Role of infinite invariant measure in deterministic subdiffusion
Takuma Akimoto
1,
*
and Tomoshige Miyaguchi
2
1
Department of Mechanical Engineering, Keio University, Yokohama 223-8522, Japan
2
Department of Applied Physics, Graduate School of Engineering, Osaka City University, Osaka 558-8585, Japan
Received 11 May 2010; revised manuscript received 31 July 2010; published 7 September 2010
Statistical properties of the transport coefficient for deterministic subdiffusion are investigated from the
viewpoint of infinite ergodic theory. We find that the averaged diffusion coefficient is characterized by the
infinite invariant measure of the reduced map. We also show that when the time difference is much smaller than
the total observation time, the time-averaged mean square displacement depends linearly on the time differ-
ence. Furthermore, the diffusion coefficient becomes a random variable and its limit distribution is character-
ized by the universal law called the Mittag-Leffler distribution.
DOI: 10.1103/PhysRevE.82.030102 PACS numbers: 05.40.Fb, 05.45.Ac, 87.15.Vv
Anomalous subdiffusion, where the mean squared dis-
placement MSD is x
2
t t
0 1, is a ubiquitous
feature of nonequilibrium phenomena. Subdiffusion has been
observed in various types of experiments: charge-carrier
transport in amorphous semiconductors 1, Brownian mo-
tion of polymers 2, chain diffusion in polymer melts 3,
aftershock diffusion in earthquakes 4, and diffusion in cells
and membranes 5–9. In particular, subdiffusion in single-
particle trajectories has recently attracted significant interest
from the biological community because of the progress in
single molecule experiments 5–9.
There are roughly three types of mechanisms that gener-
ate subdiffusion: i anti-persistence e.g., the fractional
Brownian equation, ii geometric disorder e.g., diffusion
within fractal objects such as percolation clusters, and iii
long-time trappings characterized by a power law e.g., non-
hyperbolic dynamical systems and continuous time random
walks CTRWs. Although the time-averaged MSD
TAMSD equates to the ensemble-averaged MSD
EAMSD for i and ii, it is not true for iii10–12. Thus,
for case iii, the usual ergodicity, i.e., “the time average
TA coinciding with the ensemble average EA,” is vio-
lated. However, a generalization of the usual ergodicity
would be valid i.e., ergodicity in an infinite measure space
13; it states that the normalized TA coincides with
EA Y see Eq. 5, where Y is a random variable with a
universal distribution the Mittag-Leffler distribution. This
distributional behavior is reminiscent of the random behavior
of the diffusion coefficient in cells 6–9.
EAMSD is defined as
x
m
- x
0
2
E
lim
K→
1
K
k=1
K
x
m
k
- x
0
k
2
, 1
where x
0
1
,..., x
0
K
is a set of initial points and x
m
k
is the kth
initial point at time m 14. TAMSD is defined for a single
trajectory as
x
m
- x
0
2
T
N
1
N
k=0
N-1
x
k+m
- x
k
2
. 2
For CTRW, it has been shown that TAMSD grows lin-
early in time 10,15:
x
m
- x
0
2
T
N = Dm m N , 3
and that the diffusion coefficients D are random variables
10. Additionally, the following scaling in terms of N has
been theoretically demonstrated 10,12,15,
x
m
- x
0
2
T
N
E
N
-1
, 4
where ·
E
represents the ensemble average with respect to
initial points and is the exponent in subdiffusion,
x
m
- x
0
2
E
m
. Moreover, TAMSD for the fixed time dif-
ference m has been shown to obey the Mittag-Leffler distri-
bution 15.
In this Rapid Communication, we study deterministic sub-
diffusions generated by intermittent maps from infinite er-
godic theory. Our main result is that the diffusion coefficient
of TAMSD is characterized by the infinite invariant measure
of the reduced map. We also show analytically that the dis-
tribution function of the diffusion coefficient converges to
the Mittag-Leffler distribution and that TAMSD increases
linearly in time m without taking the ensemble average.
Reviews of infinite ergodic theory. We give a brief review
here of infinite ergodic theory. Let T be a conservative 16,
ergodic 17, measure preserving transformation on a phase
space I and let be an invariant measure. Darling-Kac-
Aaronson theorem DKA theorem13,18,19 then says that
the normalized time average of the observation function
f x L
+
1
20 converges in distribution to the random
variable Y
with the normalized Mittag-Leffler distribution of
order 13,21
1
a
N
k=0
N-1
f T
k
X
0
⇒ f Y
, 5
where the initial point X
0
is a random variable, f =
I
fd,
a
N
is called the return sequence, and X ⇒ Y means that a
random variable X converges in distribution to a random
variable Y 13. Note that the normalized time average does
*
akimoto@z8.keio.jp
PHYSICAL REVIEW E 82, 030102R2010
RAPID COMMUNICATIONS
1539-3755/2010/823/0301024 ©2010 The American Physical Society 030102-1