ON THE ENERGY TRANSFER IN FPU LATTICES H. Christodoulidi ...€¦ · NX−1 i=1 X∞ k=4...
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ON THE ENERGY TRANSFER IN FPU LATTICES H. Christodoulidi, A. Ponno, S. Flach and Ch. Skokos Max-Planck-Institut f ¨ ur Physik komplexer Systeme, N ¨ othnitzer Str. 38, Dresden Abstract We study the evolution of dynamics in the Fermi-Pasta-Ulam- α model in order to classify and characterize the behavior of the system. This classification is divided into three main time intervals, called stages, in which there is a qualitative change in the dynamics of the system. We compare every stage in the FPU dynamics with those of Toda’s system. Thus we can use Toda as a tool to distinguish between the chaotic and integrable behavior in the FPU-α system. Introduction The one dimensional FPU-α lattice with fixed boundary con- ditions is described by the Hamiltonian H FPU = 1 2 N -1 X k =1 p 2 k + N -1 X k =0 [ 1 2 (q k +1 - q k ) 2 + α 3 (q k +1 - q k ) 3 ] (1) with x(0) = x(N )=0. The Toda lattice, described by the Hamiltonian function H T = N -1 X k =1 p 2 k + 1 4a 2 N -1 X k =1 e 2a(q k+1 -q k ) - N 4a 2 (2) can be regarded as an approximation of the FPU-α Hamilto- nian (1) of order α 2 , since H T = H FPU - N -1 X i=1 ∞ X k =4 2 k -2 a k -2 k ! (q i+1 - q i ) k . (3) At the harmonic limit α =0 is H T = H FPU = H 2 = N -1 X k =1 E k (4) where E k are the harmonic energies and ω k = 2 sin kπ 2N the harmonic frequencies for both systems. In the present work, we excite the first normal mode k =1 of the FPU-α and Toda systems, with initial conditions q i = A · sin πi N , ˙ q i =0 , for i =1, 2, ..., N - 1 . STAGE I The harmonic energies of both systems are characterized by a sharp power law growth in time, due to acoustic resonances. Figure 1: FPU-α and Toda systems with N = 32, α =0.33. Normalized harmonic energy evolution plotted in logarithmic scale, for different values of the total energy: i) E =0.01, ii) E =0.1, iii) E =1. Blue lines correspond to the power law E k ∝ t 2(k -1) . STAGE II The harmonic energy spectrum saturates around a constant profile. FPU-α and Toda systems exhibit in Fourier space the natu- ral packet formation, in which the total energy is exchanged among few low-frequency modes [1], and the exponential en- ergy localization of the tail modes [2], i.e. higher-frequency modes. This stage holds for FPU-α up to the time that the energy spectra of tail modes slowly grow and the system tends to reach equipartition. The exponential localization, of the averaged Toda energy spectra of the tail modes, are given by Log ( E k E )= -σk + (5) where σ = eC 0 N √ αε 1/4 (6) and = Log ( C 1 Nαε 1/2 ) (7) with C 0 3 -1/2 and C 1 6 (See Fig.4). Figure 2: FPU-α and Toda systems with N = 32, α =0.33. Normalized and time averaged harmonic energy evolution for both systems, plotted in logarithmic scale, for different values of the total energy: i) E =0.01, ii) E =0.1, iii) E =1. Figure 3: Toda system with N = 32, α =0.33. Normalized and time averaged energy spectra for total energy: i) E = 0.01, ii) E =0.1, iii) E =1. In each panel we plot the spectra at times 10 4 , 10 6 , 10 8 . The black line corresponds to Eq. (5). Figure 4: Toda system with α =0.33. Numerical evidence for the validity of Eq. (6). The plot of Log (σN ) versus -Log (ε) is a line that saturates to Eq. (6) as the degrees of freedom N increase. The red line, which corresponds to N = 1024, is -0.256Log (ε)+0.4278. STAGE III Energy is diffused in FPU-α system from the packet to the tail. We numerically estimate this diffusion by computing i) the moments of the normalized energy spectra, defined as m s = N X k =1 k s ( E k E ) s (8) ii) the sum of the harmonic energies of the last third part of the mode interval i.e. [N/3,N ], which we call tail energy η = N X k =N/3 E k E . (9) Numerical evaluations of both quantities show a power law increase in time, of the form m s ∝ D s t γ s , η ∝ Dt γ (10) for the moments and the tail energy respectively (see Figs.5 and 7 ). By m s and η we denote the moments and the tail energy, de- fined for the averaged harmonic energies E k . The exponents of these powers laws are numerically evalu- ated and appear in Figs.6 and 7(ii). Both of them fluctuate very strongly indicating a dense region of invariant objects. As E increases, these objects are destroyed and the system tends to equipartition linearly in time (η ∝ t). Figure 5: FPU-α model with N = 32, α =0.33. Evolution of moments m j , j =1, ..., 16, plotted in logarithmic scale for total energy i) E =0.01, ii) E =0.1, iii) E =1. The variation of moments, from m 1 to m 16 , is shown by the line colour, which ranges from yellow to blue. The black curves correspond to moments m j ,j =1, ..., 16. Figure 6: FPU-α model with N = 32, α =0.33. i) The slope γ j of the least squares linear fit, of moments m j , j =1, ..., 16 versus the total energy of the system E . Same line colours with Fig.5 are used. ii) The same for the slope γ j of moments m j , j =1, ..., 16. Figure 7: FPU-α model with N = 32, α =0.33. i) Evolution of the tail energies, instantaneous η and averaged η (orange curves). The black straight lines correspond to the linear fit of η in the time windows [10 4 , 3 · 10 6 ] for the total energies E =0.01, E =0.1 and [10 3 , 3 · 10 5 ] for E =1. Equipartition of the system is reached when η =1/3. ii) The slope γ of the linear fit of η and η versus E . Figure 8: Numerical computation of the 2 nd Toda integral J for both systems, for the total energies i) E =0.01, ii) E = 0.1, iii) E =1. Conclusions The energy transfer in the FPU-α model from the lower fre- quency modes to the tail modes is initially very sharp, after that stops for a certain time window [3] and then starts again with a linear in time process, that leads the system to equipar- tition. Comparison with the Toda model shows that only the last part is due to non-integrability of FPU-α. Acknowledgements We thank G. Benettin and S. Ruffo for fruitful discussions. H.Ch. appreciated the warm hospitality of the MPIPKS, Dresden. References [1] L. Berchialla, L. Galgani, A. Giorgilli DCDS 11, 855-866, (2004). [2] S. Flach, M. V. Ivanchenko, O. I. Kanakov, Phys. Rev. Lett. 95 064102 (2005). [3] S. Flach, A. Ponno Physica D 237, 908-917, (2008).
Transcript of ON THE ENERGY TRANSFER IN FPU LATTICES H. Christodoulidi ...€¦ · NX−1 i=1 X∞ k=4...
ON THE ENERGY TRANSFER IN FPU LATTICESH. Christodoulidi, A. Ponno, S. Flach and Ch. Skokos
Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Str. 38, Dresden
AbstractWe study the evolution of dynamics in the Fermi-Pasta-Ulam-α model in order to classify and characterize the behavior ofthe system. This classification is divided into three main timeintervals, called stages, in which there is a qualitative changein the dynamics of the system. We compare every stage inthe FPU dynamics with those of Toda’s system. Thus wecan use Toda as a tool to distinguish between the chaotic andintegrable behavior in the FPU-α system.
IntroductionThe one dimensional FPU-α lattice with fixed boundary con-ditions is described by the Hamiltonian
HFPU =1
2
N−1∑
k=1
p2k +
N−1∑
k=0
[1
2(qk+1 − qk)
2 +α
3(qk+1 − qk)
3] (1)
with x(0) = x(N) = 0.The Toda lattice, described by the Hamiltonian function
HT =
N−1∑
k=1
p2k +
1
4a2
N−1∑
k=1
e2a(qk+1−qk) − N
4a2(2)
can be regarded as an approximation of the FPU-α Hamilto-nian (1) of order α2, since
HT = HFPU −N−1∑i=1
∞∑
k=4
2k−2ak−2
k!(qi+1 − qi)
k. (3)
At the harmonic limit α = 0 is
HT = HFPU = H2 =
N−1∑
k=1
Ek (4)
where Ek are the harmonic energies and ωk = 2 sin kπ2N the
harmonic frequencies for both systems.In the present work, we excite the first normal mode k = 1of the FPU-α and Toda systems, with initial conditions qi =A · sin πi
N , qi = 0 , for i = 1, 2, ..., N − 1 .
STAGE IThe harmonic energies of both systems are characterized by asharp power law growth in time, due to acoustic resonances.
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Figure 1: FPU-α and Toda systems with N = 32, α = 0.33.Normalized harmonic energy evolution plotted in logarithmicscale, for different values of the total energy: i) E = 0.01, ii)E = 0.1, iii) E = 1. Blue lines correspond to the power lawEk ∝ t2(k−1).
STAGE IIThe harmonic energy spectrum saturates around a constantprofile.FPU-α and Toda systems exhibit in Fourier space the natu-ral packet formation, in which the total energy is exchangedamong few low-frequency modes [1], and the exponential en-ergy localization of the tail modes [2], i.e. higher-frequencymodes.This stage holds for FPU-α up to the time that the energyspectra of tail modes slowly grow and the system tends toreach equipartition.
The exponential localization, of the averaged Toda energyspectra of the tail modes, are given by
Log(Ek
E) = −σk + % (5)
whereσ =
eC0
N√
αε1/4(6)
and% = Log(
C1
Nαε1/2) (7)
with C0 ' 3−1/2 and C1 ' 6 (See Fig.4).
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Figure 2: FPU-α and Toda systems with N = 32, α = 0.33.Normalized and time averaged harmonic energy evolution forboth systems, plotted in logarithmic scale, for different valuesof the total energy: i) E = 0.01, ii) E = 0.1, iii) E = 1.
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Figure 3: Toda system with N = 32, α = 0.33. Normalizedand time averaged energy spectra for total energy: i) E =0.01, ii) E = 0.1, iii) E = 1. In each panel we plot thespectra at times 104, 106, 108. The black line corresponds toEq. (5).
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Figure 4: Toda system with α = 0.33. Numerical evidencefor the validity of Eq. (6). The plot of Log(σN) versus−Log(ε) is a line that saturates to Eq. (6) as the degreesof freedom N increase. The red line, which corresponds toN = 1024, is −0.256Log(ε) + 0.4278.
STAGE IIIEnergy is diffused in FPU-α system from the packet to thetail.We numerically estimate this diffusion by computing i) themoments of the normalized energy spectra, defined as
ms =
N∑
k=1
ks(Ek
E)s (8)
ii) the sum of the harmonic energies of the last third part ofthe mode interval i.e. [N/3, N ], which we call tail energy
η =
N∑
k=N/3
Ek
E. (9)
Numerical evaluations of both quantities show a power lawincrease in time, of the form
ms ∝ Dstγs, η ∝ Dtγ (10)
for the moments and the tail energy respectively (see Figs.5and 7 ).
By ms and η we denote the moments and the tail energy, de-fined for the averaged harmonic energies Ek.The exponents of these powers laws are numerically evalu-ated and appear in Figs.6 and 7(ii). Both of them fluctuatevery strongly indicating a dense region of invariant objects.As E increases, these objects are destroyed and the systemtends to equipartition linearly in time (η ∝ t).
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Figure 5: FPU-α model with N = 32, α = 0.33. Evolutionof moments mj, j = 1, ..., 16, plotted in logarithmic scalefor total energy i) E = 0.01, ii) E = 0.1, iii) E = 1. Thevariation of moments, from m1 to m16, is shown by the linecolour, which ranges from yellow to blue. The black curvescorrespond to moments mj,j = 1, ..., 16.
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Figure 6: FPU-α model with N = 32, α = 0.33. i) The slopeγj of the least squares linear fit, of moments mj, j = 1, ..., 16versus the total energy of the system E. Same line colourswith Fig.5 are used. ii) The same for the slope γj of momentsmj, j = 1, ..., 16.
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Figure 7: FPU-α model with N = 32, α = 0.33. i) Evolutionof the tail energies, instantaneous η and averaged η (orangecurves). The black straight lines correspond to the linear fitof η in the time windows [104, 3 · 106] for the total energiesE = 0.01, E = 0.1 and [103, 3 · 105] for E = 1. Equipartitionof the system is reached when η = 1/3. ii) The slope γ of thelinear fit of η and η versus E.
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Figure 8: Numerical computation of the 2nd Toda integral Jfor both systems, for the total energies i) E = 0.01, ii) E =0.1, iii) E = 1.
ConclusionsThe energy transfer in the FPU-α model from the lower fre-quency modes to the tail modes is initially very sharp, afterthat stops for a certain time window [3] and then starts againwith a linear in time process, that leads the system to equipar-tition. Comparison with the Toda model shows that only thelast part is due to non-integrability of FPU-α.
AcknowledgementsWe thank G. Benettin and S. Ruffo for fruitful discussions.H.Ch. appreciated the warm hospitality of the MPIPKS,Dresden.
References[1] L. Berchialla, L. Galgani, A. Giorgilli DCDS 11, 855-866, (2004).[2] S. Flach, M. V. Ivanchenko, O. I. Kanakov, Phys. Rev. Lett. 95 064102 (2005).[3] S. Flach, A. Ponno Physica D 237, 908-917, (2008).
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