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arXiv:0905.1429v3 [cond-mat.stat-mech] 16 Oct 2009
Paradoxical diffusion: Discriminating between normal and anomalous random walks
Bart lomiej Dybiec∗and Ewa Gudowska-Nowak†
Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research,
Jagellonian University, ul. Reymonta 4, 30–059 Krak´ow, Poland
(Dated: October 16, 2009)
Commonly, normal diffusive behavior is characterized by a linear dependence of the second cen-
tral moment on time, hx2(t)i ∝ t, while anomalous behavior is expected to show a different time
dependence, hx2(t)i ∝ tδwith δ < 1 for subdiffusive and δ > 1 for superdiffusive motions. Here we
demonstrate that this kind of qualification, if applied straightforwardly, may be misleading: There
are anomalous transport motions revealing perfectly “normal” diffusive character (hx2(t)i ∝ t), yet
being non-Markov and non-Gaussian in nature. We use recently developed framework [1, Phys. Rev.
E75, 056702 (2007)] of Monte Carlo simulations which incorporates anomalous diffusion statistics
in time and space and creates trajectories of such an extended random walk. For special choice
of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis
of paradoxical diffusion is shown to hide temporal memory effects which can be properly detected
only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov
equation).
PACS numbers: 05.40.Fb, 05.10.Gg, 02.50.-r, 02.50.Ey,
I. INTRODUCTION
Usually various types of diffusion processes are classi-
fied by analysis of the spread of the distance traveled by
a random walker. If the mean square displacement grows
like h[x−x(0)]2i ∝ tδwith δ < 1 the motion is called sub-
diffusive, in contrast to normal (δ= 1) or superdiffusive
(δ > 1) situations. For a free Brownian particle moving
in one dimension, a stochastic random force entering its
equation of motion is assumed to be composed of a large
number of independent identical pulses. If they posses
a finite variance, then by virtue of the standard Cen-
tral Limit Theorem (CLT) the distribution of their sum
follows the Gaussian statistics. However, as it has been
proved by L´evy and Khintchine, the CLT can be general-
ized for independent, identically distributed (i.i.d) vari-
ables characterized by non-finite variance or even non-
finite mean value. With a L´evy forcing characterized by
a stability index α < 2 independent increments of the
particle position sum up yielding h[x−x(0)]2i ∝ t2/α
[2], see below. Such enhanced, fast superdiffusive motion
is observed in various real situations when a test par-
ticle is able to perform unusually large jumps [3, 4, 5].
L´evy flights have been documented to describe motion
of fluorescent probes in living polymers, tracer particles
in rotating flows and cooled atoms in laser fields. They
serve also as a paradigm of efficient searching strategies
[6, 7, 8] in social and environmental problems with some
level of controversy [9].
In contrast, transport in porous, fractal-like media or
relaxation kinetics in inhomogeneous materials are usu-
ally ultraslow, i.e. subdiffusive [5, 10, 11]. The most
∗Electronic address: bartek@th.if.uj.edu.pl
†Electronic address: gudowska@th.if.uj.edu.pl
intriguing situations take place however, when both ef-
fects – occurrence of long jumps and long waiting times
for the next step – are incorporated in the same scenario
[12]. The approach to this kind of anomalous motion
is provided by continuous time random walks (CTRW)
which assume that the steps of the walker occur at ran-
dom times generated by a renewal process. In particular,
a mathematical idealization of a free Brownian motion
(Wiener process W(t)) can be then derived as a limit (in
distribution) of i.i.d random (Gaussian) jumps taken at
infinitesimally short time intervals of non-random length.
Other generalizations are also possible, e.g. W(t) can
be defined as a limit of random Gaussian jumps per-
formed at random Poissonian times. The characteris-
tic feature of the Gaussian Wiener process is the conti-
nuity of its sample paths. In other words, realizations
(trajectories) of the Wiener process are continuous (al-
though nowhere differentiable) [13]. The process is also
self-similar (scale invariant) which means that by rescal-
ing t′=λt and W′(t) = λ−1/2W(λt) another Wiener
process with the same properties is obtained. Among
scale invariant stable processes, the Wiener process is the
only one which possesses finite variance [13, 14, 15, 16].
Moreover, since the correlation function of increments
∆W(s) = W(t+s)−W(t) depends only on time differ-
ence sand increments of non-overlapping times are sta-
tistically independent, the formal differentiation of W(t)
yields a white, memoryless Gaussian process [17]:
˙
W(t) = ξ(t),hξ(t)ξ(t′)i=δ(t−t′) (1)
commonly used as a source of idealized environmental
noises within the Langevin description
dX(t) = f(X)dt +dW (t).(2)
Here f(X) stands for the drift term which in the case of
a one-dimensional overdamped motion is directly related
to the potential V(X), i.e. f(X) = −dV (X)/dX .
2
In more general terms the CTRW concept may asymp-
totically lead to non-Markov, space-time fractional noise
˜
ξ(t), and in effect, to space-time fractional diffusion. For
example, let us define ∆ ˜
W(t)≡∆X(t) = PN(t)
i=1 Xi,
where the number of summands N(t) is statistically
independent from Xiand governed by a renewal pro-
cess PN(t)
i=1 Ti6t < PN(t)+1
i=1 Tiwith t > 0. Let
us assume further that Ti,Xibelong to the domain
of attraction of stable distributions, Ti∼Sν,1and
Xi∼Sα,β , whose corresponding characteristic functions
φ(k) = hexp(ikSα,β )i=R∞
−∞ eikxlα,β (x;σ)dx, with the
density lα,β(x;σ), are given by
φ(k) = exp h−σα|k|α1−iβsignktan πα
2i,(3)
for α6= 1 and
φ(k) = exp −σ|k|1 + iβ 2
πsignklog |k|.(4)
for α= 1. Here the parameter α∈(0,2] denotes the
stability index, yielding the asymptotic long tail power
law for the x-distribution, which for α < 2 is of the
|x|−(1+α)type. The parameter σ(σ∈(0,∞)) charac-
terizes the scale whereas β(β∈[−1,1]) defines an asym-
metry (skewness) of the distribution.
Note, that for 0 < ν < 1, β= 1, the stable variable
Sν,1is defined on positive semi-axis. Within the above
formulation the counting process N(t) satisfies
lim
t→∞ Prob N(t)
(t/c)ν< x= lim
t→∞ Prob
[(t/c)νx]
X
i=1
Ti> t
= lim
n→∞ Prob
[n]
X
i=1
Ti>cn1/ν
x1/ν
(5)
= lim
n→∞ Prob
1
cn1/ν
[n]
X
i=1
Ti>1
x1/ν
= 1 −Lν,1(x−1/ν ),
where [(t/c)νx] denotes the integer part of the number
(t/c)νxand Lα,β(x) stands for the stable distribution of
random variable Sα,β, i.e. lα,β (x) = dLα,β(x)/dx. More-
over, since
lim
n→∞ Prob (1
c1n1/α
n
X
i=1
Xi< x)→Lα,β(x) (6)
and
p(x, t) = X
n
p(x|n)pn(n(t)),(7)
asymptotically one gets
p(x, t)∼(c2t)−ν/α Z∞
0
lα,β (c2t)−ν/αxτν/α lν,1(τ)τν/α dτ,
(8)
where c1and c2are constants. The resulting (in gen-
eral non-Markov) process becomes ν/α self-similar L´evy
random walk [4, 15, 18, 19, 20, 21, 22, 23], i.e.
p(x, t) = t−ν/αp(xt−ν/α,1).(9)
The asymptotic form given by Eq. (8) can be easily
derived [15, 21, 24, 25] for decoupled CTRW by applying
Tauberian theorems to the Montroll-Weiss [26] expres-
sion
p(q, u) = Z∞
0
dt Z+∞
−∞
dxe−ut+iqxp(x, t)
=1−ψ(u)
u
1
1−w(q)ψ(u)(10)
for the Laplace-Fourier transform of p(x, t). The latter
satisfies the integral (master) equation
p(x, t) = δ(x)1−Zt
0
ψ(t)dt
+Zt
0
ψ(t−s)Z∞
−∞
w(x−y)p(y, t)dyds.(11)
Here ψ(t) stands for a waiting time probability density
function (PDF) whereas w(x) denotes a jump length
PDF. With a suitable change of time and space vari-
ables, in the limit of x→ ∞,t→ ∞, the Laplace-Fourier
transform p(q, u) can be written as
p(q, u) = uν−1
uν+|q|α
=uν−1Z∞
0
ds exp [−s(uν+|q|α)] .(12)
The inverse Laplace transform of p(q, u) can be expressed
in a series form [15]:
p(q, t) = 1
2πi Zc+∞
c−∞
p(q, u)eutdu
=
∞
X
k=0
(−1)k
Γ(kν + 1) (|q|αtν)k
=Eν(−|q|αtν),(13)
where Eν(z) is the Mittag-Leffler function of order ν
Eν(z) =
∞
X
k=0
zk
Γ(kν + 1) .(14)
Further application of the inverse Fourier transform in q
yields [15] a final series representation of p(x, t):
p(x, t) = 1
π|y|tν/α
∞
X
k=0
(−1)k
|y|kα
Γ(kα + 1)
Γ(kν + 1)
×cos hπ
2(kα + 1)i(15)
3
where y=x/tν/α . The above series are divergent for
α>ν. However, for α=ν, the summation has been
shown [15] to produce the closed analytical formula
p(x, t) = 1
π|y|t
sin(πν/2)
|y|ν+|y|−ν+ 2 cos(πν /2) (16)
where (as previously) y=x/tν/α.
The exact solution of the decoupled CTRW, as given
by the infinite sum (7) of stable probability densities, has
been studied by Barkai [21] based on a special choice of
PDFs w(x) and ψ(t). In particular, in [21] the extremely
slow convergence of certain CTRW solutions to the (frac-
tional) diffusion approximation has been discussed. The
rigorous proof of equivalence between some classes of
CTRWs and fractional diffusion has been given by Hilfer
and Anton [27]
In this article we investigate CTRW scenarios which,
in an asymptotic limit, yield paradoxical diffusion, i.e.
the non-Markovian superdiffusive process taking place
under sublinear operational time. The combination of
long flights and long breaks between them is responsi-
ble for the characteristic shape of the PDF and scaling
properties of moments. In particular, for α= 2ν, the
paradoxical diffusion process exhibits the same scaling
as an ordinary Brownian motion despite its PDF is sig-
nificantly different from Gaussian.
In a forthcoming Section, the relation between frac-
tional calculus and CTRW approach is briefly reminded
and the experimental results based on numerical PDF
estimators are presented. Section III is devoted to the
discussion of scaling properties of moments. In Section
IV detection and analysis of memory effects in empirical
series of the CTRW-type realizations are proposed and
critically tested.
II. RELATION BETWEEN CTRW AND
FRACTIONAL CALCULUS
The theory of stochastic integration of a corresponding
Ito-Langevin equation with respect to a general CTRW
“measure” d˜
W(t) = dX has been developed in a series of
papers [24, 25, 28, 29]. Here, we study statistical prop-
erties of such a motion constrained to the initial posi-
tion X(0) = 0. To achieve the goals, we adhere to the
scheme of stochastic subordination [1, 29, 30], i.e. we
obtain the process of primary interest X(t) as a function
X(t) = ˜
X(St) by randomizing the time clock of the pro-
cess X(s) using a different clock St. In this approach St
stands for ν-stable subordinator, St= inf {s:U(s)> t},
where U(s) denotes a strictly increasing ν-stable pro-
cess whose distribution Lν,1yields a Laplace transform
he−kU(s)i=e−skν. The parent process ˜
X(s) is composed
of increments of symmetric α-stable motion described in
an operational time s
d˜
X(s) = −V′(˜
X(s))ds +dLα,0(s),(17)
and in every jump moment the relation U(St) = tis ful-
filled. The (inverse-time) subordinator Stis (in general)
non-Markovian hence, as it will be shown, the diffusion
process ˜
X(St) possesses also some degree of memory. The
above setup has been recently proved [1, 29, 30] to give
a proper stochastic realization of the random process de-
scribed otherwise by a fractional diffusion equation:
∂p(x, t)
∂t =0D1−ν
t∂
∂x V′(x) + ∂α
∂|x|αp(x, t),(18)
with the initial condition p(x, 0) = δ(x). In the above
equation 0D1−ν
tdenotes the Riemannn-Liouville frac-
tional derivative 0D1−ν
t=∂
∂t 0D−ν
tdefined by the relation
0D1−ν
tf(x, t) = 1
Γ(ν)
∂
∂t Zt
0
dt′f(x, t′)
(t−t′)1−ν(19)
and ∂α
∂|x|αstands for the Riesz fractional derivative
with the Fourier transform F[∂α
∂|x|αf(x)] = −|k|αˆ
f(x).
Eq. (18) has been otherwise derived from a generalized
Master equation [31]. The formal solution to Eq. (18)
can be written [31] as:
p(x, t) = Eν ∂
∂x V′(x) + ∂α
∂|x|αtνp(x, 0).(20)
For processes with survival function Ψ(t) = 1−Rt
0ψ(τ)dτ
(cf. Eq. (11)) given by the Mittag-Leffler function
Eq. (14), this solution takes an explicit form [15, 24, 25,
31, 32]
p(x, t) =
∞
X
n=0
tνn
n!E(n)
ν(−tν)wn(x) (21)
where E(n)
ν(z) = dn
dznEν(z) and wn(x)∝lα,0(x), see [24,
33].
In this paper, instead of investigating properties of an
analytical solution to Eq. (18), we switch to a Monte
Carlo method [1, 29, 30, 34, 35] which allows generating
trajectories of the subordinated process X(t) with the
parent process ˜
X(s) in the potential free case, i.e. for
V(x) = 0. The assumed algorithm provides means to ex-
amine the competition between subdiffusion (controlled
by a ν-parameter) and L´evy flights characterized by a
stability index α. From the ensemble of simulated trajec-
tories the estimator of the density p(x, t) is reconstructed
and statistical qualifiers (like quantiles) are derived and
analyzed.
As mentioned, the studied process is ν/α self-similar
(cf. Eq. (9)). We further focus on examination of a spe-
cial case for which ν/α = 1/2. As an exemplary values
of model parameters we choose ν= 1, α = 2 (Markovian
Brownian diffusion) and ν= 0.8, α = 1.6 (subordination
of non-Markovian sub-diffusion with L´evy flights). Ad-
ditionally we use ν= 1, α = 1.6 and ν= 0.8, α = 2
as Markovian and non-Markovian counterparts of main
4
cases analyzed. Fig. 1 compares trajectories for all ex-
emplary values of νand α. Straight horizontal lines (for
ν= 0.8) correspond to particle trapping while straight
vertical lines (for α= 1.6) correspond to L´evy flights.
The straight lines manifest anomalous character of diffu-
sive process.
To further verify correctness of the implemented ver-
sion of the subordination algorithm [1], we have per-
formed extensive numerical tests. In particular, some
of the estimated probability densities have been com-
pared with their analytical representations and the per-
fect match between numerical data and analytical results
have been found. Fig. 2 displays numerical estimators of
PDFs and analytical results for ν= 1 with α= 2 (Gaus-
sian case, left top panel), ν= 1 with α= 1 (Cauchy
case, right top panel), ν= 1/2 with α= 1 (left bottom
panel) and ν= 2/3 with α= 2 (right bottom panel). For
those last two cases, the expressions for p(x, t) has been
derived [15], starting from the series representation given
by Eq. (15). For ν= 1/2, α = 1 the appropriate formula
reads
p(x, t) = −1
2π3/2√texp x2
4tEi −x2
4t,(22)
while for ν= 2/3, α = 2 the probability density is
p(x, t) = 32/3
2t1/3Ai |x|
(3t)1/3.(23)
Ei(x) and Ai(x) are the integral exponential function and
the Airy function respectively. We have also compared
results of simulations and Eq. (16) for other sets of pa-
rameters ν,α. Also there, the excellent agreement has
been detected (results not shown).
-30
-20
-10
0
10
20
0 25 50 75 100
t
t
t
t
x x
x x
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
-30
-15
0
15
30
45
0 25 50 75 100
t
t
t
t
x x
x x
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
-20
-10
0
10
20
30
0 25 50 75 100
t
t
t
t
x x
x x
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
-20
-10
0
10
20
0 25 50 75 100
t
t
t
t
x x
x x
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
FIG. 1: Sample trajectories for ν= 1, α = 2 (left top panel),
ν= 1, α = 1.6 (left bottom panel), ν= 0.8, α = 2 (right top
panel) and ν= 0.8, α = 1.6 (right bottom panel). Eq. (17)
was numerically approximated by subordination techniques
with the time step of integration ∆t= 10−2and averaged
over N= 106realizations.
0
0.1
0.2
0.3
0.4
0.5
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
x
x
x
p(x,t) p(x,t)
p(x,t) p(x,t)
ν=1 α=2
t = 1
t = 5
t=20
t=30
0.1
0.2
0.3
0.4
0.5
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
x
x
x
p(x,t) p(x,t)
p(x,t) p(x,t)
ν=1 α=1
t = 1
t = 5
t=20
t=30
10-2
10-1
100
-5 -4 -3 -2 -1 0 1 2 3 4 5 x
x
x
x
p(x,t) p(x,t)
p(x,t) p(x,t)
ν=1/2 α=1
t = 1
t = 5
t=20
t=30
10-3
10-2
10-1
100
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
x
x
x
p(x,t) p(x,t)
p(x,t) p(x,t)
ν=2/3 α=2
t = 1
t = 5
t=20
t=30
FIG. 2: (Color online) PDFs for ν= 1 with α= 2 (left top
panel), ν= 1 with α= 1 (right top panel), ν= 1/2 with
α= 1 (left bottom panel) and ν= 2/3 with α= 2 (right
bottom panel). Eq. (17) was numerically approximated by
subordination techniques with the time step of integration
∆t= 10−3and averaged over N= 106realizations. Solid
lines present theoretical densities: Gaussian (left top panel),
Cauchy (right top panel) and the p(x, t) given by Eqs. (22)
(left bottom panel) and (23) (right bottom panel). Note the
semi-log scale in the bottom panels.
0
0.1
0.2
0.3
0.4
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
0
0.1
0.2
0.3
0.4
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
ν=1.0 α=2.0
ν=1.0 α=1.6
ν=0.8 α=2.0
ν=0.8 α=1.6
10-3
10-2
10-1
100
10-1 100101102103
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
10-3
10-2
10-1
100
10-1 100101102103
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
FIG. 3: (Color online) PDFs (top panel) and 1 −CDF(x, t)
(bottom panel) at t= 2 (left panel), t= 15 (right panel).
Simulation parameters as in Fig. 1. Eq. (17) was numerically
approximated by subordination techniques with the time step
of integration ∆t= 10−3and averaged over N= 106realiza-
tions. Solid lines present theoretical asymptotic x−1.6scaling
representative for α= 1.6 and ν= 1, i.e. for Markovian L´evy
flight.
Figure 3 and 4 display time-dependent probability den-
sities p(x, t) and corresponding cumulative distribution
functions (CD F (x, t) = Rx
−∞ p(x′, t)dx′) for “short” and
for, approximately, an order of magnitude “longer” times.
The persistent cusp [36] located at x= 0 is a finger-print
of the initial condition p(x, 0) = δ(x) and is typically
recorded for subdiffusion induced by the subordinator St
5
0
0.1
0.2
0.3
0.4
0.5
0.6
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
0
0.1
0.2
0.3
0.4
0.5
0.6
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
ν=1.0 α=2.0
ν=1.0 α=1.2
ν=0.6 α=2.0
ν=0.6 α=1.2
10-3
10-2
10-1
100
10-1 100101102103
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
10-3
10-2
10-1
100
10-1 100101102103
x
x
x
x
1-CDF(x,2) p(x,2)
1-CDF(x,15) p(x,15)
FIG. 4: (Color online) PDFs (top panel) and 1 −CDF(x, t)
(bottom panel) at t= 2 (left panel), t= 15 (right panel).
Simulation parameters as in Fig. 1. Eq. (17) was numerically
approximated by subordination techniques with the time step
of integration ∆t= 10−2and averaged over N= 106realiza-
tions. Solid lines present theoretical asymptotic x−1.2scaling
representative for α= 1.2 and ν= 1, i.e. for Markovian L´evy
flight.
with ν < 1. For Markov L´evy-Wiener process [37, 38] for
which the characteristic exponent ν= 1, the cusp disap-
pears and PDFs of the process ˜
X(St) become smooth at
x= 0. In particular, for the Markovian Gaussian case
(ν= 1, α= 2) corresponding to a standard Wiener diffu-
sion, PDF perfectly coincides with the analytical normal
density N(0,√t).
The presence of L´evy flights is also well visible in
the power-law asymptotic of CDF, see bottom panels of
Figs. 3 and 4. Indeed, for α < 2 independently of the
actual value of the subdiffusion parameter νand at arbi-
trary time, p(x, t)∝ |x|−(α+1) for x→ ∞. Furthermore,
all PDFs are symmetric with median and modal values
located at the origin.
III. SCALING PROPERTIES OF MOMENTS
The ν/α self-similar character of the process (cf.
Eq. (9)) is an outcome of allowed long flights and long
breaks between successive steps. In consequence, the
whole distribution scales as a function of x/tν /α with
the width of the distribution growing superdiffusively for
α < 2νand subdiffusively for α > 2ν. This tν /α scal-
ing is also clearly observable in the behavior of the stan-
dard deviation and quantiles qp(t), defined via the rela-
tion Prob {X(t)6qp(t)}=p, see Figs. 5, 6. For random
walks subject to superdiffusive, long-ranging trajectories
(α= 1.6), the asymptotic scaling is observed for suffi-
ciently long times, cf. Fig. 5. On the other hand, normal
(Gaussian) distribution of jumps superimposed on sub-
diffusive motion of trapped particles (ν= 0.8) clearly
shows rapid convergence to the ν/α law. Notably, both
10-1
100
101
102
101102103104
var(x)/t
t
ν=1.0 α=2.0
ν=1.0 α=1.6
ν=0.8 α=2.0
ν=0.8 α=1.6
FIG. 5: (Color online) Time dependence of var(x)/t. Straight
lines present t2ν/α−1theoretical scaling (see Eq. (30) and ex-
planation in the text). Simulation parameters as in Fig. 1.
sets ν= 1, α = 2 and ν= 0.8, α = 1.6 lead to the same
scaling t1/2, although in the case ν= 0.8, α = 1.6, the
process X(t) = ˜
X(St) is non-Markov, in contrast to a
standard Gaussian diffusion obtained for ν= 1, α = 2.
Thus, the competition between subdiffusion and L´evy
flights questions standard ways of discrimination between
normal (Markov, h[x−x(0)]2i ∝ t) and anomalous (gen-
erally, non-Markov h[x−x(0)]2i ∝ tδ) diffusion processes.
Indeed, for ν= 1, the process X(t) is not only 1/α
self-similar but it is also memoryless (i.e. Markovian).
In such a case, the asymptotic PDF p(x, t) is α-stable
[14, 37, 39, 40] with the scale parameter σgrowing with
time like t1/α, cf. Eq. (3). This is no longer true for
subordination with ν < 1 when the underlying process
becomes non-Markovian and the spread of the distribu-
tion follows the tν/α -scaling (cf. Fig. 6, right panels).
Some additional care should be taken when discussing
the scaling character of moments of p(x, t) [2, 32, 41].
Clearly, L´evy distribution (with α < 2) of jump lengths
leads to infinite second moment (see Eqs. (3) and (4))
hx2i=Z∞
−∞
x2lα,β(x;σ)dx =∞,(24)
irrespectively of time t. Moreover, the mean value hxi
of stable variables is finite for α > 1 only (hxi= 0 for
symmetric case under investigation). Those observations
seem to contradict demonstration of the scaling visible in
Fig. 5 where standard deviations derived from ensembles
of trajectories are finite and grow in time according to a
power law. A nice explanation of this behavior can be
given following argumentation of Bouchaud and Georges
[2]: Every finite but otherwise arbitrarily long trajectory
of a L´evy flight, i.e. the stochastic process underlying
Eq. (18) with ν= 1, is a sum of finite number of inde-
pendent stable random variables. Among all summed N
stable random variables there is the largest one, let say
6
lc(N). The asymptotic form of a stable densities
lα,β (x;σ)∝x−(1+α),(25)
together with the estimate for lc(N) allow one to estimate
how standard deviations grows with a number of jumps
N. In fact, the largest value lc(N) can be estimated from
the condition
NZ∞
lc(N)
lα,β (x)dx ≈1.(26)
which locates most of the “probability mass” in events
not exceeding the step length lc(otherwise, the relation
states that lc(N) occurred at most once in Ntrials, [2]).
Alternatively, lc(N) can be estimated as a value which
maximizes probability that the largest number chosen in
Ntrials is lc
lα,β(lc)
lc
Z
0
lα,β(x)dx
N−1
=lα,β(lc)
1−
∞
Z
lc
lα,β (x)dx
N−1
.
(27)
By use of Eqs. (26) and (25), simple integration leads to
lc(N)∝N1/α.(28)
Due to finite, but otherwise arbitrarily large number of
trials N, the effective distributions becomes restricted to
the finite domain which size is controlled by Eq. (28).
Using the estimated threshold, see Eq. (28) and asymp-
totic form of stable densities, see Eq. (25), it is possible
to derive an estimate of hx2i
hx2i ≈ Zlc
x2lα,β (x)dx ≈N1/α2−α=N2/α−1.(29)
Finally, after Njumps
hx2iN=Nhx2i ∝ N2/α.(30)
Consequently, for L´evy flights standard deviation grows
like a power law with the number of jumps N. In our
CTRW scenario incorporating competition between long
rests and long jumps, the number of jumps N=N(t)
grows sublinearly in time, N∝tν, leading effectively
to hx2iN∝t2ν/α with 0 < ν < 1 and 0 < α < 2.
Since in any experimental realization tails of the L´evy
distributions are almost inaccessible and therefore effec-
tively truncated, analyzed sample trajectories follow the
pattern of the tν/α scaling, which is well documented in
Fig. 5.
IV. DISCRIMINATING MEMORY EFFECTS
Clearly, by construction, for ν < 1 the limiting
“anomalous diffusion” process X(t) is non-Markov. This
feature is however non-transparent when discussing sta-
tistical properties of the process by analyzing ensemble-
averaged mean-square displacement for the parameters
10-1
100
101
102
103
101102103104
t
t
t
t
q... q...
q... q...
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
10-1
100
101
102
103
101102103104
t
t
t
t
q... q...
q... q...
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
10-1
100
101
102
103
101102103104
t
t
t
t
q... q...
q... q...
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
10-1
100
101
102
103
101102103104
t
t
t
t
q... q...
q... q...
ν=1 α=2
ν=1 α=1.6
ν=0.8 α=2
ν=0.8 α=1.6
FIG. 6: (Color online) Quantiles: q0.9,q0.8,q0.7,q0.6(from top
to bottom) for ν= 1, α = 2 (left top panel), ν= 1, α = 1.6
(left bottom panel), ν= 0.8, α = 2 (right top panel) and
ν= 0.8, α = 1.6 (right bottom panel). The straight line
presents theoretical tν/α scaling. Simulation parameters as in
Fig. 1.
set ν/α = 1/2, (e.g. ν= 0.8, α = 1.6) when contrary to
what might be expected, h[x−x(0)]2i ∝ t, similarly to the
standard, Markov, Gaussian case. This observation im-
plies a different problem to be brought about: Given an
experimental set of data, say time series representative
for a given process, how can one interpret its statistical
properties and conclude about anomalous (subdiffusive)
character of underlying kinetics? The similar question
has been carried out in a series of papers discussing use
of transport coefficients in systems exhibiting weak er-
godicity breaking (see [42] and references therein).
To further elucidate the nature of simulated data sets
for ν/α = 1/2, we have adhered to and tested formal
criteria [17, 43, 44, 45] defining the Markov process.
The standard formalism of space- and time- continuous
Markov processes requires fulfillment of the Chapman-
Kolmogorov equation (t1> t2> t3)
P(x1, t1|x3, t3) = X
x2
P(x1, t1|x2, t2)P(x2, t2|x3, t3).
(31)
along with the constraint for conditional probabilities
which for the “memoryless” process should not depend
on its history. In particular, for a hierarchy of times
t1> t2> t3, the following relation has to be satisfied
P(x1, t1|x2, t2) = P(x1, t1|x2, t2, x3, t3).(32)
Eqs. (31) and (32) have been used to directly verify
whether the process under consideration is of the Marko-
vian or non-Markovian type. From Eq. (31) squared
cumulative deviation Q2between LHS and RHS of the
Chapman-Kolmogorov relation summed over initial (x3)
7
and final (x1) states has been calculated [43]
Q2=X
x1,x3P(x1, t1|x3, t3)
−X
x2
P(x1, t1|x2, t2)P(x2, t2|x3, t3)2
.(33)
The same procedure can be applied to Eq. (32) leading
to
M2=X
x1,x2,x3hP(x1, t1|x2, t2)
−P(x1, t1|x2, t2, x3, t3)i2.(34)
Figure 7 presents evaluation of Q2(top panel) and M2
(bottom panel) for t1= 27 and t3= 6 as a function of
the intermediate time t2={7,8,9, . . . , 25,26}. It is seen
that deviations from the Chapman-Kolmogorov identity
are well registered for processes with long rests when sub-
diffusion wins competition with L´evy flights at the level
of sample paths. The tests based on Q2(see Eq. (33))
and M2(see Eq. (34)) have comparative character. The
deviations Q2and M2are about three order of magni-
tudes higher for the parameter sets ν= 0.8, α = 2.0 and
ν= 0.8, α = 1.6 than Q2and M2values for the Marko-
vian counterparts with ν= 1 and α= 2, α= 1.6, re-
spectively. Performed analysis clearly demonstrates non-
Markovian character of the limiting diffusion process for
ν < 1 and the findings indicate that scaling of PDF,
p(x, t) = t−1/2p(xt−1/2,1) or, in consequence, scaling
of the variance h[x−x(0)]2i ∝ tand interquantile dis-
tances (see Fig. 6) do not discriminate satisfactory be-
tween “normal” and “anomalous” diffusive motions [12].
In fact, linear in time spread of the second moment does
not necessarily characterize normal diffusion process. In-
stead, it can be an outcome of a special interplay between
subdiffusion and L´evy flights combined in the subordina-
tion X(t) = ˜
X(St). The competition between both pro-
cesses is better displayed in analyzed sample trajectories
X(t) where combination of long jumps and long trapping
times can be detected, see Fig. 1.
V. CONCLUSIONS
Summarizing, by implementing Monte Carlo simula-
tions which allow visualization of stochastic trajectories
subjected to subdiffusion (via time-subordination) and
superdiffusive L´evy flights (via extremely long jumps in
space), we have demonstrated that the standard mea-
sure used to discriminate between anomalous and nor-
mal behavior cannot be applied straightforwardly. The
mean square displacement alone, as derived from the (fi-
nite) set of time-series data does not provide full infor-
mation about the underlying system dynamics. In order
to get proper insight into the character of the motion,
10-9
10-8
10-7
10-6
10-5
8 10 12 14 16 18 20 22 24 26 28
Q2
t2
ν=1.0 α=2.0
ν=1.0 α=1.6
ν=0.8 α=2.0
ν=0.8 α=1.6
10-8
10-7
10-6
10-5
8 10 12 14 16 18 20 22 24 26 28
M2
t2
ν=1.0 α=2.0
ν=1.0 α=1.6
ν=0.8 α=2.0
ν=0.8 α=1.6
FIG. 7: (Color online) Squared sum of deviations Q2, see
Eq. (33), (top panel) and M2, see Eq. (34), (bottom panel)
for t1= 27, t3= 6 as a function of the intermediate time
t2. 2D histograms were created on the [−10,10]2domain.
3D histograms were created on the [−10,10]3domain. Due
to non-stationary character of the studied process the analy-
sis is performed for the series of increments x(t+ 1) −x(t).
Simulation parameters as in Fig. 1.
it is necessary to perform analysis of individual trajec-
tories. Subordination which describes a transformation
between a physical time and an operational time of the
system [1, 46] is responsible for unusual statistical prop-
erties of waiting times between subsequent steps of the
motion. In turn, L´evy flights are registered in instan-
taneous long jumps performed by a walker. Super- or
sub- linear character of the motion in physical time is
dictated by a coarse-graining procedure, in which frac-
tional time derivative with the index νcombines with
a fractional spatial derivative with the index α. Such
situations may occur in motion on random potential sur-
faces where the presence of vacancies and other defects
introduces both – spatial and temporal disorder [47]. We
believe that the issue of the interplay of super- and sub-
diffusion with a crossover resulting in a pseudo-normal
paradoxical diffusion may be of special interest in the
context of e.g. facilitated target location of proteins on
folding heteropolymers [48] or in analysis of single parti-
8
cle tracking experiments [42, 49, 50, 51], where the hid-
den subdiffusion process can be masked and appear as a
normal diffusion.
Acknowledgments
This project has been supported by the Marie Curie
TOK COCOS grant (6th EU Framework Program under
Contract No. MTKD-CT-2004-517186) and (BD) by the
Foundation for Polish Science. The authors acknowledge
many fruitfull discussions with Andrzej Fuli´nski, Marcin
Magdziarz and Aleksander Weron.
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