Statistical-Thermodynamics Formalism of Self-Similarity

Abstract

The self-similarity in dynamical and stochastic systems is formulated from a statistical-thermodynamical standpoint. The global structures of the self-similarity are found to be determined by one generating function which plays a role similar to the Helmholtz free energy in the equilibrium statistical mechanics. The interrelations among fractal measure theories developed for, e.g., generalized fractal dimensions of strange sets and velocity structure functions in turbulence are clarified from a unified point of view.