Computational studies of discrete breathers. Sergej Flach MPIPKS Dresden January 2003

Transcription

1 Computationa studies of discrete breathers Sergej Fach MPIPKS Dresden January 2003 CONTENT: 0. A bit on numerics of soving ODEs 1. How to observe breathers in simpe numerica runs 2. Obtaining breathers up to machine precision 3. Perturbing breathers - Foquet theory 3.1. Dynamica stabiity 3.2. Wave scattering 4. Summary Usefu on the desktop: Abramowitz/Stegun, Numerica Recipes Typeset by FoiTEX 1

2 IMPORTANT NOTE: For sake of simpicity in most cases one-dimensiona attice modes and nearest neighbor interaction are used Generaization to higher attice dimensions and arger interaction ranges is STRAIGHTFORWARD If resuts or methods are dimension or interaction range specific, we wi inform you! Typeset by FoiTEX 2

3 A few definitions first, using a simpe mode cass 1.5 H = 1 2 p2 + V (x ) + W (x x 1 ) 1 V (0)=1, W (0)=0.1 V (0) = W (0) = V (0) = W (0) = 0 V (0), W (0) 0 Equations of motion: ẋ = p, ṗ = V (x ) W, 1 + W +1, Sma ampitude pane waves: x (t) e i(ω qt q), ω 2 q = V (0)+4W (0) sin(q 2 ) ω q 0.5 V (0)=0, W (0)= q Group veocity v g (q): v g (q) = dω q dq So for N sites we are studying trajectories in a 2N-dimensiona phase space! Typeset by FoiTEX 3

4 A bit on numerics of soving ODEs ẋ = f(x, t) - Runge-Kutta 4th order, O(h 5 ), 4 f cacuations per step ẍ = f(x) - aternative sympectic Veret (Leap-Frog) i.e. x(t + h) 2x(t) + x(t h) = h 2 f(x(t)), O(h 4 ), 1 f cacuation per step Aways think hard before choosing an agorithm Take into account: tota simuation time maximum error required overa stabiity Thumb rue: short simuations: RK4 ong simuations: sympectic Typeset by FoiTEX 4

5 How do we make finite temperature simuations? Main toos: microcanonica simuation (deterministic) Nose-Hoover simuation (deterministic, N + 1) Langevin dynamics (stochastic) Monte Caro (stochastic) H = 1 2ẋ (x2 1)2 + C 2 (x x 1 ) 2 S k (t) = x (t+τ)x k (τ) τ, A(ω) = 0 cos(ωt)a(t) Thumb rue: stochastic for excitations deterministic for reaxations Nose-Hoover??? Boundary conditions! Correation ength ξ ξ 2 = d 2 dq 2S q(t = 0) q=0 2S q=0 (t = 0) Typeset by FoiTEX 5

6 Sidenotes on spatia and tempora Fourier transforms Spatia transforms: A q = e iq( k) A k Tempora Fourier transforms for correation functions: Fion s integration formua (Abramowitz/Stegun): t 2n t 0 f(t) cos(ωt)dt = h [α(ωh) (f 2n sin(ωt 2n ) f 0 sin(ωt 0 )) + β(ωh)c 2n + γ(ωh)c 2n 1 ] + O(nh 4 f (3) ) C 2n = C 2n 1 = n i=0 f 2i cos(ωt 2i ) 1 2 [f 2n cos(ωt 2n ) + f 0 cos(ωt 0 )] n i=1 f 2i 1 cos(ωt 2i 1 ) α(z) = 1 z + sin 2z 2z 2 2 sin2 z z 3, β(z) = 2 1+cos 2 z z 2 sin 2z z 3, γ(z) = 4 sin z z 3 cos z z 2 Typeset by FoiTEX 6

7 Tempora Fourier transforms for anaytica time-periodic functions: Simpe trapezoida rue does it with exponentia accuracy! A(t) = A(t + T ), ω = 2π T A(kω) = 0 T n=t/h cos(kωt)a(t)dt = h cos(kωih)a(ih) + O(e.../h ) i=1 Typeset by FoiTEX 7

8 How to observe breathers in simpe numerica runs fast workstation, or (even better) vector computer: therma equiibrium (finite temperature), interaction not too arge (?), observe permanent creation and annihiation of ocaized excitations, ife times approx. 10 times the interna osciation periods Excitation of sighty perturbed standing wave with λ 1: moduationa instabiity, wave decays into separate spatia regions(distance λ) with arge ampitude ocaized excitations! (Peyrard 1998) Typeset by FoiTEX 8

9 Targetted initia conditions simpy a osciators at rest (equiibrium) oca excitations of a few osciators rather arbitrary initia conditions: ocaized exciations exist for arge times (how ong?) INTENSITY / ARB. UNITS INTENSITY/ A.U FREQUENCY. u ENERGY FREQUENCY (u 1 +1) TIME Typeset by FoiTEX 9

10 How about simuation entertainment? A set of usefu utiities: PGPLOT (Fortran, C) JAVA MATLAB, etc Typeset by FoiTEX 10

11 How about numerica runs in higher attice dimensions? attice size pus 10 rows on each side with ineary increasing friction ẋ makes 1600 sites! Zero or infinite friction give perfect transmission or refection Optimize numericay for inearized equations with oca initia condition, measure remaining energy after some proper arge time (here t = 2000). How usefu are Poincare maps? A ot if the dynamics is evoving on a ow-dimensiona manifod Very instructive for 3-dimensiona phase space (3 = 4 1) Deas with projections of a 2d manifod on a 2d manifod embedded in a 3d space Lots of tricky effects, read the books or try by yoursef! Typeset by FoiTEX 11

12 Obtaining breathers up to machine precision Time-periodic ocaized excitations persist quasi-periodic excitations radiate Reason: resonances with ω q! H = 1 2 p2 + V (x ) + W (x x 1 ) Ansatz: x (t) = k A k e ikω b t Insert into EoM, assume ocaization, go into tais, inearize w.r.t. A k V (z) = α=2,3,... v α α zα, W (z) = α=2,3,... w α α zα ẍ = v 2 x w 2 (2x x 1 x +1 )+F n (x ) = α=3,4,... v α x α 1 + w α ((x x 1 ) α 1 (x +1 x ) α 1 ), F (n) (t) = + k= F (n) k e ikω b t k 2 ω 2 b A k = v 2 A k + w 2 (2A k A k, 1 A k,+1 ) + F (n) k Thus for kωb ωq inearized equations aow for ocaization A k, 0 This is genericay possibe for noninear spatiay discrete systems, because ω q is bounded, as opposed to spatiay continuous systems. For the same reasons quasiperiodic excitations wi typicay radiate even in spatiay discrete systems because k 1 ω b1 + k 2 ω b2 = ω q can be reaized! Typeset by FoiTEX 12

13 Designing a map Nr.1 to find soutions A (i+1) k = 1 k 2 ω 2 b (v 2 + 2w 2 )A (i) k w 2(A (i) k, 1 + A(i) k,+1 ) + F (n) k (A (i) k ), λ k = v 2 k 2 ω 2 b A (i+1) k = 1 v 2 (k 2 ω 2 b 2w 2)A (i) k + w 2(A (i) k, 1 + A(i) k,+1 ) F (n) k So choose λ > 1 for = 0, k = ±1 and λ < 1 otherwise! For ow order poynomia potentia functions e.g.: (A (i) k ), λ k = k2 ω 2 b v 2 F (n) k = α=3,4,... v α + k 1,k 2,...,k α 1 = Otherwise integrate numericay at each step: A k1 A k2...a kα 1 δ k,(k1 +k k α 1 ) Stop the iteration when e.g. F (n) k = 1 T 1 T 2 T/2 F (n) (t)e ikω 1 t dt k, A (i) k A(i 1) k < Typeset by FoiTEX 13

14 v 2 = 2, v 3 = 3, v 4 = 1, w 2 = 0.1 v 2 = 1, v 4 = 1, w 2 = 0.1 A k attice site k num.resut inearization ω b = 1.3 A k k num.resut inearization Typeset by FoiTEX 14

15 Why do we compute with doube precision the ocaized tais pretty exact down to , by just using a standard Fortran compier? Because doube precision does not mean but a cutoff after 16 digits. So if we start with initia guesses exacty zero and ocaize around zero, then the ony obstace to be perfect is the maximum power the compier reserves for the exponentia representation of foating point numbers :-))) Rotating Wave Approximation: cutoff in k-space! Exampe: ẍ = V (x), V (x) = 1 2m x2m ω = 1 2π(2m) 1 1/2m Γ(1/2m + 1/2) 2 Γ(1/2m) RWA with k = ±1 ony: ω 1 = 2 1 m (2m) 1/2 1/2m m = 2 : 2.3% error! E 1/2 1/2m (2m 1)! m!(m 1)! E1/2 1/2m V (x) = 1 4 (x2 1) 2 : FREQUENCY back soid exact red dashed k=(0),1 bue circes k=1,3 brown circes k=0,1, ENERGY Typeset by FoiTEX 15

16 A specia case of (neary) homogeneous potentia functions = 1 2 p2 + v 2 2 x2 +P OT, P OT = v 2m 2m x2m + w 2m 2m (x x 1 ) 2m, m = 2, 3, 4,... ẍ + v 2 x = v 2m x 2m 1 w 2m (x x 1 ) 2m 1 + w 2m (x +1 x ) 2m 1 Time space separation: x (t) = A G(t) G + v 2 G G 2m 1 = κ = 1 A v 2m A 2m 1 w 2m (A A 1 ) 2m 1 + w 2m (A +1 A ) 2m 1 κ > 0 is a separation parameter, can be choosen freey. Time dependence G = v 2 G κg 2m 1 : singe anharmonic osciator Spatia profie: κa = P OT x {x A }, S A = 0, S = 1 2 κ A 2 P OT ({x A }) Breathers are saddes of S! Typeset by FoiTEX 16

17 Method Nr.2: Saddes on the rim! choose direction in N-dimensiona space of a A, e.g. ( ), ( ) Start from space origin P0 A = 0, depart with sma steps in chosen direction, compute S It wi first increase and then pass through a maximum P1 Now we are on the rim Compute the gradient of S here and make a sma step in opposite direction to P2 Maximise S on the ine P0-P2 to be on the rim again. Repeat unti you reach a sadde! A very simpe and efficient way to compute different types of breathers, mutibreathers etc in arbitrary dimensiona attices Typeset by FoiTEX 17

18 Method Nr.3: Homocinic orbits (ony in d = 1 and with short range interaction)! A +1 = A + v 2m A 2m 1 + w 2m (A A 1 ) 2m 1 κa 2d map with R = (x, y ) = (A 1, A ): x +1 = y y +1 = y + v 2m y 2m 1 + w 2m (y x ) 2m 1 κy 1 2m 1 1 2m 1 Fixpoint: R F = (0, 0) (un)stabe 1d manifod: (backward) iteration converges to R F Manifod intersections: homocinic points! Iterated for- or backward yied homocinic orbits. i.e. breathers! Refection symmetry: one homocinic point on x = y depends parametricay on κ Simpe numerica search by e.g. fixing x 0 = y 0 and varying κ Can be used for a forma existence proof! Existence of mutibreathers foows from generic intersection structure Typeset by FoiTEX 18

19 Using the phase space So far: periodic orbits as soutions of agebraic equations Variabes: Fourier coefficients or simpy ampitudes Of course we can use more genera methods of soving agebraic equations, e.g. various gradient methods or Newton routines We need aways a good initia guess (start cose to a case where you know the soution!) Periodic orbit: oop in phase space Isoated periodic orbit (PO): neighbourhood in phase space free of POs with identica conserved quantities (as opposed to POs on resonant tori) Gradient methods: more sophisticated in programming Newton routines: may suffer from ong times needed to invert matrices, danger when cose to a noninvertabe case (bifurcations!) (reca: f(x = s) = 0, f(x) = f(x 0 ) + f (x 0 )(x x 0 ) + hot x n+1 = x n f(x n )/f (x n )) If besides the Hamitonian H we have another conserved quantity B, then the manifods of some isoated periodic orbits may satisfy the paraeity of gradients, i.e. grad(h) grad(b) Typeset by FoiTEX 19

20 Method Nr.4: NEWTON in phase space Integrate a given initia condition R with x (t = 0) X, p (t = 0) P over a certain time T : x (T ) I x ({X, P }, T ) p (T ) I p ({X, P }, T ) Consider the functions F x = I x X, F p = I p P If R beongs to a PO with period T then F x = F p = 0 For a Newton routine to converge: remove a degeneracies! If R beongs to the PO, then a 1d manifod of points beong to the PO Degeneracy removed by one additiona condition, e.g. P M = 0 So for N degrees of freedom zeroes of 2N 1 couped equations of 2N 1 variabes! Make sure that a zero of these 2N 1 equations with the additiona initia condition P M = 0 uniqey fixes p M (T ) = 0, e.g. through energy conservation. Typeset by FoiTEX 20

21 Define R = (X 1, X 2,..., X M,..., X N, P 1,..., P M 2, P M 1, P M+1, P M+2,..., P N ) F = (F x 1, F x 2,..., F x M,..., F x N, F p 1,..., F p M 2, F p M 1, F p M+1, F p M+2,..., F p N ) F = R(T ) R Given an initia guess R (0) expand How to compute M: R (0,m) = R (0) + e m F n ( R) = F n ( R (0) )+ m F n R m R (0)(R m R (0) m ) M nm = 1 F n( R (0,m) ) F n ( R (0) ) F ( R) = F ( R (0) ) + M( R R (0) ) M nm = F n R R (0) = R n(t ) m R R (0) δ nm m One Newton step: R such that F = 0: R = R (0) M 1 F ( R (0) ) M nm = 1 R (0,m) n (T ) R (0) n (T ) δ nm Repeat unti F or max F n < ɛ Typeset by FoiTEX 21

22 Advantages of Newton: exponentia convergence F nit +1 F nit easy to program we may use one Newton matrix for severa iterations Disadvantages of Newton: computationa time N 2 matrix inversion sensitive to bifuractions may need subte routines (singuar vaue decomposition, deaing with sparse matrices etc) Bake s representation of Newton: Method Nr.5: STEEPEST DESCENT in phase space g( R) = F x F x + F p F p ( g) n = g R n Start in phase space, go in direction opposite to the gradient! Advantages of Descent computationa time N insensitive to bifurcations Disadvantages of Descent more cumsy to program sower convergence distinguish zero minimima from neary zero minima? Typeset by FoiTEX 22

23 Some aspects of symmetries Equations of motion are invariant under some symmetry operations: continuous time-shift symmetry t t + τ time reversa symmetry t t, p p Parity symmetry x x, p p discrete transationa symmetry on the attice other discrete permutationa attice symmetries which eave the attice invariant (spatia refections etc) Each discrete symmetry impies that given a trajectory in phase space, a new trajectory is generated by appying the symmetry operation If the new manifod equas the origina one, then the trajectory is invariant under the symmetry Otherwise it is not invariant Note that the symmetry operation on the phase space does not invove time! In inear equation systems symmetry breaking is possibe ony in the presence of degeneracies In noninear equation systems symmetry breaking is common Typeset by FoiTEX 23

24 Exampe: a pane wave in a harmonic chain is not invariant under time reversa, because of degeneracy A breather is per definition not invariant under discrete transationa symmetry If it is invariant under other symmetries, this can be used to substantiay ower the numerica effort! For time-reversa breathers it is possibe to find an origin in time when x (t) = x ( t), p (t) = p ( t), saves 50% computationa time! Time-reversa parity-invariant breathers: x (t + T /2) = x (t), p (t + T /2) = p (t), saves 75% computationa time! Computing attice permutationa invariant breathers may substantiay ower the computationa effort by finding the irreducibe breather section, especiay in higher attice dimensions! Breathers which are not invariant under time reversa posess a nonzero energy fux: possibe in d=2 and arger! Typeset by FoiTEX 24

25 Perturbing breathers Suppose we found a breather soution x (t). Now we add a sma perturbation ɛ (t) to it and inearize the resuting equations for ɛ (t): ɛ = m 2 H x x m {x (t)}ɛ m This probem corresponds to a time-dependent Hamitonian H(t) H(t) = 1 2 π H {x 2 m x x (t)}ɛ ɛ m m ɛ = H, π = H π ɛ There is a conservation aw İ = 0 I = π (t)ɛ (t) π (t)ɛ (t) For simpicity we drop the attice index here. Define the matrix J J = and the evoution matrix U(t) It foows π(t) ɛ(t) = U(t) I = (π(t), ɛ(t)) J I = (π(0), ɛ(0)) U T (t)j U(t) π(0) ɛ(0) π (t) ɛ (t) U T (t)j U(t) = J Thus U(t) is sympectic! π (0) ɛ (0) Typeset by FoiTEX 25

26 Uy = λy, U T y = λy Uy = 1 λ y, y = J 1 y = J y If U is rea and (λ, y) are an eigenvaue and eigenvector, so are (λ, y ), ( 1 λ, J y), ( 1 λ, J y ) Consider now the mapping over one period for a breather, which defines the Foquet Matrix F U(t) = U(t + T b ), U(T b ) F Boch s Theorem: a Foquet eigenstates with λ = e iω νt b when taken as initia conditions correspond to A breather is ineary stabe when a Foquet eigenvaues reside on the unit circe, i.e. have absoute vaue 1 One Foquet eigenvaue is aways ocated at λ = +1 Its eigenvector is tangent to the periodic orbit As eigenvaues come in pairs, there are two eigenvaues at λ = +1. Upon changing a contro parameter the Foquet eigenvaues may move on the unit circe, coide and eave the circe Then a breather turns from ineary stabe to ineary unstabe Aubry s Band Theory does just that (heps to find Krein signatures, reate pairs or quadrupets of eigenvaues etc): ɛ (t) = e iω νt (ν) (t), (ν) (t) = (ν) (t+t b ) ɛ = m 2 H x x m {x (t)}ɛ m + Eɛ Typeset by FoiTEX 26

27 The different kinds of Foquet eigenvectors Eigenvectors (simpy the perturbations at time t = 0: F = (ɛ 1, ɛ 2,..., ɛ N, π 1, π 2,..., π N ) can be ocaized or deocaized in the attice space Finite number of ocaized states Large number 2N of deocaized states Deocaized states are just pane waves far from the breather Obtaining the Foquet matrix: simiar to the Newton matrix! Before diagonaizing: make sure to use a possibe symmetries to reduce the Foquet matrix to its noninteracting irreducibe parts! What is it usefu for? Characterizing breathers: can be stabe, unstabe Eigenvaue coisions mark bifurcations (of new periodic orbits!) Bifurcations correspond to resonances! Search for bifuractions which may yied moving breathers By choosing proper boundary conditions study scattering of pane waves by breathers! Reate the eigenvaue structure of the Foquet matrix to the scattering resuts Understand as much as possibe what is ost by inearizing the phase space fow Typeset by FoiTEX 27

28 Scattering of pane waves by discrete breathers in 1d Simuation using fu equations Initia condition: breather pus pane wave packet Packet: either gaussian or haf chain pane wave with sharp cuts (better!) Wait unti stationary pattern around breather appears N But not onger than 2max(vg) Choose ampitude of pane wave not too arge (noninear corrections) and not too sma (as compared to the breather soution errors) e KG chain with q = 0.2π: LATTICE SITE NUMBER t q/π Typeset by FoiTEX 28

29 Computing transmission up to machine precision Scattering goes through the extended Foquet states: ɛ (t) = k= ω +3Ω q ω +2Ω q ω q+ωb b b e k e i(ω q+kω b )t Scattering by exact breather soutions: eastic one-channe or ineastic f-channe scattering (number of open channes ess or equa to the number of DOF per attice site) Minimum attice size: check the ocaization ength of a cosed channes (can be quite different from the ocaization ength of the breather!) ω q ω q ω Ω q b ω 2Ω q b ω q Numerica Scheme for one-channe scattering: zeroes of G: G( ɛ(0), ɛ(0)) = ɛ(0) ɛ(0) e iω qt b ɛ(t b ) ɛ(t b ) ω 3Ω q b Typeset by FoiTEX 29

30 Lattice: FPU chains: N, ( N + 1),..., 1, 0, 1,...(N 1), N Boundary conditions: ɛ N+1 = e iω qt, ɛ N 1 = (A + ib)e iω qt Step Nr.1: Fixing for the moment A, B, use standard Newton to find zeroes of G. Step Nr.2: find vaues for A, B such that ɛ N = e iq iω qt Use the notation ɛ (t) = ζ (t)e iω qt. Then the transmission coefficient is given by t t q = 4 sin 2 q (A + ib)e iq ζ N π q Typeset by FoiTEX 30

31 Instead of a cosing Dissipative systems: Breather famiies get quantized into attractors which are surrounded by basins of attraction Phonons are damped out, thus possibe quasiperiodic and even chaotic in time breathers, and even moving breathers Possibe hysteresis between different breather states upon ooping contro parameters Quantum systems Eigenvaue probems pus eigenfunctions Provide with information on breather tunneing Eigenfunction can be mapped onto cassica phase space to observe correspondence with cassica trajectories What a this coud be good for in the future: Everything concerning nonequiibrium processes Reaxation in quantum systems Energy concentration via reaxation Description of chemica processes Typeset by FoiTEX 31