4 - Random walks and chaos

Summary

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 biological 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 functions 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 mathematical 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 dynamical equations of Newton.