Aging generates regular motions in weakly chaotic systems

Abstract

Using intermittent maps with infinite invariant measures, we investigate the universality of time-averaged observables under aging conditions. According to Aaronson-Darling-Kac theorem, in non-aged dynamical systems with infinite invariant measures, the distribution of the normalized time averages of integrable functions converge to the Mittag-Leffler distribution. This well known theorem holds when the start of observations coincides with the start of the dynamical processes. Introducing a concept of the aging limit where the aging time $t_a$ and the total measurement time $t$ goes to infinity while the aging ratio $t_a/t$ is a constant, we obtain a novel distributional limit theorem of time-averaged observables integrable with respect to the infinite invariant density. Applying the theorem to the Lyapunov exponent in intermittent maps, we find that regular motions and a weakly chaotic behavior coexist in the aging limit. This mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in Hamiltonian systems. The probability of finding regular motions in non-aged processes is zero, while in the aging regime it is finite and it increases when system ages.

7 pages, 5 figures

References in corpus (2)

• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Pesin-Type Identity for Weak Chaos

Cited by in corpus (20)

• Infinite Ergodic Theory for Heterogeneous Diffusion Processes
• Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes
• Ageing underdamped scaled Brownian motion: ensemble and time averaged particle displacements, non-ergodicity, and the failure of the overdamping approximation
• Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes
• Infinite ergodic theory meets Boltzmann statistics
• Scale invariant Green-Kubo relation for time averaged diffusivity
• Infinite invariant density in a semi-Markov process with continuous state variables
• Distributional behavior of time averages of non-$L^1$ observables in one-dimensional intermittent maps with infinite invariant measures
• Anomalous diffusion and the Moses effect in a model of aging
• $1/f^{\beta }$ noise for scale-invariant processes: How long you wait matters
• Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory
• Aging arcsine law in Brownian motion and its generalization
• Generalized Lyapunov exponent as a unified characterization of dynamical instabilities
• Numerical estimate of infinite invariant densities: application to Pesin-type identity
• Brownian particles in periodic potentials: coarse-graining versus fine structure
• Non-normalizable quasi-equilibrium solution of the Fokker-Planck equation for nonconfining fields
• Singularity of Levy walks in the lifted Pomeau-Manneville map
• Local equilibrium properties of ultraslow diffusion in the Sinai model
• Transition to anomalous dynamics in a simple random map
• Does an intermittent dynamical system remain (weakly) chaotic after drilling in a hole?