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observations of human mobility and the new generation of mobility models, pre-
senting to what extent real human mobility patterns deviate from those expected
from simple diffusion processes.
15.1.1 Motion Models: Brownian Motion and Levy Flights
In 1827, while he was studying sexual relations of plants, botanist Robert Brown
noticed that granules contained in grains of pollen were in constant motion, and
that this motionwas not caused by currents in the fluid or evaporation. He thought
at first that they were jiggling around because they were alive or because of the
organic nature of the matter. So, he did the same experiment with dead organic
and inorganic matter, finding there was just as much jiggling. The movement
evidently had nothing to do with the substance ever being alive or dead, and
this left him and his contemporaries with a puzzling question: What is this
mysterious perpetual motion that keeps the pollen moving?
A possible explanation for the so-called
Brownian motion
1
is that all the
molecules in the fluid are in vigorous motion, and these tiny granules are moved
around by this constant battering from all sides as the fluid molecules bounce
off. Imagine we are in the middle of a crowd and there is a big balloon. As the
individuals move around, they push the balloon from all directions: sometimes
the balloon will move to the left, occasionally to the right, overall displaying a
random, jittery motion like the paths in Figure
15.1
. A particle of pollen behaves
like a really huge balloon in the midst of a dense crowd.
Such an atomic-molecular thesis was described by Einstein, who in 1905
published a theoretical analysis of Brownian motion and showed that the mean
distance reached by particles from the first collision point must grow with the
square root of time. It means, for example, that after 4 seconds, the distance is
only twice (
√
4
=
2) the one found after a second, and not four times as insight
would suggest. Einstein's calculations were confirmed experimentally in 1908
by physicist Jean Baptiste Perrin, who convinced even the most skeptical about
the validity of the atomic-molecular hypothesis.
Before Einstein, Louis Bachelier derived independently several mathematical
properties of Brownianmotion, including the equation for the probability
P
(
x,t
)
for the position
x
of a Brownian random walker at time
t
, when the walker starts
as the origin at time
t
=
0. The equation for
P
(
x,t
) in one dimension is given by
the
diffusion equation
, with a Gaussian solution. Therefore, a Brownian motion
is basically a random walk with a normal distribution for the position of the
random walker after a time
t
, with the variance proportional to
t
. It means that
1
The first observation of Brownian motion was reported in 1785 by the Dutch physician Jan Ingen-
haysz. However, Brown was the first to discover the ubiquity of the phenomenon.
