probing ergodicity and nonergodicity€¦ · 1. intermittent nonlinear many-body dynamics 2....

Probing Ergodicity and Nonergodicity

Sergej FLACH

Center for Theoretical Physics of Complex Systems Institute for Basic Science

Daejeon South Korea

1.  Intermittent Nonlinear Many-Body Dynamics

2.  Discrete Time Quantum Walks

1.  Intermittent Nonlinear Many-Body Dynamics

2.  Discrete Time Quantum Walks

Probing Ergodicity and Nonergodicity

Sergej FLACH

Center for Theoretical Physics of Complex Systems Institute for Basic Science

Daejeon South Korea

Intermittent Nonlinear Many Body Dynamics

1.  Warm Up

2.  Equipartition and Ergodicity

3.  Fermi, Pasta, Ulam

4.  Klein-Gordon

5.  DNLS

6.  Outlook

Test: what is integrability?

Warm Up

Liouville integrability: phase space dimension: 2N there exists a maximal set of N Poisson commuting invariants – functions on phase space whose Poisson bracket with H vanish there exist special canonical sets: action-angle variables dynamics happens on N-dimensional tori, each with N fixed actions, and angles evolving linear in time

Warm Up

Test: what is Anderson Localization?

Warm Up

Eigenvalues:

Width of EV spectrum:

Asymptotic decay:

Localization volume of NM: L

l

Localization length:

Anderson localization

in

Anderson (1958)

Test: what is Many Body Localization?

Warm Up

It is Anderson localization in Fock space of EFs of noninteracting problem Take a fermionic system, assume that all single particle states are localized Add local (two body) interaction Gauge the ground state energy to zero Consider excited states with finite energy densities T States below Tc1 are in MBL phase (zero conductivity) States above Tc1 but below Tc2 are in nonergodic metallic state (nonzero conductivity, fractal wavefunctions) You can not explain this transition using finite energy excitations above the ground state This is probably an example of quantizing Arnold diffusion and fractal phase space flow structure of a corresponding classical interacting wave problem It pays off to look into nonlinear dynamical systems close to integrability

Warm Up

Equipartition and Ergodicity

What a nice and stable (integrable?) nonergodic system!

But: There is a small chance that the Earth and Venus could collide in the next 5 billion years (Illustration: J Vidal-Madjar/NASA/IMCCE-CNRS) Jaques Laskar , Paris Observatory



J. Laskar's work spans various field of fundamental astronomy, his main interest being the study of motions in planetary systems. He devoted large efforts to obtain accurate solutions for the long-term motion of planets in the Solar System that are used as the world reference for paleoclimate studies. In pursuing this work, he demonstrated that the orbital motion of the planets of the Solar System is chaotic, with exponential divergence of the orbits of a factor of 10 every 10 million years, making it impossible to predict its motion beyond 60 million years. He showed that planetary perturbations create a large chaotic zone for the spin axis motion of all the terrestrial planets. He demonstrated that without the presence of the Moon, the Earth’s axis would be highly unstable, and could vary from 0 to about 85 degrees. He also demonstrated that the spin axis of Mars is chaotic, and can vary between 0 and 60 degrees, inducing high climatic variations on its surface. In order to improve the long-term ephemeris for the Solar System, he initiated the development of the INPOP planetary ephemerides


Variations in the Earth’s orbit and spin vector are a primary control on insolation and climate; their recognition in the geological record has revolutionized our understanding of palaeoclimate dynamics, and has catalysed improvements in the accuracy and precision of the geological timescale. Yet the secular evolution of the planetary orbits beyond 50 million years ago remains highly uncertain, and the chaotic dynamical nature of the Solar System predicted by theoretical models has yet to be rigorously confirmed by well constrained (radioisotopically calibrated and anchored) geological data. Here we present geological evidence for a chaotic resonance transition associated with interactions between the orbits of Mars and the Earth, using an integrated radioisotopic and astronomical timescale from the Cretaceous Western Interior Basin of what is now North America. This analysis confirms the predicted chaotic dynamical behaviour of the Solar System …


Classical ergodicity: •  visit all parts of phase space under constraints due to integrals of motion •  all microstates have the same weight •  time averages equal phase space averages •  close to integrable limits: adiabatic invariants, KAM, Arnold diffusion

Quantum ergodicity: •  wave function present in all parts of Hilbert space •  close to integrable limits: nonergodic wave functions, MBL •  choice of the Hilbert basis? •  Is MBL a result of quantizing the Arnold web?

noninteracting (single particle) systems: pathological interacting few body systems: chaos, stickiness, mixed phase space interacting many body systems: è no strict ergodicity (on the order of the universe lifetime for any reasonable system example) è exponential growth of microstates with algebraic decay of transition times (e.g. Gaveau and Schulman 2015) still: many macroscopic systems behave ‘good’ or ‘ergodic’ yet: more and more systems are getting on the list of ‘bad’ or ‘nonergodic’ è many body localization è nonergodic metals è glasses è interacting many body systems on lattices with bounded sp spectra they all have in common being close to some integrable limit Need novel approaches to predict loss of ergodicity!


At the very integrable limit: a set of frozen actions, i.e. integrals of motion Close to the integrable limit: additional coupling network between the actions destroys integrability and unfreezes actions What are the simplest qualitatively different network classes? è Short range networks with countable set of actions è Long range networks with countable set of actions Long range networks: è Translationally invariant systems, any interactions low energy densities, homogeneous slowing down Short range networks: è Disordered systems with Anderson Localization, local interactions low energy densities, inhomogeneous slowing down, MBL è Systems with local interactions, short range hoppings high energy densities, inhomogeneous slowing down, MBL And there can be many other intermediate interpolating classes! Glasses????


Goal: run a system in an ergodic parameter region, quantitatively characterize its distance from a nonergodic parameter region Idea: compute statistics of fluctuations instead of correlation functions Why: fluctuations of interest are well defined, can be traced back to their microscopic dynamics, and analyzed Method: choose observable f (should be sensitive to nonergodic fluctuations close to the integrable limit, i.e sensitive to adiabatic invariants) obtain <f> - defines a generalized Poincare equilibrium manifold f=<f> if system is ergodic, trajectory will pierce infinitely many times measure excursion times between piercings compute probability distribution functions (PDF) Note: if tail of PDF is proportional to x-Υ then: 1st and 2nd moments diverge if γ ≤ 2 , ergodicity is broken !



Symplectic Integration scheme

Assume integrabls A, B and


O(τ2)

Integrable corrector: :

O(τ4)

Symplectic Integration scheme

Test: what are the integrals of motion of various models in various energy density limits?



models, integrable limits:

H =PN

n=1

hp2n2 + V (qn) +W (qn+1 � qn)

i

FPU: V (q) = 0 , W (q) = 12q

2 + 13↵q

3 + 14�q

4

KG: V (q) = 12q

2 + 14q

4 , W (q) = k2 q

2

JJs (rotors): V (q) = 0 , W (q) = (1� cos(q))

DNLS (BH): H =PN

n=1

⇥�( n

⇤n+1 + cc) + g

4 | n|4⇤

Example: DNLS Symplectic Integration scheme


Origin of equipartion and ergodicity? Wave interactions!

Fermi, Pasta, Ulam

1955

FPU problem: N=32, excited mode q=1 did not observe equipartition energy stays localized in few modes recurrences after more integrations thresholds in energy, system size etc

two time scales T1: formation of exponentially localized packets in normal mode space T2: gradual destruction of exponential localization, and equipartition

xn(t) =NX

q=1

Qq(t) sin(⇡qn

N + 1) , !q = 2 sin(

⇡q

2(N + 1))

Q̈q + !

2qQq = � ↵p

2(N + 1)

X

q1,q2=1

N!q!q1!q2Bq,q1,q2Qq1Qq2

Bq,q1,q2 =X

±(�q±q1±q2,0 � �q±q1±q2,2(N+1))

Fermi, Pasta, Ulam

selective but long range network In some sense translationally invariant systems are similar to huge quantum dots with zero level spacing

Galgani and Scotti (1972): exponential localization after short transient Galgani, Giorgilli, Benettin, Ponno, Penati, and many many others (… much later …): slow delocalization in tails, equipartition After potentially very long second time scale Casetti, Cerruti-Sola, Pettini, Cohen (1997): scaling of second time scale T2 ? Ponno, Christodoulidi, Skokos, SF (2011): energy diffusion from core to tail modes, indication of divergence of T2 at KAM threshold ?

Theory for T2? Theory for equipartition? Where is KAM regime? Relation to turbulence?

Hunting T2 (Danieli,Campbell,SF)

criterium for reaching T2: entropy (similar to Casetti et al):

Eq

(t) = (Q̇2q

+ !2q

Q2q

)/2 , E =P

q

Eq

⌫q

= Eq

/E , S = �P

q

⌫q

ln(⌫q

) , Smax

= lnN

⌘(t) = S(t)�S

max

S(0)�S

max

, 0 ⌘ 1

Weakly coupled normal modes in Gibbs equilibrium:

⌘eq

= 1��

lnN

, � ⇡ 0, 5772 (Euler constant)

FPU with N = 32 : ⌘eq

= 0.1218ηeq defines a phase space separating manifold which we can use similar to a Poincare sectioning surface for arbitrary trajectories !

FPU: time to reach equipartition Danieli, Campbell, SF PRE 95 060202R (2017)

PRE 95 060202R (2017)

Danieli, Campbell, SF PRE 95 060202R (2017)

Blue open squares: Casetti Black filled circles: We Casetti: FPU T2=1011 1-3 days of CPU We: FPU T2 = 1015 14-80 years of CPU 3 months – 1 year on GPU cluster Unless KAM hits

PRE 95 060202R (2017)

Measuring distributions of return times at equipartition

Power law tails t - γ

1st moment finite, 2nd diverges Scale free relaxation !

Long excursions in phase space Stickiness to regular orbits ε = 0.05661

FPU : intermittent equipartition PRE 95 060202R (2017)

Measuring distributions of return times at equipartition


1st moment finite, 2nd diverges Also in correlation functions Scale free relaxation !


FPU : intermittent equipartition PRE 95 060202R (2017)

Indeed it is stickiness to regular orbits! ε = 0.05661

A consistent quantitative way to study relaxations at equilibrium, and stickiness, in high-dimensional nonlinear dynamical systems

here we stick è to a q-torus è with high frequency è

ç here we stick ç to a q-breather with high frequency ç

PRE 95 060202R (2017)

Strong Nonlinearity: Discrete Breathers

Nonlinear wave interaction generates localization by frequency detuning

Exciting a plane wave in a two-dimensional lattice Time-periodic spatially localized exact solutions

•  Josephson junction networks •  Coupled nonlinear optical waveguides •  BEC in optical lattices •  Driven micromechanical cantilever arrays •  Antiferromagnetic layered structures (C2H5NH3)2CuCl4 •  Poly-γ-benzyl-L-glutamate (PBLG) •  H on Si(111), CO on Ru(001) •  PtCl based crystals, α-Uranium

Discrete Breathers: experimentally observed and studied in

•  Josephson junction networks

•  Coupled nonlinear optical waveguides

•  BEC in optical lattices

•  Driven micromechanical cantilever arrays

•  Antiferromagnetic layered structures (C2H5NH3)2CuCl4

•  Poly-γ-benzyl-L-glutamate (PBLG)

•  H on Si(111), CO on Ru(001)

•  PtCl based crystals, α-Uranium

Ustinov Silberberg, Segev Oberthaler Sievers, Sato Sievers, Sato Hamm Guyott-Syonnest, Jakob Swanson

Klein-Gordon



1st moment finite, 2nd diverges Scale free relaxation !

KG : intermittent equipartition PRE 95 060202R (2017)

DNLS, from N=32 to N=4096

Large N: sensitivity of observable to fluctuations e.g. participation number: average of order N variance of order N1/2

need fluctuation of order N1/4 thus: choose many observables: simply the integrals of motion (actions) of the integrable limit DNLS: |ψn|2

Simultaneously define N equilibrium manifolds, track piercings through all of them!

DNLS (BH): H =PN

n=1

⇥�( n

⇤n+1 + cc) + g

4 | n|4⇤

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6

y

x

Non-Gibbsian

Gibbsian

β = ∞β=0

y=2x+x2/2

norm density 10-12

10-10

10-8

10-6

10-4

10-2

100

1 2 3 4 5

PDF

Log10tr

N = 1024

(a)

PDF+(tr)PDF-(tr)

10-12

10-8

10-4

1 2 3 4 5

PDF +

(t r)

Log10tr

N=512N=1024N=2048N=4096

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6

α

x

y=2x+x2/2y=x2

-0.02

0

0.02

0.04

0.06

0.08

0.1

1 2 3 4 5 6 7 8

log 1

0Λ(t)

log10t

x =2, y=3x =2, y=4

x =2, y=5.79x =2, y=6

DNLS, from N=32 to N=4096

Main results

•  1st moment of excursion time PDFs diverges close to integrable limit

•  Long range networks: ergodicity breaks only at the limit, all relaxation times diverge at the limit

•  Short range networks: ergodicity breaks at finite distance to limit, some relaxation times diverge, some stay finite at a finite distance to the limit •  We have a quantitative tool to probe nonergodicity

C. Danieli D.K. Campbell Y. Kati Mithun T

probing ergodicity and nonergodicity€¦ · 1. intermittent nonlinear many-body dynamics 2....

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