Weak ergodicity breaking with deterministic dynamics

Abstract

The concept of weak ergodicity breaking is defined and studied in the context of deterministic dynamics. We show that weak ergodicity breaking describes a weakly chaotic dynamical system: a nonlinear map which generates subdiffusion deterministically. In the non-ergodic phase non-trivial distribution of the fraction of occupation times is obtained. The visitation fraction remains uniform even in the non-ergodic phase. In this sense the non-ergodicity is quantified, leading to a statistical mechanical description of the system even though it is not ergodic.

Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

“…separation of trajectories is sub-exponential, we have a strong indication that the usual Boltzmann-Gibbs statistical mechanics is not valid. Indeed it was found that certain systems with zero Lyapunov exponents break ergodicity [8,9]. Classical entropy theory is also not applicable in this case [6,7], particularly the entropy and average algorithmic complexity grow non linearly in time [7], while for a system with a positive Lyapunov exponent they increase linearly in time.…”

“…Classical entropy theory is also not applicable in this case [6,7], particularly the entropy and average algorithmic complexity grow non linearly in time [7], while for a system with a positive Lyapunov exponent they increase linearly in time. Still the situation is not hopeless from the point of view of statistical mechanics and one may consider distributions of time average observables [8,9,10,11,12].…”

“…Its generalizations attracted vast research using different methods such as continuous time random walks [20,21] and periodic orbit theory [22] to name a few. Sojourn times of trajectories in the vicinity of the unstable fixed point are described by power law statistics leading to aging [23] and non Gaussian fluctuations [7], which are related to weak ergodicity breaking [8,9]. In the non ergodic phase z > 2 the density function of the map is concentrated on the unstable fixed point in the long time limit.…”

Fluctuation-Dissipation Theorem for Event-Dominated Processes

“…Herein we shift the focus from variables to events, an event being a collision that may produce an abrupt change in the value of a variable. We assume these events to be the renewal, non-Poisson, and nonergodic events studied by Barkai and co-workers [8], that A B S and that the variable S has only two distinct values, S 1. Finally, we assume that the external perturbation changes the prescription that determines the times at which events occur, namely, the times at which the variable S may, or may not, change its sign.…”

“…This final assumption reveals the lack of a Hamiltonian formalism, a condition shared by Refs. [5,7].We define the nonstationary autocorrelation function where t; t 0 is the waiting-time distribution density of age t 0 [8,9]. Note that Eq.…”