law tail to match; the corrections to scaling of Appendix § 7 apply verbatim with $\alpha\to q_c$. We verified Equations (43) and (44) by exact sampling of the model ($2 \times10^6$ trajectories, $p_2 = 3/2$, $t_0 = 1$, $\sigma= 1$): over the tail-resolved fit window of § 3.1, the fitted slopes for $q = 1,\ldots,6$ are $\{0.47, 0.93, 1.49, 2.42, 3.45, 4.47\}$, against $\max(q/2, q-p_2) = \{0.5, 1, 1.5, 2.5, 3.5, 4.5\}$, with the largest deviations adjacent to the crossover $q_c = 3$, as the corrections of Appendix § 7 prescribe; the amplitudes of Equation (43) are matched within a few percent.

The renewal analogue is the Lévy walk with flight time sampled from a Lévy distribution with exponent $p_3 > 2$: its bulk is Gaussian and diffusive ($\theta= 1/2$), while the tail of the distribution is characterized by $\zeta= 1$ and $\beta= p_3 - 1$, $\alpha= p_3$ so that $q_c = 2(p_3 - 1)$, as reported in the literature [12, 31, 32, 33, 34]. Therefore, according to Corollary 1, the bulk and the tail are separated by a region where $\mathcal{P}$ is negligible.

Conclusions

We have employed methods of asymptotic matching, widely used in the context of hydrodynamic boundary layers and other scaling theories, to deal with the large-distance cutoff of the PDF of the displacement in strong anomalous diffusion. We established that strong anomalous transport emerges whenever the PDF features algebraic tails; conversely, in Appendix § 9 we prove that strong anomalous diffusion only arises if the distribution has power-law, or more slowly decaying, tails.

PDFs giving rise to strong anomalous diffusion are characterized by three exponents: $\theta$ describes how the width of the distribution broadens in time, $\zeta$ describes the scaling of the cutoff, and $\alpha+ 1$ is the exponent of the power-law tails. The scaling of the moments of displacement is fully characterized by hyper-scaling relations based solely on the moment order $q$ and these exponents. Equation (6) relates the time dependence of the moments to $q$, $\alpha$, $\zeta$, and $\theta$; this prediction rests purely on the constraints arising from matching the bulk and the tail of the distribution. The slope $\theta$ of the lower part of the spectrum reflects the scaling of the bulk of the distribution, which in most applications is entailed by large-deviation theory; the slope $\zeta$ of the upper part reflects the scaling of the cutoff, which always arises for processes with bounded speed. The crossover between the two regimes is set by the exponent $\alpha$ characterizing the power-law tail of the PDF, and the moment $q=\alpha$ grows as $t^{\alpha\theta}\ln t$, cf. Equation (7)—a result relevant, for instance, to the $t\ln t$ scaling of the MSD in the Lorentz gas with infinite horizon [56, 57, 58]. The exponent $\beta$ describing the time scaling of the tail distribution is expressed in terms of the other exponents by the hyper-scaling relation $\beta=\alpha\left(\zeta-\theta\right)$, Equation (3d). Finally, Equation (A1) provides the leading-order corrections to the power-law growth of the moments, and Equation (A2) the time scale that must be surpassed to clearly discriminate the leading-order power law from its sub-leading corrections.

The slowly decaying corrections to the power-law growth of the moments often give rise to severe systematic errors in the estimated exponents, as demonstrated in Appendix § 7. An informed way to determine $\alpha$, $\zeta$, and $\theta$, based on the simultaneous scaling analysis of the bulk and of the tails of the PDF, was described in Appendix § 8. Whenever the crossover time (A2) exceeds the available observation time, fits of the moments are performed in the sub-leading regime, and the resulting estimates of $\alpha$ can deviate substantially from the true value; the drifting apparent exponents of § 4.2 are the extreme form of this phenomenon.

We further determined what changes when the distribution is Gaussian, or, more generally, has no power-law tails. In that case the spectrum of moments is single-linear ($\alpha=\infty$), and we proved (Proposition 1) that this is no accident: asymptotic matching between two different self-similar branches forces the intermediate asymptotics to be a power law, so that power-law tails are necessary, and not only sufficient, for a time-persistent bi-linear spectrum. When ballistic excursions are exponentially rare rather than power-law distributed, they produce only a transient imitation of strong anomalous diffusion, which exists provided their persistence time exceeds the diffusive scale of the background ($\tau>2\sigma^2$), and which even satisfies the hyper-scaling relation at any fixed observation time. The Gaussian background is the marginal case: for heavier, stretched-exponential bulk tails the imitation even terminates at a finite time (Remark 1). An exactly solvable deterministic realization is provided by the Slicer Map itself, once its slicer sequence decays faster than any power law: the offset of the apparent second branch of the spectrum is then the running exponent of the survival function (§ 5.1). Letting the members diffuse, rather than stop, at the end of their flight yields the fly-and-diffuse model, which interpolates between the two scenarios and realizes the Gaussian minimal model as a particle system (§ 5.3). Its fingerprints are an apparent crossover moment that drifts linearly with the observation time, crossover times growing linearly with the moment order, and the impossibility of collapsing the tails of the distribution with any fixed exponent $\beta$. Conversely, power-law flights over a Gaussian bulk do produce time-persistent strong anomalous diffusion, even though the two branches of the distribution never match: the crossover moment is then set by the crossing of the branches, $q_c = \beta/(\zeta- \theta)$—twice the tail exponent, for a diffusive bulk—rather than by the tail exponent itself, and the logarithmic enhancement at the crossover is absent (§ 5.4).

The hyper-scaling relations have been verified analytically for the Slicer Map and numerically for the spatial diffusion of cold atoms—where the non-ballistic cutoff exponent $\zeta= 3/2$ puts the full three-exponent fitting scheme to work—and for Lévy walks, and they are consistent with the exponents reported in the literature for the systems collected in Table 1. With respect to Ref. [38], which is summarized in § 2 and in Appendix § 7 and Appendix § 8, the novel contributions of the present work are the complete analysis of the SDCA, based on newly generated data and on the finite-sampling criterion for tail-dominated moments, and the converse theory of § 4, with its exactly solvable deterministic realizations. We conclude that our approach provides a unified perspective on the analysis of the moments of displacement in strong anomalous diffusion, based on hyper-scaling relations and on a thorough analysis of the time scaling of the PDF of the displacement. We expect it to be a robust tool for the analysis of experimental and numerical data whenever power-law fits of individual moments are unreliable.

Author Contributions: Conceptualization, methodology and formal analysis; writing—original draft preparation, review and editing. All authors have read and agreed to the published version of the manuscript.

Funding: This research was performed under the auspices of the Italian National Group of Mathematical Physics (GNFM) of INdAM.

Data Availability Statement: The numerical data presented in this study are available on request from the corresponding author.

Conflicts of Interest: The authors declare no conflicts of interest.

Corrections to Scaling

This appendix recalls, from Ref. [38], the leading-order corrections to the asymptotic scaling of the moments, and the resulting limits on power-law fits of individual moments.

Equation (7) establishes that there are logarithmic corrections to the power-law behavior for $q = \alpha$. The resulting very weak time dependence features prominently in the discussion of the moments of displacement for the Lorentz gas with infinite horizon [10, 18, 38, 56, 59] and of the asymptotic scaling of its PDF [58, 60].

To assess the systematic errors made when fitting exponents for $q \ne\alpha$, we observe that Equation (6a) predicts that the $q^{\mathrm{th}}$ moment of displacement is always a superposition of two power laws. The leading-order correction to the asymptotic power-law scaling involves the ratio of the two prefactors,

For a reliable estimate of the exponents we demand that the leading-order term be larger than the correction term by a factor $\Phi$. This requires that the power-law fits be performed for times well beyond the critical time $t_q$ defined by

with

The time scale $t_q$ is model specific, but it can easily become very large: by Equation (A2) it grows without bounds as $q \to\alpha$, and the exponent $\tau$ is large whenever $\zeta-\theta$ is small. For the SDCA of § 3.1, for instance, $\zeta-\theta= 3/2 - 1/p_6$ vanishes as $p_6 \to2/3$, the boundary of the strong anomalous phase, and the fitted exponents of Figure 2b scatter accordingly around the predicted collapse, most severely for the moments closest to the crossover.

Whether the fitted exponents are biased upward or downward depends on the signs of $N_q$ and $I_q$, i.e. on whether $\mathcal{F}$ and $\mathcal{J}$ approach the matching power law from below or from above (cf. § 2.2); in either case the bias persists as long as the subdominant power law in Equation (6a) is not substantially smaller than the leading order, i.e. up to the times (A2).

In general, estimates of the moments of displacement will be hard whenever the difference between $\theta$ and $\zeta$ is small, or when $q$ is close to the crossover moment $\alpha$. In order to avoid these issues, the exponents $\zeta$, $\theta$, and $\alpha$ are determined from a scaling analysis of the PDF of the displacement, as described in Appendix B.

Data Analysis of the PDFs

This appendix recalls the scheme of Ref. [38] for determining the three exponents directly from the scaling of the PDF; § 3.1 applies it to the SDCA.

In order to extract the scaling exponents directly from the scaling of the PDF, we note that for small arguments $\mathcal{J}(y)$ takes exactly the same functional form as $\mathcal{F}(z)$ for large arguments (cf. Equations (3b) and (3e)). In Figure 3a this is demonstrated by showing the scaling plots of the bulk and of the tails of the PDF in the same coordinate system, rather than plotting them separately as in Figure 1. One clearly sees that the large-$z$ data of $\mathcal{F}(z) = t^{\theta} P(r,t)$ lie on the same power law as the small-$y$ data of $\mathcal{J}(y) = t^{\zeta+\beta} P(r,t)$.

The three exponents characterizing the distribution can therefore be found by a data collapse of the time evolution of the PDFs of the displacement:

$\theta$ is used to collapse the height of the bulk distribution $\mathcal{F}(z) = t^{\theta} P(r,t)$ for small $z = r/t^{\theta}$. This one-parameter fit provides a data collapse for the bulk and the power-law tail of the distribution. It is robust because it involves a vertical shift of horizontal lines such that they all meet at the same level. For the data in Figure 3 this provides the estimate $\theta= 0.845 \pm0.006$, for a theoretically expected value of $\theta= 1/p_6 = 6/7 \simeq0.857$. The resulting data collapse is shown in Figure 1a.

$\zeta$ is used to collapse the position of the cutoff value $y_{\max}$ of the tail distribution $\mathcal{J}(y) = t^{\zeta+\beta} P(r,t)$ for the largest admissible values of $y$; the vertical position of the data is adjusted by the choice of $\beta$. The fit of $\zeta$ is robust because it involves the horizontal shift of very sharp peaks such that they all lie at the same position; or rather, the minima at the end of the algebraic decay should lie at the same height. In many circumstances $\zeta$ need not be fitted, because it must be unity when the large-scale cutoff is dominated by ballistic flights; the SDCA of § 3.1 is a case in which $\zeta$ must be fitted, since the fastest atoms have velocities that themselves diffuse, and $\zeta= 3/2$. One obtains a plot of the data as shown in Figure 1b.

$\alpha$ must provide the slope of the simultaneous asymptotic scaling of both the large-$z$ asymptotics of $\mathcal{F}(z)$ and the small-$y$ asymptotics of $\mathcal{J}(y)$. According to Equation (3d) it is related to $\beta$ by the hyper-scaling relation $\beta= \alpha(\zeta-\theta)$. A joint plot of both distributions is shown in Figure 3a. The quality of the fit is best judged via a reduced plot displaying the deviation from the suggested power law: the power-law scaling of $\mathcal{F}$ applies for $z \gtrsim z_c = 2$ and that of $\mathcal{J}$ for $y \lesssim y_c = 0.25$, and a reduced plot of the data in this range is shown in Figure 3b. The exponent $\beta$ must be adjusted such that all data points take the same value. The effect of varying $\alpha$ for fixed $\zeta$ and $\theta$ is demonstrated in Figure 3c,Figure 3d. One clearly sees that the simultaneous matching of the slopes of both line segments provides a significantly better grip on the data than a mere power-law fit. In the next section we show that this data analysis provides a reliable characterization of numerical data.

Light tails rule out strong anomalous diffusion

In this appendix we show that weak anomalous diffusion emerges when the distribution $P(r,t)$ features scaling for small values of $r/t^{\theta}$ and decays faster than any power for large $r$. Scaling of the bulk of the distribution implies that there are positive real numbers $\iota> \theta> 0$ and $c > 0$ such that for large times

At the same time, a decay of the tail faster than any power implies that for every $q > 0$ there are a constant $k_q$ and a time $t_q$ such that

We consider the following lower bound, $L_q(t)$, for the $q^{\mathrm{th}}$ moment of the distribution,

which applies because the integrand is positive. The time evolution of $L_q$ is governed by the scaling of the bulk of the PDF,

where $N_q$ is a finite positive number if the moment exists. Further, the fast decay (A3b) implies an upper bound on $D_q(t)$,

which tends to zero for large times, since $\iota> \theta$. The upper and lower bounds hence converge in the long-time limit, and the moment must scale in time in the same way as $L_q(t)$:

For $\theta= 1/2$ the system features normal diffusion, and otherwise weak anomalous diffusion: no strong anomalous diffusion can arise when the tails decay faster than any power law.

From Slicer Sequences to Survival Functions

This appendix derives (35) and (26) from the map (23). Two structural facts make the derivation elementary: the map never changes the coordinate $x$, and the slicer sequence decreases monotonically.

Consider the right-moving stream, $x \in(1/2,1]$, and let $u := 1-x \in[0,1/2)$; since $\ell_0 = 1/2$, this whole stream moves to cell 1 at the first step. A member arriving in cell $j$—which happens at time $j$, one cell per step—proceeds to cell $j+1$ if and only if $x > 1-\ell_j$, i.e. $u < \ell_j$; otherwise it is sent back, and it then bounces between cells $j-1$ and $j$ forever, since it cleared the slicer of cell $j-1$ but not that of cell $j$. A member is therefore still flying at time $m$ if and only if it has cleared every cell up to $m$:

where the equivalence holds by monotonicity: only the last, lowest slicer matters. With $x$ uniformly distributed, each of the two symmetric streams carries a mass $\ell_m$ of survivors, so that the flying fraction is

conversely, the sequence $\ell_{\lvert j\rvert} = W(j)/2$ realizes any non-increasing survival function with $W(0)=1$. Each member is thus characterized by a life time $t_c$ with $\operatorname{Prob}(t_c > s)=W(s)$—in the fly-and-die form, $t_c(x_0)=W^{-1}(x_0)$ with $x_0$ uniform on $(0,1]$, the inverse-transform sampling of $W$—and its displacement is, up to the bounded bouncing,

Everything follows from this representation.

To obtain (35), let us define the ensemble average at time $t$ of an arbitrary observable $\varphi$ as its integral over the label of the ensemble—the initial coordinate $x_0$, the only random element of the deterministic dynamics:

where it is assumed that the initial distribution of points is uniform on $(0,1]$. Changing variables from $x_0$ to the life time, $t_c = W^{-1}(x_0)$—monotone, with $\mathrm{d}x_0 = -\mathrm{d}W(t_c)$—and splitting the members into those that are dead at time $t$, with $t_c \le t$ and frozen at $r=t_c$, and those that are still flying, with $t_c > t$ and located at $r=t$, one finds

since $-W'(s)\,\mathrm{d}s$ is the life-time probability element, $\operatorname{Prob}(t_c \in[s,s+\mathrm{d}s))$. The integration variable $s$ has therefore three equivalent readings: it is the life time $t_c$ labelling the members—the fundamental one, being the ensemble measure pushed forward from $x_0$; it is the frozen position $r$ of the dead members, since $r=t_c$ at unit speed; and its range $(0,t)$ is the support of the frozen record. As $\varphi$ is arbitrary, the distribution of $R(t)$ is

which is (35): the frozen record inherits the life-time density—position equals time of death—while the flying cohort rides the front. Normalization is automatic, $[W(0)-W(t)]+W(t)=1$. If $W$ is not differentiable, $-W'(r)\,\mathrm{d}r$ is read as the Stieltjes measure $-\mathrm{d}W(r)$. In particular, for the map at integer times $W$ is a step function, constant on the intervals $[k,k+1)$: its classical derivative vanishes almost everywhere, while $-\mathrm{d}W$ is the purely atomic measure $\sum_{k}[W(k-1)-W(k)]\delta(r-k)\,\mathrm{d}r$, so that (35) becomes

the exact distribution of the frozen cells, whose weights are twice the strip lengths $\Delta_k$ of Ref. [44]. The smooth form of Equation (25) describes the equivalent continuous-time fly-and-die process, of which the map is the integer-time sampling, and the difference between the two descriptions affects the moments only at relative order $1/t$.

To obtain (36), multiply (35) by $r^q$ and integrate: the $\Theta$ term restricts the integration to the frozen record, the $\delta$ term evaluates it at the front, and

which depends on $t$ in three places: the upper limit of the integral, and both factors of the flying term. Differentiating, we get:

In the interval $\mathrm{d}t$ the dying mass $-W'(t)\,\mathrm{d}t$ leaves the front and is deposited in the record at the very same position, $r=t$, so that death, by itself, changes no moment; the only net effect is the ballistic transport of the still-flying mass. Integration from $R(0)=0$ then gives the compact form of (36), and the logarithmic slope (27) is its immediate consequence.

To obtain Equation (38), rescale $s=ut$ in the compact form of Equation (36), so that

and substitute it into (27): the prefactor $qt^q$ cancels between numerator and denominator, leaving

where we have:

Therefore, wherever $\varepsilon$ varies little over logarithmic time scales, we can write $\varepsilon(s)\simeq\varepsilon(t)$ for $s\in(ut,t)$, and $W(ut)/W(t)\simeq u^{-\varepsilon(t)}$, which implies:

The $u$-integral converges at $u=0$ if and only if $q>\varepsilon(t)$: this is the origin of the restriction in the preceding display. The result is the upper branch of Equation (38), with $\zeta=1$. For the pure power law, we have $W\propto t^{-p_2}$, hence $\varepsilon\equiv p_2$, and the ratio $W(ut)/W(t)=u^{-p_2}$ holds identically, so that the upper branch is exact at all times.

For $q<\varepsilon(t)$, instead, the $u$-integral draws its mass from the neighborhood of $u=0$—small $u$ corresponds to early times, $s=ut$—and the moment stops growing: $\langle r^q\rangle_t$ saturates to the finite limit $\int_0^\infty r^q(-\mathrm{d}W(r))$, the $q^{\rm th}$ moment of the lifetime, since every member is eventually frozen at $r=t_c$. Meanwhile $\varepsilon(t)$ exceeds $q$ from some time on, so the numerator $qt^qW(t)$ of (27) vanishes as $t$ grows; equivalently, the rescaled $u$-integral grows without bound, and its inverse—the logarithmic slope—tends to zero. This is the lower branch of Equation (38). The knee sits where the two behaviors exchange dominance, at $q\approx\varepsilon(t)$.

An independent route to Equation (36) is the one that generalizes beyond single-time quantities as in Ref.[44]: writing $R(t)=\int_0^t\Theta(t_c-s)\,\mathrm{d}s$ and raising it to the $q^{\rm th}$ power,

since $\langle\Theta(t_c-s_1)\cdots\Theta(t_c-s_q)\rangle=\operatorname{Prob}(t_c>\max_i s_i)=W(\max_i s_i)$, the average being the $x_0$-integral defined above, and the shell $\max_i s_i\in(u,u+\mathrm{d}u)$ has volume $\mathrm{d}(u^q)$. The same representation extends to multi-time quantities; for in stance, $\langle R(m) R(n)\rangle= \int_{0}^{m} \int_{0}^{n} W(\max(s,s'))\,\mathrm{d}s\,\mathrm{d}s'$, whose power-law case is studied in [44].

Note also that Equation (36) involves only $W$ itself, never its derivative, and that its compact form is exact at integer times for the step function as well: on each unit cell,

and the sum over the cells reproduces, upon rearrangement of the terms, the moments of the atomic distribution (A6), $\langle r^q\rangle_n = \sum_{k\le n} k^q[W(k-1)-W(k)] + n^q W(n)$. Indeed, the one-step increment of the latter mirrors the continuous-time cancellation above:

the discrete twin of $q\,t^{q-1}W(t)\,\mathrm{d}t$: the set of points dying during the step appears in the record at $(n+1)^q$ and cancels against its disappearance from the front, leaving only the ballistic advance of the mass $W(n)$ that flies through the step.

Deterministic Diffusion in the Fly-and-Diffuse Model

This appendix spells out the deterministic chaotic version of the fly-and-diffuse model of Section 5.3 and derives its exact displacement statistics. The phase space is $[0,2)\times[0,1]\times\mathbb{Z}$: a member carries the Slicer coordinate $x$, the Baker coordinate $y$, and its cell $m$. Flying members, $x\in[0,1)$, follow the Slicer Map (33), advancing one cell per step with $y$ untouched; at the slicing event they are marked once and for all by $x\mapsto x+1$. Dead members, $x\in[1,2)$, evolve by

the Baker transformation of the unit square, translated to the strip $x\in[1,2)$ and extended by one cell jump per step: it is invertible, it preserves the Lebesgue measure, and its stretching rate gives the Lyapunov exponent $\ln2$, against the exponent zero of the flight phase, which is a translation. The first equation of (A8) plays no role in the displacement: the pair $(y,m)$ evolves autonomously, and every statistic below involves $y$ alone. It is what makes the dead-phase dynamics invertible—exponents $\pm\ln2$, as in a volume-preserving system—while the mark $x\mapsto x+1$ records death as part of the state, keeping the whole evolution a single map on the phase space.

The jumps are governed by the binary digits of $y$. Writing $y=\sum_{i\ge1}b_i2^{-i}$ with $b_i\in\{0,1\}$, the doubling $y\mapsto2y\bmod1$ deletes the leading digit, so that the $i$-th diffusive step moves the member by $\sigma_i=2b_i-1=\pm1$. With $y$ uniformly distributed, the digits are independent fair coins: the event $\{b_1=\beta_1,\ldots,b_j=\beta_j\}$ is a dyadic interval of length $2^{-j}$, whatever the $\beta_i$. In the language of symbolic dynamics, the digit string is the itinerary of $y$ over the partition $\{[0,1/2),[1/2,1)\}$, on which the doubling acts as the shift, and the Lebesgue measure reads as the Bernoulli $(1/2,1/2)$ measure on the symbols: the deterministic map and the cointossing process become probabilistically—i.e. in a coarse-grained sense—the same object under two names. We have thus introduced a diffusive phase as a simple random walk,

with zero mean and variance $j$: in the notation of § 5.3, $\sigma^2=1$, i.e. $D=1/2$, in units of cells per step.

Independence completes the solution. The life time depends on $x$ alone, the walk on $y$ alone, and the initial measure is a product, so the displacement law of the right-moving stream is the exact convolution

the life-time atoms of (A6) smeared by the walk accumulated since each death; the left-moving stream is its mirror image. The computation of the second moment

together with

shows that the fly-and-diffuse identity (42) holds exactly at integer times, with $2D=1$. By the de Moivre–Laplace theorem for binomial distributions, $\Pi_j$ converges to the Gaussian kernel of variance $j$ at relative order $1/j$, so all the formulas of § 5.3 are recovered. We verified (A10) and the MSD by iterating the map in exact dyadic arithmetic ($2\times10^5$ members, $\tau=10$, $n=200$), with agreement to statistical accuracy.