THERMODYNAMICS OF RARE EVENTS

Scaling laws of physics are derived from extreme value distributions. Small jump processes that comprise a compound Poisson distribution generate the asymptotic distributions of stable laws. These extreme value distributions, or their tails, can be expressed in terms of the entropy decrease. As an example, the scaling law for the radius of gyration of a polymer is derived which is comparable to Flory's formula. The entropy is identified by its property of concavity, which is shown to coincide with Boltzmann's probabilistic definition for first passage in a random walk. A more general definition is required for nonintegral dimensions. The relation to mean-field theory of the kinetic Weiss-Ising model is shown and this distribution of the order parameter is governed by an asymptotic distribution for the smallest value rather than a normal distribution. Finally, the logarithm of the sample size is shown to be the yardstick for the decrease in entropy.