#  Non-Gaussian Normal Diffusion in Low Dimensional Systems

Qingqing Yin1, Yunyun Li1,†, Fabio Marchesoni1,2,†, Shubhadip Nayak3, and
Pulak K. Ghosh3† Center for Phononics and Thermal Energy Science, Shanghai Key
Laboratory of Special Artificial Microstructure Materials and Technology,
School of Physics Science and Engineering, Tongji University, Shanghai 200092,
China
Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
Department of Chemistry, Presidency University, Kolkata 700073, India
Corresponding authors. E-mail<EMAIL_ADDRESS>†
<EMAIL_ADDRESS>†<EMAIL_ADDRESS>
###### Abstract

Brownian particles suspended in disordered crowded environments often exhibit
non-Gaussian normal diffusion (NGND), whereby their displacements grow with
mean square proportional to the observation time and non-Gaussian statistics.
Their distributions appear to decay almost exponentially according to
“universal” laws largely insensitive to the observation time. This effect is
generically attributed to slow environmental fluctuations, which perturb the
local configuration of the suspension medium. To investigate the microscopic
mechanisms responsible for the NGND phenomenon, we study Brownian diffusion in
low dimensional systems, like the free diffusion of ellipsoidal and active
particles, the diffusion of colloidal particles in fluctuating corrugated
channels and Brownian motion in arrays of planar convective rolls. NGND
appears to be a transient effect related to the time modulation of the
instantaneous particle’s diffusivity, which can occur even under equilibrium
conditions. Consequently, we propose to generalize the definition of NGND to
include transient displacement distributions which vary continuously with the
observation time. To this purpose, we provide a heuristic one-parameter
function, which fits all time-dependent transient displacement distributions
corresponding to the same diffusion constant. Moreover, we reveal the
existence of low dimensional systems where the NGND distributions are not
leptokurtic (fat exponential tails), as often reported in the literature, but
platykurtic (thin sub-Gaussian tails), i.e., with negative excess kurtosis.
The actual nature of the NGND transients is related to the specific
microscopic dynamics of the diffusing particle.

Non-Gaussian Normal Diffusion, transport phenomena, stochastic process, active
matter

##  I Introduction

Possibly misinterpreting the original works of Albert Einstein and Marian
Smoluchowski on Brownian motion, one tends to associate the normal diffusion
of an ideal Brownian particle with the Gaussian distribution of its spatial
displacements. Recent observations Granick1 ; Granick2 ; Bhatta ; Sung1 ;
Sung2 ; Granick3  of Brownian motion in fluctuating crowded environments led
to question the generality of this notion. Indeed, it implicitly assumes
Fick’s diffusion Gardiner , whereby the directed displacements of an
overdamped particle, say, in the $x$ direction, $\Delta x(t)=x(t)-x(0)$, would
grow according to the asymptotic Einstein law, $\langle\Delta
x^{2}(t)\rangle=2Dt$, and with Gaussian statistics. The probability density
function (pdf) of the rescaled observable, $\delta_{t}=\Delta x/\sqrt{t}$,
would thus be a stationary Gaussian function with half-variance $D$.

However, there are no fundamental reasons why the diffusion of a physical
Brownian tracer should be of the Fickian type. For instance, in real
biophysical systems, displacement pdf’s have been reported, which retain
prominent exponential tails over extended intervals of the observation time,
even after the tracer has attained the asymptotic condition of normal
diffusion. Such an effect, often termed non-Gaussian normal diffusion (NGND),
disappears only for exceedingly long observation times (possibly inaccessible
to real experiments Granick1 ), when the displacement distributions eventually
turn Gaussian, as dictated by the central limit theorem, without changes of
the diffusion constant. Persistent diffusive transients of this type have been
detected in diverse experimental setups Granick1 ; Granick2 ; Bhatta ; other1
; other3 ; other4 . Extensive numerical simulations confirmed the occurrence
of NGND in crowded environments featuring slowly diffusing or changing
microscopic constituents (filaments Granick1 ; Bhatta , large hard spheres
Sung1 ; Sung2 ; Granick3 , clusters Kegel ; Kob , and other heterogeneities
Tong ; Cherstvy ).

The current interpretation of this phenomenon postulates the existence of one
or more fluctuating processes affecting composition and geometry of the
particle’s suspension medium Granick1 . It seems reasonable that, for
observation times comparable with the relevant environmental relaxation
time(s), the tracer displacements may obey a non-Gaussian statistics. The
rescaled pdf’s, $p(\delta_{t})$, are expected to be Gaussian for both much
shorter and much larger observation times, but with different half-variance:
the free diffusion constant, $D_{0}$, for $t\to 0$ (no crowding effect) and
the asymptotic diffusion constant, $D$, introduced above, for $t\to\infty$
(central limit theorem). The mechanism how the tracer’s normal diffusion sets
in and the constant $D$ remains unaltered through the entire non-Gaussian
transient, varies, instead, from case to case. In summary, key features of the
NGND phenomenon appear to be: (i) its transient nature, whereby the
observables taken into account are intrinsically non-stationary; (ii) a time-
modulated instantaneous diffusivity of the tracer. As discussed in the
following, these conditions can occur even in the absence of external (non-
equilibrium) perturbations of the Brownian dynamics.

A simple heuristic explanation Granick1  of the NGND phenomenon models the
effects of the slowly fluctuating environment in terms of an ad hoc
distribution of the tracer’s diffusion constant. Imposing an exponential
distribution of the diffusion constant with average $D$, a straightforward
superstatistical procedure yields the exponential (Laplace) rescaled
distribution, $p(\delta_{t})=\exp(-\delta_{t}/\alpha)/2\alpha$, with
$\alpha^{2}=D$. A more suggestive NGND paradigm is provided by the notion of
diffusing diffusivity Slater , whereby the asymptotic particle’s diffusion
constant is replaced by a time fluctuating auxiliary observable, $D(t)$.
Regarding $D(t)$ as a continuous stochastic process with average $D$ and time
constant $\tau$, the distribution $p(\delta_{t})$ changes from exponential for
$t\ll\tau$ to Gaussian for $t\gg\tau$; in both time regimes, the displacement
diffusion is normal, with $\langle\Delta x^{2}(t)\rangle=2Dt$ Slater . Refined
variations of these paradigms Metzler2 ; Jain1 ; Jain2 ; Tyagi ; Luo ; Sokolov
; Metzler , predict different exponential decays of the transient
distributions. These phenomenological approaches have two major limitations,
namely: (i) They fail to incorporate the free Gaussian diffusion detected in
most real and numerical experiments at very short observation times, $t\to 0$,
when crowding plays no role. This is because these approaches purportedly
ignore the microscopic details of the actual diffusion mechanisms; (ii) They
generally aim at “universal” non-Gaussian transient pdf’s, namely, at
functions $p(\delta_{t})$ insensitive to $t$ over extended domains, $t<\tau$.
[In the superstatistical models $\tau=\infty$.] However, even if this strategy
may appear to agree with NGND observations for complex systems Granick1 ;
Granick2 ; Bhatta ; Sung1 ; Sung2 ; Granick3 , it is obvious that to reproduce
the exponential-Gaussian crossover, the transient rescaled distributions must
assume the form $p(\delta_{t},t)$, i.e., they must depend explicitly on $t$.

This study focuses on the microscopic mechanisms responsible for NGND. To this
purpose, motivated by a preliminary study PRR , we investigated, both
numerically and analytically, directed diffusion of different idealized
tracers in confined geometries. We selected low dimensional systems mostly
inspired to cell biology RMP_BM ; Lipowski . For appropriately short
observation times, NGND emerges as a transient effect of the time modulation
of the tracers’ microscopic diffusivity. This effect can occur even in the
absence of environmental fluctuations. It suffices to require that the
tracer’s dynamics be governed by two concurring diffusion mechanisms, at least
one of them characterized by a finite relaxation time, $\tau$. During
transients times of the order of $\tau$, the displacement distributions can
deviate from their asymptotic Gaussian profile also after normal diffusion has
set in. Moreover, such deviations do not necessarily imply the emergence of
“fatter” exponential tails (leptokurtic transients), but under certain
conditions, the distribution tails can get “thinner” (platykurtic transients).

This observation suggests typical NGND features are to be found in much wider
a class of diffusion systems. Indeed, contrary to experimental and numerical
observations on extended systems, the NGND transient displacement
distributions in low dimensional models, are found to depend on the
observation time. This led us to address the question of phenomenological
fitting functions capable of reproducing the $t$-dependence of the rescaled
pdf’s, $p(\delta_{t},t)$. We also noticed that the $t$ dependence of the
transient rescaled pdf’s can be suppressed, though not completely, by
considering models where the onset of normal diffusion is controlled by some
intrinsic time constant, which can be taken much shorter than the upper bound,
$\tau$, of the non-Gaussian transient. This provides us with a criterion to
formulate low dimensional models that better capture the known NGND
phenomenology in complex systems.

The present paper is organized as follows. In Sec. II we elaborate on a toy
discrete model of NGND proposed first in Ref. Slater  and then revisited in
Ref. PRR . The purpose of this section is to single out key NGND aspects, like
the time scales regulating the diffusion mechanisms and the nature of the non-
Gaussian transients, that is, lepto- versus platykurtic. In Sec. III we
consider the diffusion of a two dimensional (2D) ellipsoidal Brownian particle
in a highly viscous, homogeneous and isotropic fluid in thermal equilibrium.
For observation times shorter than its rotational relaxation time, the
particle does undergo normal diffusion. However, its instantaneous diffusivity
in a given direction is modulated in time due to its elongated shape. This
results in exponentially decaying transient distributions of the particle’s
directional displacements. In Sec. IV we analyze the diffusion of a 2D self-
propelling symmetric particle in a homogeneous and isotropic active medium
with finite orientational relaxation time. NGND is characterized here by thin
tails of the transient displacement distributions. In both cases, however,
transients are governed by one time scale, only, their orientational diffusion
time, $\tau$: on increasing the observation time, normal diffusion just
anticipates the onset of the Gaussian statistics of the particle’s
displacements. In Sec. V we introduce a phenomenological fitting function,
$p_{\beta}(\delta_{t})$ for the rescaled displacement distributions, with only
one adjustable parameter, $\beta$. This function is designed ad hoc to ensure
normal diffusion with the observed diffusion constant, $D$, at any time, while
$\beta$ encodes the $t$-dependence of the rescaled displacement distributions.
Relevant values of the fitting parameter $\beta$ are $\beta=2$ for a Gaussian
pdf, $\beta=1$ for a Laplace (exponential) pdf, $\beta<2$ for a leptokurtic
pdf, and $\beta>2$ for a platykurtic pdf. In Sec. VI we analyze the NGND
phenomenon in a narrow corrugated channel chemphyschem ; PNAS  with
fluctuating pores PRR . NGND occurs for time-correlated pore fluctuations,
random and periodic, alike, and, more importantly, for observation times
comprised between two distinct, controllable time scales. The correlation time
of the pore fluctuations sets the transient time scale, $\tau$, whereas the
average pore-crossing time governs the onset of normal diffusion. Upon
choosing the former much larger than the latter, the NGND transient is made
grow wider and the $t$-dependence of $\beta$ weaker. As a practical
application of the tools introduced thus far, in Sec. VII we investigate the
diffusion of a passive Brownian tracer in a periodic array of planar counter-
rotating convection rolls. The peculiarity of this model is that, by tuning
its dynamical parameters, transients can change from lepto- to platykurtic.
Two are the systems’s characteristic time scales: the mean time for the
particle to first exit a convection roll and its average revolution period
inside the roll. At low (high) temperatures, the former (latter) time scale is
larger and thus plays the role of transient time, $\tau$; accordingly, the
NGND transients are leptokurtic (platykurtic). Finally, in Sec. VIII we
summarize the main conclusions of our approach to NGND.

##  II A Discrete NGND Model

Though sounding exotic to some readers, the phenomenon of NGND turns out to be
way more general than the more familiar Fickian diffusion. To make this point,
we elaborate now on a coarse grained model, first proposed in Ref. Slater ,
which serves well the purpose of illustrating NGND in continuous systems of
any dimensionality.

Let us coarse grain the trajectory of a tagged particle in the $x$ direction
as the sum of small random steps, $\Delta x_{i}$, taken at fixed discrete
times, $t_{i}=i\Delta t$, where $i=1,\dots N$ and $\Delta t=1$, for
simplicity. Accordingly, the position of the particle at time $N$ is
$x_{N}=\sum_{i=1}^{N}\Delta x_{i}$. A stochastic average over the particle’s
steps $\Delta x_{i}$ yields the mean square displacement at time $N$,

$\langle x_{N}^{2}\rangle=\sum_{i=1}^{N}\langle\Delta
x_{i}^{2}\rangle+2\sum_{i\neq j}{}^{\prime}\langle\Delta x_{i}\Delta
x_{j}\rangle,$ (1)

where $\sum_{i\neq j}^{\prime}$ stays for $\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}$.
Sufficient conditions to establish normal diffusion are that: (1) the step
directions are uncorrelated, $\langle\Delta x_{i}\Delta x_{j}\rangle=0$, that
is, for any given $\Delta x_{i}$, displacements $\Delta x_{j}$ and $-\Delta
x_{j}$, are equiprobable; (2) the variance, $\langle\Delta x_{i}^{2}\rangle$,
of the step probabilities, $p(\Delta x_{i})$, are of the same order of
magnitude, though not necessarily identical. These requirements are less
stringent than the assumptions implicit in the standard random walker model
for Brownian motion Gardiner .

Indeed, for the sake of generality, one should not rule out finite
correlations of the step lengths Slater . For instance, we can assume that
during each unit time step the particle’s diffusion is normal with time-
dependent constant, $D_{i}$, i.e.,

$p(\Delta x_{i})=(4\pi D_{i})^{-1/2}\exp(-\Delta x_{i}^{2}/4D_{i}).$ (2)

This assumption guarantees that the directions of the particle’s steps are
uncorrelated, while their length correlation is controlled by the auto-
correlation of the time sequence of the constants $D_{i}$, which, in turn, is
specific to the system at hand. It follows immediately that

$\langle x_{N}^{2}\rangle=2\langle D\rangle N,$ (3)

and

$\displaystyle\langle x_{N}^{4}\rangle-3\langle
x_{N}^{2}\rangle^{2}=12\mu_{D}\langle D\rangle^{2}N+24{\sum_{i\neq
j}}^{\prime}C_{ij},$ (4)

with $\mu_{D}=(\langle D^{2}\rangle-\langle D\rangle^{2})/\langle
D\rangle^{2}$ and $C_{ij}=\langle D_{i}D_{j}\rangle-\langle
D_{i}\rangle\langle D_{j}\rangle$. For any given stationary model, there
exists an appropriate distribution of the constants $D_{i}$, $p(D_{i})$, so
that $\langle D_{i}\rangle\equiv\langle D\rangle$.

Suppose now that two particle steps, $\Delta x_{i}$ and $\Delta x_{j}$ are
statistically uncorrelated only for large time differences, i.e., $\langle
D_{i}D_{j}\rangle=\langle D_{i}\rangle\langle D_{j}\rangle$ for $|i-j|>\tau$.
We then distinguish two limiting cases,

(i) $N\gg\tau$, where

$\mu_{x}=\frac{\langle x_{N}^{4}\rangle-3\langle x_{N}^{2}\rangle^{2}}{\langle
x_{N}^{2}\rangle^{2}}=\frac{3\mu_{D}}{N}\rightarrow 0.$ (5)

A vanishing excess kurtosis, $\mu_{x}$, hints at a Gaussian $x_{N}$
distribution. This is the asymptotic limit of the displacement distributions
predicted by the central limit theorem.

(ii) $N<\tau$, where

$\mu_{x}=\frac{\langle x_{N}^{4}\rangle-3\langle x_{N}^{2}\rangle^{2}}{\langle
x_{N}^{2}\rangle^{2}}=3\bar{\mu}_{D}.$ (6)

with $\bar{\mu}_{D}=(2/N^{2}\langle D^{2}\rangle)\sum_{i\neq
j}{}^{\prime}C_{ij}$. Eqs. (3) and (6) embody the definition of NGND. The
finite excess kurtosis, $\mu_{x}$ depends on the actual auto-correlation of
the constants $D_{i}$. For instance, on assuming $\langle
D_{i}D_{j}\rangle=\langle D^{2}\rangle$ for all $i$ and $j$ with $|i-j|<\tau$,
we obtain $\bar{\mu}_{D}=\mu_{D}$. In particular, for the exponential
distribution $p(D_{i})=\exp(-D_{i}/\langle D\rangle)/\langle D\rangle$ assumed
in the diffusing diffusivity model of Ref. Slater , $\mu_{D}=1$. Not
surprisingly, the resulting value of the excess kurtosis, $\mu_{x}=3$,
corresponds to a Laplace distribution of the total displacement $x_{N}$ Slater
.

Of course a more realistic choice of the correlator $C_{ij}$ can yield
different values of $\mu_{x}$. In most applications $C_{ij}$ is definite
positive and decays to zero with time, i.e., with $|i-j|$; hence
$0<\mu_{x}<3$, Accordingly, the corresponding $x_{N}$ distributions are
leptokurtic, with tails decaying slower than those of a Gaussian distribution,
but typically faster than exponentially. On the other hand, we cannot exclude
the possibility that $C_{ij}$ decays to zero oscillating. This implies that,
in principle, $\mu_{D}$ can assume negative values, so that the corresponding
transient distribution of $x_{N}$ may be platykurtic. In Refs. nonG1 ; nonG2 ;
nonG3  the present approach has been extended also to microscopically non-
Gaussian diffusive processes [where the $\Delta x$ distribution of Eq.(2) does
not apply].

We conclude this section with a final remark about the time scales involved in
this discrete model. One time scale has been introduced explicitly, namely the
characteristic decay time, $\tau$, of the correlator $C_{ij}$ or,
equivalently, the correlation time of the step lengths, $\Delta x_{i}$. A
second one is implicit in our choice for the step distribution, $p(\Delta
x_{i})$. In Eq. (2) the coarse grained diffusion was assumed to be normal over
the time step $\Delta t=1$. This implies that in the corresponding continuum
system normal diffusion is expected to have occurred at some intrinsic time
scale much shorter than $\tau$. Of course the discrete model of this section
cannot reproduce the diffusion properties at times shorter than the
discretization time scale, $\Delta t$.

##  III Diffusion of an Ellipsoidal Particle

We consider first the simple case of a 2D ellipsoidal particle of semiaxes $a$
and $b$, with $a>b$, diffusing in a highly viscous, homogeneous and isotropic
medium, subject to equilibrium thermal fluctuations. This is a well-known
problem in biological physics Berg . The particle’s elongation causes a
dissipative coupling between the center of mass translational degrees of
freedom, $x$ and $y$ in the laboratory frame, and the rotational degree of
freedom, $\theta$. As sketched in Fig. 1(b), the angle $\theta$ defines the
orientation of the particle’s long axis with respect to the horizontal $x$
axis. The physical consequences of such a mechanism were first recognized by
F. Perrin Perrin . An ellipsoidal particle tends to diffuse independently in
directions parallel and perpendicular to its long axis, that is along its
principal axes. The relevant diffusion constants in the body frame are denoted
here by $D_{a}$ and $D_{b}$, with $D_{a}\geq D_{b}$. In 2D, rotational
diffusion is governed by an additional diffusion constant, $D_{\theta}$, which
will be handled here as unrelated to the translational constants, $D_{a}$ and
$D_{b}$, to avoid unnecessary complications involving hydrodynamic effects and
fabrication issues Berg ; Perrin . Over the angular relaxation time
$\tau=1/D_{\theta}$, random diffusion erases any directional memory of the
particle’s motion. Related to this mechanism is the crossover between
anisotropic diffusion with constants $D_{a}$ and $D_{b}$ at short observation
times, $t\ll\tau$, and isotropic diffusion with constant $D=(D_{a}+D_{b})/2$
at long observation times, $t>\tau$ ellipsoid .

The anisotropic-isotropic crossover can be numerically investigated by
integrating the Langevin equations Kloeden  describing the roto-translational
motion of a free ellipsoidal Brownian particle,

$\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle\xi_{x}(t),~{}~{}~{}\dot{y}=\xi_{y}(t),$ (7)
$\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle\xi_{\theta}(t),$
(8)

where the translational noises, $\xi_{i}(t)$ with $i=x,y$, and the rotational
noise, $\xi_{\theta}(t)$, model three independent stationary Gaussian
fluctuation sources with zero means and autocorrelation functions
$\langle\xi_{i}(t)\xi_{j}(0)\rangle=2D_{ij}\delta(t)$ and
$\langle\xi_{\theta}(t)\xi_{\theta}(0)\rangle=2D_{\theta}\delta(t)$. The
matrix $D_{ij}$ encodes the dissipative roto-translational coupling, namely
ellipsoid

$D_{ij}=({1}/{2})[(D_{a}+D_{b})\delta_{ij}+(D_{a}-D_{b})M_{ij}(\theta)],$ (9)

with ${\bf M}=\begin{pmatrix}\cos 2\theta&\sin 2\theta\\\ \sin 2\theta&-\cos
2\theta\end{pmatrix}$.

Figure 1: Overdamped 2D ellipsoidal particle of semi-axes $a=0.5$ and $b=0.05$
diffusing in a homogeneous medium with Langevin Eqs. (7)-(9): (a), (c)
displacement pdf’s for different initial orientations [uniform $\theta(0)$
distribution in (a), and $\theta(0)=0$ in (c)] and increasing observation
times, $t$, see legends; (b) $\langle\Delta x^{2}\rangle$ vs. $t$ for the
initial conditions (i.c.) of (a) (empty symbols) and (c) (filled symbols).
Simulation parameters are: $D_{a}=1$, $D_{b}=(b/a)D_{a}$ and $D_{r}=0.01$.
Asymptotic diffusion in (b) follows the normal diffusion law, $2Dt$ with
$D=(D_{a}+D_{b})/2$ (dashed line), independent of the i.c. At very short
times, the diffusion constant depends on $\theta(0)$ (see sketch). The pdf’s
have been fitted by means of Eq. (17) for $D$ fitting the large-$t$ simulation
data of (b) and $\beta$ as reported in the legends.

Despite their apparent simplicity, the analytical solution of the Langevin
Eqs. (7)-(9) is rather cumbersome Pecora ; Franosch . The diffusion properties
of a typical ellipsoidal particle are summarized in Fig. 1. The mean square
displacement, $\langle\Delta x^{2}(t)\rangle$, plotted in panel (b) as a
function of the observation time $t$, was first computed under assuming a
uniform distribution of $\theta(0)$. This initial condition (i.c.) was
justified with the practical difficulty of measuring the particle
instantaneous orientation and with the isotropy of the suspension medium. The
resulting asymptotic diffusion constant, numerically determined as

$D=\lim_{t\to\infty}\langle\Delta x^{2}(t)\rangle/2t,$

agrees with the expected value, $D=(D_{a}+D_{b})/2$, obtained by averaging
$D_{ij}(\theta)$ in Eq. (9) with respect to the isotropic equilibrium
distribution of $\theta$. This result is indeed an effect of our choice for
the i.c. of $\theta$. A stochastic average over a uniform $\theta(0)$
distribution is equivalent to imposing isotropic particle’s diffusion, that is
establishing Einstein law at any time. In contrast, by setting $\theta(0)=0$,
the numerical data for $\langle\Delta x^{2}\rangle$ versus $t$, also shown in
Fig. 1(b), bridge two linear laws with different diffusion constant: $D=D_{a}$
for $t\ll\tau$ and $D=(D_{a}+D_{b})/2$ for $t\gg\tau$.

More revealing are the distributions of the unidirectional displacements,
$\Delta x$, for increasing observation times, $t$, plotted in panels (a) and
(c), respectively for a uniform initial angular distribution and
$\theta(0)=0$. As first theoretically predicted by Prager Prager  and
numerically confirmed by the authors of Ref. Franosch , for both i.c. the
rescaled displacement pdf’s do approach the Gaussian profile of Fickian
diffusion with half-variance $D=(D_{a}+D_{b})/2$, but only for $t\gg\tau$,
that is well after the anisotropic-isotropic crossover took place. Most
remarkably, for $\theta(0)=0$, in panel (c), the displacement distributions
approach a Gaussian profile both for $t\ll\tau$ and $t\gg\tau$, each with the
corresponding half-variance $D$ shown in panel (b), respectively, $D_{a}$ and
$(D_{a}+D_{b})/2$. The short-$t$ “reentrant” Gaussian distribution does not
appear in panel (a), due to the randomized i.c.. The explanation of this
behavior is simple. In panel (c), the particle’s long axis was initially
oriented parallel to the $x$ axis, $\theta(0)=0$. Therefore, it started
diffusing in the $x$ direction like a one dimensional Brownian particle, with
diffusion constant $D_{a}$. Subsequently, angular fluctuations mixed diffusion
along the two symmetry axes with time constant $\tau$. This argument can be
extended to any choice of $\theta(0)$: based on Eq. (9), the short-$t$
diffusion constant is expected to be
$D=(1/2)[(D_{a}+D_{b})+(D_{a}-D_{b})\cos\theta(0)]$, see inset of Fig. 1(b).
Of course, the i.c. only influence the anisotropic diffusion regime at short
$t$.

There is only one characteristic time scale in this model, namely, the angular
relaxation time, $\tau=1/D_{\theta}$. However, normal diffusion turns out to
set in for shorter observation times, $t\sim\tau$, than the displacement
Gaussian statistics. To explain this behavior, we notice that during the
transient time, $\tau$, a maximum mean square displacement, $2D_{a}\tau$,
occurs parallel to the major axis; observing the same displacement in the
perpendicular direction would take a larger time,
$\tau_{*}=\tau(D_{a}/D_{b})$. The onset of the Gaussian $\Delta x$ statistics
is thus delayed to larger observation times with $t>\tau_{*}$.

In conclusion, this simple model of equilibrium Brownian motion exhibits NGND.
On decreasing the observation time, $t$, the rescaled displacement
distribution in a fixed laboratory direction, changes from Gaussian for
$t\gg\tau$, to a leptokurtic distribution with fat exponential tails for
$t\sim\tau$, independently of the i.c.. This behavior is consistent with the
phenomenological picture of Sec. II. This is apparent in the case of uniform
initial orientation. Normal diffusion is ensured by the fact that, after the
particle has taken a step $\Delta x_{i}$ at the discrete time $t_{i}=i\Delta
x$, it will next take a step $\pm\Delta x_{j}$ at time $t_{j}$, with equal
probability. On the contrary, the step lengths $\Delta x_{i}$ and $\Delta
x_{j}$ are correlated for $|t_{j}-t_{i}|<\tau$. Indeed, the effective half-
width of the diffusing particle parallel to the $x$ axis, varies randomly
between $b$ at $\theta=0,\pi$ and $a$ at $\theta=\pm\pi/2$. Accordingly, the
particle’s instantaneous diffusion constant fluctuates between $D_{a}$ and
$D_{b}$; its fluctuations are exponentially time correlated with time constant
$\tau$. As discussed in Sec. II, this leads to a rescaled pdf,
$p(\delta_{t},t)$, with positive excess kurtosis.

##  IV Diffusion of a Janus Particle

We consider next the case of a pointlike particle undergoing persistent
Brownian motion, namely, a 2D artificial microswimmer. Typical artificial
microswimmers are Brownian particles capable of self-propulsion in an active
medium Granick ; Muller . Like in the foregoing section, the suspension medium
can be taken homogeneous, isotropic and highly viscous. Such particles are
designed to harvest environmental energy by converting it into kinetic energy.
The simplest class of artificial swimmers investigated in the literature are
the so-called Janus particles (JP), mostly spherical colloidal particles with
two differently coated hemispheres, or “faces” Marchetti ; Gompper . Recently,
artificial micro- and nanoswimmers of this class have been the focus of
pharmaceutical (e.g., smart drug delivery smart ) and medical research (e.g.,
robotic microsurgery Wang ). Relevant to the present work is the observation
that their function is governed, in time and space, by their diffusive
properties through complex environments, which are often spatially patterned
Bechinger  or confined ourPRL .

Figure 2: Symmetric Janus particle diffusing in a homogeneous medium with
$D_{0}=1$, $v_{0}=1$, $D_{\theta}=0.01$, and uniform distributions of the
particle’s initial position and orientation, Eqs. (10)-(11): (a) displacement
pdf’s at different observation times, $t$; (b) diffusion law, $\langle\Delta
x^{2}\rangle$ vs. $t$. The numerical data agree well with the analytical law
of Eq. (12) (solid curve); the normal diffusion limits at large and short $t$,
Eqs. (13) and (14), are drawn for a comparison (dashed lines); the pdf’s in
(a) have been fitted by means of Eq. (17) with $D$ fitting the large-$t$ data
in (b) and an appropriate choice of $\beta$ (see legend). The pdf’s with
$\beta=2$ at the shortest and largest $t$, panel (a), are Gaussian curves with
half-variance $D_{0}$ and $D_{0}+D_{s}$, respectively.

The overdamped dynamics of a pointlike active JP can be formulated by means of
two translational and one rotational Langevin equation

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
v_{0}\cos\theta+\xi_{x}(t),~{}~{}~{}\dot{y}=v_{0}\sin\theta+\xi_{y}(t),$ (10)
$\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle\xi_{\theta}(t),$
(11)

where $x$ and $y$ are the coordinates of the particle’s center of mass, and
the self-propulsion velocity has constant modulus, $v_{0}$, and orientation
$\theta$, taken with respect to the $x$ axis, see sketch in Fig. 2(b). The
translational noises in the $x$ and $y$ directions, $\xi_{x}(t)$ and
$\xi_{y}(t))$, and the rotational noise, $\xi_{\theta}(t)$, are stationary,
independent, delta-correlated Gaussian noises,
$\langle\xi_{i}(t)\xi_{j}(0)\rangle=2\delta_{ij}D_{i}\delta(t)$ with
$i,j=x,y,\theta$. The noise strengths $D_{x}=D_{y}=D_{0}$ (isotropic
translational fluctuations) and $D_{\theta}$ are assumed here to be unrelated
for generality (e.g., to account for different self-propulsion mechanisms
ourPRL ). The reciprocal of $D_{\theta}$ is the correlation (or angular
persistence) time, $\tau$, of the self-propulsion velocity. For simplicity, we
ignore chiral effects due to unavoidable fabrication defects Wang ; Loewen ;
chiral1 . It is worthy comparing the Langevin Eqs. (7)-(8) and (10)-(11): for
the ellipsoidal particle anisotropy is geometric, i.e., due to its elongated
shape, whereas, for a pointlike JP anisotropy is dynamical, i.e., associated
with the instantaneous orientation of its self-propulsion velocity.

A detailed analytical treatment of the Langevin Eqs. (10)-(11) is to be found
in Ref. Franosch_SciRep . The unidirectional diffusion of a free JP in 2D
reads Golestanian07 ; Loewen09 ; chiral2 ,

$\displaystyle\langle\Delta
x^{2}(t)\rangle=2(D_{0}+D_{s})t+D_{s}\tau(e^{-|t|/\tau}-1),$ (12)

which approaches the Einstein law,

$\displaystyle\langle\Delta x^{2}(t)\rangle=2(D_{0}+D_{s})t,$ (13)

only for $t\gg\tau$. Here, the unidirectional diffusion constant, $D$,
consists of two distinct contributions, a translational, $D_{0}$, and a self-
propulsion term, $D_{s}=v_{0}^{2}/2D_{\theta}$. Instead, for short observation
times Eq. (12) tends to

$\displaystyle\langle\Delta x^{2}(t)\rangle=2D_{0}t,$ (14)

that is, to the normal diffusion law of a passive particle with $v_{0}=0$. The
analytical law of Eq. (12) and its normal limits for large and small
observation times compare well with our simulation results in Fig. 2(b).

The displacement distribution, $p(\delta_{t},t)$, exhibits a Gaussian profile
both for $t\to 0$ and $t\to\infty$, but with different half-variances,
respectively $D_{0}$ and $D=D_{0}+D_{s}$, see Fig. 2(a). The crossover between
these two Gaussian limits is characterized by platykurtic transient pdf’s with
fast decaying tails. Experimental evidence of this phenomenon has been
reported in Ref. Loewen2 . In the limit $t\to 0$, the displacement
distributions become sensitive to the particle’s initial orientation. For a
uniform distribution of $\theta(0)$, shown in Fig. 2(a), the rescaled pdf’s
approach a Gaussian function with half-variance $D_{0}$, as to be expected for
an isotropic persistent Brownian motion in the ballistic regime, $t\ll\tau$.
However, for a fixed value of $\theta(0)$, say, $\theta(0)=0$, the pdf is
still a Gaussian with the same half-variance, $D_{0}$, but its center moves to
higher $\Delta x$ values, with $\langle\Delta x(t)\rangle=v_{0}t$ sperm  (not
shown). For intermediate observation times, $t\simeq\tau$, the displacement
pdf’s develop two symmetric maxima a distance of the order of the persistence
length, $\Delta x\sim v_{0}\tau$, from their centers Loewen2 .

The different nature of the diffusion transients of ellipsoidal and active
JP’s can be easily explained in terms of the coarse grained model of Sec. II.
The orientation of a JP is time correlated; from Eqs. (10)-(11),
$\langle\cos\theta(t)\cos\theta(0)\rangle=\langle\sin\theta(t)\sin\theta(0)\rangle=(1/2)\exp(-|t|/\tau)$.
This implies that both the orientation and the length of the discrete steps in
the $x$ direction, $\Delta x_{i}$ are time correlated; given any pair of
steps, $\Delta x_{i}$ and $\Delta x_{j}$, both their time correlations vanish
asymptotically only for $|t_{j}-t_{i}|\gg\tau$. However, on comparing panels
(a) and (b) of Fig. 2 we notice the existence of a rather wide range of
observation times, where $\langle\Delta x^{2}(t)\rangle$ has approached a
linear function of $t$, Eq. (13), while the rescaled $\Delta x$ distributions
are still apparently platykurtic.

The platykurtic nature of this transient is consistent with the coarse grained
model of Sec. II, because the angular correlation of the self-propulsion
velocity vector amounts to an oscillatory behavior of $D_{i}-\langle
D_{i}\rangle$, a necessary condition to observe a negative excess kurtosis,
$\mu_{x}<0$. It remains to explain why, like in Sec. III, the onset of normal
diffusion anticipates the onset of the Gaussian $\Delta x$ statistics. We know
Loewen2  that the self-propulsion mechanism of Eqs. (10)-(11) is responsible
for the non-Gaussian profile of $p(\delta_{t})$, an effect mitigated by the
translational noise as long as $2D_{0}t>l_{\theta}^{2}$, where
$l_{\theta}=v_{0}\tau$ is the JP persistence length. Therefore, the Gaussian
statistics of the unidirectional JP displacements is expected to emerge only
for $t>\tau_{*}$, with $\tau_{*}=\tau(D_{s}/D_{0})$. Note that in the
simulations of Fig. 2 we set $\tau_{*}>\tau$.

The results presented in this section lead us to conclude that we are in the
presence of another manifestation of the NGND phenomenon.

##  V Transient Displacement Distributions

As mentioned in Sec. I, the notion of NGND is commonly associated with the
existence of a wide interval of observation times, where diffusion follows a
normal law with fixed constant, $D$, and the rescaled displacement
distribution, $p(\delta_{t})$, decays (almost) exponentially independently of
$t$. The exponential to Gaussian crossover is hardly accessible to direct
observation Granick1 . In the low dimensional systems investigated here,
instead, such a transition takes place over a relatively narrower $t$
interval, which led us to look for a phenomenological function
$p(\delta_{t},t)$ fitting our simulation data from transient up asymptotic $t$
values.

Contrary to the diffusing diffusivity models, where the limiting Laplace and
Gaussian distributions are functions of the sole diffusion constant, $D$, a
more realistic fitting procedure needs at least one additional parameter,
$\beta$, to capture the $t$-dependence of the transient pdf’s. Inspired by the
numerical findings of Secs. III and IV, we started from the compressed
exponential function

$p(\delta_{t})=p_{0}e^{-(\delta_{t}/\delta_{0})^{\beta}},$ (15)

where $\beta\geq 1$. The scaling factor, $\delta_{0}$, and the normalization
constant, $p_{0}$, have been computed by imposing the conditions

$\int_{0}^{\infty}p(\delta_{t})d\delta_{t}=1,~{}~{}~{}\int_{0}^{\infty}\delta_{t}^{2}p(\delta_{t})d\delta_{t}=2D,$
(16)

to obtain the one-parameter ad hoc fitting function,

$p_{\beta}(\delta_{t})=\frac{\beta}{\Gamma(\frac{1}{\beta})^{\frac{3}{2}}}\left[\frac{\Gamma(\frac{3}{\beta})}{2D}\right]^{\frac{1}{2}}\exp\left[-\left(\frac{\delta_{t}^{2}}{2D}\frac{\Gamma(\frac{3}{\beta})}{\Gamma(\frac{1}{\beta})}\right)^{\frac{\beta}{2}}\right].$
(17)

This function has been derived phenomenologically starting from the standard
stretched exponential distribution,
$p_{\beta}(\delta_{t})=A\exp(-B\delta_{t}^{\beta})$. The constants $A$ and $B$
have then be determined by normalizing $p_{\beta}(\delta_{t})$ to one and
ensuring that its second moment yields $\langle\delta_{t}^{2}\rangle=2D$ for
any value of the free parameter $\beta$. In view of its derivation, the
heuristic distribution (17) may apply also to the transients of
microscopically non-Gaussian diffusion models nonG1 ; nonG2 ; nonG3 . The
fitting parameter $\beta$ is allowed to vary with $t$; it assumes values in
the range $1\leq\beta\leq 2$ for leptokurtic distributions (positive excess
kurtosis) and $\beta\geq 2$ for platykurtic distributions (negative excess
kurtosis).

The fits of the pdf’s drawn in panels (a),(c) of Figs. 1 and (a) of Fig. 2
have been generated from Eq. (17) by setting $D$ equal to the diffusion
constants that best fitted the large-$t$ diffusion data in the respective
panels (b) and, then, computing $\beta$ to get the best fit of the rescaled
displacement distributions at different $t$. The same fitting procedure has
been applied in Figs. 3 and 4 of the forthcoming sections.

Our phenomenological formula (17) fits rather closely the numerical pdf’s
reported in Secs. III and IV, at least for sufficiently large observation
times. As a matter of fact, the heuristic argument leading to the fitting
function $p_{\beta}(\delta_{t})$ assumes normal diffusion at any $t$. This is
consistent with the diffusive dynamics of the ellipsoidal Brownian particle
with isotropic i.c., displayed in Figs. 1(a)-(b). However, this cannot be the
case, for instance, of the active JP of Fig. 2, whose diffusion law for
$t<\tau$ clearly deviates from the asymptotic law of Eq. (13). A comparison
with the simulation output confirms that the proposed fitting procedure works
well for both systems in the transient regime, $t>\tau$.

##  VI Diffusion in a Time Modulated Channel

In most numerical and experimental investigations Granick1 ; Granick2 ; Bhatta
; Sung1 ; Sung2 ; Granick3  the transient distributions of $\delta_{t}$ are
presented as sort of universal functions, $p(\delta_{t})$, which decay with
(almost) exponential law independently of $t$. Sometimes the transient
interval is so wide that the exponential-Gaussian crossover is not accessible
to direct observation. The question then rises as to what extent the low
dimensional systems addressed in this work may share that property. In the
notation of Sec. V, this corresponds to determining conditions for the fitting
parameter $\beta$ to be constant with $\beta\neq 2$ (non-Gaussian transient)
over a wide range of $t$. Note that in the models of Secs. III and IV the
onset of the normal diffusion and the Gaussian statistics regimes, which
delimit the NGND transient, are governed by the sole angular relaxation time
$\tau$ ($\tau_{*}$ being proportional to $\tau$). In the standard formalism of
the central limit theorem this would correspond to saying that the higher
cumulants of the displacement distribution vanish slower with the observation
time than the second moment approaches its linear growth Feller . In this
regard, more interesting are systems where the NGND transients are delimited
by two distinct time constants.

Figure 3: Diffusion of a pointlike overdamped particle in a randomly
fluctuating channel, Eq. (19) with $\varepsilon(t)$ representing the Ornstein-
Uhlenbeck process of Eq. (20). Simulation parameters are: $y_{L}=1$,
$x_{L}=\pi$, $D_{0}=1$, $D_{\varepsilon}=3$, $\tau=50$, and random $x(0)$,
$y(0)$ and $\varepsilon(0)$. In the main panel, rescaled displacement pdf’s
are shown for increasing observation times, $t$, in the normal diffusion
regime, see data for $\langle\Delta x^{2}\rangle$ vs. $t$ in the inset. The
fitting $\beta$ values have been obtained from Eq. (17) with $D=0.335$. At
short $t$, the statistics of our data is not good enough to resolve the $t$
dependence of $\beta$.

A study case is represented by the diffusion of a standard Brownian particle
in a confined geometry chemphyschem , the simplest example being a chainlike
structure of cavities connected by narrow pores PRR . In 2D, the dynamics of
an overdamped symmetric Brownian particle in a channel is modeled by two
simple Langevin equations

$\dot{x}=\xi_{x}(t),~{}~{}~{}\dot{y}=\xi_{y}(t),$ (18)

where $x$ and $y$ are the coordinates of the particle’s center of mass and the
translational fluctuations $\xi_{x}(t)$ and $\xi_{y}(t)$ are zero-mean, white
Gaussian noises with autocorrelation functions
$\langle\xi_{i}(t)\xi_{j}(0)\rangle=2\delta_{i,j}D_{0}\delta(t)$ and
$i,j=x,y$. The strength of $\xi_{i}(t)$ coincides with the free-particle
diffusion constant, $D_{0}$, which is typically proportional to the
temperature of the suspension fluid. However, contrary to Secs. III and IV,
the particle is now confined to diffuse inside a narrow corrugated channel
with axis oriented along $x$ and symmetric walls, $y=\pm w(x,t)$. Following
Refs. chemphyschem , we assumed for simplicity the sinusoidally modulated
channel half-width

$w(x,t)=(y_{L}/2)[\varepsilon^{2}+(1-\varepsilon^{2})\sin^{2}(\pi x/x_{L})].$
(19)

Here $y_{L}$ and $x_{L}$ are respectively the maximum width and the length of
the unit channel cell, sketched in Fig. 3, and $\varepsilon^{2}y_{L}$ is the
fluctuating width of the pores located at $x{\rm mod}(x_{L})=0$. In the case
of a pointlike particle, hydrodynamic effects PNAS  can be ignored. Moreover,
let the width of the channel pores be time modulated without affecting the
particle’s free diffusion constant, $D_{0}$, for instance, by applying a
tunable external gating potential. Therefore, when integrating the Langevin
Eq. (18), we neglected the particle radius with respect to $x_{L}$ and $y_{L}$
(pointlike particle approximation) and imposed reflecting boundary conditions
at the walls ourPRL .

In Ref. PRR  we considered the case when $\varepsilon(t)$ is an Ornstein-
Uhlenbeck process

$\dot{\varepsilon}=-\varepsilon/\tau+\sqrt{D_{\varepsilon}/\tau^{2}}~{}\xi_{\varepsilon}(t),$
(20)

where $\xi_{\varepsilon}(t)$ is another Gaussian zero-mean valued noise,
independent of $\xi_{x}(t)$ and $\xi_{y}(t)$ and delta-correlated,
$\langle\xi_{\varepsilon}(t)\xi_{\varepsilon}(0)\rangle=2\delta(t)$. The
channel pores open and close randomly in time with average width
$\langle\varepsilon^{2}\rangle y_{L}$, where $\langle\varepsilon^{2}\rangle$
coincides with the variance of $\varepsilon(t)$,
$D_{\varepsilon}/\tau=(\pi/2)\langle|\varepsilon|\rangle^{2}$.

This channel model manifests prominent NGND, as illustrated in Fig. 3. The
displacement distributions are Gaussian for both very short (not shown, see
Ref. PRR ) and asymptotically long observation times. Indeed, the particle
diffuses freely with constant $D_{0}$ inside each channel’s cell for
$t<\tau_{L}$, with $\tau_{L}=x_{L}^{2}/8D_{0}$, before escaping into an
adjacent cell after a mean exit time
$\tau_{0}=\tau_{L}/\langle|\varepsilon|\rangle$ PRR . For $t\gg\tau_{0}$, the
$x$ directed diffusion process can thus be described as a random walker with
spatial step $x_{L}$ and time constant $\tau_{0}$; memory of the i.c. adopted
in our simulations is completely erased. The ensuing mean square displacement
then follows the Einstein law with approximated diffusion constant
$D=x_{L}^{2}/2\tau_{0}$ JCP137 . The displacement distribution assumes its
asymptotic Gaussian profile only for observation times much larger than the
correlation time of the pore fluctuations, $t\gg\tau$. In the simulations of
Fig. 3, we set $\tau\gg\tau_{0}$, which thus defines a NGND transient
interval, $(\tau_{0},\tau)$, where diffusion is normal, but the displacement
distributions are non-Gaussian. By taking such interval wide enough, the
transient pdf’s, $p(\delta_{t},t)$, grow insensitive to $t$, and so do the
fitted $\beta$ values. This way, we mimic the situation reported in the
literature Granick1 ; Granick2 ; Bhatta ; Sung1 ; Sung2 ; Granick3  for more
complex systems.

This prescription for NGND control is independent of the detailed statistics
of the pore fluctuations. For instance, one can consider the case of a
periodically time modulated pore width with

$\varepsilon(t)=\delta_{\varepsilon}\cos(t/\tau).$ (21)

Figure 4: Diffusion of a pointlike overdamped particle in a corrugated
channel, Eq. (19), with $\varepsilon(t)$ representing the periodically time
modulation of Eq. (21). Simulation parameters are: $y_{L}=1$, $x_{L}=\pi$,
$D_{0}=1$, $\delta^{2}_{\varepsilon}=0.03$, and (a) $\tau=100$, (b)
$\tau=500$. The initial conditions were set by imposing uniform distributions
of the particle’s initial position and $\varepsilon(t)$ initial phase. In the
main panels, rescaled displacement pdf’s are plotted for increasing $t$, in
the normal diffusion regime. The values of $\beta$ in the legend have been
obtained by fitting Eq. (17) with $D$ computed numerically from the
asymptotes, $\langle\Delta x^{2}\rangle=2Dt$, drawn in the insets.

The simulation data plotted in Fig. 4(a) confirm the existence of the NGND
transient interval $(\tau_{0},\tau)$, where $\tau$ is now the period of the
sinusoidal function $\varepsilon(t)$ of Eq. (21) and
$\langle\varepsilon^{2}\rangle=\delta_{\varepsilon}^{2}/2=(\pi^{2}/8)\langle|\varepsilon|\rangle^{2}$.
For a quantitative comparison, in Figs. 3 and 4(a)
$\langle\varepsilon^{2}\rangle$ have been assigned the same value, so that for
the simulation parameters of Fig. 4 both time scales, $\tau_{0}$ and $\tau$,
are larger by approximately the same factor two.

The NGND phenomenon in Fig. 4(a) is apparent. The rescaled pdf’s shown there
are clearly non-Gaussian. Fat oscillating tails arise for $t>\tau$, as an
effect of the spatial periodicity of the channel. Indeed, the oscillation
period of the plotted distributions is of the order $x_{L}/\sqrt{t}$. This
effect, detectable also in Figs. 3 and 4(b), plays here a marginal role.
Indeed, on disregarding such oscillations, the tails of the non-Gaussian
distributions can still be fitted by the function of Eq. (17) for an
appropriate choice of the free parameter $\beta$.

However, in Fig. 4(a) and in contrast with Fig. 3, the deterministic nature of
the pore time modulation allowed us to resolve a weak $t$-dependence of
$\beta$, without increasing the statistical accuracy of our simulation runs.
It is important to remark that such a residual $t$-dependence of the transient
rescaled pdf’s can be further suppressed by widening the transient interval
$(\tau_{0},\tau)$. An example is shown in panel (b) of Fig. 4, where the
simulation parameters are the same as in panel (a), except for the modulation
period, $\tau$, which is five times larger. We remark that, for the simple
model at hand, this result is analytically predictable upon reformulating the
particle’s dynamics in the dimensionless units, $x\to x/x_{L}$, $y\to y/x_{L}$
and $t\to t/\tau$.

##  VII Diffusion in Convection Rolls

We finally address the reasons why transients under NGND conditions can be
either lepto- or platykurtic. In Secs. III and IV we looked at two simple
models, which exhibit distributions of the one or the other type,
respectively, with $1\leq\beta\leq 2$ and $\beta\geq 2$. We consider now a
slightly more complicated 2D system, which can undergo both transients,
depending on the choice of its dynamical parameters. The numerical analysis of
its diffusion properties will help us shed light on the different microscopic
mechanisms responsible for these two type of transients, thus justifying the
generalization of the NGND notion proposed in this work.

Figure 5: Diffusion in the periodic convective flow pattern of Eq. (22): (a)
Flow cell unit consisting of four counter-rotating subcells; (b) The
asymptotic diffusion constant, $D$ vs. $D_{0}$: the numerical data (dots) are
compared with the analytical prediction discussed in the text, see Eq. (24).
The stream function parameters are $U_{0}=1$ and $L=2\pi$, and the diffusion
scale is $D_{L}=U_{0}L/2\pi$.

To this purpose we investigated the diffusion of a pointlike overdamped
particle of coordinates $x$ and $y$, suspended in a stationary planar laminar
flow with periodic center-symmetric stream function Gollub1 ; Gollub2 ; Rosen
; Pomeau ; Saintillan ; Vulpiani ; Neufeld .

$\psi(x,y)=({U_{0}L}/{2\pi})\sin({2\pi x}/{L})\sin({2\pi y}/{L}),$ (22)

where $U_{0}$ is the maximum advection speed and $L$ the wavelength of the
flow unit cell. The ensuing particle’s dynamics can be formulated in terms of
two driven Langevin equations,

$\displaystyle\dot{x}=u_{x}+\xi_{x}(t),~{}~{}~{}~{}\dot{y}=u_{y}+\xi_{y}(t),$
(23)

with the vector $(u_{x},u_{y})=(\partial_{y},-\partial_{x})\psi$ representing
the local advection velocity. As illustrated in Fig. 5(a), this defines four
counter-rotating flow subcells, also termed convection rolls. The
translational noises, $\xi_{i}(t)$ with $i=x,y$ are stationary, independent
Gaussian noises with auto-correlation functions
$\langle\xi_{i}(t)\xi_{j}(0)\rangle=2D_{0}\delta_{ij}\delta(t)$. They can be
regarded as modeling homogeneous, isotropic thermal fluctuations. In our
simulations, the flow parameters, $U_{0}$ and $L$ were kept fixed, as they
define the natural length and time units, $L$ and $\Omega_{L}^{-1}=L/2\pi
U_{0}$, respectively. Therefore, the only tunable parameter left is the noise
strength, $D_{0}$. Having in mind a stationary system, we assumed uniform
distributions of the initial particle’s coordinates, $x(0)$ and $y(0)$.
Indeed, due to the incompressibility of $(u_{x},u_{y})$, in the presence of
translational noise, a particle’s trajectory eventually fills up the flow unit
cell uniformly. To this regard, we remind that, in the absence of noise, the
advection period tends to diverge as the closed trajectory of a dragged
particle runs close the subcell boundaries Weiss ; hence, for $D_{0}=0$ the
particle gets trapped in a convection roll Rosen ; Pomeau ; Neufeld .

Figure 6: Diffusion mechanisms in the periodic flow pattern of Eq. (22): mean
first-exit time, $T_{D}$, vs. thermal noise, $D_{0}$. Convection flow
parameters are $U_{0}=1$ and $L=2\pi$. The asymptotic solid lines on the left
and right are respectively $T_{0}$ and $T_{0}/4$, with $T_{0}$ given in Eq.
(25); the horizontal dashed line represents the advection period, $T_{L}$.
Three $D_{0}$ intervals. shaded in different colors, are separated by $D_{*}$,
obtained by imposing $T_{D}=T_{L}$, and $D_{L}$, defined in Fig. 5. In each
interval, the range of $\beta$ is reported for reader’s convenience; no NGND
was detected for $D_{0}>D_{L}$.

Particle transport in such a flow pattern has been studied under diverse
conditions and a rich phenomenology has emerged Gollub1 ; Gollub2 ; Rosen ;
Pomeau ; Saintillan ; Vulpiani ; Neufeld . We focus here on the Brownian
diffusion of a passive particle under the simultaneous action of translational
fluctuations and advective drag. A first important feature of this system is
illustrated in Fig. 5(b), where we plotted the dependence of the asymptotic
diffusion constant, $D$, on the noise intensity (and particle’s no-flow
diffusion constant), $D_{0}$. The mean square displacement is an
asymptotically linear function of time for any choice of $D_{0}$. However, on
increasing $D_{0}$, the diffusion constant, $D$, changes from

$D=\kappa\sqrt{D_{L}D_{0}},$ (24)

for $D_{0}<D_{L}$ (advective diffusion), to $D=D_{0}$, for $D_{0}>D_{L}$
(thermal diffusion), an abrupt crossover occurring at $D_{0}\simeq D_{L}$,
with $D_{L}=U_{0}L/2\pi$ RR2 . The constant $\kappa$ depends on the geometry
of the flow cells Rosen ; Pomeau ; for a 2D array of square counter-rotating
convection rolls, $\kappa\simeq 1.06$ Rosen . This property can be explained
with the fact that for $D<D_{L}$ the spatial diffusion occurs along the
separatrices delimiting the four subcells of the stream function, $\psi(x,y)$,
of Fig. 5(a). Stated otherwise, the diffusion process is regulated by the
advection velocity field ourPOF . Vice versa, for $D_{0}>D_{L}$ the effects of
advection on the particle’s diffusion become negligible. Not surprisingly, we
detected NGND only for $D_{0}<D_{L}$.

The diffusion process is governed by two competing mechanisms: (i) Particle’s
circulation inside the counter-rotating subcells of $\psi(x,y)$. The
corresponding vorticity, $\nabla\times{\mathbf{u}}=-\nabla^{2}\psi$, has a
maximum, $\Omega_{L}=2\pi U_{0}/L$, at the center of the subcells. This
defines the time scale, $T_{L}=2\pi/\Omega_{L}$, for the advection period,
that is an estimate of the average time taken by advection to drag the
particle around a convection roll; (ii) Diffusion across the convection rolls.
The mean first-exit time, $T_{D}$, of a Brownian particle out of a unit
convection cell of $\psi(x,y)$, can be easily computed for $D_{0}\gg D_{L}$
simply by ignoring advection Gardiner ,

$T_{0}=\frac{1}{D_{0}}\left(\frac{L}{2\pi}\right)^{2}\left(\frac{4}{\pi}\right)^{4}~{}\sum_{m,n}^{(\rm
odd)}\frac{1}{m^{2}}\frac{1}{n^{2}}\frac{1}{m^{2}+n^{2}},$ (25)

where the summation is restricted to the odd values of $m$ and $n$. In the
opposite limit, $D_{0}\ll D_{L}$, $T_{D}$ is just one fourth of $T_{0}$,
because, as anticipated above, at very low noise levels, the exit process
consists of a slow activation mechanism, which takes the particle from the
center of a subcell to its boundaries, followed by a relatively faster flow-
driven propagation along the grid formed by the subcell separatrices.

Figure 7: Leptokurtic, $D_{0}<D^{*}$ (a), and platykurtic transients,
$D^{*}<D_{0}<D_{L}$ (b), in a periodic array of 2D convection rolls. $D_{0}$,
$t$ and $\beta$ are reported in the legends; convection flow parameters are
$U_{0}=1$ and $L=2\pi$. All transient pdf’s were taken in the regime of normal
diffusion, see insets.

Thanks to thermal fluctuations, the Brownian particle jumps from roll to roll,
thus diffusing in the $x,y$ plane. Its coarse-grained motion can be modeled as
a discrete random walker with time constant $T_{D}$ Gardiner . Therefore, for
large observation times, $t>T_{D}$, the particle executes normal diffusion.

For the simulation parameters adopted in Fig. 6, the crossover between low-
and high-noise estimates of $T$, respectively $T_{D}=T_{0}$ and
$T_{D}=T_{0}/4$, occurs in the region of advective diffusion, $D_{0}<D_{L}$.
More remarkably, it appears to correspond to the condition, $T_{D}=T_{L}$,
namely, when the two competing time scales of the particle’s dynamics inside a
convection roll coincide. Such a condition defines a unique $D_{0}$ value,
$D_{*}$, which splits the advective diffusion domain into two distinct
subdomains, respectively, $D_{0}<D_{*}$ and $D_{*}<D_{0}<D_{L}$.

Numerical simulation clearly shows evidence of NGND for $t>T_{D}$, in close
analogy with the models of Secs. III and IV, except for an important
peculiarity: The transient displacement distributions displayed in Fig. 7,
turn out to be leptokurtic, with $1\leq\beta\leq 2$, for $D_{0}<D_{*}$, and
platykurtic, with $\beta\geq 2$, for $D_{*}<D_{0}<D_{L}$. This can be
explained with the fact that here the displacement length correlation, see
Sec. II, is dominated by thermal noise in the lower $D_{0}$ interval, where
$T_{D}>T_{L}$, and by advection in the larger $D_{0}$ interval, where
$T_{L}>T_{D}$. Accordingly, in the formulation of Secs. III and IV, the role
of transient time, $\tau$, is played respectively by $T_{D}$ for $D_{0}<D_{*}$
and by $T_{L}$ for $D_{0}>D_{*}$.

For $D_{*}\ll D_{0}<D_{L}$, the onset of normal diffusion and the exponential-
Gaussian transitions are thus regulated by two distinct time scales,
respectively, $T_{0}$ and $T_{L}$. Indeed, the slowest time modulation of the
particle’s dynamics is due to the advective drag inside the convection rolls.
By generalizing our discussion for the NGND of a free JP, Sec. IV, we conclude
that such a rotational dynamics must be responsible for the negative excess
kurtosis of the unidirectional particle’s displacements reported in Fig. 7(b).
The range of the $\beta$ values, fitted according to the procedure of Sec. V,
is shown in Fig. 6 for each $D_{0}$ interval.

In conclusion, the NGND transients of this model can change from leptokurtic
to platykurtic simply by raising the strength of the internal noise. Most
remarkably, this and related diffusive systems are easily accessible to direct
experimental demonstration Gollub2 ; Saintillan .

##  VIII Conclusions

In this work we have investigated NGND transients Granick1 ; Granick2 ; Bhatta
; Sung1 ; Sung2 ; Granick3  in low dimensional stochastic processes. These
become apparent when the Einstein law, which characterizes normal diffusion,
sets in for observation times, $t$, shorter than the asymptotic Gaussian
displacement statistics, predicted by the central limit theorem. A wide class
of low dimensional systems manifest NGND under the condition that their local
dynamics is subjected to time correlated modulations.

Time modulation can affect the effective particle geometry (e.g., its cross-
section in the diffusion direction, Sec. III), its dynamics (e.g., its
isotropic self-propulsion mechanism, Sec. IV), or its confinement geometry
(e.g., the cross-section of the directed channel containing the particle, Sec.
VI). In all cases discussed here the system’s modulation is time correlated
with time constant, $\tau$, larger than any other microscopic dynamical time
scale. We then noticed that NGND becomes more prominent when the onset times
of normal diffusion and the Gaussian displacement statistics are well
separated, with the former much lower than the latter. This situation is well
illustrated by the fluctuating narrow channel of Sec. VI, where normal
diffusion occurs for $t$ larger than the mean pore crossing time and the
Gaussian statistics sets in for $t$ larger that the tunable correlation time
of the pore modulation.

In low dimensional systems, NGND features exhibit a smooth dependence on the
observation time. The transient rescaled displacement distributions are not
“universal” over large $t$ intervals, in sharp contrast with the extended
disordered systems first studied in the literature Granick1 ; Granick2 ;
Bhatta ; Sung1 ; Sung2 ; Granick3 . To quantify the $t$-dependence of the
transient pdf’s we introduced an ad hoc fitting function,
$p_{\beta}(\delta_{t})$, which, by construction, reproduces the normal
diffusion law, with diffusion constant obtained by direct observation, and
fits the numerical curves $p(\delta_{t},t)$ by tuning only one free parameter,
$\beta$. Actually, in Sec. VI we noticed that by increasing the gap between
the two distinct time scales defining the transient interval, the
$t$-dependence of $\beta$ is suppressed, with $\beta$ tending to one (Laplace
distribution). This situation closer resembles the current description of the
NGND phenomenon in complex systems.

However, NGND in low dimensional systems has the advantage of being easily
controllable by tuning the time modulation of the microscopic dynamics. For
instance, the two simplest models discussed here, the free ellipsoidal and
Janus particles, exhibit remarkably different transient distributions,
respectively with fat, $1\leq\beta\leq 2$, and thin tails, $\beta\geq 2$.
Platykurtic transient distributions are peculiar to systems with rotational
modulation of the diffusion process, because, as discussed in Sec. II, this
can cause negative time correlations of the unidirectional displacement
lengths; hence the negative values of the excess kurtosis.

This conclusion is corroborated by Brownian diffusion in the periodic array of
2D convection rolls discussed in Sec. VII. In contrast with the elementary
models of Secs. III and IV, there the physical mechanism determining the
transient time varies depending on the strength of the thermal fluctuations.
At low temperatures, the transient dynamics of the particle is governed by
isotropic random jumps from convection roll to convection roll, largely
insensitive to the details of its trajectory inside the individual rolls. On
the contrary, at higher temperatures, but still in the advective diffusion
regime, roll jumping grows faster compared with the circulation inside the
rolls; transients are then dominated by a rotational dynamics, which causes a
negative excess kurtosis of the particle’s displacements.

We conclude now mentioning a number of open issues we intend to address in the
next future.

(i) We showed that low-dimensional systems exhibit NGND transients for
observation times not too much larger than their largest intrinsic relaxation
time. It remains to be seen how one can make such transient time intervals
wider, for instance, by means of a hierarchy of additional stochastic degrees
of freedom.

(ii) We wonder to what extent our discussion of discrete NGND in Sec. II is
related to the formalism of the large deviations theory LDT . This might
provide an alternate phenomenological description of the NGND transients, also
applicable to higher dimensional systems.

(iii) NGND transients in laminar flows are of great relevance in
microfluidics. This results reviewed in Sec.7 will be published in a more
detailed report to appear soon JFM . We showed that leptokurtic (platykutic)
transients are an effect of the mostly thermal (advective) tracer’s diffusion.
The question then rises as how this explanation translates in the cases of
turbulent flows, a recurrent problem in biological systems.

(iv) Finally, it is conceivable that persistent NGND transients impact how
active micro-swimmers interact with each other or with confining walls or
other obstacles, to form all kinds of clustered structures. The implications
of such a mechanism in the technology of active matter need further
investigation.

###### Acknowledgements.

Y.L. is supported by the NSF China under grant No. 11875201 and No. 11935010.
P.K.G. is supported by SERB Start-up Research Grant (Young Scientist) No.
YSS/2014/000853 and the UGC-BSR Start-Up Grant No. F.30-92/2015.

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