SLIDE 1 １次元間欠写像における 長時間平均のふるまい
Department of Applied Physics, Advanced School of Science and Engineering, Waseda University
1. Introduction (Ergodic problems and Intermittent phenomena ) 2. Universal Distributions
(Mittag Leffler, Generalized Arcsine, Stable)
秋元 琢磨
What is ergodicity of the non-equilibrium state?
第3回 九州大学 産業技術数理研究センターワークショップ（兼 第3回連成シニュレーション フォーラム） 「自然現象における階層構造と数理的アプローチ」 2008年3月6日
SLIDE 2
Ergodic Theory
Question
SLIDE 3 Intermittent Phenomena
- M. Bottiglieri and C. Godano, On-off intermittency in earthquake occurrence,
Phys.Rev. E 75, 026101 (2007).
Off state Threshold (the Southern California catalog, 1973-2003)
SLIDE 4 Nonergodicity
Power law, 1/f spectrum, Non-stationarity, Non-ergodicity Characteristics X.Brokmann et al., Statistical aging and nonergodicity in the fluorescence
- f single nanocrystals, Phys. Rev. Lett. 90, 120601 (2003).
- n state
- ff state
SLIDE 5 Nonergodicity and Non-stationarity
Nonstationary and non-ergodic behavior
Time average does not converge to constant value.
SLIDE 6 Mushroom Billiard
1
- 1
- T. Miyaguchi, Escape time statistics for mushroom billiard, Phy. Rev. E 75
066215 (2007). (Aizawa lab. Waseda univ. Satoru Tsugawa)
SLIDE 7
Remark
SLIDE 8 Purpose
To make clear the non-stationary phenomena and the foundation of the ergodicity in the non-equilibrium state Analyzing the distribution of the time average of some
- bservation functions in infinite measure systems which
are related to intermittent phenomena.
SLIDE 9
Conditions
SLIDE 10 Infinite measure systems
Invariant density can not be normalized. Indefiniteness of the invariant density
The residence time distribution obeys the Log-Weibull distribution.
SLIDE 11
Conservative and Dissipative
SLIDE 12 DKA Limit Theorem
Darling – Kac – Aaronson Limit Theorem (1981)
Random variable (Mittag-Leffler distribution)
Lyapunov exponent
SLIDE 13 Mittag-Leffler Distribution
Lyapunov exponent for Boole transformation Lyapunov exponent for MB map
SLIDE 14 Skew Modified Bernoulli Map
Invariant measure (Infinite measure) Skew Modified Bernoulli map The skew modified Bernoulli map is closely related to the intermittent phenomena. (Rayleigh-Benard convection, Lorentz model)
MB map (B=3.0,c=0.3)
Indifferent fixed points
SLIDE 15 Lamperti-Thaler Generalized Arcsine Law
- M. Thaler, A limit theorem for sojourns near indifferent fixed points of
- ne-dimensional maps, Ergod. Th. & Dynam. Sys. 22 1289-1312 (2002).
SLIDE 16
Generalized Arcsine Law
SLIDE 17
Remark on the invariant density and mean
(On state) (Off state)
The invariant density is not symmetric.
SLIDE 18 Universal Distributions
Time Average of the observation function Distributional Limit Theorems for the Time Average
[1] T. Akimoto, Generalized Arcsine Law and Stable Law in an Infinite Measure System, arXiv:0801.1382v. Random variables
SLIDE 19 with finite mean
Examples in the MB map Definition Finite mean Locally integrable
SLIDE 20
Generalized Arcsine Law
Generalized Arcsine distribution
SLIDE 21 Numerical Simulations
Distributions of the time average
Observation function
As the value of B becomes large, the middle peak becomes low and edge peaks become high.
Time average p.d.f. of the time average
SLIDE 22
Numerical Simulations
SLIDE 23 Application to Correlation Function
Correlation function is intrinsically random (Generalized Arcsine distribution) and never decays.
- Remark. The convergence becomes slow as n becomes large.
SLIDE 24
Correlation Function
SLIDE 25
Remark on Wiener-Khintchine Theorem
SLIDE 26
Power Spectrum
Question
SLIDE 27 Distribution of
Generalized arcsine distribution
SLIDE 28
Numerical Simulations
(Gamma distribution)
SLIDE 29 with infinite mean
Definition Infinite mean Locally integrable
SLIDE 30
Stable Distributions
Power law phenomena Earthquake, fluorescence intermittency of nanocrystals, motion of bacteria, chaotic dynamics, finance
SLIDE 31
Theorem and Conjecture
SLIDE 32
Finite Measure Case ( )
Invariant density
SLIDE 33
Infinite Measure Case ( )
SLIDE 34
Convergence to the invariant density
SLIDE 35
Numerical Simulations
SLIDE 36
The Scaling Exponent
Assumption
SLIDE 37 Concluding Remarks
In infinite measure dynamical systems the time average
- f some observation functions converges in distribution
(Generalized Arcsine Law, Stable Law).
Non-stationary time series (Fluorescence of nanocrystals) Stationary random variables
Time average
SLIDE 38
Concluding Remarks
Ergodicity of non-equilibrium state in dynamical system is related to infinite measure systems.
