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PHYSICAL REVIEW E 66, 015101(R) (2002)
ALLEGRINI et al.
Let us see how Bob builds up his single generating tra-
jectory. For a generic time 1, let us consider the time IN fitting
the property that lN= 7T + 71 + TN- 1 < 1, while IN+ 7N> 1. In
this condition, Bob's generating trajectory is given by
y(1) = [oo + 1 + ... + N-1SN-1 + (1- IN)SN],
(5)
with Wplaying here the role of velocity intensity. Let us call
ith event the random selection of the pair { T7i, si}. The walker
starts moving immediately after the occurrence of the first
event and spends the whole time T70 in a condition of uniform
motion, laminar phase [5], before the occurrence of the sec-
ond event, at which time the motion direction can also be
inverted, and so on. In this case, the adoption of infinitely
many trajectories x(t) corresponds to creating a stationary
condition, with the walkers staying in the first laminar phase
with a time distribution corresponding to equilibrium [10].
Before illustrating the crucial result of this paper, based on
the stationary condition, let us discuss briefly the nonstation-
ary case [11] when Bob has really at his disposal infinitely
many sequences { T1i,s}, and consequently many trajectories
y(1). This would be equivalent to an out-of-equilibrium con-
dition, which would relax to the equilibrium condition with a
relaxation prescription a 1/1L 1 [12]. How many events will
have been realized by Bob for any of his walker up to time 1?
For 1> (7), the number of events, N, is expected to be N
=1/(r). Actually, we can set all this on a rigorous basis
using a theorem by Feller [13]. At any instant of time 1 the
number of random walkers for which I~' 1 Tl 1 and
N 1 Ti>1, is not fixed, and its mean value, (N), is given by
(N)- 1 +(3 1 (6)
Note that each event implies a fixed amount of entropy in-
crease, due to the random prescriptions adopted to realize an
event. Thus, Eq. (6) shows that the rate of entropy increase is
not constant. It is constant either in the exponential case
(ordinary statistical mechanics, with A = oc) or in the
asymptotic time limit, namely, in the scaling regime compat-
ible with the perspective of thermodynamic equilibrium. One
might be tempted to consider Eq. (6) to reflect a nonstation-
ary condition that in the case /t <3 would live forever.
It is not so. First of all, the relaxation to equilibrium is
faster than the memory effect, 1/1 '-1 vs 1/1-2. A careful
study of the stationary condition confirms this remark. The
Levy walk realized by Bob can be described as the solution
of the differential equation dx/dt= ((t), where ( t) is a sto-
chastic velocity keeping the value W(- W) for a time T1i,
with or without a change of sign at the end of this sojourn
time, as a result of the coin tossing. In the stationary case the
correlation function (g(t1) ((t2)( 2) depends on t=l t
-t2, and it is denoted by 0 g(t). Using the renewal theory
[5,14], C D(t) is related to f(7) byIt is remarkable that this correlation function has the same
asymptotic properties as the correction term to the condition
of constant rate of entropy increase established by Eq. (6).
We expect p(x, t) to become a Levy stable distribution for
t-- oo, according to the GCLT prediction of Eq. (3), as fully
confirmed by numerical simulation (see, for instance, Ref.
[15]). However, this transition process is infinitely slow. In
fact, we note that at any time t a finite number of Bob's
trajectories x(t) are still in the same laminar region where
they were at t= 0. These trajectories are moving by uniform
motion with velocity Wand - W, thus establishing peaks of
decreasing intensity and an abrupt truncation of the PDF, at
its right and left border, respectively. To evaluate this num-
ber, or the probability that a trajectory contributes to the
propagation front, Ip(t), we must refer ourselves to the prob-
ability distribution 'as(T). This is the alternating signs dis-
tribution, or distribution of times through which the trajec-
tory keeps moving in the positive or negative direction, a
distribution not coinciding with /i(7r), due to the random
choice of sign. The two distributions are related to one an-
other through their Laplace transforms, 'as(s) and /(s), re-
spectively, by means of [11]ss (s)
2 - #(s)(8)
Using the renewal theory [5] we prove that
1 +o
Ip(t) =( (t' - t) as(t') dt'.
(ast(9)
These trajectories keep moving by ballistic motion and thus
contribute to the propagation fronts signaled in the numerical
treatment by two ballistic peaks. It becomes thus evident that
a very plausible form for the PDF is given by
p(x, t)= K( t)pL(X, t) 8( Wt- X ) + S(IX X- Wt)Ip( t),
(10)
where 0(. ) denotes the Heaviside step function. pL(x, t) is a
distribution that for t--o becomes identical to the anti-
Fourier transform of Eq. (3), and K(t) is a time-dependent
factor ensuring the normalization of the distribution p(x, t),
thereby taking the form1-Ip(t)
K( t) W
1 -2 pL(x,t)dx(11)
Using the method of Laplace transform, it is straigthforward
to prove that lim tK(t)= 1 and that(12)
lim[Ip( t) - ( t)] = 0
On the basis of these arguments we reach the conclusion that
in the asymptotic time limit Eq. (10) becomes identical to015101-2
1 + T
S( t) = t ( t' - t) ( t') dt' - . (7)
1 t (7) t, T+ t u-
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Allegrini, Paolo; Bellazzini, Jacopo; Bramanti, G.; Ignaccolo, Massimiliano; Grigolini, Paolo & Yang, J. Scaling Breakdown: A Signature of Aging, article, July 12, 2002; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc67630/m1/2/: accessed April 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.