Statistical Mechanics of Dynamical Systems

Abstract

A statistical-mechanical formalism of chaos based on the geometry of invariant sets in phase space is discussed to show that chaotic dynamical systems can be treated by a formalism analogous to that of thermodynamic systems if one takes a relevant coarse-grained quantity, but their statistical laws are quite different from those of thermodynamic systems. This is a generalization of statistical mechanics for dealing with dissipative and hamiltonian (i.e., conservative) dynamical systems of a few degrees of freedom.Thus the sum of the local expansion rate of nearby orbits along a relevant orbit over a long but finite time has been introduced in order to describe and characterize (1) a drastic change of the structure of a chaotic attractor at a bifurcation and anomalous phenomena associated, (2) a critical scaling of chaos in the neighborhood of a critical point for the bifurcation to a nonchaotic state, and a self-similar temporal structure of a critical orbit on the critical 2~ attractor and the critical golden tori without mixing, (3) the critical KAM torus, diffusion and repeated sticking of a chaotic orbit to a critical torus in hamiltonian systems. Here a q-phase transition, analogous to the ferromagnetic phase transition, plays an important role. They are illustrated numerically and theoretically by treating the driven damped pendulum, the driven Duffing equation, the Henon map, and the dissipative and conservative standard maps.This description of chaos breaks the time-reversal symmetry of hamiltonian dynamical laws analogously to statistical mechanic!~ of irreversible processes. The broken timereversal symmetry is brought about by orbital instability of chaos.

Some Open Points in Nonextensive Statistical Mechanics

“…The case q sen < 1 yields, in (21), a power-law behavior ξ ∝ t 1/(1−qsen) in the limit t → ∞. This power-law asymptotics were since long known in the literature [Grassberger and Scheunert, 1981;Schneider et al, 1987;Anania & Politi, 1988;Hata et al, 1989;Mori et al, 1989]. The case q sen < 1 is in fact more complex than indicated in Eqs.…”

Two Stories Outside Boltzmann-Gibbs Statistics: Mori’s Q-Phase Transitions and Glassy Dynamics at the Onset of Chaos

“…As a function of the running variable −∞ < q < ∞ the q-Lyapunov coefficients become a function λ(q) with two steps located at q = q = 1 − ln 2/(z − 1) ln α(z) and q = Q = 2 − q. In this manner contact can be established with the formalism developed by Mori and coworkers [12] and the q-phase transition obtained in Refs. [13].…”

“…Notably, the appearance of a specific value for the q index (and actually also that for its conjugate value Q = 2 − q) works out [7] to be due to the occurrence of Mori's 'q-phase transitions' [12] between 'local attractor structures' at µ c . As shown in Fig.…”

“…For the onset of chaos at µ c (z = 2) a q-phase transition was numerically determined [12,13]. According to ψ(λ) above we obtain a conjugate pair of q-phase transitions that correspond to trajectories linking two regions of the attractor, the most crowded and most sparse.…”

“…i) The finding [7] that the dynamics at the onset of chaos is made up of an infinite family of Mori's q-phase transitions [12,13], each associated to orbits that have common starting and finishing positions located at specific regions of the attractor. Every one of these transitions is related to a discontinuity in the σ function of 'diameter ratios' [14], and this in turn implies a qexponential ξ t and a spectrum of q-Lyapunov coefficients equal to the Tsallis rate of entropy production for each set of attractor regions.…”