Comment on "Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem"

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by  Igor Goychuk

Comment on “Non-Normalizable Densities in Strong Anomalous Diffusion:
Beyond the Central Limit Theorem”
Igor Goychuk1
1Institute for Physics and Astronomy, University of Potsdam,
Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
A comment on the Letter by A. Rebenshtok, S. Denisov, P. H¨anggi, and E. Barkai, Phys. Rev.
Lett. 112, 110601 (2014).
Ina recentLetter [1], anattempt is madeto justify Pv(α)(v¯α,t)dv¯α = 1. Hence, P∞(α)(v¯α) :=
5 and to provide a further support to earlier erroneous lRimt→∞Pv(α)(v¯α,t) is also normalized. At the first
01 claims, see e. g. in [2], that particle distributions look, our P∞(α)(v¯α) may seem to be their ICD. How-
or,equivalently,probabilitydensitiescanbecomenon- ever, this is not so. The function I (v¯) in Eq.
2 cd
normalizable in the case of anomalous L´evy walk dif- (9) in [1] is not a PDF of v¯ ≡ v¯1, which would
n fusion,andthis anomalycanreflectaphysicalreality. (1)
a be limt→∞Pv (v¯1,t) = limt→∞tP(x = v¯t,t) =
In this Comment, I show that such claims cannot be
J justified and supported. P∞(1)(v¯1), but a special construct. It can be consid-
8 ered as the t → ∞ limit of the function I (v¯,t) :=
To begin with, any probability density function or cd
2 PDF must be non-negative and normalized. Both a tα−1Pv(1)(v¯1,t). Its normalization,

] common sense and the postulates of probability the-
h
ory fix these important properties, once and forever,
c I (v¯,t)dv¯=tα−1, (2)
e see e.g. Ch. 2 in [3]. In Ref. [1], Eq. (8) is derived Z cd
m from the correct result in Eq. (2) therein by taking
- an asymptotic limit t → ∞. The PDF in Eq. (2) is indeed diverges for α > 1 in the limit t → ∞. Of
t
a obviously normalized for any particular model within course, at any fixed and finite observation time t, the
t the L´evy walk model considered. The authors seem
s (generalized) moment averages
. perfectly understandthis,seetheir Eq. (3). However,
t
a already this fact implies that Eq. (8) cannot be cor-
m rect being applied to the whole range of x variations. h|v¯|qi = |v¯|qIcd(v¯,t)dv¯ (3)
- Indeed, Eq. (8) describes a part of the distribution, R Icd(v¯,t)dv¯
d
for a sufficiently large x, however, not the entire dis- R
n = |v¯|qP(1)(v¯,t)dv¯
o tribution. Inthis respect,Eq. (1)in[1]alsodescribes Z v
c only a tail of the time distribution ψ(τ). Clearly, the
[ distribution in Eq. (1) in [1] is non-normalizable, if = t1−αZ |v¯|qIcd(v¯,t)dv¯

1 by mistake to extend it to small τ → 0. However, to
v claim this for possible were absurd. An example of
6 PDF can be readily given. It is Pareto distribution, can be found either with Icd(v¯,t), or with PDF
99 ψ(τ)=α/[τ0(1+τ/τ0)1+α],whereaconstantτ0 sepa- Pv(1)(v¯,t). However,it is incorrectto use for this pur-
6 rates small, τ ≪τ0, and large, τ ≫τ0, time intervals pose Icd(v¯) obtained in the formal limit of infinite t,
0 betweenthescatteringevents. Bythesametoken,the astheauthorsdointheirEq. (13). Itisgenerally,e.g.
. claim that the distribution P (x,t) in their Eq. (8) for q <α, wrong. The limit t→∞ must be taken in
1 A
is not normalizablebecause P (x,t)∼|x|−1−α in the our Eq. (3). Importantly, stochastic numerics can be
0 A
5 vicinity of x→0 is obviouslywrong. The correctdis- doneonlyatfinite tandτ0. Inthisrespect,whichcon-
1 tribution does not behave in this way for |x|<v τ . crete PDF ψ(τ) was used in the numerics done in [1]
0 0
v: After deriving incorrect result in Eq. (8), an at- isdim. Mostobviously,anydecentexperiment,either
i tempttofurthersubstantiateitismadein[1]byusing real or numerical, done at finite t will yield Icd(v¯,t),
X
the conceptofinfinite covariantdensity(ICD). Below whichisperfectlynormalizable,andnotIcd(v¯). Hence
r Iexplainwhythisattemptfails. Forthis,letusclarify ICD cannot correspondto a physicalreality,contrary
a
the origin of ICD in their Eq. (9). It is not a PDF, to the claims in [1].
but a function concocted to be non-normalizable for
α >1. Indeed, let us change random variable from x
to v¯ = x/tα, where t is just a parameter, and not a
α
random variable. Then, the corresponding PDF is
[1] A. Rebenshtok, S. Denisov, P. H¨anggi, and E. Barkai,
Phys. Rev.Lett. 112, 110601 (2014).
Pv(α)(v¯α,t)=Z δ(v¯α−x/tα)P(x,t)dx (1) [2] E. Lutz and F. Renzoni, NaturePhys. 9, 615 (2013).
[3] A. Papoulis, Probability, Random Variables, and
=tαP(x=v¯αtα,t). Stochastic Processes, 3d ed. (McGraw-Hill, New York,
1991).
Most obviously, it is normalized for any t,