### Mathematics in Engineering

2019, Issue 2: 359-377. doi: 10.3934/mine.2019.2.359
Research article Special Issues

# Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

• Received: 30 November 2018 Accepted: 26 February 2019 Published: 12 April 2019
• We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral $J$ in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, $J$ exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, $J(t)$ is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5, 2.5]$. For 'generic data' initial conditions, instead, $J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.

Citation: Helen Christodoulidi, Christos Efthymiopoulos. Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral[J]. Mathematics in Engineering, 2019, 1(2): 359-377. doi: 10.3934/mine.2019.2.359

### Related Papers:

• We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral $J$ in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, $J$ exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, $J(t)$ is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5, 2.5]$. For 'generic data' initial conditions, instead, $J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.

 [1] Bambusi D, Ponno A (2006) On metastability in FPU. Commun Math Phys 264: 539–561. doi: 10.1007/s00220-005-1488-1 [2] Berman GP, Izrailev FM (2005) The Fermi-Pasta-Ulam problem: 50 years of progress. Chaos 15: 015104. doi: 10.1063/1.1855036 [3] Benettin G, Livi R, Ponno A (2009) The Fermi-Pasta-Ulam problem: Scaling laws vs. initial conditions. J Stat Phys 135: 873–893. [4] Benettin G, Ponno A (2011) Time-scales to equipartition in the Fermi-Pasta-Ulam problem: Finite-size effects and thermodynamic limit. J Stat Phys 144: 793–812. doi: 10.1007/s10955-011-0277-9 [5] Benettin G, Christodoulidi H, Ponno A (2013) The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J Stat Phys 152: 195–212. doi: 10.1007/s10955-013-0760-6 [6] Benettin G, Pasquali S, Ponno A (2018) The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J Stat Phys 171: 521–542. doi: 10.1007/s10955-018-2017-x [7] Carati A, Galgani L, Giorgilli A, et al. (2007) Fermi-Pasta-Ulam phenomenon for generic initial data. Phys Rev E 76: 022104. [8] Carati A, Galgani L (1999) On the specific heat of Fermi-Pasta-Ulam Systems and their glassy behavior. J Stat Phys 94: 859–869. doi: 10.1023/A:1004531032623 [9] Carati A, Maiocchi A, Galgani L, et al. (2015) The Fermi-Pasta-Ulam system as a model for glasses. Math Phys Anal Geom 18: 31. doi: 10.1007/s11040-015-9201-x [10] Carati A, Ponno A (2018) Chopping time of the FPU $\alpha$-model. J Stat Phys 170: 883–894. doi: 10.1007/s10955-018-1962-8 [11] Casetti L, Cerruti-Sola M, Pettini M, et al. (1997) The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems. Phys Rev E 55: 6566–6574. doi: 10.1103/PhysRevE.55.6566 [12] Christodoulidi H, Efthymiopoulos C, Bountis T (2010) Energy localization on q-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences. Phys Rev E 81: 016210/1–16. [13] Christodoulidi H, Efthymiopoulos C (2013) Low-dimensional q-Tori in FPU lattices: Dynamics and localization properties. Phys D 261: 92–113. doi: 10.1016/j.physd.2013.07.007 [14] Christodoulidi H (2017) Extensive packet excitations in FPU and Toda lattices. EPL 119: 40005. doi: 10.1209/0295-5075/119/40005 [15] Chirikov BV (1960) Resonance processes in magnetic traps. Soviet J At Energy 6: 464–470. doi: 10.1007/BF01483352 [16] Chirikov BV (1979) A universal instability of many-dimensional oscillator systems. Phys Rep 52: 263–379. doi: 10.1016/0370-1573(79)90023-1 [17] Danieli C, Campbell DK, Flach S (2017) Intermittent many-body dynamics at equilibrium. Phys Rev E 95: 060202(R). [18] Danieli C, Mithun T, Kati Y, et al. (2018) Dynamical glass in weakly non-integrable many-body systems. [arxiv:1811.10832]. [19] Dauxois T (2008) Fermi, Pasta, Ulam, and a mysterious lady. Phys Today 61: 55–57. [20] Ferguson WE Jr, Flaschka H, McLaughlin DW (1982) Nonlinear normal modes for the Toda Chain. J Comput Phys 45: 157–209. doi: 10.1016/0021-9991(82)90116-4 [21] Fermi E, Pasta J, Ulam S (1995) Studies of non linear problems. Los Alamos report No LA-1940. [22] Flach S, Ivanchenko MV, Kanakov OI (2005) q-Breathers and the Fermi-Pasta-Ulam problem. Phys Rev Lett 95: 064102. doi: 10.1103/PhysRevLett.95.064102 [23] Flach S, IvanchenkoMV, Kanakov OI (2006) q-Breathers in Fermi-Pasta-Ulam chains: Existence, localization, and stability. Phys Rev E 73: 036618. doi: 10.1103/PhysRevE.73.036618 [24] Flach S, Ponno A (2008) The Fermi-Pasta-Ulam problem Periodic orbits, normal forms and resonance overlap criteria. Phys D 237: 908–917. doi: 10.1016/j.physd.2007.11.017 [25] Flaschka H (1974) The Toda lattice. II. Existence of integrals. Phys Rev B 9: 1924–1925. [26] Goldfriend T, Kurchan T (2019) Equilibration of Quasi-Integrable Systems. Phys Rev E 99: 022146. doi: 10.1103/PhysRevE.99.022146 [27] Izrailev FM, Chirikov BV (1966) Statistical properties of a nonlinear string. Dokl Akad Nauk SSSR 166: 57–59. [28] Gallavotti G (2008) The Fermi-Pasta-Ulam Problem: A Status Report. Springer, Berlin- Heidelberg, vol. 728. [29] Genta T, Giorgilli A, Paleari S, et al. (2012) Packets of resonant modes in the Fermi-Pasta-Ulam system. Phys Lett A 376: 2038–2044. doi: 10.1016/j.physleta.2012.05.006 [30] Hénon M (1974) Integrals of the Toda lattice. Phys Rev B 9: 1921–1923. doi: 10.1103/PhysRevB.9.1921 [31] Kantz H, Livi R, Ruffo S (1994) Equipartition thresholds in chains of anharmonic oscillators. J Stat Phys 76: 627–643. doi: 10.1007/BF02188678 [32] Kantz H (1989) Vanishing stability thresholds in the thermodynamic limit of nonintegrable conservative systems. Phys D 39: 322–335. doi: 10.1016/0167-2789(89)90014-6 [33] Kruskal MD, Zabusky NJ (1965) Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15: 240–243. [34] Parisi G (1997) On the approach to equilibrium of a Hamiltonian chain of anharmonic oscillators. EPL 40: 357–362. doi: 10.1209/epl/i1997-00471-9 [35] Penati T, Flach S (2007) Tail resonances of Fermi-Pasta-Ulam q-breathers and their impact on the pathway to equipartition. Chaos 17: 023102/1-16. [36] Paleari S, Penati T (2005) Equipartition times in a Fermi-Pasta-Ulam system. Discrete Contin Dyn S 2005: 710–719. [37] Ponno A, Bambusi D (2005) Korteweg-de Vries equation and energy sharing in Fermi-Pasta- Ulam. Chaos 15: 015107. doi: 10.1063/1.1832772 [38] Ponno A, Christodoulidi H, Skokos Ch, et al. (2011) The two-stage dynamics in the Fermi-Pasta- Ulam problem: From regular to diffusive behavior. Chaos 21: 043127. doi: 10.1063/1.3658620 [39] Livi R, Pettini M, Ruffo S, et al. (1985) Equipartition threshold in nonlinear large Hamiltonian systems The Fermi-Pasta-Ulam model. Phys Rev A 31: 1039–1045. [40] Lvov YV, Onorato M (2018) Double scaling in the relaxation time in the $\beta$-Fermi-Pasta-Ulam- Tsingou model. Phys Rev Lett 120: 144301. doi: 10.1103/PhysRevLett.120.144301 [41] Shepelyansky DL (1997) Low-energy chaos in the Fermi-Pasta-Ulam problem. Nonlinearity 10: 1331–1338. doi: 10.1088/0951-7715/10/5/017 [42] Toda M (1970) Waves in Nonlinear Lattice. Prog Theor Phys Suppl 45: 174–200. doi: 10.1143/PTPS.45.174
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沈阳化工大学材料科学与工程学院 沈阳 110142

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