Probing Ergodicity and Nonergodicity

Transcription

1 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

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

3 Intermittent Nonlinear Many Body Dynamics 1. Warm Up 2. Equipartition and Ergodicity 3. Fermi, Pasta, Ulam 4. Klein-Gordon 5. DNLS 6. Outlook

4 Warm Up Test: what is integrability?

5 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

6 Warm Up Test: what is Anderson Localization?

7 Anderson localization Anderson (1958) in Eigenvalues: Width of EV spectrum: Asymptotic decay: Localization length: Localization volume of NM: L l

8 Anderson localization Anderson (1958) in Eigenvalues: Width of EV spectrum: Asymptotic decay: Localization length: Localization volume of NM: L l

9 Warm Up Test: what is Many Body Localization?

10 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 T c1 are in MBL phase (zero conductivity) States above T c1 but below T c2 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

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

12 Equipartition and Ergodicity 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

13 Equipartition and Ergodicity 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

14 Equipartition and Ergodicity 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

15 Equipartition and Ergodicity 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?

16 Equipartition and Ergodicity 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!

17 Equipartition and 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????

18 Equipartition and Ergodicity 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: 1 st and 2 nd moments diverge if γ 2, ergodicity is broken!

19 Equipartition and Ergodicity Symplectic Integration scheme Assume integrabls A, B and

20 Equipartition and Ergodicity Symplectic Integration scheme O(τ 2 ) Integrable corrector: : O(τ 4 )

21 Equipartition and Ergodicity Test: what are the integrals of motion of various models in various energy density limits?

22 Equipartition and Ergodicity models, integrable limits: H = P N n=1 h i p 2 n 2 + V (q n)+w (q n+1 q n ) FPU: V (q) =0,W(q) = 1 2 q q q4 KG: V (q) = 1 2 q q4,w(q) = k 2 q2 JJs (rotors): V (q) = 0,W(q) =(1 cos(q)) DNLS (BH): H = P N n=1 ( n n+1 + cc)+ g 4 n 4

23 Equipartition and Ergodicity Example: DNLS Symplectic Integration scheme

24 Fermi, Pasta, Ulam Origin of equipartion and ergodicity? Wave interactions! 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

25 Fermi, Pasta, Ulam x n (t) = Q q +! 2 qq q = NX q=1 Q q (t)sin( qn N +1 ),! q q =2sin( 2(N + 1) ) p 2(N + 1) X q 1,q 2 =1 N! q! q1! q2 B q,q1,q 2 Q q1 Q q2 B q,q1,q 2 = X ± ( q±q1 ±q 2,0 q±q 1 ±q 2,2(N+1)) selective but long range network In some sense translationally invariant systems are similar to huge quantum dots with zero level spacing

26

27 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?

28 FPU: time to reach equipartition Hunting T2 (Danieli,Campbell,SF) Danieli, Campbell, SF PRE R (2017) criterium for reaching T2: entropy (similar to Casetti et al): E q (t) =( Q 2 q +! 2 qq 2 q)/2,e = P q E q q = E q /E, S = P q q ln( q ),S max =lnn (t) = S(t) S max S(0) S max,0apple apple1 Weakly coupled normal modes in Gibbs equilibrium: eq = 1 ln N, 0, 5772 (Euler constant) FPU with N = 32 : eq = η eq defines a phase space separating manifold which we can use similar to a Poincare sectioning surface for arbitrary trajectories!

29 PRE R (2017)

30 Danieli, Campbell, SF PRE R (2017)

31 PRE R (2017) Blue open squares: Casetti Black filled circles: We Casetti: FPU T2= days of CPU We: FPU T2 = years of CPU 3 months 1 year on GPU cluster Unless KAM hits

32 FPU : intermittent equipartition PRE R (2017) Measuring distributions of return times at equipartition Long excursions in phase space Stickiness to regular orbits ε = Power law tails t - γ 1 st moment finite, 2 nd diverges Scale free relaxation!

33 FPU : intermittent equipartition PRE R (2017) Measuring distributions of return times at equipartition Long excursions in phase space Stickiness to regular orbits ε = Power law tails t - γ 1 st moment finite, 2 nd diverges Also in correlation functions Scale free relaxation!

34 Indeed it is stickiness to regular orbits! ε = PRE R (2017) here we stick è ç here we stick to a q-torus è ç to a q-breather with high frequency è with high frequency ç A consistent quantitative way to study relaxations at equilibrium, and stickiness, in high-dimensional nonlinear dynamical systems

35 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

36 Discrete Breathers: experimentally observed and studied in Josephson junction networks Ustinov Coupled nonlinear optical waveguides Silberberg, Segev BEC in optical lattices Oberthaler Driven micromechanical cantilever arrays Sievers, Sato Antiferromagnetic layered structures (C2H5NH3)2CuCl4 Sievers, Sato Poly-γ-benzyl-L-glutamate (PBLG) Hamm H on Si(111), CO on Ru(001) Guyott-Syonnest, Jakob PtCl based crystals, α-uranium Swanson

37 Klein-Gordon

38 KG : intermittent equipartition PRE R (2017) Long excursions in phase space Stickiness to regular orbits ε = Power law tails t - γ 1 st moment finite, 2 nd diverges Scale free relaxation!

39 DNLS, from N=32 to N=4096 DNLS (BH): H = P N n=1 ( n n+1 + cc)+ g 4 n 4 Large N: sensitivity of observable to fluctuations e.g. participation number: average of order N variance of order N 1/2 need fluctuation of order N 1/4 thus: choose many observables: simply the integrals of motion (actions) of the integrable limit DNLS: ψ n 2 Simultaneously define N equilibrium manifolds, track piercings through all of them!

40 DNLS, from N=32 to N= y Non-Gibbsian norm density β = β=0 y=2x+x 2 /2 Gibbsian x PDF (a) N = 1024 PDF + (t r ) PDF - (t r ) PDF + (t r ) N=512 N=1024 N=2048 N= Log 10 t r Log 10 t r α y=2x+x 2 /2 y=x x x =2, y=3 x =2, y=4 x =2, y=5.79 x =2, y=6 log 10 Λ(t) log 10 t

41 Main results 1 st 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