Analytical approach to Lyapunov time: Universal scaling and thermalization

Liu, Yue; He, Dahai

doi:10.1103/physreve.103.l040203

Cited by 5 publications

(1 citation statement)

References 55 publications

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“…As far as the question of if and when the FPUT chain thermalizes is concerned, there have been many studies done to investigate this. Some of the methods include the closeness to integrability and soliton solutions [3], stochasticity threshold [4,5], Lyapunov exponent studies [6][7][8], breather solutions [9][10][11][12] and local equilibration studies [13]. More recently the formalism of wave turbulence [14,15] was applied to the FPUT chain [16][17][18][19] in order to explain the dependence of equilibration time on the nonlinearity parameter.…”

Section: Introductionmentioning

confidence: 99%

Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Ganapa¹

2023

Preprint

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The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α−FPUT system in more detail. We aim to reconstruct the quasiperiodic behaviour observed in the original paper from the canonical transformation used to remove the three wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work we show that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α−FPUT system, and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α−FPUT problem. We argue that certain aspects of the α−FPUT system such as presence of the stochasticity threshold, thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.

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Section: Introductionmentioning

confidence: 99%

Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Ganapa¹

2023

Preprint

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show abstract

Effective Stochastic Model for Chaos in the Fermi–Pasta–Ulam–Tsingou Chain

Goldfriend

2023

J Stat Phys

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Quasiperiodicity in the α -Fermi–Pasta–Ulam–Tsingou problem revisited: An approach using ideas from wave turbulence

Ganapa

2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The Fermi–Pasta–Ulam–Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work, we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α-FPUT system in more detail. We aim to reconstruct the quasiperiodic behavior observed in the original paper from the canonical transformation used to remove the three-wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work, we find that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three-wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α-FPUT system and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α-FPUT problem. We argue that certain aspects of the α-FPUT system such as thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.

show abstract

Analytical approach to Lyapunov time: Universal scaling and thermalization

Cited by 5 publications

References 55 publications

Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Effective Stochastic Model for Chaos in the Fermi–Pasta–Ulam–Tsingou Chain

Quasiperiodicity in the α -Fermi–Pasta–Ulam–Tsingou problem revisited: An approach using ideas from wave turbulence

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