Universal glassy dynamics at noiseperturbed onset of chaos

1
Aging at the edge of chaos
Alberto Robledo Instituto de Física Universidad
Nacional Autónoma de México
2
Unimodal maps

• Universality of deterministic irregularities
• - Period-doubling route to chaos
• - Mechanism for intermittency

• Late 70s to early 80s
• - Analogy with critical phenomena
• - Metaphor for turbulence

• Late 90s to early 00s
• - Bona fide example of nonextensive
  statistics
• - Analogy with glassy dynamics

3
Phenomenology of supercooling and glass
formation (presumed manifestations of ergodicity
breaking)

• Two-step relaxation
• - Power-law decay towards and away from a
  plateau

• Adam-Gibbs formula
• - Connection between kinetics and
  thermodynamics

• Aging
• - History-dependent relaxation

4
Universal RG dynamics in logistic maps

• Trajectories inside Feigenbaums attractor
• - Intertwined power-law time subsequences

• q-generalized Lyapunov coefficient
• - Weakly divergent sensitivity to initial
  conditions

• Pesin theorem at onset of chaos
• - Weakly decreasing loss of information

5
Noisy period-doubling transitions to chaos

• Bifurcation gap
• - Noise fluctuations smear sharp features

• Suppression of subharmonics
• - Connection with noise-free parameter

• Scaling of Lyapunov exponent with noise
  amplitude
• - Analogy with finite magnetization

6
RG dynamics at noise-perturbed onset of chaos

• Structure of trajectories
• - Wandering through power-law pathways

• Duration of the plateau
• - Running into the bifurcation gap

• ß and a relaxation processes
• - Falling into the attractor merging of bands

7
Map counterpart of Adam-Gibbs law aging at the
edge of chaos

• Entropy of noisy trajectories
• - Link between landscape and dynamical
  properties

• Relaxation through intricate trajectories
• - Waiting-time scaling

• Attractor position subsequences
• - A waiting time for each subsequence

8
Concluding remarks
9
(No Transcript)
10
(No Transcript)
11
- Robledo, A., The renormalization group and
optimization of non-extensive entropy
criticality in non-linear one-dimensional maps
Physica A 314, 437-441 (2002). - Baldovin, F.
and Robledo, A., Sensitivity to initial
conditions at bifurcations in one-dimensional
non-linear maps rigorous non-extensive
solutions Europhysics Letters 60, 518-524
(2002). - Baldovin, F. and Robledo, A., RG
universal dynamics at the onset of chaos in
logistic maps and non- extensive statistical
mechanics Physical Review E 66, 045104-1 -
045104-4 (R) (2002). - Latora, V., Rapisarda, A.
and Robledo, A., Revisiting disorder and
Tsallis statistics Letters to the Editor,
Science 300, 250 (2003). - Baldovin, F. and
Robledo, A. Non-extensive Pesin identity Exact
RG analytical results for the dynamics at the
edge of chaos of the logistic map
cond-mat/0304410. - Robledo, A. Universal
glassy dynamics at noise-perturbed onset of
chaos. A route to ergodicity breakdown
cond-mat/0307285.
12
(No Transcript)
13
Adam-Gibbs formula
tx A exp (B/TSc)
tx is a relaxation time (or, equivalently, the
viscosity) and Sc, the configurational entropy,
is related to the number of minima of the
systems multidimensional potential energy
surface the energy landscape.
14
Aging in coupled rotors
Montemurro, Tamarit and Anteneodo, PRE 67, 031106
(2003)
15
Two orbits at the edge of chaos
16
Sensitivity to initial conditions
17
q-Pesin identity
18
(No Transcript)
19
Bifurcation gap
20
Scaling of Lyapunov exponent
,
21

• Unperturbed orbit with

and
where
For
we find
.
or
,
and
Similarly for all time subsequences

• Noise-perturbed orbit with

where
and
For
we find
,
22
Crossover time / contact with bifurcation gap
Largest number of bands at noise amplitude s
with
Sampling the energy landscape
,
moving on a plateau with
crossover to
a chaotic regime with
23
Falling into the attractor
24
Entropy at contact with the bifurcation gap
where
and since
one obtains
scaling relations
or
Diverging relaxation time as entropy vanishes
Ergodicity failure as s ? 0
25
Master orbit
26
Attractor position subsequences
A waiting time for each subsequence



Aging scaling relation
or
- Time translation invariance for t gt tx (s) -
Containment of aging for increasing noise
amplitude s gt 0
27
(No Transcript)
28
Pitchfork tangent bifurcations