#  Superstatistical approach of the anomalous exponent
for scaled Brownian motion

M. A. F. dos Santos1, L. Menon Jr.1 and D. Cius2 1 Department of Physics, PUC-
Rio, Rua Marquês de São Vicente 225, 22451-900, Rio de Janeiro, RJ, Brazil  2
Programa de Pós-Graduação Ciências/Física, Universidade Estadual de Ponta
Grossa, 84030-900 Ponta Grossa, Paraná, Brazil

###### Abstract

Anomalous diffusion phenomenon is an intriguing process that tracer diffusion
presents in numerous complex systems. Current experimental and theoretical
investigations have reported the emergence of random diffusivity scenarios
accompanied by the heterogeneity of the $\alpha$-anomalous diffusion
exponents. In this framework, we investigate a heterogeneous ensemble of
tracers governed by scaled Brownian motion (sBm). The heterogeneous features
are considered on anomalous diffusion exponent and diffusivity. We introduce
two superstatistics of anomalous exponent, the truncated-Gaussian and
truncated $\chi^{2}$-Gamma distributions. In this way, we show how the
statistics of anomalous exponent affect the spreading of a mixture of
particles governed by sBm. We also analyse the role of different temporal
scales of sBm in superstatistics. Furthermore, we investigate the effects of
coupling between diffusivity and anomalous exponent on superstatistics of sBm.
The investigation provides a thorough analysis of simulation and analytical
approach. The results imply rich classes of anomalous diffusion processes
accompanied by non-Gaussian diffusion.

Keywords: Superstatistics; Anomalous diffusion; Non-Gaussian distribution;
Heterogeneous systems

##  I Introduction

The Brownian motion (Bm) was firstly observed in the erratic motion of tiny
particles in a simple fluid Brown (1828a, b). After that, it was modelled by
different ways Einstein (1905); Sutherland (1905); Von Smoluchowski (1906);
Langevin (1908) that were experimentally validated by Perrin Perrin (1908,
1909). Since then, it is known that the tracer diffusion of Brownian particles
has two remarkable features: $(i)$ linear mean square displacement at
$t$-time, i.e. $\overline{x^{2}}=2Dt$, in which $D$ is the diffusivity; and
$(ii)$ Gaussian shape for tracer distribution in space. Numerous experiments
on tracer diffusion reported systems that may break one or both Brownian
features. Particularly, when the linear time growth for MSD is violated, we
obtain the anomalous diffusion process, namely Metzler and Klafter (2000)

$\displaystyle\overline{x^{2}}\propto D_{\alpha}t^{\alpha},$ (1)

in which $\alpha$ anomalous diffusion exponent and $D_{\alpha}$ diffusion
coefficient $[\text{length}]^{2}/[\text{time}]^{\alpha}$. The second moment is
defined by $\overline{x^{2}}=\int_{\Omega}p(x,t)x^{2}dx$, with $p(x,t)$ the
probability distribution of tracers, and $\Omega$ the spatial variable domain.
More specifically, for $\alpha=1$ we recover the usual Brownian diffusion, for
$\alpha<1$ we have the sub-diffusion, and for $\alpha>1$, the super-diffusion.
The anomalous diffusion emerges in many theoretical and experimental
investigations, each one with their mechanisms Metzler and Klafter (2000); dos
Santos (2019); Oliveira et al. (2019).

Anomalous diffusion may be accompanied by non-Gaussian probability density
function (PDF) in space for the tracer particles. For instance, fractional
diffusion processes and nonlinear diffusion dos Santos (2019, 2019, 2018);
Sene (2019); Hristov (2011); Bologna et al. (2000); Tsallis and Bukman (1996),
such diffusion process breaks both Brownian features. There are also the
formalisms that imply anomalous diffusion accompanied by Gaussian shape, such
as the scaled Brownian motion (sBm) Lim and Muniandy (2002); Bodrova et al.
(2019); Jeon et al. (2014); Safdari et al. (2015); Tawfik and Abdelhamid
(2021) and fractional Brownian motion (fBm) Mandelbrot and Van Ness (1968);
Deng and Barkai (2009); Jeon and Metzler (2010). Particularly, the research
Deng and Barkai (2009) on ergodicity analyse was corrected Wang et al. (2020)
and generalised Cherstvy et al. (2021). Currently, experimental data with
single-tracking particles revealed a diffusion class whose diffusion has a
non-Gaussian shape for tracer displacements in space; however, the MSD
increases linearly with time Wang et al. (2012). Such systems have been
investigated through the superstatistical approach to Brownian particles
Chechkin et al. (2017); Metzler (2017); Chakraborty and Roichman (2020);
Metzler (2020); Ślezak et al. (2019); Lanoiselée and Grebenkov (2019); Molina-
García et al. (2016); Maćkała and Magdziarz (2019). In a general overview,
superstatistics was proposed in the non-equilibrium statistical physics
context. Nowadays, it is useful to describe many complex systems Gravanis and
Akylas (2021); Thompson and Katul (2008); Chung et al. (2021); Hassanabadi et
al. (2021); Agahi and Khalili (2020); Agahi and Alipour (2020); Schäfer et al.
(2018); dos Santos et al. (2021); dos Santos (2020); dos Santos and Junior
(2020); Akıllı et al. (2021); Vitali et al. (2018). In the random mobility
framework, there are also other formalisms that have been considered, as the
“diffusing diffusivity” model Chubynsky and Slater (2014); Jain and Sebastian
(2016), and the subordination model Fogedby (1994); Chechkin and Sokolov
(2021).

The diffusion of Brownian particles in heterogeneous scenarios allows us to
consider a probability density function (PDF) for diffusivities $f(D)$ to
approach the problem from two different point of views Metzler (2020):
heterogeneous ensemble of tracers and identical tracers in a heterogeneous
medium with each particle diffusing into their own patch. The heterogeneous
environment interpretation corresponds to the superstatistics approach only
when particles do not hit on the boundary of the patch. If particles explore
other patches, the system undergoes a Gaussian statistic Metzler (2020); dos
Santos and Junior (2021). However, regardless of superstatistic
interpretations, the system presents a non-Gaussian shape and usual diffusion.
Particularly, the superstatistics for the heterogeneous ensemble of particles
governed by sBm and fBm was addressed in Ref. dos Santos et al. (2021). Beyond
of the distributed diffusivities for particles (or patches), the particles may
have their own anomalous diffusion exponent $\alpha$ (see Eq. (1)), such as
discussed in Ref. Itto and Beck (2021). This formulations imply a statistic
for anomalous exponents, i.e., a PDF $g(\alpha)$. Experiments with single-
particle tracking in complex biological media have presented heterogeneity of
$\alpha$ exponent, for instance in tracer-particles diffusing in the cytoplasm
of mammalian cells Janczura et al. (2021); Sabri et al. (2020); Etoc et al.
(2018), heterogeneous population of amoeboid cells Cherstvy et al. (2018) and
polydisperse vacuoles in highly motile amoeboid cells Thapa et al. (2019).
Others problems which are not connected to biological scenarios and exhibit
distinct anomalous exponents can be found in the diffusion of trajectories in
chaotic Hamiltonian systems Palmero et al. (2021); Manos and Robnik (2014).

In the anomalous-exponent statistics framework, our proposal here is to
motivate the two most usual PDFs of anomalous exponent that define novel
superstatistics, consequently showing the non-Gaussian and MSD behaviour for
tracer diffusion governed by sBm. During our analysis, we will show the
importance of changing the time scale to address cases where anomalous
diffusion occurs for the complete ensemble. Another important point is to
introduce the conditioned case between the anomalous exponent and diffusivity,
this type of feature has been reported in some works in recent years Cherstvy
et al. (2019); Speckner and Weiss (2021); Janczura et al. (2021); Sabri et al.
(2020); Etoc et al. (2018); Cherstvy et al. (2018); Thapa et al. (2019).

This article is organised in the following way: in Section (II), we introduce
the general setup on the statistical approach of anomalous exponent, showing
what has been investigated in theoretical and experimental scenarios.
Furthermore, we show how the superstatistical approach provides the necessary
tools to describe a heterogeneous ensemble of tracers governed by sBm. In
Section (III), we analyse carefully the role of the anomalous exponent
statistics for a mixture of anomalous tracers. Then, we present two truncated
superstatistics for the anomalous exponent as well as how such insights may be
applied to obtain different classes of the anomalous non-Gaussian diffusion
processes. In Section (IV), we open a discussion about the importance of a
generalised temporal scaling in the tracers (sBm) as well as the conditioned
distribution of anomalous exponent and diffusivity. Finally, in Section (V),
we present the conclusions and how our approach contributes to random
diffusivity scenarios.

##  II  General setup on statistics of anomalous tracers

Recent experiments and simulations have reported tracers diffusion with
heterogeneous mobilities in which each trace presents different
$\alpha$-anomalous exponents, implying a PDF $g(\alpha)$. For instance, such
features have been reported for particles diffusing in biological matter
Bräuchle et al. (2002); Seisenberger et al. (2001); Janczura et al. (2021);
Sabri et al. (2020); Weiss (2019); Etoc et al. (2018); Golan and Sherman
(2017); Korabel et al. (2021), diffusion in agitated fluids Gires et al.
(2020), heterogeneous process Cherstvy and Metzler (2015); Xu et al. (2020),
and random walks with correlated noise Kepten et al. (2013); Itto and Beck
(2021). In this scenario, there are some sophisticated developments to
describe the mixture of heterogeneous tracers in complex diffusion. A current
way is by considering $\alpha$ anomalous diffusion exponent as a random
variable, allowing us to approach systems whose anomalous exponent is randomly
chosen, i.e., each tracer has its own diffusivity and anomalous exponent such
as in superstatistics interpretation Itto (2014); Itto and Beck (2021).
Investigation of dynamics with random anomalous exponent emerges also in
different theoretical landscapes, such as: fractional Brownian motion (fBm)
Kepten et al. (2011); Burnecki et al. (2012), the continuous time random
walkers Massignan et al. (2014), and the heterogeneous-diffusion process with
position-dependent diffusion coefficient Xu et al. (2020). These systems open
the way to the investigation of generalised pictures. In this work, we focus
our attention on the superstatistics approach of anomalous exponent $\alpha$
and generalised diffusivity $D$ of heterogeneous tracers governed by the
scaled Brownian motion, i.e., $dx/dt=\sqrt{2\mathcal{D}(t)}\eta_{t}$, in which
$\eta_{t}$ is the white noise and $\mathcal{D}(t)=D\alpha t^{\alpha-1}$ is the
time-dependent diffusion coefficient. Different names for $\mathcal{D}(t)$ and
$D$ can be found in literature Jeon et al. (2014), for instance in the
superstatistics framework, in Ref. dos Santos et al. (2021) the authors wrote
$\mathcal{D}(t)=\mu\lambda(t)$, where $\mu$ is mobility and $\lambda(t)$
represent a general time scale. Here we call $\mathcal{D}(t)$ and $D$ by time-
dependent diffusion coefficient and diffusivity, respectively. Recalling that
$D$ is a generalised diffusion coefficient due to the physical dimensions, see
Eq. (1).

The ensemble of heterogeneous particles (or tracers) will imply a distribution
of anomalous exponents and diffusivities that define the two-variable
superstatistical approach Itto and Beck (2021). Here, we consider that all
tracers are governed by sBm, and each tracer with its anomalous exponent and
diffusivity. Hence, the overall PDF of the tracers is given by

$\displaystyle
p(x,t)=\int_{\Omega_{\alpha}}\int_{\Omega_{D}}\pi(\alpha,D)\mathcal{G}_{\alpha}(x,t|D)dDd\alpha,$
(2)

in which $\alpha$ is the anomalous diffusion exponent, and $D$, the
diffusivity, with $\Omega_{\alpha}$ and $\Omega_{D}$ being the domains,
respectively. The PDF $\pi(\alpha,D)$ is the joint distribution that defines a
superstatistic, such PDF contains all information about $\alpha$ and $D$
variables. Moreover, $\mathcal{G}_{\alpha}(x,t|D)$ is the PDF of tracers with
indexed $\alpha$ and $D$ parameters, which in the framework of sBm, has a
Gaussian shape. Indeed, we are superposing many different subsets of anomalous
tracers each one with an indexed $\alpha$ exponent. Thereby, the random
diffusivity term is related to the set of generalised diffusion coefficients
$\\{D\\}$ that implies heterogeneity of the tracers. This superstatistical
approach provides an efficient way to address experimental data tracking
associated with a mixture of tracers. There are considerations associated to
different approximations on PDF of parameters, i.e. $\pi(\alpha,D)$. For
instance, $\pi(\alpha,D)=g(\alpha)f(D)$ corresponds to the case in which
$\alpha$ and $D$ are uncorrelated parameters, considering $g(\alpha)$ and
$f(D)$ as the PDFs of anomalous exponents and diffusivities, respectively.
Another complex scenario occurs when the anomalous diffusion exponent is
conditioned to diffusivity, i.e. $\pi(\alpha,D)=g(\alpha|D)f(D)$, as it was
approached in superstatistical treatment on two variables. Current
applications about it were observed in data of protein diffusing within a
bacteria Itto and Beck (2021). There are also experimental indicators of
correlations between diffusivities and anomalous exponents in the spreading
dynamics of amoeboid cells Cherstvy et al. (2018); Thapa et al. (2019). In
such situations the authors found an exponential adjustment which connects
both quantities, e.g., $D_{\alpha}\sim e^{-c_{1}\alpha+c_{2}}$, for more
details see Ref. Cherstvy et al. (2018). The relation between $\alpha$ and $D$
have been observed in others recent researches Cherstvy et al. (2019);
Speckner and Weiss (2021).

The overall PDF of tracers depends on interpretation, heterogeneous anomalous
tracers or identical tracer-particles moving in their own patch with indexed
mobility. All tracers associated to fixed $(\alpha,D)$ values that imply a
particular average $\overline{x^{2}_{\alpha}}$. Therefore, by mixing the MSD
profiles with different $\alpha$ and $D$ the overall MSD in the
superstatistics sense is given by

$\displaystyle\left\langle\overline{x^{2}}\right\rangle=\int_{\Omega_{\alpha}}\int_{\Omega_{D}}\pi(\alpha,D)\overline{x^{2}_{\alpha}}dDd\alpha,$
(3)

in which
$\left\langle\overline{x^{2}}\right\rangle=\left\langle\overline{x^{2}_{\alpha}}\right\rangle$.
Keeping in mind that $\overline{x^{2}_{\alpha}}$ emerges from an arbitrary
generalised dynamic that implies power-law behaviour for time, i.e.,
$\overline{x^{2}_{\alpha}}\sim Dt^{\alpha}$, or similar ones. Figure. (1)
shows $30$ trajectories generated by scaled Brownian process
($dx/dt=\sqrt{2\alpha t^{\alpha-1}}\eta_{t}$, in which $\eta_{t}$ is the white
noise) for $dt=0.1$, step number $N=1000$ and truncated Gaussian PDF of
$\alpha$ values with two different parameters. The truncated Gaussian of
anomalous exponents is defined by
$g_{n}(\alpha)=\mathcal{N}^{-1}\exp(-(\alpha-\alpha_{p})^{2}/2\sigma^{2})$
with $\alpha\in(0,2]$, in which $\mathcal{N}$ is the normalisation constant
and $\sigma$ controls the shape of PDF of the anomalous exponents. In Figure
(1-a), we consider that $\alpha$ is randomly chosen from a truncated Gaussian
distribution with $\sigma=0.5$, see the Eq. (5). Fig. (1-b) considers a
similar situation but with $\sigma=0.15$. Such graphical representations
reveal to us a rich scenario for the mixed scaled Brownian motion, the
truncated Gaussian PDF, as well as other anomalous exponent statistics, will
be carefully investigated in the next sections. The sBm may admit yet a
general time-dependent diffusion coefficient Bodrova et al. (2016) which allow
us to introduce a general temporal scale $\psi(t)$ thought the relation
$\mathcal{D}(t)=D\frac{d\ }{dt}\psi(t)\propto\frac{d\
}{dt}\overline{x^{2}_{\alpha}}$, the role of this general scale will be
studied in this work. It is noteworthy that scaled Brownian motion establishes
an anomalous diffusion process based on the overdamped Langevin equation
Molina-García et al. (2016); Maćkała and Magdziarz (2019); Bodrova et al.
(2016).

Figure 1: Figures show scaled Brownian tracers, each one with their own
$\alpha$-anomalous exponent randomly chosen from the truncated Gaussian
distribution (5), with $\alpha\in(0,2]$ and $\alpha_{p}=1$. Fig. (a) shows
trajectories for $\sigma=0.5$ exponent. Fig. (b) shows trajectories for
$\sigma=0.15$ exponent.

##  III Heterogeneous anomalous exponent for scaled diffusion

Under the general setup of $\alpha$-statistics previously discussed, we have
an open way to new theoretical diffusion approaches. In this section, we
analyse the $\alpha$-statistics effects on a tracer diffusion in a complex
environment. To do this, we disregard the correlation effect which might exist
experimentally between $\alpha$ and $D$, so that random diffusivity does not
overshadow the effect of the fluctuations of the anomalous exponent on sBm.
Consequently, we have the approximation for the $\pi(\alpha,D)$ statistic of
parameters

$\displaystyle\pi(\alpha,D)=g(\alpha)f(D),$ (4)

such PDF describes the scenarios in which the anomalous diffusion exponent is
not conditioned to diffusivity.

The PDF $g(\alpha)$ of the anomalous exponent was investigated through the
overall MSD in Eq. (3) with $f(D)=\delta(D-D_{0})$. For different statistics
of $\alpha$-exponent, see Ref. Li (2021). A relevant feature here is that for
$\alpha$ contained in an infinite domain, the limit
$\lim_{\alpha\to\infty}g(\alpha)$ must converge to zero, so that the overall
MSD $\left\langle\overline{x^{2}}\right\rangle$ cannot diverge from any time
value. To avoid the divergence effects at MSD calculus, we suggest two
truncated $\alpha$ statistics. The first one, is the truncated-normal
statistics (or truncated-Gaussian statistics) of the anomalous diffusion
exponent, defined as follows

$\displaystyle
g_{n}(\alpha)=\begin{cases}\displaystyle\mathcal{N}^{-1}\exp\left(-\frac{(\alpha-\alpha_{p})^{2}}{2\sigma^{2}}\right)&\quad\text{if
}0\leq\alpha\leq\alpha_{\text{max}}\\\ 0&\quad\text{ otherwise}\end{cases}$
(5)

in which
$\mathcal{N}=\sqrt{\frac{\pi}{2}}\sigma\left(\text{erf}(\alpha_{p}(\sqrt{2}\sigma)^{-1})+\text{erf}((\alpha_{\text{max}}-\alpha_{p})(\sqrt{2}\sigma)^{-1})\right)$
is the normalisation factor, $\sigma$ controls the distribution width, and
$\alpha_{p}$ defines where the peak of the distribution is. The Fig. (2-a)
exemplifies the qualitative features of $g(\alpha)$ distribution. Thereby, for
lower values of $\sigma$-variance the PDF (5) leads to a concentrated delta
distribution, i.e. $\lim_{\sigma\to 0}g(\alpha)\sim\delta(\alpha-\alpha_{p})$
as well as the standard scaled Brownian diffusion with $\alpha_{p}$. The
Gaussian statistic in Eq. (5) for anomalous exponents was approached in
experiments on single tracking particles diffusing within an environment with
micro pillars Chakraborty and Roichman (2020), particles diffusing in a
heterogeneous environment Xu et al. (2020); Cherstvy and Metzler (2015), and
biological specimen diffusing in soft-matter Weiss (2019).

Now, we consider the second statistic, which is the truncated-$\chi^{2}$ gamma
distribution, given by

$\displaystyle
g_{\chi}(\alpha)=\begin{cases}\displaystyle\mathcal{N}^{-1}\alpha^{\gamma-1}\exp\left(-\frac{\alpha}{a}\right)&\quad\text{if
}0\leq\alpha\leq\alpha_{\text{max}}\\\ 0&\quad\text{ otherwise}\end{cases}$
(6)

in which
$\mathcal{N}=a^{\gamma}\left(\Gamma(\gamma)-\Gamma\left(\gamma,\alpha_{\text{max}}a^{-1}\right)\right)$
is the normalisation condition, $\gamma$ and $a$ are free parameters that
control the shape of distribution, see Fig. (2). The $\chi^{2}$-Gamma PDF in
superstatistics was applied in protein movement in biological matter Itto and
Beck (2021). There are other problems where $\chi^{2}$ PDF for the anomalous
exponent seems present in experimental or numerical data of tracking
particles, such as telomeres diffusing in the nucleus of mammalian cells
Stadler and Weiss (2017), heterogeneous diffusion states in the cytoplasm
Janczura et al. (2021) and heterogeneous diffusion processes Cherstvy et al.
(2013).

Both $\alpha$-statistics have truncation, which is crucial to ensure that the
overall MSD behaviour does not go to infinity for a fixed time.
Mathematically, the integration (3) on $\alpha$ values to a fixed time cannot
diverge; therefore, truncation ensures the MSD convergence. From a physical
point of view, the truncation condition guarantees us that there are not
individual tracers with unrealistic values of $\alpha$. Thereby, there are
individual tracers with anomalous diffusion exponents that are sub or
superdiffusive, but never with $\alpha>\alpha_{\text{max}}$, in which
$\alpha_{\text{max}}$ was imposed. In most experimental situations, tracers
with $\alpha>3$ represent a negligible number; therefore, the truncation works
well under this condition.

Figure 2: Figures present PDF of the anomalous diffusion exponent for (a)
truncated Gaussian statistic (with $\alpha_{p}=1$) and (b) truncated
$\chi^{2}$-Gamma statistic. The dashed lines represent the analytical approach
(see Eqs. (5) and (6)). The coloured curves were obtained from simulations.

The model considers heterogeneous tracers with their own indexed $\alpha$
exponent, i.e., $\alpha_{i}$ exponent that remains fixed the whole time.
Individually, each tracer is generated by the following scaled-stochastic
process

$\displaystyle\frac{dx_{i}}{dt}=\sqrt{2D_{i}\alpha_{i}t^{\alpha_{i}-1}}\eta(t),$
(7)

in which $t>0$. The $\alpha_{i}$ and $D_{i}$ are different for each
$i$-indexed tracer. The parameter $D_{i}$ is the generalised diffusion
coefficient with dimension $[\text{length}]^{2}/[\text{time}]^{\alpha}$. The
tracer’s displacement on an individual level obeys a Gaussian diffusion
process, given by

$\displaystyle\mathcal{G}_{\alpha_{i}}(x,t|D_{i})=\frac{1}{\sqrt{4\pi
D_{i}t^{\alpha_{i}}}}\exp\left(-\frac{x^{2}}{4D_{i}t^{\alpha_{i}}}\right).$
(8)

This distribution has the same mathematical form of the PDF associated with
fractional Brownian motion Cherstvy et al. (2021). To analyse the PDF of
tracers in space only with anomalous exponent effects, we approximate
$f(D)\approx\delta(D-D_{0})$ to remove the diffusivity effects. Thereby, we
consider two $\alpha$ distributions of the anomalous diffusion exponent that
are not conditioned to diffusivity and have a finite domain. The mix of all
tracers is defined by the overlapping of the heterogeneous sBm, as shown in
Eq. (2). In this situation, we reduced our problem to a one-variable
superstatistics, which implies

$\displaystyle
P(x,t)\simeq\int_{0}^{\alpha_{max}}\mathcal{G}_{\alpha}(x,t|D_{0})g(\alpha)d\alpha.$
(9)

This PDF of tracers allow us to define a pure superstatistics on the anomalous
exponent, i.e., $f(D)\approx\delta(D-D_{0})$. The MSD in Eq. (3) for this case
is given by

$\displaystyle\left\langle\overline{x^{2}}\right\rangle=2D_{0}\int_{0}^{\alpha_{max}}t^{\alpha}g(\alpha)d\alpha.$
(10)

recalling that we adopt the notation $\langle
x^{2}\rangle=\left\langle\overline{x^{2}}\right\rangle$ for the figures. For
$g(\alpha)$ different from Dirac-delta distribution, we have non-Gaussian
diffusion.

Now, we may check the match between analytical and simulation approaches. In
order to do so, we consider two PDFs of the anomalous exponent introduced in
Eqs. (5) and (6), and $10^{4}$ tracers generated by Eq. (7). Consequently,
figures (3-a) and (3-b) present the PDF of tracers with truncated Gaussian
statistic for the anomalous exponents; the figures consider different times
and parameters. Moreover, Figs. (3-c) and (3-d) show the influence of $\sigma$
and $\alpha_{p}$ values on MSD. Figure (4) repeats the approach to the
truncated $\chi^{2}$-Gamma distribution of anomalous exponent. Figs. (4-a) and
(4-b) show the PDF of displacements in space. Figs. (4-c) and (4-d) show the
influence of $\gamma$ and $a$ parameters on MSD, respectively. Both
superstatistics present a perfect agreement between analytical and simulation
results. An important feature here is that through the straight lines in cyan
colour in Figs. (4-c) and (4-d), we notice that the MSD curves do not follow a
perfect power-law behaviour in time, i.e. $\langle x^{2}\rangle\nsim t^{q}$.
Particularly, for $a=2$ (green curve), as shown Fig. (4-d), the MSD curve for
long times is approximated to a power-law time behaviour due to the bigger
concentration of anomalous exponent values around $\alpha\sim 2$. However, for
the Gaussian PDF of the anomalous exponent, the overall MSD presents an exact
power-law behaviour in time when $\sigma\to 0$.

Figure 3: Figures show overall PDFs of tracers in space and MSD behaviour in
time. All figures consider truncated Gaussian statistics for the anomalous
exponent $\alpha\in(0,2]$. The dashed curves represent the analytical approach
that were obtained through of Eqs. (9) and (10) for PDF and MSD, respectively.
The symbolic curves represent the simulation results obtained by the scaled
Brownian process, see Eq. (7). Fig. (a) shows the PDF of tracers for
$\alpha_{p}=1.0$ and different times. Fig. (b) shows the PDF of tracers for
$\alpha_{p}=1.0$ and different $\sigma$ values. Figs. (c) and (d) show the
overall MSD of tracers in time for different $\sigma$ and $\alpha_{p}$ values,
respectively. Figure 4: Figures show overall PDFs of tracers in space and MSD
behaviour in time. All figures consider truncated $\chi^{2}$-Gamma statistics
for the anomalous exponent $\alpha\in(0,2]$. The dashed curves represent the
analytical approach that were obtained through of Eqs. (9) and (10) for PDF
and MSD, respectively. The symbolic curves represent the simulation results
obtained by the scaled Brownian process, see Eq. (7). Fig. (a) shows the PDF
of tracers for $a=0.1$ and different times. Fig. (b) shows the PDF of tracers
for $a=0.1$ and different $\gamma$ values. Fig. (c) shows the MSD for $a=0.1$
and different $\gamma$ values. Fig. (d) shows the MSD for $\gamma=10$ and
different $a$ values.

##  IV Other temporal scaling into the SBM and two-variable superstatistics

A heterogeneous ensemble of tracers with statistics over $\alpha$-anomalous
exponent and diffusivity $D$ was considered in the sBm context in the previous
section. Such systems with heterogeneity only in diffusivity were investigated
to different landscapes Molina-García et al. (2016); Vitali et al. (2018); dos
Santos and Junior (2021, 2020). Motivated by cases investigated in the
previous section, we introduce two new features in this section: firstly, a
general temporal scaling; secondly, we consider the entanglement between the
$\alpha$ and $D$ parameters to emulate correlated situations.

To introduce the idea of general temporal scaling in superstatistics, let us
introduce a motivation connected to our problem. The MSD for the previous
model increases in time according with Eq. (3). For simplified cases of
$\alpha$-superstatistics, the MSD may be calculated analytically Li (2021).
For instance, we consider an uncoupled $\pi(\alpha,D)=f(D)g(\alpha)$
distribution with $f(D)=\delta(D-D_{0})$ and $g(\alpha)$ being a uniform
distribution with domain $\alpha\in[\alpha_{\text{min}},\alpha_{\text{max}}]$,
applying this statistics in Eq. (3) and considering a long time limit, we
obtain

$\displaystyle\lim_{t\to\infty}\left\langle\overline{x^{2}}\right\rangle$
$\displaystyle=$
$\displaystyle\lim_{t\to\infty}2D_{0}\int^{\alpha_{\text{max}}}_{\alpha_{\text{min}}}t^{\alpha}d\alpha$
(11) $\displaystyle\simeq$ $\displaystyle
2D_{0}t^{\alpha_{\text{max}}}(\log(t))^{-1}.$

This limit shows that there is a logarithmic factor that avoids a pure power
law growth Li (2021). If we consider a Poisson-like form of fluctuations for
the $\alpha$-anomalous exponent Itto (2014, 2012), i.e.
$g(\alpha)\propto\exp(\lambda\alpha)$ for $0\leq\alpha\leq 1$, the
superstatistics of MSD imply $\left\langle\overline{x^{2}}\right\rangle\propto
t/\log(t)$ for long times. A mixing of fractional-time continuous random walk
was considered in Ref. Itto (2014). Then, we have a logarithmic factor in Eq
(11) that is implicitly present in numerical simulation cases approached in
the previous section, see Figs. (3) and (4). Thereby, it is not so easy to
obtain a power-law time behaviour for superstatistics of anomalous exponents
by mixing power-law profiles, i.e.
$\int_{\Omega_{\alpha}}t^{\alpha}g(\alpha)\sim t^{\alpha_{\text{eff}}}$ with
$\alpha_{\text{eff}}>0$. However, most of the experimental investigations
present a MSD behaviour that increases as a power-law function of time; this
lead us to consider an appropriate time scaling.

In the previous section, a power-law time temporal scale was considered for
the time-dependent diffusion coefficient. However, different profiles for
$\mathcal{D}(t)$ can emerge due to intrinsic or extrinsic factors, related to
memory and aging effects or due to changes in the spatial structure of the
medium Sen (2004); Latour et al. (1994); Safdari et al. (2015); Cherstvy et
al. (2021). Then, other functions of the diffusion coefficient may configure
different classes of temporal scales. Thus, we consider a general function for
temporal scaling $\psi_{\alpha}(t)$ dos Santos et al. (2021); Lim and Muniandy
(2002); Cherstvy et al. (2021); dos Santos and Junior (2020), in this
perspective the general temporal scale embraces the power-law time scale as a
particular case. Moreover, different profiles of temporal scales beyond of
power-law time function, e.g., logarithmic and exponential profiles, have been
motivated by the biophysical application of the diffusion models with time-
dependent diffusion coefficient, namely, the diffusion of water molecules in
brain tissues Cherstvy et al. (2021). The overall MSD of the tracer is given
by

$\displaystyle\left\langle\overline{x^{2}}\right\rangle=\int_{\Omega_{D}}\int_{\Omega_{\alpha}}2D\psi_{\alpha}(t)\pi(\alpha,D)d\alpha
dD,$ (12)

recalling that we adopt the notation $\langle
x^{2}\rangle=\left\langle\overline{x^{2}}\right\rangle$ for the figures. The
time-dependent diffusivity is defined by
$\mathcal{D}(t)=\psi_{\alpha}^{\prime}(t)D$, see Ref. dos Santos et al.
(2021).

In this framework, we can choose the temporal scale $\psi(t)$ to change the
effective diffusivity, allowing us to control the short and long times limits
of overall MSD. In Ref. Cherstvy et al. (2021), the authors explore two
different profiles of time-dependent diffusivity following logarithmic or
exponential forms. We consider a multiplicative logarithmic factor in scaling-
law of sBm to obtain an exact power-law behaviour of MSD for long times.
Thereby, we propose the following explicit form for temporal scale
$\psi_{\alpha}(t)=\left(1+t/\tau\right)^{\alpha}\log\left(1+t/\tau\right)$, in
which $\tau$ is a temporal constant $[\text{time}]$. We have the following
limits: for short times, ($t\ll\tau$) $\psi_{\alpha}(t)\sim t$; for long
times, ($t\gg\tau$) $\psi_{\alpha}(t)\sim t^{\alpha}\log t$. It is noteworthy
that for short times the effect of the $\alpha$ exponent is negligible.
Particularly, our first choice for the temporal scale have special particular
cases, such as $\psi(t)\propto t\log[t/\tau]$ that was approached in Ref.
Cherstvy et al. (2021), and the time scale $\psi(t)\propto\log(1+t/\tau)$
investigated in Ref. Bodrova et al. (2016) motivated by ultraslow diffusion
processes. However, we did this chosen because the logarithmic term is
important to cancel the term $(\log(t/\tau))^{-1}$ that will appear in the
overall MSD, such as appeared in the example on Eq. (11). Thereby, the
superstatistics over the $\alpha$ variable in Eq. (12) imply two limits which
may be determined separately. For short times,
$\left\langle\overline{x^{2}}\right\rangle\simeq 2D_{\text{eff}}t$, in which
the effective diffusivity is given by

$\displaystyle
D_{\text{eff}}=\int_{\Omega_{D}}\int_{\Omega_{\alpha}}D\pi(\alpha,D)d\alpha
dD.$ (13)

For long time limits, we have as a general result
$\left\langle\overline{x^{2}}\right\rangle\propto t^{\alpha_{\text{eff}}}$, in
which $\alpha_{\text{eff}}$ depends on the $\alpha$ statistic. In another way,
we can consider a different temporal scale $\psi_{\alpha}(t)$, which presents
a usual diffusion over long times. To obtain a transition between two
different diffusive process that finishes in usual behaviour, we consider
$\psi_{\alpha}(t)=te^{-\tau/t}+t^{\alpha}e^{-t/\tau}$, that implies a
crossover from anomalous to usual diffusion, e.g.
$\left\langle\overline{x^{2}}\right\rangle\simeq 2D_{\text{eff}}t$ for long
times; $D_{\text{eff}}$ is defined in Eq. (13). This kind of crossover out of
the superstatistical context was considered in Ref. Safdari et al. (2015); dos
Santos and Junior (2021). Both temporal scales considered here are in Fig. (7)
for a general situation that admits a coupling between $\alpha$ and $D$. Now
we will explore in more detail the coupled situations between $\alpha$ and
$D$.

To understand the effects of $\alpha$ on $D$ or vice versa, we consider a
coupled situation for the PDF of the anomalous exponent, introducing an
$\alpha$-PDF conditioned to diffusivity values, namely

$\displaystyle\pi(\alpha,D)=\tilde{g}(\alpha|D)f(D),$ (14)

considering $f(D)=\int_{\Omega_{\alpha}}\pi(\alpha,D)d\alpha$, and
$\int_{\Omega_{\alpha}}\tilde{g}(\alpha|D)d\alpha=1$. This setup to
conditioned distribution emulates a correlation between fluctuating
parameters, such as discussed in two-variable superstatistics treatment Itto
and Beck (2021). The distribution of anomalous exponents may be obtained by
the following integration $g(\alpha)=\int_{\Omega_{\alpha}}\pi(\alpha,D)dD$.
Here, we also consider the conditioned $\tilde{g}(\alpha|D)$ PDF as a Gaussian
superstatistic for the anomalous exponent, as follows

$\displaystyle\pi(\alpha,D)=\mathcal{N(D)}^{-1}\exp\left(-\frac{(\alpha-\alpha_{p})^{2}}{2\sigma_{D}^{2}}\right)f(D),$
(15)

in which
$\mathcal{N}(D)=\int_{\Omega}\exp\left(-(\alpha-\alpha_{p})^{2}(2\sigma_{D}^{2})^{-1}\right)d\alpha$.
The $\sigma_{D}$ contain the correlations between $\alpha$ and $D$. For
$\sigma=\text{const}$ regardless of $D$ we recover the standard cases
approached previously. Thereby, $\sigma(D)$ explicitly depends on the
diffusivity, because the $\pi$-superstatistics in Eq. (15) may emulate systems
whose correlations between the anomalous exponent and diffusivity are unknown,
but exist.

The PDF of diffusivities $f(D)$ may assume numerous shapes as reported in
Chechkin et al. (2017). The main goal now is to introduce a PDF $f(D)$ to
exemplify how the distribution of diffusivity impacts the distribution of
anomalous exponent. To do that, we chose the most famous distribution, which
is the exponential PDF, namely

$\displaystyle f(D)$ $\displaystyle=$
$\displaystyle\frac{1}{D_{0}}\exp\left(-\frac{D}{D_{0}}\right).$ (16)

The superstatistical approach to Gaussian packets with exponential PDF for
diffusivities implies a Laplace diffusion. Therefore, the exponential
diffusivity distribution was reported in a large number of investigations Wang
et al. (2009, 2012); Metzler (2017); Chubynsky and Slater (2014). Sometimes
the Laplace diffusion occurs only when the observable time is shorter than the
correlation time of the system. Here, we extend such analysis to a general
situation in which the anomalous exponent has a distribution conditioned to
diffusivity values.

To continue our discussion, we consider Eq. (16) to describe the heterogeneous
diffusivities of the tracer, and (15) for the PDF of the anomalous exponents
with

$\displaystyle\sigma_{D}=\sigma_{0}(D/d_{0})^{a},$ (17)

in which $\sigma_{D}$ controls the shape of $g(\alpha)$ PDF, $\sigma_{0}$ and
$d_{0}$ are constant parameters, and $a\in\Re$ is the coupling parameter. For
$a=0$, we recover the uncoupled case. Fig. (5-a) show PDF $f(D)$ for different
$D_{0}$ values and Fig. (5-b) shows the influence of the $D_{0}$ values in
$g(\alpha)=\int_{\Omega_{D}}\pi(\alpha,D)dD$. This figure exemplifies the
$D_{0}$ influence on the effective diffusivity (13) for different coupling
values in anomalous exponent distribution. For the lower values of $a$ and
$D$, we have constant behaviour for $D_{\text{eff}}$, and for higher values
for $a$ and $D$, the $D_{\text{eff}}$ presents an upper bound. However, we
obtain different results for $a<0$ and large $D_{0}$ values; the effective
diffusivity becomes smaller. Another interesting effect happens when $a$ is
large and $D_{0}$ is small, because the diffusivity of particles chosen from
$f(D)$ are always small with large $a$ values, which implies that $\alpha$-PDF
becomes more and more concentrated around $\alpha\sim\alpha_{p}$, and in such
circumstances $D_{\text{eff}}$ gets smaller.

Fig. (6) complements our analysis by showing us the effects of coupling the
variable $a$ over the effective diffusivity and distribution of anomalous
exponents, in Figs. (6-a) and (6-b), respectively. These results show how the
choices of diffusivity values can change the choices of the anomalous
diffusion; the $(\alpha,D)$-coupling may be explored for other superstatistics
that are connected with different experimental scenarios.

With this background of a conditioned PDF for $\alpha$ and $D$, Fig. (7) shows
different temporal scales for $\psi_{\alpha}$. In Fig. (7-a) and (7-b), the
superstatistics on the anomalous exponent is relevant for long times and short
times, respectively. Thereby, Fig. (7-a) and Fig. (7-b) reveal that the
coupling parameter $a=1$ presents a strong break of power-law time relation in
MSD behaviour for $\sigma_{0}\sim 0.07$ ($\sigma\simeq 0.07D$) in long and
short temporal scales, respectively. Moreover, there is an evident crossover
between short and long times in which, for $\sigma_{0}=0.01$, we may observe a
power-law behaviour.

Figure 5: Fig. (a) shows PDF of diffusivities for different $D_{0}$ values.
Fig. (b) shows the influence of $D_{0}$ values on effective diffusivity,
resulted by Eq. (15) into (13). Both figures consider
$\sigma_{0}/d_{0}^{a}=1$, $\alpha_{p}=1$ and $\alpha\in(0,2]$.  Figure 6: Fig.
(a) shows effective diffusivity for different coupling parameters $a$. Fig.
(b) shows the influence of $a$ values on PDF of anomalous exponents, obtained
by integration $g(\alpha)=\int_{\Omega_{\alpha}}\pi(\alpha,D)dD$. Both figures
consider $\sigma_{0}/d_{0}^{a}=1$, $\alpha_{p}=1$ and $\alpha\in(0,2]$.
Figure 7: Figures show MSD for different temporal scale but with same values
of $\sigma_{0}$ in the PDF defined by Eq. (15). Fig. (a) considers
$\psi_{\alpha}(t)=\left(1+t/\tau\right)^{\alpha}\log\left(1+t/\tau\right)$
into Eq. (12), in which $\tau=0.1$ and $\alpha=0.5$. Fig. (b) considers
$\psi_{\alpha}(t)=te^{-\tau/t}+t^{\alpha}e^{-t/\tau}$ into Eq. (12), in which
$\tau=0.1$ and $\alpha=1.5$. Both figures consider same values for $a=1$,
$D_{0}=1$, $d_{0}=1$, $\alpha_{p}=1$ and $\alpha\in(0,2]$. Particularly, the
dashed curves represent the uncoupled situations between $\alpha$ and $D$,
i.e. $a=0$ in Eq. (17).

##  V Conclusion

Ensembles of anomalous tracers have been reported in different experimental
settings and can be associated with different heterogeneous complex systems.
More recently, the two-variable superstatistics of fractional Brownian walkers
was proposed to describe protein diffusion in bacteria Itto and Beck (2021).
Further, current experimental investigations with single-particle tracking
technique in tracers have reported statistics of anomalous exponents and
diffusivities, for instance in endosomes trajectories inside living eukaryotic
cells Korabel et al. (2021), spreading dynamics of amoeboid cells Cherstvy et
al. (2018), diffusion of tracers in mucin hydrogels Cherstvy et al. (2019) and
beads diffusion in Xenopus extracts Speckner and Weiss (2021). In this
framework, our investigation promotes advances in theoretical models that
reveal anomalous non-Gaussian diffusion as a macroscopic behaviour which is a
consequence of superstatistics of the individual anomalous tracers.

We have discussed the current scenarios that embrace the PDF of the anomalous
diffusion exponents and diffusion coefficients for tracers diffusing in
complex environments. Then, we proposed two different truncated
superstatistics for the $\alpha$-anomalous exponent to apply in scaled
Brownian motion. Numerous anomalous exponent distributions in literature are
contained in the considered cases, which are truncated Gaussian and
$\chi^{2}$-Gamma distributions. Firstly, the situations without diffusivity
effects were considered, i.e. $f(D)\simeq\delta(D-D_{0})$, so that it does not
overshadow the effects of the anomalous exponent. Thereby, we have realised
numerical simulations of heterogeneous scaled Brownian equation to present a
match with the superstatistical model. This allowed us to explore the PDF in
space and MSD in time, which presents non-Gaussian distribution without the
effects of diffusivities.

Furthermore, we also have considered other temporal scaling $\psi(t)$ in sBm,
to obtain power-law time behaviours for MSD. Particularly, we discuss why a
logarithmic time factor is necessary in temporal scaling to obtain an
effective anomalous exponent for long time behaviours. We have also described
how the heterogeneous tracers are affected when the truncated Gaussian PDF for
the anomalous exponent is conditioned to diffusivity values, i.e.
$\pi(\alpha,D)\simeq f(D)\tilde{g}(\alpha|D)$. This investigation opens new
interesting approaches that may be useful for describing particles diffusing
in disordered environments. A large diversity of new situations can be
approached for different $\alpha$ distributions as well as for other temporal
scales $\psi_{\alpha}(t)$ in sBm.

## Acknowledgements

We thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
(CAPES), Grant No. 001.

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