Dynamical order in chaotic Hamiltonian system with many degrees of freedom

Transcription

1 1 Dynamical order in chaotic Hamiltonian system with many degrees of freedom Tetsuro KONISHI Dept. of Phys., Nagoya University, Japan Sep. 22, 2006 at SM& FT 2006, Bari (Italy), September 20-22, 2006

2 2 Acknowledgements Organizers Collaborators Hiroko Koyama (Waseda Univ.) Stefano Ruffo (Firenze Univ.) Toshio Tsuchiya (Kyoto Univ.) Naoteru Goda (NAO) Kunihiko Kaneko (Tokyo Univ.)

3 3 Abstract In this talk we introduce several systems which have ordered structure such as clusters by its own dynamics. The systems consists of particles with longrange interaction, just like many-body systems in astrophysics. Since the systems are Hamiltonian systems, the spatial structures thus formed are not attractors or asymptotic states we observe in the infinite future. Rather the states are observed in transiency or in the course of itinerancy among several quasi-stationary states.

4 4 1 Part I : Introduction (my) basic, very primitive question: What the world is made of? [picture: Andromeda galaxy] [picture: hemoglobin molecule] Various structures exist in every scale. The world is not uniform.

5 5 2 statistical description Way to describe structure most successful : (equilibrium) statistical mechanics (+ phase transition theory)

6 6 3 basis of statistical mechanics (total system) = (system to be observed)+(a large system called hea (1) total system is closed, conserved requiring principle of equal weight for total system canonical prob. distribution for the system to be observed (note: equal weight in the total system equipartition of energy p2 = k B T for the system to be observed.) 2m 2

7 3.1 ergodicity def. time average = phase space average Ā = A (2) equivalent def. set). invariant set is the whole phase space ifself (or empty (regardless of initial condition one can visit any state.) [Arnold and Avez] sometimes, ergodicity fails to be satisfied... 7

8 8 4 demonstration 1: standard map standard map (x, p) (x, p ) p = p + K 2π sin(2πx) x = x + p (mod 1) (K > 0) visiting phase point in very much non-uniform way (1/f-type fluctuation, sticking/stagnant motion)

9 9 5 discrete standard map : investigating non-uniformity of phase space Rannou(1984), TK(2006) standard map (I x, I p ) (I x, I p ), I xetc. 0, 1, 2,, M [ I p = I p + M K ] 2π sin(2πi x M ) I x = I x + I p ([ ]: nearest integer) 1 to 1 every orbit is periodic (mod M) (K > 0)

10 1x 1x 10 K=1.0 K= p p 1 orbit orbit discrete standard map : left:low resolution (M = 32), right: increased resolution (M = 1024)

11 statistics of orbits: histgram of period for mesh=4096 longest period "dsm-4096-hist.data" "numorb.data" using 1:3 3*x longest period period/(longest period) Tue Jun 27 13:26: mesh Tue Jun 27 13:15: (left) distribution of periods of orbits (resolution=4096) (right) longest period vs. resolution no dominant orbit appears 11

12 12 dynamics: K=1.0, mesh=4096; longest periodic orbit : id= "dsm-4096-longest-orbit-xp.data" using ($2/4096):($1/4096) 0.8 K=1.0, mesh=4096; 3rd-longest periodic orbit : id=3673,period= "dsm rd-longest-orbit-xp.data" using ($2/4096):($1/4096) x x Tue Jun 27 13:32: p Tue Jun 27 13:37: (left) longest periodic orbit (M = 4096) (right) 3rd longest periodic orbit (M = 4096) p

13 13 6 many body system failure of ergodicity in small system: pathology caused by limitation of deg. of freedom? many body system helps?

14 14 7 failure of ergodicity in many body systems 7.1 trivial example free particle system H = N i=1 p 2 i 2m i (3) harmonic chain H = N i=1 p 2 i 2m + N i=1 k 2 (x i+1 x i ) 2 (4) every mode energy is conserved : E k (t) = E k (0)

15 less trivial example 1-dim. linear chain with quartic or quadratic interaction (AKA Fermi Pasta Ulam model) H = N i=1 p 2 i 2m + N i=1 k 2 (x i+1 x i ) 2 + β N i=1 1 4 (x i+1 x i ) 4 (5) However, numerical simulation tells us that the mode energies are recurrent for the system (no relaxation).

16 16 E 1 (t = 0) = E total, E k (t = 0) = 0 (k 2) (wrong) E 1 = E 2 = = E N (true) E 1 (T ) E 0, E k << E 0 (k 2) Later analysis revealed that the model (5) is quite close to an integrable equation of 1-dim field modified KdV equation.

17 8 Statistical mechanics: successful application to explain structure Anyway, removing some difficulties, statistical mechanics succeeded in explaining order and structures: ferromagnetic transition for Ising model and many other spin systems helix-coil transition for long chain of molecules huge number of examples.. 17

18 18 Statistical mechanics assumes: the system is in thermal equilibrium there may be some perturbations appplied to the system, but they relaxes quickly: in other words, there exists a characteristic relaxation time τ relax and the perturbations relaxes in time as ( exp t ) τ relax downside : fails to describe structures which are not stationary

19 19 Examples: dynamics of water molecules (Ohmine group, Sasai group) Elliptical galaxies 10Research/10Dynamics/Gallery/index.phtml.en M. Sasai et al., J. Chem. Phys. 96(1992) 3045.

20 20 Purpose of this talk: As we have seen, spatial structures are not necessarily in equilibrium Then what can we do? The structures are created from their own dynamics, so we hope we can understand something from dynamics By presenting some examples where spatial structure are created from dynamics, we will search for methods and concepts to understand the dynamical origin of spatial structures.

21 21 Part II : Clustered motion in globally coupled symplectic map Refs. TK and K. Kaneko, J. Phys. A 25 (1992) 6283 K. Kaneko and TK, Physica D 71 (1994) 146 model clustered motion and random motion Lyapunov spectra minimum exponent? Fluctuation property of Lyapunov exponent Phase space structure order supported by chaos

22 22 Model (x i, p i ) ( ) x i, p i, i = 1, 2,, N, N p i = p i + K sin 2π(x j x i ), (6) j=1 K > 0 : attractive, K < 0 : repulsive x i = x i + p i.

23 two phases of motion (with same K) 1 N=12,K=0.2 (Z=5.2) clustered : x uniformly random: x 0 0 t rand.a.dat t p i = p i + K N sin 2π(x j x i ), K : real j=1 23

24 24 Lyapunov spectra Lyapunov spectrum : N=8, K=0.1 Both phases are chaos. difference of Lyap. exp. is observed only at the λ N 1. This subspace represents instability of cluster? value of exponents uniformly random clustered state i/n

25 25 Fluctuation property of Lyapunov exponent Variance of the maximum Lyap.exp. clustered state strong temporal correlation non-clustered state no temporal correlation variance 10 3 T 0.74 T 1.1 uniformly random state clustered state N=8, K= length of each interval T

26 phase space structure init. cond. for clustered motion (white) local λ 1 ( black stable) 0.5 K = K = 0.2 P2 P P Z P Lyapunov Exponent 0.2

27 Self-similar structures in the phase space common to most Hamiltonian systems chaos supports the order 27

28 28 Part III : Emergence of power-law correlation in the mass-sheet model Refs. H. Koyama and TK, Phys. Lett. A vol. 295 (2002) 109 H. Koyama and TK, Europhys. Lett., vol. 58(2002) 356. H. Koyama and TK, Phys. Lett. A vol.279 (2001) 226.

29 29 model x1 x2 xn x Fig. 1 A schematic picture of the model (7) H = N i=1 p 2 i 2m + 2πGm2 i>j x i x j, < x i <, (7)

30 30 formation of fractal structure dynamics seen in µ-space size=2**15 : initial condition = uniformly random: velocity fluctuation vamp=0.0 size=2**15 : initial condition = uniformly random: velocity fluctuation vamp=0.0 size=2**15 : initial condition = uniformly random: velocity fluctuation vamp=0.0 size=2**15 : initial condition = uniformly random: velocity fluctuation vamp= v 0 v 0 v 0 v x x x -0.6 "<awk $1==120{print $3,$4} n ph-lin.dat" x Fig. 2 Formation of fractal structure. Initial condition : x i : uniformly random in [0,1], u i = 0, N = Time are ,4.6875, , and

31 31 box-counting dimension log2(number of occupied boxes) box-counting dimension for (x,v) space 3 x+3 2 x* x+1 "< awk {print $1,log($3)/log(2)} n bc2d-2.data" k: number of division for length = 2^k Fig. 3 Box counting dimension of the µ-space distribution shown in the bottom of Fig.2. Sample orbits with the same class of initial condition with different random number gives dimension D = 1.1 ± 0.04 for 24 samples. Lines with D = 1 are also shown for comparison.

32 32 2-body correlation function ξ(r) dp = ndv (1 + ξ(r)) (8) ξ(r) r α (9) 10 2-point correlation function : file=n15-011:mesh=10^5 "n corr-mesh data" xi(r) e r Fig. 4 two-point correlation function ξ(r) for t = in Fig. 2. Exponent α of ξ r α is α = 0.20 ± 0.03 for 24 samples.

33 33 development of power-law structure spatial scale : small large ( hierarchical clustering ) two-point correlation function e-06 1e Fig. 5 Evolution of correlation function ξ(r) at t = 5 l, l = 3, 4,, (from bottom to top). System size N = 2 14, Initial condition: x i =random, v i = 0.

34 34 relaxation of the structure Actually, the power-law structure ξ(r) r α decays, but long after virialization Vr t Fig. 6 time dependence of the exponent α(t) (left) and the virial ratio2e kin /E pot (right)(9)

35 35 Part IV : equilibrium and nonequilibrium behavior of excitation in HMF model Koyama, Ruffo, TK (2006) Koyama, talk at this conference H = N i=1 p 2 i N N i,j=1 (1 cos(θ i θ j )) (10) cluster formation excitation of high-energy particle

36 36 trapping ratio : R N LowEnergyP article N total (11) trapping ratio is well described by equilibrium statistical mechanics on the other hand, lifetime of fully-clustered state (state without any high energy particles) obeys power law

37 37 Summary Spatial structures can emerge in non-equilibrium situations. Their origins are non-uniformity of the phace space, where dynamical properties of the system is reflected: phase space phase space thermal equilibrium itinerancy among quasi equilibria equilibrium structure (left) and dynamical order(right) chaos does not necessarily imply mess nor darkness : they can be rich sources of dynamic order.

38 long-range interaction may be important in creating structure/order which are dynamically changing. 38