PHYSICAL REVIEW E 95, 022126 (2017)
Aging ballistic L´ evy walks
Marcin Magdziarz
*
and Tomasz Zorawik
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology,
Wyspianskiego 27, 50-370 Wroclaw, Poland
(Received 21 December 2016; published 22 February 2017)
Aging can be observed for numerous physical systems. In such systems statistical properties [like probability
distribution, mean square displacement (MSD), first-passage time] depend on a time span t
a
between the
initialization and the beginning of observations. In this paper we study aging properties of ballistic L´ evy
walks and two closely related jump models: wait-first and jump-first. We calculate explicitly their probability
distributions and MSDs. It turns out that despite similarities these models react very differently to the delay t
a
.
Aging weakly affects the shape of probability density function and MSD of standard L´ evy walks. For the jump
models the shape of the probability density function is changed drastically. Moreover for the wait-first jump
model we observe a different behavior of MSD when t
a
≪ t and t
a
≫ t .
DOI: 10.1103/PhysRevE.95.022126
I. INTRODUCTION
Suppose we begin observations of a system which was
initialized at t = 0 after some time t
a
. There can be many
reasons why we would want to do this, from technical
restrictions of a measuring device to sheer curiosity. In many
cases the delay largely changes statistical properties of the
observed process. Such a phenomenon is called aging, a term
which was originally used in the area of glassy materials [1–4].
Aging was also reported for blinking nanocrystals [5–8], where
the changes between on and off state for a single, illuminated
quantum dot during time interval [t
a
,t + t
a
] were measured.
It turned out that when the aging time t
a
increases, more
long on-state and off-state periods can be observed. Similar
behavior is displayed by potassium channels dynamics [9].
For more examples see [10] and references therein.
In a recent paper [11] the authors studied statistical
properties of aging continuous time random walks (CTRWs).
In this article we develop a similar theory for a different, very
useful model for anomalous diffusion—L´ evy walk (LW). This
model can be used, for instance, to describe the dynamics
of the already mentioned blinking nanocrystals [5]. Other
striking and sometimes very beautiful examples of applications
include: migration of swarming bacteria [12], light transport
in special optical materials (L´ evy glass) [13], and foraging
patterns of animals [14–16]. More examples are described
in a review paper devoted to this model [17], see also [18].
The particle which perform L´ evy walk moves with a constant
velocity v (in this article for simplicity we set v = 1) for a time
period which follows a power law ψ (τ ) ∝ τ
1+α
with α> 0.
Then it chooses randomly a new direction of the motion [17].
Here we focus on the case α ∈ (0,1) which leads to a ballistic
regime [19].
L´ evy walks L(t ) can also be analyzed in a context of
coupled continuous time random walks: the so-called wait-first
L
WF
(t ) and jump-first models L
JF
(t )[17]. The particle which
performs wait-first L´ evy walk instead of moving with the
*
Corresponding author: marcin.magdziarz@pwr.edu.pl
constant velocity v for time T remains motionless for time T
and then executes a jump which length equals v · T . As a result
trajectories are discontinuous, contrary to the standard LW, see
Fig. 1. The jump-first scenario differs from the wait-first case
with the changed order of waiting and jumping moments. The
CTRW approach to L´ evy walks was analyzed in Refs. [20,21].
Although the jump models and the standard L´ evy walk appear
to be very similar, they have very different statistical properties.
In Ref. [19] a method to find probability density functions
p(x,t ) (PDFs) for all these models in the ballistic regime was
proposed by Froemberg et al. For another approach to this
problem for the jump models see [22]. It is also worth to
mention that PDFs of multidimensional isotropic L´ evy walks
were found in Refs. [23,24]. We emphasize that all the results
we present here are calculated for the diffusion limits of LW
and two other coupled CTRW models [25,26].
In what follows we assume aging time t
a
> 0 and analyze
aging L´ evy walk X
a
(t ), wait-first Y
a
(t ), and jump-first Z
a
(t )
models, where
X
a
(t ) = L(t + t
a
) − L(t
a
),
Y
a
(t ) = L
WF
(t + t
a
) − L
WF
(t
a
), (1)
Z
a
(t ) = L
JF
(t + t
a
) − L
JF
(t
a
).
We propose a method to compute distributions p(dx,t
a
,t ) of
aging LWs and calculate ensemble and time averaged MSDs.
One should underline here that the distribution function of the
time averaged intensity correlation function for standard LWs
was computed in Ref. [5].
II. AGING CONTINUOUS L
´
EVY WALKS
In this section we calculate the distribution p(dx,t
a
,t ) of
aging L´ evy walk X
a
(t ). The forward renewal time W
t
a
after t
a
for the non-aging process L(t ) has the distribution [11,27–30]
f (t
a
,w) =
sin(πα)
π
t
α
a
w
a
(t
a
+ w)
1
0w
, (2)
where 1
A(w)
denotes the indicator function which equals 1
when A(w) is true and 0 otherwise. This issue can be also
2470-0045/2017/95(2)/022126(6) 022126-1 ©2017 American Physical Society