  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.
