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4
Random walks and chaos
In the late nineteenth century, it was believed that a continuous function such as those
describing physical processes must have a derivative “almost everywhere.” At the
same time some mathematicians wondered whether there existed functions that were
everywhere continuous, but which did not have a derivative at any point (continuous
everywhere but differentiable nowhere). Perhaps you remember our discussion of such
strange things from the first chapter. The motivation for considering such pathological
functions was initiated by curiosity within mathematics, not in the physical or biologi-
cal sciences where one might have expected it. In 1872, Karl Weierstrass (1815-1897)
gave a lecture to the Berlin Academy in which he presented functions that had the
remarkable properties of continuity and non-differentiability. Twenty-six years later,
Ludwig Boltzmann (1844-1906), who connected the macroscopic concept of entropy
with microscopic dynamics, pointed out that physicists could have invented such func-
tions in order to treat collisions among molecules in gases and fluids. Boltzmann had a
great deal of experience thinking about such things as discontinuous changes of particle
velocities that occur in kinetic theory and in wondering about their proper mathemati-
cal representation. He had spent many years trying to develop a microscopic theory of
gases and he was successful in developing such a theory, only to have his colleagues
reject his contributions. Although kinetic theory led to acceptable results (and provided
a suitable microscopic definition of entropy), it was based on the time-reversible dynam-
ical equations of Newton. That is a fundamental problem because entropy distinguishes
the past from the future, whereas the equations of classical mechanics do not. This
basic inconsistency between analytic dynamics and thermodynamics remains unre-
solved today, although there are indications that the resolution of this old chestnut lies
in microscopic chaos.
It was assumed in the kinetic theory of gases that molecules are not materially
changed as a result of scattering by other molecules, and that collisions are instanta-
neous events as would occur if the molecules were impenetrable and perfectly elastic.
As a result, it seemed quite natural that the trajectories of molecules would some-
times undergo discontinuous changes. As mentioned in the discussion on classical
diffusion, Robert Brown observed the random motion of a speck of pollen immersed
in a water droplet. The discontinuous changes in the speed and direction of the
motion of the pollen mote observed are shown in Figure
3.2
. Jean Baptiste Perrin,
of the University of Paris, experimentally verified Einstein's predictions on the clas-
sical diffusion of Brown's pollen mote, and received the Nobel Prize for his work
