Application of classical statistical mechanics to multifractals and dynamical systems

TL;DR

This paper extends classical Gibbs-Boltzmann statistical mechanics to complex systems like multifractals and non-Hamiltonian dynamics, introducing effective thermolization and a governing potential formalism.

Contribution

It applies classical statistical mechanics to complex, non-Hamiltonian systems, developing a formalism for effective thermolization and governing potential.

Findings

Successful application to multifractals and non-Hamiltonian systems

Introduction of effective thermolization of stochastic noise

Development of a formal governing potential

Abstract

Classical, self-consistent theory of statistical mechanics was developed for the thermodynamic and conservative Hamiltonian systems. Later there were many attempts (Sinai-Bowen-Ruelle's temperature, Tsallis' non-extensive theory) to apply similar formalism to non-Hamiltonian dynamical systems. Although these theories reveal aspects of complex behavior, they have limited applicability. This paper applies the classical Gibbs-Boltzmann statistical mechanics to complex systems such as i.i.d. processes, multifractals, and non-Hamiltonian dynamical systems with strange attractors. The effective thermolization of stochastic noise in the system is introduced and the formalism of a ruling (governing, free energy) potential is developed.